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Re: Eternal Universe With New Cosmological Model

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  • sauwelios
    The interpretation I gave is my rendition of Laurence Lampert s interpretation in his most important book, *Nietzsche s Teaching*. Lampert argues there that
    Message 1 of 24 , Aug 3, 2010
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      The interpretation I gave is my rendition of Laurence Lampert's interpretation in his most important book, *Nietzsche's Teaching*. Lampert argues there that the dwarf makes it "too easy on himself" because he asserts that time is a circle, but does not ponder the *implications* of that assertion. Zarathustra then sets forth those implications, thereby silencing the dwarf, as the dwarf cannot stand those implications.

      Lampert interprets the black snake not as a symbol of eternal recurrence, but of Zarathustra's disgust at the eternal recurrence even of the small man (see *TSZ*, 'The Convalescent'). Zarathustra's vision of the shepherd is a pre-vision, a vision of his own future: namely of himself in 'The Convalescent'.


      --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@...> wrote:
      >
      >
      >
      > I had always struggled to grasp that part of The Vision and the Riddle, how he repudiated the circle of the dwarf while seeming to argue for the eternal return. Is the black snake that crawled into the shepherds throat the doctrine of the eternal return itself, and is Zarathustra urging him to reject it by biting off the head of the snake? Could it be that Nietzsche wanted to propagate the doctrine so it could one day be overcome...a breeding agent that would destroy the world weary while providing the highest types one more mountain to surmount?
      >
      > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
      > >
      > > I think we will always keep running into the infinity/nothingness problem.
      > >
      > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
      > >
      > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
      > >
      > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
      > > ['The Three Evil Things', 1.]
      > >
      > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
      > >
      > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
      > > [*WP* 1067.]
      > >
      > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
      > >
      > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
      > > [*WP* 1062.]
      > >
      > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
      > >
      > >
      > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
      > > >
      > > >
      > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
      > > >
      > > >
      > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
      > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
      > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
      > > > • The universe experiences phases of both acceleration and deceleration.
      > > >
      > > > New life for a scientific consideration of the eternal recurrence?
      > > >
      > > > http://www.physorg.com/news199591806.html
      > > >
      > >
      >
    • Ian
      Hey, this is actually my first post on this forum. Actually, Nietzche did present an argument for eternal recurrence in the final chapter of The Will to Power
      Message 2 of 24 , Sep 6, 2010
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        Hey, this is actually my first post on this forum.

        Actually, Nietzche did present an argument for eternal recurrence in the final chapter of The Will to Power (the one in which he draws several conclusions from the science known at the time).

        "1059: The thought of eternal recurrence: its first principles, which must necessarily be true if it were true

        [...]

        1061: The two extremes of thought--the materialistic and the platonic--are reconciled in eternal recurrence: both are regarded as ideals.

        [...]

        1063: The principle of the conservation of energy inevitably involves eternal recurrence.

        [After showing that reality cannot reach equilibrium, and refuting arguments that the universe cannot be eternal:]

        1066: [...] If the universe may be conceived as a definite quantity of energy, as a definite number of centres of energy--and every other concept remains indefinite and therefore useless--it follows therefrom that the universe must go through a calculable number of combinations in the great game of chance which constitutes its existence. In infinity, at some moment or other, every possible combination must once have been realised; not only this, but it must have been realised an infinite number of times. And inasmuch as between every one of these combinations would determine the whole series in the same order, a circular movement of absolutely identical series is thus demonstrated: the universe is thus shown to be a circular movement which has already repeated itself an infinite number of times, and which plays its game for all eternity."

        Whether or not what he says is still relevant today, he seems to have been seriously arguing for the eternal recurrence of the same, although he initially ( in e.g. TGS) probably only intended it as a guideline: live your life as though you would be forced to repeat that exact series of events an infinite number of times. He also doesn't seem to have thought of it as literally that time was circular, just that in an infinite amount of time, and with a finite number of particles, the sequence of events we are living now will necessarily happen an infinite number of times, and necessarily has happened an infinite number of times before.

        Well, I'm sorry that that post was really long. He actually says a couple more things, but this was long enough as it is.

        Take care,

        ~Ian

        --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
        >
        > I think we will always keep running into the infinity/nothingness problem.
        >
        > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
        >
        > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
        >
        > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
        > ['The Three Evil Things', 1.]
        >
        > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
        >
        > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
        > [*WP* 1067.]
        >
        > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
        >
        > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
        > [*WP* 1062.]
        >
        > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
        >
        >
        > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
        > >
        > >
        > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
        > >
        > >
        > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
        > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
        > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
        > > • The universe experiences phases of both acceleration and deceleration.
        > >
        > > New life for a scientific consideration of the eternal recurrence?
        > >
        > > http://www.physorg.com/news199591806.html
        > >
        >
      • sauwelios
        You re right, but then, what I said was: [A]s far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*,
        Message 3 of 24 , Sep 6, 2010
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          You're right, but then, what I said was:

          "[A]s far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way."

          *The Will to Power* is not a 'published work'; it's a compilation from the notebooks. ;)


          --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@...> wrote:
          >
          >
          >
          >
          > Hey, this is actually my first post on this forum.
          >
          > Actually, Nietzche did present an argument for eternal recurrence in the final chapter of The Will to Power (the one in which he draws several conclusions from the science known at the time).
          >
          > "1059: The thought of eternal recurrence: its first principles, which must necessarily be true if it were true
          >
          > [...]
          >
          > 1061: The two extremes of thought--the materialistic and the platonic--are reconciled in eternal recurrence: both are regarded as ideals.
          >
          > [...]
          >
          > 1063: The principle of the conservation of energy inevitably involves eternal recurrence.
          >
          > [After showing that reality cannot reach equilibrium, and refuting arguments that the universe cannot be eternal:]
          >
          > 1066: [...] If the universe may be conceived as a definite quantity of energy, as a definite number of centres of energy--and every other concept remains indefinite and therefore useless--it follows therefrom that the universe must go through a calculable number of combinations in the great game of chance which constitutes its existence. In infinity, at some moment or other, every possible combination must once have been realised; not only this, but it must have been realised an infinite number of times. And inasmuch as between every one of these combinations would determine the whole series in the same order, a circular movement of absolutely identical series is thus demonstrated: the universe is thus shown to be a circular movement which has already repeated itself an infinite number of times, and which plays its game for all eternity."
          >
          > Whether or not what he says is still relevant today, he seems to have been seriously arguing for the eternal recurrence of the same, although he initially ( in e.g. TGS) probably only intended it as a guideline: live your life as though you would be forced to repeat that exact series of events an infinite number of times. He also doesn't seem to have thought of it as literally that time was circular, just that in an infinite amount of time, and with a finite number of particles, the sequence of events we are living now will necessarily happen an infinite number of times, and necessarily has happened an infinite number of times before.
          >
          > Well, I'm sorry that that post was really long. He actually says a couple more things, but this was long enough as it is.
          >
          > Take care,
          >
          > ~Ian
          >
          > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
          > >
          > > I think we will always keep running into the infinity/nothingness problem.
          > >
          > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
          > >
          > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
          > >
          > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
          > > ['The Three Evil Things', 1.]
          > >
          > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
          > >
          > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
          > > [*WP* 1067.]
          > >
          > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
          > >
          > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
          > > [*WP* 1062.]
          > >
          > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
          > >
          > >
          > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
          > > >
          > > >
          > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
          > > >
          > > >
          > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
          > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
          > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
          > > > • The universe experiences phases of both acceleration and deceleration.
          > > >
          > > > New life for a scientific consideration of the eternal recurrence?
          > > >
          > > > http://www.physorg.com/news199591806.html
          > > >
          > >
          >
        • Ian
          Oops, you re right; you did say his published works. :0 However, we should keep in mind that at the time he wrote these notebooks, he was working on the
          Message 4 of 24 , Sep 7, 2010
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            Oops, you're right; you did say "his published works." :0

            However, we should keep in mind that at the time he wrote these notebooks, he was working on the second book in his (intended) four-part series "The Transvaluation of All Values" (of which The Antichrist was to be the first book). In any case, my main point was that it seems that he was, while writing his notebooks, indeed trying to prove the eternal recurrence of the same, if only to himself.

            It is also worth mentioning that Nietzche himself said in (I think) Ecce Homo that his philosophy evolved over time, so, while he was initially (in e.g. TGS) proposing the eternal recurrence as a thought experiment, or as a guideline for living one's life, he may have eventually come to the conclusion that the eternal recurrence will indeed happen. That and he said in BGE that he would never attempt to mold his philosophy into a complete system.

            So it seems that what we have here are a bunch of works that appear to have been written by someone with ADD...

            Also, I notice that this topic is about a new theory of cosmology that may give credibility to the idea of eternal recurrence. I'd just like to point out another such theory based on the transactional interpretation of quantum mechanics: http://sites.google.com/site/jdquirk/articles/circular-causality

            ~Ian

            --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
            >
            > You're right, but then, what I said was:
            >
            > "[A]s far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way."
            >
            > *The Will to Power* is not a 'published work'; it's a compilation from the notebooks. ;)
            >
            >
            > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
            > >
            > >
            > >
            > >
            > > Hey, this is actually my first post on this forum.
            > >
            > > Actually, Nietzche did present an argument for eternal recurrence in the final chapter of The Will to Power (the one in which he draws several conclusions from the science known at the time).
            > >
            > > "1059: The thought of eternal recurrence: its first principles, which must necessarily be true if it were true
            > >
            > > [...]
            > >
            > > 1061: The two extremes of thought--the materialistic and the platonic--are reconciled in eternal recurrence: both are regarded as ideals.
            > >
            > > [...]
            > >
            > > 1063: The principle of the conservation of energy inevitably involves eternal recurrence.
            > >
            > > [After showing that reality cannot reach equilibrium, and refuting arguments that the universe cannot be eternal:]
            > >
            > > 1066: [...] If the universe may be conceived as a definite quantity of energy, as a definite number of centres of energy--and every other concept remains indefinite and therefore useless--it follows therefrom that the universe must go through a calculable number of combinations in the great game of chance which constitutes its existence. In infinity, at some moment or other, every possible combination must once have been realised; not only this, but it must have been realised an infinite number of times. And inasmuch as between every one of these combinations would determine the whole series in the same order, a circular movement of absolutely identical series is thus demonstrated: the universe is thus shown to be a circular movement which has already repeated itself an infinite number of times, and which plays its game for all eternity."
            > >
            > > Whether or not what he says is still relevant today, he seems to have been seriously arguing for the eternal recurrence of the same, although he initially ( in e.g. TGS) probably only intended it as a guideline: live your life as though you would be forced to repeat that exact series of events an infinite number of times. He also doesn't seem to have thought of it as literally that time was circular, just that in an infinite amount of time, and with a finite number of particles, the sequence of events we are living now will necessarily happen an infinite number of times, and necessarily has happened an infinite number of times before.
            > >
            > > Well, I'm sorry that that post was really long. He actually says a couple more things, but this was long enough as it is.
            > >
            > > Take care,
            > >
            > > ~Ian
            > >
            > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
            > > >
            > > > I think we will always keep running into the infinity/nothingness problem.
            > > >
            > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
            > > >
            > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
            > > >
            > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
            > > > ['The Three Evil Things', 1.]
            > > >
            > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
            > > >
            > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
            > > > [*WP* 1067.]
            > > >
            > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
            > > >
            > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
            > > > [*WP* 1062.]
            > > >
            > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
            > > >
            > > >
            > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
            > > > >
            > > > >
            > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
            > > > >
            > > > >
            > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
            > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
            > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
            > > > > • The universe experiences phases of both acceleration and deceleration.
            > > > >
            > > > > New life for a scientific consideration of the eternal recurrence?
            > > > >
            > > > > http://www.physorg.com/news199591806.html
            > > > >
            > > >
            > >
            >
          • Ian
            By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski s extension,
            Message 5 of 24 , Sep 24, 2010
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              By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.

              --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
              >
              > I think we will always keep running into the infinity/nothingness problem.
              >
              > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
              >
              > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
              >
              > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
              > ['The Three Evil Things', 1.]
              >
              > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
              >
              > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
              > [*WP* 1067.]
              >
              > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
              >
              > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
              > [*WP* 1062.]
              >
              > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
              >
              >
              > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
              > >
              > >
              > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
              > >
              > >
              > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
              > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
              > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
              > > • The universe experiences phases of both acceleration and deceleration.
              > >
              > > New life for a scientific consideration of the eternal recurrence?
              > >
              > > http://www.physorg.com/news199591806.html
              > >
              >
            • sauwelios
              I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche s theory of the eternal recurrence).
              Message 6 of 24 , Sep 24, 2010
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                I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).


                --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@...> wrote:
                >
                > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                >
                > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                > >
                > > I think we will always keep running into the infinity/nothingness problem.
                > >
                > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                > >
                > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                > >
                > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                > > ['The Three Evil Things', 1.]
                > >
                > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                > >
                > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                > > [*WP* 1067.]
                > >
                > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                > >
                > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                > > [*WP* 1062.]
                > >
                > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                > >
                > >
                > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                > > >
                > > >
                > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                > > >
                > > >
                > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                > > > • The universe experiences phases of both acceleration and deceleration.
                > > >
                > > > New life for a scientific consideration of the eternal recurrence?
                > > >
                > > > http://www.physorg.com/news199591806.html
                > > >
                > >
                >
              • Ian
                [I]t *is* necessarily true if space/time is all there is. Says who? According to Einstein s theory of General Relativity (which is based heavily on
                Message 7 of 24 , Sep 25, 2010
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                  "[I]t *is* necessarily true if space/time is all there is."

                  Says who? According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe, and without there having to be "nothingness" inside the sphere. According to Minkowskian geometry, which forms the other main geometric basis of GR, the same arguments can be applied to time. The *only reason* that this seems counter-intuitive to us is that, in our everyday lives, we don't encounter things that behave like spacetime does.

                  Granted, GR doesn't work on small scales, and doesn't explain how spacetime is bent by forces other than gravity (and attempts thus far to explain this, e.g. supergravity, have failed). However, the theory is uncannily accurate on large scales (which *is* what we're talking about when we're talking about scales on which things like eternal recurrence become important), since gravity is the only force of any importance on this scale (since the nuclear forces only work on small scales, and electric charges nearly cancel out perfectly on this scale).

                  ~Ian

                  --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
                  >
                  > I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).
                  >
                  >
                  > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                  > >
                  > > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                  > >
                  > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                  > > >
                  > > > I think we will always keep running into the infinity/nothingness problem.
                  > > >
                  > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                  > > >
                  > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                  > > >
                  > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                  > > > ['The Three Evil Things', 1.]
                  > > >
                  > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                  > > >
                  > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                  > > > [*WP* 1067.]
                  > > >
                  > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                  > > >
                  > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                  > > > [*WP* 1062.]
                  > > >
                  > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                  > > >
                  > > >
                  > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                  > > > >
                  > > > >
                  > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                  > > > >
                  > > > >
                  > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                  > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                  > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                  > > > > • The universe experiences phases of both acceleration and deceleration.
                  > > > >
                  > > > > New life for a scientific consideration of the eternal recurrence?
                  > > > >
                  > > > > http://www.physorg.com/news199591806.html
                  > > > >
                  > > >
                  > >
                  >
                • sauwelios
                  I don t think you ve thought through the infinity/nothingness problem. Thus you say: According to Einstein s theory of General Relativity (which is based
                  Message 8 of 24 , Sep 25, 2010
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                    I don't think you've thought through the infinity/nothingness problem. Thus you say:

                    "According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe".

                    If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere -- 'nothingness'?


                    --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@...> wrote:
                    >
                    > "[I]t *is* necessarily true if space/time is all there is."
                    >
                    > Says who? According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe, and without there having to be "nothingness" inside the sphere. According to Minkowskian geometry, which forms the other main geometric basis of GR, the same arguments can be applied to time. The *only reason* that this seems counter-intuitive to us is that, in our everyday lives, we don't encounter things that behave like spacetime does.
                    >
                    > Granted, GR doesn't work on small scales, and doesn't explain how spacetime is bent by forces other than gravity (and attempts thus far to explain this, e.g. supergravity, have failed). However, the theory is uncannily accurate on large scales (which *is* what we're talking about when we're talking about scales on which things like eternal recurrence become important), since gravity is the only force of any importance on this scale (since the nuclear forces only work on small scales, and electric charges nearly cancel out perfectly on this scale).
                    >
                    > ~Ian
                    >
                    > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                    > >
                    > > I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).
                    > >
                    > >
                    > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                    > > >
                    > > > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                    > > >
                    > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                    > > > >
                    > > > > I think we will always keep running into the infinity/nothingness problem.
                    > > > >
                    > > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                    > > > >
                    > > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                    > > > >
                    > > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                    > > > > ['The Three Evil Things', 1.]
                    > > > >
                    > > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                    > > > >
                    > > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                    > > > > [*WP* 1067.]
                    > > > >
                    > > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                    > > > >
                    > > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                    > > > > [*WP* 1062.]
                    > > > >
                    > > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                    > > > >
                    > > > >
                    > > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                    > > > > >
                    > > > > >
                    > > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                    > > > > >
                    > > > > >
                    > > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                    > > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                    > > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                    > > > > > • The universe experiences phases of both acceleration and deceleration.
                    > > > > >
                    > > > > > New life for a scientific consideration of the eternal recurrence?
                    > > > > >
                    > > > > > http://www.physorg.com/news199591806.html
                    > > > > >
                    > > > >
                    > > >
                    > >
                    >
                  • Ian
                    I don t think you ve thought through the infinity/nothingness problem. Actually, I have thought this through, and I knew what, in essence, your argument was.
                    Message 9 of 24 , Oct 3, 2010
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                      "I don't think you've thought through the infinity/nothingness problem."

                      Actually, I have thought this through, and I knew what, in essence, your argument was. GR predicts many phenomena that seem counter-intuitive to us humans, simply because we are used to energy scales where these phenomena aren't a major part of our lives. Quantum mechanics is even "stranger" according to our common sense.

                      On the large energy/time scales that these theories deal with, our common sense fails. Massively. We conclude, therefore, that our common sense is not a good tool to use on these scales. For example, some quantum gravity theories (e.g. loop quantum gravity) predict that time is discrete, not continuous. This contradicts everything we intuitively know about time, because the smallest time unit is too damn small for us to notice.

                      GR, which works almost perfectly on the timescales we're talking about here, is based on a non-Euclidean geometry (Euclidean geometry is the geometry that intuitively makes sense to us), which means that a space-time surface can be curved, without having to be curved "within" anything else; curvature is a metric property of space-time. I would recommend that you do some reading on Riemannian geometry, so that you can see this for yourself (although the mathematics are pretty advanced). With this in mind, questions like

                      "If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere [...]?"

                      become as utterly meaningless as "What is the absolute velocity of the Earth?"

                      ~Ian

                      --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
                      >
                      > I don't think you've thought through the infinity/nothingness problem. Thus you say:
                      >
                      > "According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe".
                      >
                      > If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere -- 'nothingness'?
                      >
                      >
                      > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                      > >
                      > > "[I]t *is* necessarily true if space/time is all there is."
                      > >
                      > > Says who? According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe, and without there having to be "nothingness" inside the sphere. According to Minkowskian geometry, which forms the other main geometric basis of GR, the same arguments can be applied to time. The *only reason* that this seems counter-intuitive to us is that, in our everyday lives, we don't encounter things that behave like spacetime does.
                      > >
                      > > Granted, GR doesn't work on small scales, and doesn't explain how spacetime is bent by forces other than gravity (and attempts thus far to explain this, e.g. supergravity, have failed). However, the theory is uncannily accurate on large scales (which *is* what we're talking about when we're talking about scales on which things like eternal recurrence become important), since gravity is the only force of any importance on this scale (since the nuclear forces only work on small scales, and electric charges nearly cancel out perfectly on this scale).
                      > >
                      > > ~Ian
                      > >
                      > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                      > > >
                      > > > I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).
                      > > >
                      > > >
                      > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                      > > > >
                      > > > > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                      > > > >
                      > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                      > > > > >
                      > > > > > I think we will always keep running into the infinity/nothingness problem.
                      > > > > >
                      > > > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                      > > > > >
                      > > > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                      > > > > >
                      > > > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                      > > > > > ['The Three Evil Things', 1.]
                      > > > > >
                      > > > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                      > > > > >
                      > > > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                      > > > > > [*WP* 1067.]
                      > > > > >
                      > > > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                      > > > > >
                      > > > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                      > > > > > [*WP* 1062.]
                      > > > > >
                      > > > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                      > > > > >
                      > > > > >
                      > > > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                      > > > > > >
                      > > > > > >
                      > > > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                      > > > > > >
                      > > > > > >
                      > > > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                      > > > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                      > > > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                      > > > > > > • The universe experiences phases of both acceleration and deceleration.
                      > > > > > >
                      > > > > > > New life for a scientific consideration of the eternal recurrence?
                      > > > > > >
                      > > > > > > http://www.physorg.com/news199591806.html
                      > > > > > >
                      > > > > >
                      > > > >
                      > > >
                      > >
                      >
                    • sauwelios
                      Well, as you have not grounded any of your assertions, there is nothing for me to go into. I don t know what you mean by common sense . The
                      Message 10 of 24 , Oct 3, 2010
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                        Well, as you have not grounded any of your assertions, there is nothing for me to go into.

                        I don't know what you mean by "common sense". The infinity/nothingness problem is based on *logic*.

                        You say: "a space-time surface can be curved, without having to be curved 'within' anything else". I think a curve must *by definition* exist within something else. Merriam-Webster's Online Dictionary, for instance, defines "curve" (noun) in terms of "curve" (verb), which it in turn defines as:

                        "to have or take a turn, change, or deviation from a straight line or plane surface without sharp breaks or angularity"
                        http://www.merriam-webster.com/dictionary/curve

                        If you can define "curve" without doing so in terms of Euclidean geometry (e.g., straight lines or plane surfaces), please do so. If you can't, a space-time surface apparently cannot be thought of as "curved without having to be curved within anything else"; it can then at most be thought of as "'curved' without having to be curved"---whatever that means.



                        --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@...> wrote:
                        >
                        > "I don't think you've thought through the infinity/nothingness problem."
                        >
                        > Actually, I have thought this through, and I knew what, in essence, your argument was. GR predicts many phenomena that seem counter-intuitive to us humans, simply because we are used to energy scales where these phenomena aren't a major part of our lives. Quantum mechanics is even "stranger" according to our common sense.
                        >
                        > On the large energy/time scales that these theories deal with, our common sense fails. Massively. We conclude, therefore, that our common sense is not a good tool to use on these scales. For example, some quantum gravity theories (e.g. loop quantum gravity) predict that time is discrete, not continuous. This contradicts everything we intuitively know about time, because the smallest time unit is too damn small for us to notice.
                        >
                        > GR, which works almost perfectly on the timescales we're talking about here, is based on a non-Euclidean geometry (Euclidean geometry is the geometry that intuitively makes sense to us), which means that a space-time surface can be curved, without having to be curved "within" anything else; curvature is a metric property of space-time. I would recommend that you do some reading on Riemannian geometry, so that you can see this for yourself (although the mathematics are pretty advanced). With this in mind, questions like
                        >
                        > "If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere [...]?"
                        >
                        > become as utterly meaningless as "What is the absolute velocity of the Earth?"
                        >
                        > ~Ian
                        >
                        > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                        > >
                        > > I don't think you've thought through the infinity/nothingness problem. Thus you say:
                        > >
                        > > "According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe".
                        > >
                        > > If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere -- 'nothingness'?
                        > >
                        > >
                        > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                        > > >
                        > > > "[I]t *is* necessarily true if space/time is all there is."
                        > > >
                        > > > Says who? According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe, and without there having to be "nothingness" inside the sphere. According to Minkowskian geometry, which forms the other main geometric basis of GR, the same arguments can be applied to time. The *only reason* that this seems counter-intuitive to us is that, in our everyday lives, we don't encounter things that behave like spacetime does.
                        > > >
                        > > > Granted, GR doesn't work on small scales, and doesn't explain how spacetime is bent by forces other than gravity (and attempts thus far to explain this, e.g. supergravity, have failed). However, the theory is uncannily accurate on large scales (which *is* what we're talking about when we're talking about scales on which things like eternal recurrence become important), since gravity is the only force of any importance on this scale (since the nuclear forces only work on small scales, and electric charges nearly cancel out perfectly on this scale).
                        > > >
                        > > > ~Ian
                        > > >
                        > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                        > > > >
                        > > > > I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).
                        > > > >
                        > > > >
                        > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                        > > > > >
                        > > > > > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                        > > > > >
                        > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                        > > > > > >
                        > > > > > > I think we will always keep running into the infinity/nothingness problem.
                        > > > > > >
                        > > > > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                        > > > > > >
                        > > > > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                        > > > > > >
                        > > > > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                        > > > > > > ['The Three Evil Things', 1.]
                        > > > > > >
                        > > > > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                        > > > > > >
                        > > > > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                        > > > > > > [*WP* 1067.]
                        > > > > > >
                        > > > > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                        > > > > > >
                        > > > > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                        > > > > > > [*WP* 1062.]
                        > > > > > >
                        > > > > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                        > > > > > >
                        > > > > > >
                        > > > > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                        > > > > > > >
                        > > > > > > >
                        > > > > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                        > > > > > > >
                        > > > > > > >
                        > > > > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                        > > > > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                        > > > > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                        > > > > > > > • The universe experiences phases of both acceleration and deceleration.
                        > > > > > > >
                        > > > > > > > New life for a scientific consideration of the eternal recurrence?
                        > > > > > > >
                        > > > > > > > http://www.physorg.com/news199591806.html
                        > > > > > > >
                        > > > > > >
                        > > > > >
                        > > > >
                        > > >
                        > >
                        >
                      • sauwelios
                        Here is a thought I had some years ago. If the space of the universe is a 4D hypersphere, it is unthinkable that it be enclosed by 4D hypernothingness. It must
                        Message 11 of 24 , Oct 3, 2010
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                          Here is a thought I had some years ago. If the space of the universe is a 4D hypersphere, it is unthinkable that it be enclosed by 4D hypernothingness. It must be enclosed by other 4D hyperobjects, though not an infinite amount of them, as that, too, is unthinkable. Perhaps 4D hyperspace is hypercurved. But this cannot be an infinite regress, as that, too, is unthinkable. Anyway...

                          Even if there is only a fraction of a hypersecond between the creation and the destruction of {the 4D hypersphere that is the space of the universe} by other 4D hyperobjects in hypertime, that may still mean that eternal recurrence is a fact: namely, if "eternal" only refers to time, and not to hypertime. If the ring of recurrence exists for only a hypermoment, time (though not necessarily hypertime!) is indeed circular. However, because this is all very abstract, I have a hard time imagining the ramifications for the 3D universe of the destruction of the 4D hypersphere that is the space of the universe. Apparently, imagining it all those years ago led me to reject the thought I have just expressed.

                          Suppose that the 4D hypersphere that is the space of the universe is cut in half in 4D hyperspace. Then there is at least a hypermoment in which time is not a circle, but two semicircles---and only *one* of the semicircles is the one that contains *us*. And how can we know the condition of {the 4D hyperobject that is the space of our universe} at this hypermoment? We might be located on a timeline segment that ends at what we call "tomorrow"!


                          --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
                          >
                          > Well, as you have not grounded any of your assertions, there is nothing for me to go into.
                          >
                          > I don't know what you mean by "common sense". The infinity/nothingness problem is based on *logic*.
                          >
                          > You say: "a space-time surface can be curved, without having to be curved 'within' anything else". I think a curve must *by definition* exist within something else. Merriam-Webster's Online Dictionary, for instance, defines "curve" (noun) in terms of "curve" (verb), which it in turn defines as:
                          >
                          > "to have or take a turn, change, or deviation from a straight line or plane surface without sharp breaks or angularity"
                          > http://www.merriam-webster.com/dictionary/curve
                          >
                          > If you can define "curve" without doing so in terms of Euclidean geometry (e.g., straight lines or plane surfaces), please do so. If you can't, a space-time surface apparently cannot be thought of as "curved without having to be curved within anything else"; it can then at most be thought of as "'curved' without having to be curved"---whatever that means.
                          >
                          >
                          >
                          > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                          > >
                          > > "I don't think you've thought through the infinity/nothingness problem."
                          > >
                          > > Actually, I have thought this through, and I knew what, in essence, your argument was. GR predicts many phenomena that seem counter-intuitive to us humans, simply because we are used to energy scales where these phenomena aren't a major part of our lives. Quantum mechanics is even "stranger" according to our common sense.
                          > >
                          > > On the large energy/time scales that these theories deal with, our common sense fails. Massively. We conclude, therefore, that our common sense is not a good tool to use on these scales. For example, some quantum gravity theories (e.g. loop quantum gravity) predict that time is discrete, not continuous. This contradicts everything we intuitively know about time, because the smallest time unit is too damn small for us to notice.
                          > >
                          > > GR, which works almost perfectly on the timescales we're talking about here, is based on a non-Euclidean geometry (Euclidean geometry is the geometry that intuitively makes sense to us), which means that a space-time surface can be curved, without having to be curved "within" anything else; curvature is a metric property of space-time. I would recommend that you do some reading on Riemannian geometry, so that you can see this for yourself (although the mathematics are pretty advanced). With this in mind, questions like
                          > >
                          > > "If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere [...]?"
                          > >
                          > > become as utterly meaningless as "What is the absolute velocity of the Earth?"
                          > >
                          > > ~Ian
                          > >
                          > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                          > > >
                          > > > I don't think you've thought through the infinity/nothingness problem. Thus you say:
                          > > >
                          > > > "According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe".
                          > > >
                          > > > If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere -- 'nothingness'?
                          > > >
                          > > >
                          > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                          > > > >
                          > > > > "[I]t *is* necessarily true if space/time is all there is."
                          > > > >
                          > > > > Says who? According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe, and without there having to be "nothingness" inside the sphere. According to Minkowskian geometry, which forms the other main geometric basis of GR, the same arguments can be applied to time. The *only reason* that this seems counter-intuitive to us is that, in our everyday lives, we don't encounter things that behave like spacetime does.
                          > > > >
                          > > > > Granted, GR doesn't work on small scales, and doesn't explain how spacetime is bent by forces other than gravity (and attempts thus far to explain this, e.g. supergravity, have failed). However, the theory is uncannily accurate on large scales (which *is* what we're talking about when we're talking about scales on which things like eternal recurrence become important), since gravity is the only force of any importance on this scale (since the nuclear forces only work on small scales, and electric charges nearly cancel out perfectly on this scale).
                          > > > >
                          > > > > ~Ian
                          > > > >
                          > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                          > > > > >
                          > > > > > I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).
                          > > > > >
                          > > > > >
                          > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                          > > > > > >
                          > > > > > > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                          > > > > > >
                          > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                          > > > > > > >
                          > > > > > > > I think we will always keep running into the infinity/nothingness problem.
                          > > > > > > >
                          > > > > > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                          > > > > > > >
                          > > > > > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                          > > > > > > >
                          > > > > > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                          > > > > > > > ['The Three Evil Things', 1.]
                          > > > > > > >
                          > > > > > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                          > > > > > > >
                          > > > > > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                          > > > > > > > [*WP* 1067.]
                          > > > > > > >
                          > > > > > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                          > > > > > > >
                          > > > > > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                          > > > > > > > [*WP* 1062.]
                          > > > > > > >
                          > > > > > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                          > > > > > > >
                          > > > > > > >
                          > > > > > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                          > > > > > > > >
                          > > > > > > > >
                          > > > > > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                          > > > > > > > >
                          > > > > > > > >
                          > > > > > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                          > > > > > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                          > > > > > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                          > > > > > > > > • The universe experiences phases of both acceleration and deceleration.
                          > > > > > > > >
                          > > > > > > > > New life for a scientific consideration of the eternal recurrence?
                          > > > > > > > >
                          > > > > > > > > http://www.physorg.com/news199591806.html
                          > > > > > > > >
                          > > > > > > >
                          > > > > > >
                          > > > > >
                          > > > >
                          > > >
                          > >
                          >
                        • Ian
                          Well, as you have not grounded any of your assertions, there is nothing for me to go into. Do the words Riemannian geometry ring a bell here? I don t know
                          Message 12 of 24 , Oct 3, 2010
                          • 0 Attachment
                            "Well, as you have not grounded any of your assertions, there is nothing for me to go into."

                            Do the words "Riemannian geometry" ring a bell here?

                            "I don't know what you mean by 'common sense'. The infinity/nothingness problem is based on *logic* [emphasis original]."

                            So is Riemannian geometry.

                            Besides, if that were really true, you should be able to present your argument in a sound syllogism, or at least a paragraph proof. If you were to do so, I would then be able to tell exactly what, if anything, is wrong with your logic.

                            If you can define "curve" without doing so in terms of Euclidean geometry (e.g., straight lines or plane surfaces), please do so. If you can't, a space-time surface apparently cannot be thought of as "curved without having to be curved within anything else"; it can then at most be thought of as "'curved' without having to be curved"---whatever that means.


                            I don't understand why it cannot make reference to Euclidean geometry. If you mean in reference to a linear shape (a straight line or plane), then I can certainly do so. If you mean in reference to the geometry itself, that's impossible, as curvature has no meaning without straightness. I certainly don't understand how defining it in relation to the geometry itself means that "it can then at most be thought of as 'curved' without having to be 'curved' [...]."

                            Before I give my definition, however, I will make the distinction, which I probably should have done from the outset (as you and I seem to be talking apples and oranges here, and I blame myself for this), between *intrinsic* and *extrinsic* curvature.

                            You seem to be talking as though space-time could be extrinsically curved. This is the definition that your dictionary is using, as well as any layman. Be assured that from now on, when I use the word "curvature" without any label, I am not using this definition. The reason, in fact, that I brought up "common sense" in my last post is because our intuitive definition of curvature is the *extrinsic* definition, not the one that I will be using here.

                            When we are talking about the "curvature of space-time," however, we are necessarily talking about *intrinsic* curvature. This definition is *not* the same as our intuitive definition. Indeed, the only reason we use the same word is because we can apply it to space and time being cyclical, hyperbolic, etc.

                            With that in mind, my definition is:

                            (Intrinsic) curvature is a measure of the extent to which the metric tensor of a Riemannian manifold is locally non-isometric to a Euclidean manifold.

                            Finally, I again grace you with the suggestion that you actually do some reading on Riemannian geometry before you post again. This time, I will even be nice enough to provide some links:

                            https://secure.wikimedia.org/wikipedia/en/wiki/Riemannian_geometry
                            http://comet.lehman.cuny.edu/sormani/research/riemgeom.html
                            http://www.mathpages.com/rr/s5-07/5-07.htm

                            These pages give fairly accurate descriptions for the layman, and they are not that long.

                            ~Ian


                            --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
                            >
                            > Well, as you have not grounded any of your assertions, there is nothing for me to go into.
                            >
                            > I don't know what you mean by "common sense". The infinity/nothingness problem is based on *logic*.
                            >
                            > You say: "a space-time surface can be curved, without having to be curved 'within' anything else". I think a curve must *by definition* exist within something else. Merriam-Webster's Online Dictionary, for instance, defines "curve" (noun) in terms of "curve" (verb), which it in turn defines as:
                            >
                            > "to have or take a turn, change, or deviation from a straight line or plane surface without sharp breaks or angularity"
                            > http://www.merriam-webster.com/dictionary/curve
                            >
                            > If you can define "curve" without doing so in terms of Euclidean geometry (e.g., straight lines or plane surfaces), please do so. If you can't, a space-time surface apparently cannot be thought of as "curved without having to be curved within anything else"; it can then at most be thought of as "'curved' without having to be curved"---whatever that means.
                            >
                            >
                            >
                            > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                            > >
                            > > "I don't think you've thought through the infinity/nothingness problem."
                            > >
                            > > Actually, I have thought this through, and I knew what, in essence, your argument was. GR predicts many phenomena that seem counter-intuitive to us humans, simply because we are used to energy scales where these phenomena aren't a major part of our lives. Quantum mechanics is even "stranger" according to our common sense.
                            > >
                            > > On the large energy/time scales that these theories deal with, our common sense fails. Massively. We conclude, therefore, that our common sense is not a good tool to use on these scales. For example, some quantum gravity theories (e.g. loop quantum gravity) predict that time is discrete, not continuous. This contradicts everything we intuitively know about time, because the smallest time unit is too damn small for us to notice.
                            > >
                            > > GR, which works almost perfectly on the timescales we're talking about here, is based on a non-Euclidean geometry (Euclidean geometry is the geometry that intuitively makes sense to us), which means that a space-time surface can be curved, without having to be curved "within" anything else; curvature is a metric property of space-time. I would recommend that you do some reading on Riemannian geometry, so that you can see this for yourself (although the mathematics are pretty advanced). With this in mind, questions like
                            > >
                            > > "If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere [...]?"
                            > >
                            > > become as utterly meaningless as "What is the absolute velocity of the Earth?"
                            > >
                            > > ~Ian
                            > >
                            > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                            > > >
                            > > > I don't think you've thought through the infinity/nothingness problem. Thus you say:
                            > > >
                            > > > "According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe".
                            > > >
                            > > > If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere -- 'nothingness'?
                            > > >
                            > > >
                            > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                            > > > >
                            > > > > "[I]t *is* necessarily true if space/time is all there is."
                            > > > >
                            > > > > Says who? According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe, and without there having to be "nothingness" inside the sphere. According to Minkowskian geometry, which forms the other main geometric basis of GR, the same arguments can be applied to time. The *only reason* that this seems counter-intuitive to us is that, in our everyday lives, we don't encounter things that behave like spacetime does.
                            > > > >
                            > > > > Granted, GR doesn't work on small scales, and doesn't explain how spacetime is bent by forces other than gravity (and attempts thus far to explain this, e.g. supergravity, have failed). However, the theory is uncannily accurate on large scales (which *is* what we're talking about when we're talking about scales on which things like eternal recurrence become important), since gravity is the only force of any importance on this scale (since the nuclear forces only work on small scales, and electric charges nearly cancel out perfectly on this scale).
                            > > > >
                            > > > > ~Ian
                            > > > >
                            > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                            > > > > >
                            > > > > > I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).
                            > > > > >
                            > > > > >
                            > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                            > > > > > >
                            > > > > > > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                            > > > > > >
                            > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                            > > > > > > >
                            > > > > > > > I think we will always keep running into the infinity/nothingness problem.
                            > > > > > > >
                            > > > > > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                            > > > > > > >
                            > > > > > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                            > > > > > > >
                            > > > > > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                            > > > > > > > ['The Three Evil Things', 1.]
                            > > > > > > >
                            > > > > > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                            > > > > > > >
                            > > > > > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                            > > > > > > > [*WP* 1067.]
                            > > > > > > >
                            > > > > > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                            > > > > > > >
                            > > > > > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                            > > > > > > > [*WP* 1062.]
                            > > > > > > >
                            > > > > > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                            > > > > > > >
                            > > > > > > >
                            > > > > > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                            > > > > > > > >
                            > > > > > > > >
                            > > > > > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                            > > > > > > > >
                            > > > > > > > >
                            > > > > > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                            > > > > > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                            > > > > > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                            > > > > > > > > • The universe experiences phases of both acceleration and deceleration.
                            > > > > > > > >
                            > > > > > > > > New life for a scientific consideration of the eternal recurrence?
                            > > > > > > > >
                            > > > > > > > > http://www.physorg.com/news199591806.html
                            > > > > > > > >
                            > > > > > > >
                            > > > > > >
                            > > > > >
                            > > > >
                            > > >
                            > >
                            >
                          • sauwelios
                            ... I did some reading on it. That didn t help to ground your assertions. ... Of course. ... Below are syllogisms. In each case, a is the first premise, b
                            Message 13 of 24 , Oct 3, 2010
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                              --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@...> wrote:
                              >
                              > "Well, as you have not grounded any of your assertions, there is nothing for me to go into."
                              >
                              > Do the words "Riemannian geometry" ring a bell here?
                              >

                              I did some reading on it. That didn't help to ground your assertions.


                              > "I don't know what you mean by 'common sense'. The infinity/nothingness problem is based on *logic* [emphasis original]."
                              >
                              > So is Riemannian geometry.
                              >

                              Of course.



                              > Besides, if that were really true, you should be able to present your argument in a sound syllogism, or at least a paragraph proof. If you were to do so, I would then be able to tell exactly what, if anything, is wrong with your logic.
                              >

                              Below are syllogisms. In each case, "a" is the first premise, "b" is the second premise, and "c" is the conclusion.

                              1
                              a. To think is to form or have in the mind: http://www.merriam-webster.com/dictionary/think
                              b. The mind is finite: this is a fact of experience.
                              c. Whatever is thinkable is finite.

                              2
                              a. Whatever is thinkable is finite: see 1.
                              b. Infinity is not finite: by definition.
                              c. Infinity is not thinkable.

                              3
                              a. To think is to form or have in the mind: see 1.
                              b. Only *things* (i.e., existents) can be formed or had: forming and having both require an object.
                              c. Whatever is thinkable is a thing.

                              4
                              a. Whatever is thinkable is a thing: see 3.
                              b. 'Nothing' is no thing: by definition.
                              c. 'Nothing' is not thinkable.

                              5
                              a. Existence is all that exists: by definition.
                              b. Existence is finite: suppose.
                              c. Existence is enclosed by 'nothing' as by a boundary.

                              6
                              a. Existence is enclosed by 'nothing' as by a boundary: see 5.
                              b. 'Nothing' is unthinkable: see 4.
                              c. Existence is unthinkable.

                              7
                              a. Existence is infinite: suppose.
                              b. Infinity is unthinkable: see 2.
                              c. Existence is unthinkable.

                              8
                              a. Existence is unthinkable as finite: see 6.
                              b. Existence is unthinkable as infinite: see 7.
                              c. Existence is unthinkable.


                              > If you can define "curve" without doing so in terms of Euclidean geometry (e.g., straight lines or plane surfaces), please do so. If you can't, a space-time surface apparently cannot be thought of as "curved without having to be curved within anything else"; it can then at most be thought of as "'curved' without having to be curved"---whatever that means.
                              >
                              >
                              > I don't understand why it cannot make reference to Euclidean geometry. If you mean in reference to a linear shape (a straight line or plane), then I can certainly do so. If you mean in reference to the geometry itself, that's impossible, as curvature has no meaning without straightness. I certainly don't understand how defining it in relation to the geometry itself means that "it can then at most be thought of as 'curved' without having to be 'curved' [...]."
                              >

                              You haven't reproduced my quotation marks right in your last sentence. Anyway, that doesn't matter anymore:


                              > Before I give my definition, however, I will make the distinction, which I probably should have done from the outset (as you and I seem to be talking apples and oranges here, and I blame myself for this), between *intrinsic* and *extrinsic* curvature.
                              >
                              > You seem to be talking as though space-time could be extrinsically curved. This is the definition that your dictionary is using, as well as any layman. Be assured that from now on, when I use the word "curvature" without any label, I am not using this definition. The reason, in fact, that I brought up "common sense" in my last post is because our intuitive definition of curvature is the *extrinsic* definition, not the one that I will be using here.
                              >
                              > When we are talking about the "curvature of space-time," however, we are necessarily talking about *intrinsic* curvature. This definition is *not* the same as our intuitive definition. Indeed, the only reason we use the same word is because we can apply it to space and time being cyclical, hyperbolic, etc.
                              >
                              > With that in mind, my definition is:
                              >
                              > (Intrinsic) curvature is a measure of the extent to which the metric tensor of a Riemannian manifold is locally non-isometric to a Euclidean manifold.
                              >

                              With this definition, you withdrawn too far into jargon for me to follow you.


                              > Finally, I again grace you with the suggestion that you actually do some reading on Riemannian geometry before you post again. This time, I will even be nice enough to provide some links:
                              >
                              > https://secure.wikimedia.org/wikipedia/en/wiki/Riemannian_geometry
                              > http://comet.lehman.cuny.edu/sormani/research/riemgeom.html
                              > http://www.mathpages.com/rr/s5-07/5-07.htm
                              >
                              > These pages give fairly accurate descriptions for the layman, and they are not that long.
                              >
                              > ~Ian
                              >
                              >
                              > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                              > >
                              > > Well, as you have not grounded any of your assertions, there is nothing for me to go into.
                              > >
                              > > I don't know what you mean by "common sense". The infinity/nothingness problem is based on *logic*.
                              > >
                              > > You say: "a space-time surface can be curved, without having to be curved 'within' anything else". I think a curve must *by definition* exist within something else. Merriam-Webster's Online Dictionary, for instance, defines "curve" (noun) in terms of "curve" (verb), which it in turn defines as:
                              > >
                              > > "to have or take a turn, change, or deviation from a straight line or plane surface without sharp breaks or angularity"
                              > > http://www.merriam-webster.com/dictionary/curve
                              > >
                              > > If you can define "curve" without doing so in terms of Euclidean geometry (e.g., straight lines or plane surfaces), please do so. If you can't, a space-time surface apparently cannot be thought of as "curved without having to be curved within anything else"; it can then at most be thought of as "'curved' without having to be curved"---whatever that means.
                              > >
                              > >
                              > >
                              > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                              > > >
                              > > > "I don't think you've thought through the infinity/nothingness problem."
                              > > >
                              > > > Actually, I have thought this through, and I knew what, in essence, your argument was. GR predicts many phenomena that seem counter-intuitive to us humans, simply because we are used to energy scales where these phenomena aren't a major part of our lives. Quantum mechanics is even "stranger" according to our common sense.
                              > > >
                              > > > On the large energy/time scales that these theories deal with, our common sense fails. Massively. We conclude, therefore, that our common sense is not a good tool to use on these scales. For example, some quantum gravity theories (e.g. loop quantum gravity) predict that time is discrete, not continuous. This contradicts everything we intuitively know about time, because the smallest time unit is too damn small for us to notice.
                              > > >
                              > > > GR, which works almost perfectly on the timescales we're talking about here, is based on a non-Euclidean geometry (Euclidean geometry is the geometry that intuitively makes sense to us), which means that a space-time surface can be curved, without having to be curved "within" anything else; curvature is a metric property of space-time. I would recommend that you do some reading on Riemannian geometry, so that you can see this for yourself (although the mathematics are pretty advanced). With this in mind, questions like
                              > > >
                              > > > "If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere [...]?"
                              > > >
                              > > > become as utterly meaningless as "What is the absolute velocity of the Earth?"
                              > > >
                              > > > ~Ian
                              > > >
                              > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                              > > > >
                              > > > > I don't think you've thought through the infinity/nothingness problem. Thus you say:
                              > > > >
                              > > > > "According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe".
                              > > > >
                              > > > > If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere -- 'nothingness'?
                              > > > >
                              > > > >
                              > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                              > > > > >
                              > > > > > "[I]t *is* necessarily true if space/time is all there is."
                              > > > > >
                              > > > > > Says who? According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe, and without there having to be "nothingness" inside the sphere. According to Minkowskian geometry, which forms the other main geometric basis of GR, the same arguments can be applied to time. The *only reason* that this seems counter-intuitive to us is that, in our everyday lives, we don't encounter things that behave like spacetime does.
                              > > > > >
                              > > > > > Granted, GR doesn't work on small scales, and doesn't explain how spacetime is bent by forces other than gravity (and attempts thus far to explain this, e.g. supergravity, have failed). However, the theory is uncannily accurate on large scales (which *is* what we're talking about when we're talking about scales on which things like eternal recurrence become important), since gravity is the only force of any importance on this scale (since the nuclear forces only work on small scales, and electric charges nearly cancel out perfectly on this scale).
                              > > > > >
                              > > > > > ~Ian
                              > > > > >
                              > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                              > > > > > >
                              > > > > > > I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).
                              > > > > > >
                              > > > > > >
                              > > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                              > > > > > > >
                              > > > > > > > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                              > > > > > > >
                              > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                              > > > > > > > >
                              > > > > > > > > I think we will always keep running into the infinity/nothingness problem.
                              > > > > > > > >
                              > > > > > > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                              > > > > > > > >
                              > > > > > > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                              > > > > > > > >
                              > > > > > > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                              > > > > > > > > ['The Three Evil Things', 1.]
                              > > > > > > > >
                              > > > > > > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                              > > > > > > > >
                              > > > > > > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                              > > > > > > > > [*WP* 1067.]
                              > > > > > > > >
                              > > > > > > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                              > > > > > > > >
                              > > > > > > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                              > > > > > > > > [*WP* 1062.]
                              > > > > > > > >
                              > > > > > > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                              > > > > > > > >
                              > > > > > > > >
                              > > > > > > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                              > > > > > > > > >
                              > > > > > > > > >
                              > > > > > > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                              > > > > > > > > >
                              > > > > > > > > >
                              > > > > > > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                              > > > > > > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                              > > > > > > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                              > > > > > > > > > • The universe experiences phases of both acceleration and deceleration.
                              > > > > > > > > >
                              > > > > > > > > > New life for a scientific consideration of the eternal recurrence?
                              > > > > > > > > >
                              > > > > > > > > > http://www.physorg.com/news199591806.html
                              > > > > > > > > >
                              > > > > > > > >
                              > > > > > > >
                              > > > > > >
                              > > > > >
                              > > > >
                              > > >
                              > >
                              >
                            • Ian
                              Thank you for taking the time to write those syllogisms. Some problems with it are: 1 (a): To think is to form or have in the mind I agree with this
                              Message 14 of 24 , Oct 3, 2010
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                                Thank you for taking the time to write those syllogisms.  Some problems with it are:

                                1

                                   (a): "To think is to form or have in the mind"

                                I agree with this definition, as long as we remember that when we say that something is "in the mind," we don't mean that we actually have that thing in the mind; rather, we have a concept or representation of that thing in the mind.  I would thus amend that definition to

                                To think is to form or have a concept in the mind

                                simply to make this clear.  However, you can still logically derive "Whatever is thinkable is a thing" from that definition, as long as you use a precise and appropriate definition of "concept," so the soundness of your argument is unaffected here.

                                   (b):  "The mind is finite [...]."

                                I'm not going to dispute this premise myself, but it should be made clear that some would dispute this claim on several bases, such as a cosmic consciousness or some other assumption for which there is no evidence. Because there is no evidence for these claims at present, however, I don't dispute this premise at all (Incidentally, this premise also destroys the famous Ontological argument for God's existence, but I won't go there).

                                5 and 6, meanwhile, are invalid syllogisms.

                                In the case of 5, we suppose that existence is finite and derive the conclusion that space-time is enclosed by "nothingness."  I'm disputing the validity of this syllogism on the grounds of Riemannian geometry, which I will get to in a moment.

                                However, it depends on the precise definition of nothingness; or rather, of a "thing."  Would you consider space-time a "thing?"  For purposes such as GR, physicists consider it to be a property rather than a thing, rather like spin, velocity, etc.  On what grounds do you dispute this (if at all)?  If you do not dispute this, on the other hand, then the fact that space-time is finite does not mean that it is enclosed by "nothingness" at all.

                                I have a more fundamental objection, however:  What do you mean when you say that "existence is finite/infinite?"  What is it about existence that contains a predicate measuring whether it is finite or infinite?

                                In the case of 6, this syllogism seems rather intuitively true, until we write it in predicate form:

                                LET e="existence" AS String
                                    n="nothingness" AS String
                                    U(x)="x is unthinkable" AS Proposition
                                    x B y="x is enclosed by y as by a boundary" AS Operator

                                -1. e B n [Given 5 (c)]
                                -2. U(n) [Given 4 (c)]
                                 C. U(e) [From (1) and (2)]

                                The problem here is, B is not a well-defined operator.  Predicates like U(x) don't just transfer between two variables just because you can put both of them in a syllogism together.  Define the operator B for me, preferably by truth tables, and show how this corresponds to something "being enclosed" by something else, and I'll be able to actually evaluate the validity of this syllogism.

                                I might add, however, that if B *did* work that way, the following syllogism wouldn't just be valid, but sound.

                                -1.  The Earth is enclosed by the atmosphere as by a boundary [Experience].
                                -2.  The atmosphere is transparent [Experience].
                                 C.  The Earth is transparent [From (1) and (2)].



                                "With this definition, you [have] withdrawn too far into jargon for me to follow you."

                                It honestly would have really helped if you pointed out precisely which terms you did not understand here.  However, I will assume a basic high school literacy of mathematics, and particularly geometry, and try to break this definition down.  I'm going to attempt to do so in a way that is both precise and understandably, but does not over-simplify it.

                                (Intrinsic) curvature is a measure of the extent to which the metric tensor of a Riemannian manifold is locally non-isometric to a Euclidean manifold.

                                Metric tensor:  actually fairly simple:  it is a tensor that measures metrics.

                                To understand what a tensor is, I'll draw an analogy to scalars, vectors, matrices, and the empty set.  You, presumably, know that a matrix is a two dimensional array of numbers, such as:

                                [ 3 5 9]
                                [-2 0 1]
                                [ 7 9 0]

                                A vector, or a quantity with both magnitude and direction, can be represented by a one-dimensional array of numbers, such as (-3, 8, 0).

                                A scalar (a term for just a number) can be represented by a zero dimensional array, as it only contains one number.

                                The empty set, on the other hand, is -1 dimensional; it is represented as {}.

                                You may have already guessed that a tensor is the general name for these things.  The empty set is a -1 tensor, a scalar is a 0-tensor, a vector is a 1-tensor, and a matrix is a 2-tensor.  You can go beyond two, by the way, and talk about 3-tensors, 4-tensors, or even (hypothetically) 1,789,032,853-tensors, though I can't think of a possible use for that last one.

                                Metric comes from the greek word for "distance-measurement."  A metric function is one that defines a distance between elements in a set (and not just distance in space; it is also used to measure angle and time).

                                A metric tensor, then, is one that, given two elements, produces a quantity which defines the distance (in space, time, or angle) between them.  This tensor is always a 2-tensor (a matrix).

                                Riemannian manifold:  For our purposes, space-time.

                                Local:  The region just barely around a point (or set of points).  Put another way, given a point x0 and another point x1, which is a distance of dx away from x0, the local region around x0 is the region as dx gets arbitrarily small.

                                Isometric:  This one is the hardest for a layman to understand, and, unfortunately, the crux of the whole definition.  For now, this definition will have to do.  If you can't understand it, I'll try to use a better one, but this is a lot harder than it looks.

                                Suppose we have two metric spaces (sets of points in which the notion of distance exists; e.g. space-time) X and Y.  These spaces are said to be isomorphic iff (if and only if)  there exists a one-to-one function f from X to Y such that for any a,b in X

                                d(f(a),f(b))=d(a,b).

                                In other words, the distance between a and b in space X is equal to the distance between corresponding points f(a) and f(b) in space Y.

                                Non-isometric spaces, then, are spaces that do not meet this criteria, i.e. there is no such function.

                                Euclidean manifold:  Basically, a space-time in which Euclidean geometry holds exactly.  This is basically the "zero-point" of my curvature measurement; just like temperature and mass have to have a point defined as "zero," so does curvature.

                                The definition thus becomes:

                                A space X is curved insofar as the distance between two arbitrarily close points in X cannot be related by a one-to-one function to the distance between two arbitrarily close points in a Euclidean space Y.

                                Hope that definition helps,

                                ~Ian

                                --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
                                >
                                >
                                >
                                > --- In human_superhuman@yahoogroups.com, "Ian" ianmathwiz7@ wrote:
                                > >
                                > > "Well, as you have not grounded any of your assertions, there is nothing for me to go into."
                                > >
                                > > Do the words "Riemannian geometry" ring a bell here?
                                > >
                                >
                                > I did some reading on it. That didn't help to ground your assertions.
                                >
                                >
                                > > "I don't know what you mean by 'common sense'. The infinity/nothingness problem is based on *logic* [emphasis original]."
                                > >
                                > > So is Riemannian geometry.
                                > >
                                >
                                > Of course.
                                >
                                >
                                >
                                > > Besides, if that were really true, you should be able to present your argument in a sound syllogism, or at least a paragraph proof. If you were to do so, I would then be able to tell exactly what, if anything, is wrong with your logic.
                                > >
                                >
                                > Below are syllogisms. In each case, "a" is the first premise, "b" is the second premise, and "c" is the conclusion.
                                >
                                > 1
                                > a. To think is to form or have in the mind: http://www.merriam-webster.com/dictionary/think
                                > b. The mind is finite: this is a fact of experience.
                                > c. Whatever is thinkable is finite.
                                >
                                > 2
                                > a. Whatever is thinkable is finite: see 1.
                                > b. Infinity is not finite: by definition.
                                > c. Infinity is not thinkable.
                                >
                                > 3
                                > a. To think is to form or have in the mind: see 1.
                                > b. Only *things* (i.e., existents) can be formed or had: forming and having both require an object.
                                > c. Whatever is thinkable is a thing.
                                >
                                > 4
                                > a. Whatever is thinkable is a thing: see 3.
                                > b. 'Nothing' is no thing: by definition.
                                > c. 'Nothing' is not thinkable.
                                >
                                > 5
                                > a. Existence is all that exists: by definition.
                                > b. Existence is finite: suppose.
                                > c. Existence is enclosed by 'nothing' as by a boundary.
                                >
                                > 6
                                > a. Existence is enclosed by 'nothing' as by a boundary: see 5.
                                > b. 'Nothing' is unthinkable: see 4.
                                > c. Existence is unthinkable.
                                >
                                > 7
                                > a. Existence is infinite: suppose.
                                > b. Infinity is unthinkable: see 2.
                                > c. Existence is unthinkable.
                                >
                                > 8
                                > a. Existence is unthinkable as finite: see 6.
                                > b. Existence is unthinkable as infinite: see 7.
                                > c. Existence is unthinkable.
                                >
                                >
                                > > If you can define "curve" without doing so in terms of Euclidean geometry (e.g., straight lines or plane surfaces), please do so. If you can't, a space-time surface apparently cannot be thought of as "curved without having to be curved within anything else"; it can then at most be thought of as "'curved' without having to be curved"---whatever that means.
                                > >
                                > >
                                > > I don't understand why it cannot make reference to Euclidean geometry. If you mean in reference to a linear shape (a straight line or plane), then I can certainly do so. If you mean in reference to the geometry itself, that's impossible, as curvature has no meaning without straightness. I certainly don't understand how defining it in relation to the geometry itself means that "it can then at most be thought of as 'curved' without having to be 'curved' [...]."
                                > >
                                >
                                > You haven't reproduced my quotation marks right in your last sentence. Anyway, that doesn't matter anymore:
                                >
                                >
                                > > Before I give my definition, however, I will make the distinction, which I probably should have done from the outset (as you and I seem to be talking apples and oranges here, and I blame myself for this), between *intrinsic* and *extrinsic* curvature.
                                > >
                                > > You seem to be talking as though space-time could be extrinsically curved. This is the definition that your dictionary is using, as well as any layman. Be assured that from now on, when I use the word "curvature" without any label, I am not using this definition. The reason, in fact, that I brought up "common sense" in my last post is because our intuitive definition of curvature is the *extrinsic* definition, not the one that I will be using here.
                                > >
                                > > When we are talking about the "curvature of space-time," however, we are necessarily talking about *intrinsic* curvature. This definition is *not* the same as our intuitive definition. Indeed, the only reason we use the same word is because we can apply it to space and time being cyclical, hyperbolic, etc.
                                > >
                                > > With that in mind, my definition is:
                                > >
                                > > (Intrinsic) curvature is a measure of the extent to which the metric tensor of a Riemannian manifold is locally non-isometric to a Euclidean manifold.
                                > >
                                >
                                > With this definition, you withdrawn too far into jargon for me to follow you.
                                >
                                >
                                > > Finally, I again grace you with the suggestion that you actually do some reading on Riemannian geometry before you post again. This time, I will even be nice enough to provide some links:
                                > >
                                > > https://secure.wikimedia.org/wikipedia/en/wiki/Riemannian_geometry
                                > > http://comet.lehman.cuny.edu/sormani/research/riemgeom.html
                                > > http://www.mathpages.com/rr/s5-07/5-07.htm
                                > >
                                > > These pages give fairly accurate descriptions for the layman, and they are not that long.
                                > >
                                > > ~Ian
                                > >
                                > >
                                > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                                > > >
                                > > > Well, as you have not grounded any of your assertions, there is nothing for me to go into.
                                > > >
                                > > > I don't know what you mean by "common sense". The infinity/nothingness problem is based on *logic*.
                                > > >
                                > > > You say: "a space-time surface can be curved, without having to be curved 'within' anything else". I think a curve must *by definition* exist within something else. Merriam-Webster's Online Dictionary, for instance, defines "curve" (noun) in terms of "curve" (verb), which it in turn defines as:
                                > > >
                                > > > "to have or take a turn, change, or deviation from a straight line or plane surface without sharp breaks or angularity"
                                > > > http://www.merriam-webster.com/dictionary/curve
                                > > >
                                > > > If you can define "curve" without doing so in terms of Euclidean geometry (e.g., straight lines or plane surfaces), please do so. If you can't, a space-time surface apparently cannot be thought of as "curved without having to be curved within anything else"; it can then at most be thought of as "'curved' without having to be curved"---whatever that means.
                                > > >
                                > > >
                                > > >
                                > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                                > > > >
                                > > > > "I don't think you've thought through the infinity/nothingness problem."
                                > > > >
                                > > > > Actually, I have thought this through, and I knew what, in essence, your argument was. GR predicts many phenomena that seem counter-intuitive to us humans, simply because we are used to energy scales where these phenomena aren't a major part of our lives. Quantum mechanics is even "stranger" according to our common sense.
                                > > > >
                                > > > > On the large energy/time scales that these theories deal with, our common sense fails. Massively. We conclude, therefore, that our common sense is not a good tool to use on these scales. For example, some quantum gravity theories (e.g. loop quantum gravity) predict that time is discrete, not continuous. This contradicts everything we intuitively know about time, because the smallest time unit is too damn small for us to notice.
                                > > > >
                                > > > > GR, which works almost perfectly on the timescales we're talking about here, is based on a non-Euclidean geometry (Euclidean geometry is the geometry that intuitively makes sense to us), which means that a space-time surface can be curved, without having to be curved "within" anything else; curvature is a metric property of space-time. I would recommend that you do some reading on Riemannian geometry, so that you can see this for yourself (although the mathematics are pretty advanced). With this in mind, questions like
                                > > > >
                                > > > > "If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere [...]?"
                                > > > >
                                > > > > become as utterly meaningless as "What is the absolute velocity of the Earth?"
                                > > > >
                                > > > > ~Ian
                                > > > >
                                > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                                > > > > >
                                > > > > > I don't think you've thought through the infinity/nothingness problem. Thus you say:
                                > > > > >
                                > > > > > "According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe".
                                > > > > >
                                > > > > > If the space of the universe is the surface of a 4D hypersphere, what encloses the hypersphere -- 'nothingness'?
                                > > > > >
                                > > > > >
                                > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                                > > > > > >
                                > > > > > > "[I]t *is* necessarily true if space/time is all there is."
                                > > > > > >
                                > > > > > > Says who? According to Einstein's theory of General Relativity (which is based heavily on Riemannian geometry), the space of the universe is cyclical (or, more accurately, is the surface of a 4D hypersphere), without there having to be anything other than the universe, and without there having to be "nothingness" inside the sphere. According to Minkowskian geometry, which forms the other main geometric basis of GR, the same arguments can be applied to time. The *only reason* that this seems counter-intuitive to us is that, in our everyday lives, we don't encounter things that behave like spacetime does.
                                > > > > > >
                                > > > > > > Granted, GR doesn't work on small scales, and doesn't explain how spacetime is bent by forces other than gravity (and attempts thus far to explain this, e.g. supergravity, have failed). However, the theory is uncannily accurate on large scales (which *is* what we're talking about when we're talking about scales on which things like eternal recurrence become important), since gravity is the only force of any importance on this scale (since the nuclear forces only work on small scales, and electric charges nearly cancel out perfectly on this scale).
                                > > > > > >
                                > > > > > > ~Ian
                                > > > > > >
                                > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                                > > > > > > >
                                > > > > > > > I agree, but it *is* necessarily true if space/time is all there is (which is the case according to Nietzsche's theory of the eternal recurrence).
                                > > > > > > >
                                > > > > > > >
                                > > > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@> wrote:
                                > > > > > > > >
                                > > > > > > > > By the way, it is not necessarily true that there is nothingness in the middle and the outside of curved, even circular, space (and, by Minkowski's extension, time). I believe we have Bernhard Riemann, one of the greatest mathematicians of all time, to thank for proving that theorem.
                                > > > > > > > >
                                > > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@> wrote:
                                > > > > > > > > >
                                > > > > > > > > > I think we will always keep running into the infinity/nothingness problem.
                                > > > > > > > > >
                                > > > > > > > > > If time or causality forms a circle, there must be nothingness around it and in the middle. But to say that there is nothingness in the middle is to say that there is *nothing* in the middle, which makes it something different from a circle. And to say that there is nothingness around it is to say that it is bounded by *nothing*, which means it is infinite. And infinity is unthinkable.
                                > > > > > > > > >
                                > > > > > > > > > My contemplations on this problem, which I have here reported very briefly, drive me toward the conclusion that, contrary to the will to power, the eternal recurrence is not meant to be regarded as a fact. This seems to be supported by the fact that, as far as I know, Nietzsche never presented an argument for it in his published works, except perhaps in *TSZ*, which also stands out in every other way. There he provides an argument for it, but only in order to vanquish the dwarf ('The Vision and the Enigma'). And that only works because the dwarf has already asserted that time be a circle. Zarathustra is merely pointing out the *ramifications* of that idea to him, which is what then defeats him. But Zarathustra omits one of the two premises of his argument, which he only provides later:
                                > > > > > > > > >
                                > > > > > > > > > "[My laughing, wide-awake day-wisdom, which mocketh at all 'infinite worlds',] saith: "Where force is, there becometh number the master: it hath more force.""
                                > > > > > > > > > ['The Three Evil Things', 1.]
                                > > > > > > > > >
                                > > > > > > > > > This premise, that force be finite, leads inevitably to the infinity/nothingness problem:
                                > > > > > > > > >
                                > > > > > > > > > "This world: [...] enclosed by 'nothingness' as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be 'empty' here or there, but rather as force throughout[.]"
                                > > > > > > > > > [*WP* 1067.]
                                > > > > > > > > >
                                > > > > > > > > > The phrase here translated as "'nothingness'" is *'das Nichts'*, "'the Nothing'". But what is the difference between 'the Nothing' and *nothing*? How does this not amount to saying that this world is bounded by *nothing*, i.e., not bounded at all? Unless it amounts to saying that, though there is no empty space *within* this world, there is an infinite empty space *outside* it. But if this were the case, its finite force would inevitably disperse into that infinite emptiness. In any case, neither emptiness nor infinity nor nothingness is thinkable, so we still arrive at the same problem Nietzsche tried to solve:
                                > > > > > > > > >
                                > > > > > > > > > "[T]he world, as force, may not be thought of as unlimited, for it *cannot* be so thought of[.]"
                                > > > > > > > > > [*WP* 1062.]
                                > > > > > > > > >
                                > > > > > > > > > Likewise, it cannot be thought of as being enclosed by 'nothingness' as by a boundary. Hence I conclude that the eternal recurrence is not a fact but a *value*.
                                > > > > > > > > >
                                > > > > > > > > >
                                > > > > > > > > > --- In human_superhuman@yahoogroups.com, "Fred" <nietzschefred@> wrote:
                                > > > > > > > > > >
                                > > > > > > > > > >
                                > > > > > > > > > > New model of the universe does away with the singularity, beginning and end and describes physical laws as constantly changing and evolving...there are no cosmological constants.
                                > > > > > > > > > >
                                > > > > > > > > > >
                                > > > > > > > > > > • The speed of light and the gravitational "constant" are not constant, but vary with the evolution of the universe.
                                > > > > > > > > > > • Time has no beginning and no end; i.e., there is neither a big bang nor a big crunch singularity.
                                > > > > > > > > > > • The spatial section of the universe is a 3-sphere [a higher-dimensional analogue of a sphere], ruling out the possibility of a flat or hyperboloid geometry.
                                > > > > > > > > > > • The universe experiences phases of both acceleration and deceleration.
                                > > > > > > > > > >
                                > > > > > > > > > > New life for a scientific consideration of the eternal recurrence?
                                > > > > > > > > > >
                                > > > > > > > > > > http://www.physorg.com/news199591806.html
                                > > > > > > > > > >
                                > > > > > > > > >
                                > > > > > > > >
                                > > > > > > >
                                > > > > > >
                                > > > > >
                                > > > >
                                > > >
                                > >
                                >
                              • sauwelios
                                ... Well, for one thing, I d say a concept *is* a thing. I don t mean that all things can be formed or had in the mind. ... Thank you. I would indeed not mind
                                Message 15 of 24 , Oct 4, 2010
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                                  --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@...> wrote:
                                  >
                                  > Thank you for taking the time to write those syllogisms. Some problems
                                  > with it are:
                                  >
                                  > 1
                                  >
                                  > (a): "To think is to form or have in the mind"
                                  >
                                  > I agree with this definition, as long as we remember that when we say
                                  > that something is "in the mind," we don't mean that we actually have
                                  > that thing in the mind; rather, we have a concept or representation of
                                  > that thing in the mind. I would thus amend that definition to
                                  >
                                  > To think is to form or have a concept in the mind
                                  >
                                  > simply to make this clear. However, you can still logically derive
                                  > "Whatever is thinkable is a thing" from that definition, as long as you
                                  > use a precise and appropriate definition of "concept," so the soundness
                                  > of your argument is unaffected here.
                                  >

                                  Well, for one thing, I'd say a concept *is* a thing. I don't mean that all things can be formed or had in the mind.


                                  > (b): "The mind is finite [...]."
                                  >
                                  > I'm not going to dispute this premise myself, but it should be made
                                  > clear that some would dispute this claim on several bases, such as a
                                  > cosmic consciousness or some other assumption for which there is no
                                  > evidence. Because there is no evidence for these claims at present,
                                  > however, I don't dispute this premise at all (Incidentally, this premise
                                  > also destroys the famous Ontological argument for God's existence, but I
                                  > won't go there).
                                  >

                                  Thank you. I would indeed not mind offending religious minds with that statement.

                                  A rational argument might be that the mind is a function of the brain, and that the brain is finite: therefore, the mind is finite.


                                  > 5 and 6, meanwhile, are invalid syllogisms.
                                  >
                                  > In the case of 5, we suppose that existence is finite and derive the
                                  > conclusion that space-time is enclosed by "nothingness." I'm disputing
                                  > the validity of this syllogism on the grounds of Riemannian geometry,
                                  > which I will get to in a moment.
                                  >
                                  > However, it depends on the precise definition of nothingness; or rather,
                                  > of a "thing." Would you consider space-time a "thing?" For purposes
                                  > such as GR, physicists consider it to be a property rather than a thing,
                                  > rather like spin, velocity, etc. On what grounds do you dispute this
                                  > (if at all)? If you do not dispute this, on the other hand, then the
                                  > fact that space-time is finite does not mean that it is enclosed by
                                  > "nothingness" at all.
                                  >

                                  I did not say "space-time", but "existence". And I did not mean existence as a property, as in "the existence of that existent", but as the sum of all existents. At first, I wrote "the All".

                                  I think properties, like velocity, are completely abstract concepts. Like Nietzsche, I don't think there are things that 'have' properties; I think that if one would strip a thing of all its properties, nothing would remain.

                                  A property, then, is not a thing. This seems in line with what I said, as one cannot think of a property, e.g. velocity, in itself; one can only think of an object moving at a certain velocity, for instance.


                                  > I have a more fundamental objection, however: What do you mean when you
                                  > say that "existence is finite/infinite?" What is it about existence
                                  > that contains a predicate measuring whether it is finite or infinite?
                                  >

                                  This is already a bit too abstract language for my liking: "contains a predicate measuring". Anyway, I guess to be finite is to be bounded, and to be infinite is to not be bounded. Does that answer your question somewhat?


                                  > In the case of 6, this syllogism seems rather intuitively true, until we
                                  > write it in predicate form:
                                  >
                                  > LET e="existence" AS String
                                  > n="nothingness" AS String
                                  > U(x)="x is unthinkable" AS Proposition
                                  > x B y="x is enclosed by y as by a boundary" AS Operator
                                  >
                                  > -1. e B n [Given 5 (c)]
                                  > -2. U(n) [Given 4 (c)]
                                  > C. U(e) [From (1) and (2)]
                                  >
                                  > The problem here is, B is not a well-defined operator. Predicates like
                                  > U(x) don't just transfer between two variables just because you can put
                                  > both of them in a syllogism together. Define the operator B for me,
                                  > preferably by truth tables, and show how this corresponds to something
                                  > "being enclosed" by something else, and I'll be able to actually
                                  > evaluate the validity of this syllogism.
                                  >
                                  > I might add, however, that if B *did* work that way, the following
                                  > syllogism wouldn't just be valid, but sound.
                                  >
                                  > -1. The Earth is enclosed by the atmosphere as by a boundary
                                  > [Experience].
                                  > -2. The atmosphere is transparent [Experience].
                                  > C. The Earth is transparent [From (1) and (2)].
                                  >

                                  Well, the atmosphere is not *completely* transparent: which is why we see it as blue (due to the oxygen). In the time of the dinosaurs, we would have seen it as orange (due to methane).

                                  Anyway, that doesn't harm your argument. And I don't know how to define something by truth tables.

                                  The Earth is indeed like the atmosphere insofar as the atmosphere is part of the Earth. That part of the Earth is then 'transparent', for instance, and so the statement "the Earth is transparent" is true (though "the *whole* Earth is transparent" would be false). This seems to already come somewhat close to what I meant.

                                  If existence *is* enclosed by 'nothing', then existence as it *is* cannot be conceived without conceiving it as enclosed by 'nothing'; and as 'nothing' cannot be conceived, existence as it is cannot be conceived.


                                  >
                                  >
                                  > "With this definition, you [have] withdrawn too far into jargon for me
                                  > to follow you."
                                  >
                                  > It honestly would have really helped if you pointed out precisely which
                                  > terms you did not understand here. However, I will assume a basic high
                                  > school literacy of mathematics, and particularly geometry, and try to
                                  > break this definition down. I'm going to attempt to do so in a way that
                                  > is both precise and understandably, but does not over-simplify it.
                                  >
                                  > (Intrinsic) curvature is a measure of the extent to which the metric
                                  > tensor of a Riemannian manifold is locally non-isometric to a Euclidean
                                  > manifold.
                                  >

                                  "metric"
                                  "tensor"
                                  "Riemannian manifold"
                                  "(non-)isometric"
                                  "locally (non-)isometric"
                                  "Euclidean manifold"

                                  I may go into the rest of your post later. I knew what the Greek word "metrikos" would mean, by the way.


                                  > Metric tensor: actually fairly simple: it is a tensor that measures
                                  > metrics.
                                  >
                                  > To understand what a tensor is, I'll draw an analogy to scalars,
                                  > vectors, matrices, and the empty set. You, presumably, know that a
                                  > matrix is a two dimensional array of numbers, such as:
                                  >
                                  > [ 3 5 9]
                                  > [-2 0 1]
                                  > [ 7 9 0]
                                  >
                                  > A vector, or a quantity with both magnitude and direction, can be
                                  > represented by a one-dimensional array of numbers, such as (-3, 8, 0).
                                  >
                                  > A scalar (a term for just a number) can be represented by a zero
                                  > dimensional array, as it only contains one number.
                                  >
                                  > The empty set, on the other hand, is -1 dimensional; it is represented
                                  > as {}.
                                  >
                                  > You may have already guessed that a tensor is the general name for these
                                  > things. The empty set is a -1 tensor, a scalar is a 0-tensor, a vector
                                  > is a 1-tensor, and a matrix is a 2-tensor. You can go beyond two, by
                                  > the way, and talk about 3-tensors, 4-tensors, or even (hypothetically)
                                  > 1,789,032,853-tensors, though I can't think of a possible use for that
                                  > last one.
                                  >
                                  > Metric comes from the greek word for "distance-measurement." A metric
                                  > function is one that defines a distance between elements in a set (and
                                  > not just distance in space; it is also used to measure angle and time).
                                  >
                                  > A metric tensor, then, is one that, given two elements, produces a
                                  > quantity which defines the distance (in space, time, or angle) between
                                  > them. This tensor is always a 2-tensor (a matrix).
                                  >
                                  > Riemannian manifold: For our purposes, space-time.
                                  >
                                  > Local: The region just barely around a point (or set of points). Put
                                  > another way, given a point x0 and another point x1, which is a distance
                                  > of dx away from x0, the local region around x0 is the region as dx gets
                                  > arbitrarily small.
                                  >
                                  > Isometric: This one is the hardest for a layman to understand, and,
                                  > unfortunately, the crux of the whole definition. For now, this
                                  > definition will have to do. If you can't understand it, I'll try to use
                                  > a better one, but this is a lot harder than it looks.
                                  >
                                  > Suppose we have two metric spaces (sets of points in which the notion of
                                  > distance exists; e.g. space-time) X and Y. These spaces are said to be
                                  > isomorphic iff (if and only if) there exists a one-to-one function f
                                  > from X to Y such that for any a,b in X
                                  >
                                  > d(f(a),f(b))=d(a,b).
                                  >
                                  > In other words, the distance between a and b in space X is equal to the
                                  > distance between corresponding points f(a) and f(b) in space Y.
                                  >
                                  > Non-isometric spaces, then, are spaces that do not meet this criteria,
                                  > i.e. there is no such function.
                                  >
                                  > Euclidean manifold: Basically, a space-time in which Euclidean geometry
                                  > holds exactly. This is basically the "zero-point" of my curvature
                                  > measurement; just like temperature and mass have to have a point defined
                                  > as "zero," so does curvature.
                                  >
                                  > The definition thus becomes:
                                  >
                                  > A space X is curved insofar as the distance between two arbitrarily
                                  > close points in X cannot be related by a one-to-one function to the
                                  > distance between two arbitrarily close points in a Euclidean space Y.
                                  >
                                  > Hope that definition helps,
                                  >
                                  > ~Ian
                                  >
                                  > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@>
                                  > wrote:
                                  > >
                                  > >
                                  > >
                                  > > --- In human_superhuman@yahoogroups.com, "Ian" ianmathwiz7@ wrote:
                                  > > >
                                  > > > "Well, as you have not grounded any of your assertions, there is
                                  > nothing for me to go into."
                                  > > >
                                  > > > Do the words "Riemannian geometry" ring a bell here?
                                  > > >
                                  > >
                                  > > I did some reading on it. That didn't help to ground your assertions.
                                  > >
                                  > >
                                  > > > "I don't know what you mean by 'common sense'. The
                                  > infinity/nothingness problem is based on *logic* [emphasis original]."
                                  > > >
                                  > > > So is Riemannian geometry.
                                  > > >
                                  > >
                                  > > Of course.
                                  > >
                                  > >
                                  > >
                                  > > > Besides, if that were really true, you should be able to present
                                  > your argument in a sound syllogism, or at least a paragraph proof. If
                                  > you were to do so, I would then be able to tell exactly what, if
                                  > anything, is wrong with your logic.
                                  > > >
                                  > >
                                  > > Below are syllogisms. In each case, "a" is the first premise, "b" is
                                  > the second premise, and "c" is the conclusion.
                                  > >
                                  > > 1
                                  > > a. To think is to form or have in the mind:
                                  > http://www.merriam-webster.com/dictionary/think
                                  > > b. The mind is finite: this is a fact of experience.
                                  > > c. Whatever is thinkable is finite.
                                  > >
                                  > > 2
                                  > > a. Whatever is thinkable is finite: see 1.
                                  > > b. Infinity is not finite: by definition.
                                  > > c. Infinity is not thinkable.
                                  > >
                                  > > 3
                                  > > a. To think is to form or have in the mind: see 1.
                                  > > b. Only *things* (i.e., existents) can be formed or had: forming and
                                  > having both require an object.
                                  > > c. Whatever is thinkable is a thing.
                                  > >
                                  > > 4
                                  > > a. Whatever is thinkable is a thing: see 3.
                                  > > b. 'Nothing' is no thing: by definition.
                                  > > c. 'Nothing' is not thinkable.
                                  > >
                                  > > 5
                                  > > a. Existence is all that exists: by definition.
                                  > > b. Existence is finite: suppose.
                                  > > c. Existence is enclosed by 'nothing' as by a boundary.
                                  > >
                                  > > 6
                                  > > a. Existence is enclosed by 'nothing' as by a boundary: see 5.
                                  > > b. 'Nothing' is unthinkable: see 4.
                                  > > c. Existence is unthinkable.
                                  > >
                                  > > 7
                                  > > a. Existence is infinite: suppose.
                                  > > b. Infinity is unthinkable: see 2.
                                  > > c. Existence is unthinkable.
                                  > >
                                  > > 8
                                  > > a. Existence is unthinkable as finite: see 6.
                                  > > b. Existence is unthinkable as infinite: see 7.
                                  > > c. Existence is unthinkable.
                                  > >
                                  > >
                                  > > > If you can define "curve" without doing so in terms of Euclidean
                                  > geometry (e.g., straight lines or plane surfaces), please do so. If you
                                  > can't, a space-time surface apparently cannot be thought of as "curved
                                  > without having to be curved within anything else"; it can then at most
                                  > be thought of as "'curved' without having to be curved"---whatever that
                                  > means.
                                  > > >
                                  > > >
                                  > > > I don't understand why it cannot make reference to Euclidean
                                  > geometry. If you mean in reference to a linear shape (a straight line
                                  > or plane), then I can certainly do so. If you mean in reference to the
                                  > geometry itself, that's impossible, as curvature has no meaning without
                                  > straightness. I certainly don't understand how defining it in relation
                                  > to the geometry itself means that "it can then at most be thought of as
                                  > 'curved' without having to be 'curved' [...]."
                                  > > >
                                  > >
                                  > > You haven't reproduced my quotation marks right in your last sentence.
                                  > Anyway, that doesn't matter anymore:
                                  > >
                                  > >
                                  > > > Before I give my definition, however, I will make the distinction,
                                  > which I probably should have done from the outset (as you and I seem to
                                  > be talking apples and oranges here, and I blame myself for this),
                                  > between *intrinsic* and *extrinsic* curvature.
                                  > > >
                                  > > > You seem to be talking as though space-time could be extrinsically
                                  > curved. This is the definition that your dictionary is using, as well
                                  > as any layman. Be assured that from now on, when I use the word
                                  > "curvature" without any label, I am not using this definition. The
                                  > reason, in fact, that I brought up "common sense" in my last post is
                                  > because our intuitive definition of curvature is the *extrinsic*
                                  > definition, not the one that I will be using here.
                                  > > >
                                  > > > When we are talking about the "curvature of space-time," however, we
                                  > are necessarily talking about *intrinsic* curvature. This definition is
                                  > *not* the same as our intuitive definition. Indeed, the only reason we
                                  > use the same word is because we can apply it to space and time being
                                  > cyclical, hyperbolic, etc.
                                  > > >
                                  > > > With that in mind, my definition is:
                                  > > >
                                  > > > (Intrinsic) curvature is a measure of the extent to which the metric
                                  > tensor of a Riemannian manifold is locally non-isometric to a Euclidean
                                  > manifold.
                                  > > >
                                  > >
                                  > > With this definition, you withdrawn too far into jargon for me to
                                  > follow you.
                                  > >
                                  > >
                                  > > > Finally, I again grace you with the suggestion that you actually do
                                  > some reading on Riemannian geometry before you post again. This time, I
                                  > will even be nice enough to provide some links:
                                  > > >
                                  > > > https://secure.wikimedia.org/wikipedia/en/wiki/Riemannian_geometry
                                  > > > http://comet.lehman.cuny.edu/sormani/research/riemgeom.html
                                  > > > http://www.mathpages.com/rr/s5-07/5-07.htm
                                  > > >
                                  > > > These pages give fairly accurate descriptions for the layman, and
                                  > they are not that long.
                                  > > >
                                  > > > ~Ian
                                  > > >
                                  > > >
                                  > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@>
                                  > wrote:
                                  > > > >
                                  > > > > Well, as you have not grounded any of your assertions, there is
                                  > nothing for me to go into.
                                  > > > >
                                  > > > > I don't know what you mean by "common sense". The
                                  > infinity/nothingness problem is based on *logic*.
                                  > > > >
                                  > > > > You say: "a space-time surface can be curved, without having to be
                                  > curved 'within' anything else". I think a curve must *by definition*
                                  > exist within something else. Merriam-Webster's Online Dictionary, for
                                  > instance, defines "curve" (noun) in terms of "curve" (verb), which it in
                                  > turn defines as:
                                  > > > >
                                  > > > > "to have or take a turn, change, or deviation from a straight line
                                  > or plane surface without sharp breaks or angularity"
                                  > > > > http://www.merriam-webster.com/dictionary/curve
                                  > > > >
                                  > > > > If you can define "curve" without doing so in terms of Euclidean
                                  > geometry (e.g., straight lines or plane surfaces), please do so. If you
                                  > can't, a space-time surface apparently cannot be thought of as "curved
                                  > without having to be curved within anything else"; it can then at most
                                  > be thought of as "'curved' without having to be curved"---whatever that
                                  > means.
                                  > > > >
                                  > > > >
                                  > > > >
                                  > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@>
                                  > wrote:
                                  > > > > >
                                  > > > > > "I don't think you've thought through the infinity/nothingness
                                  > problem."
                                  > > > > >
                                  > > > > > Actually, I have thought this through, and I knew what, in
                                  > essence, your argument was. GR predicts many phenomena that seem
                                  > counter-intuitive to us humans, simply because we are used to energy
                                  > scales where these phenomena aren't a major part of our lives. Quantum
                                  > mechanics is even "stranger" according to our common sense.
                                  > > > > >
                                  > > > > > On the large energy/time scales that these theories deal with,
                                  > our common sense fails. Massively. We conclude, therefore, that our
                                  > common sense is not a good tool to use on these scales. For example,
                                  > some quantum gravity theories (e.g. loop quantum gravity) predict that
                                  > time is discrete, not continuous. This contradicts everything we
                                  > intuitively know about time, because the smallest time unit is too damn
                                  > small for us to notice.
                                  > > > > >
                                  > > > > > GR, which works almost perfectly on the timescales we're talking
                                  > about here, is based on a non-Euclidean geometry (Euclidean geometry is
                                  > the geometry that intuitively makes sense to us), which means that a
                                  > space-time surface can be curved, without having to be curved "within"
                                  > anything else; curvature is a metric property of space-time. I would
                                  > recommend that you do some reading on Riemannian geometry, so that you
                                  > can see this for yourself (although the mathematics are pretty
                                  > advanced). With this in mind, questions like
                                  > > > > >
                                  > > > > > "If the space of the universe is the surface of a 4D
                                  > hypersphere, what encloses the hypersphere [...]?"
                                  > > > > >
                                  > > > > > become as utterly meaningless as "What is the absolute velocity
                                  > of the Earth?"
                                  > > > > >
                                  > > > > > ~Ian
                                  > > > > >
                                  > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                  > <sauwelios@> wrote:
                                  > > > > > >
                                  > > > > > > I don't think you've thought through the infinity/nothingness
                                  > problem. Thus you say:
                                  > > > > > >
                                  > > > > > > "According to Einstein's theory of General Relativity (which
                                  > is based heavily on Riemannian geometry), the space of the universe is
                                  > cyclical (or, more accurately, is the surface of a 4D hypersphere),
                                  > without there having to be anything other than the universe".
                                  > > > > > >
                                  > > > > > > If the space of the universe is the surface of a 4D
                                  > hypersphere, what encloses the hypersphere -- 'nothingness'?
                                  > > > > > >
                                  > > > > > >
                                  > > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@>
                                  > wrote:
                                  > > > > > > >
                                  > > > > > > > "[I]t *is* necessarily true if space/time is all there is."
                                  > > > > > > >
                                  > > > > > > > Says who? According to Einstein's theory of General
                                  > Relativity (which is based heavily on Riemannian geometry), the space of
                                  > the universe is cyclical (or, more accurately, is the surface of a 4D
                                  > hypersphere), without there having to be anything other than the
                                  > universe, and without there having to be "nothingness" inside the
                                  > sphere. According to Minkowskian geometry, which forms the other main
                                  > geometric basis of GR, the same arguments can be applied to time. The
                                  > *only reason* that this seems counter-intuitive to us is that, in our
                                  > everyday lives, we don't encounter things that behave like spacetime
                                  > does.
                                  > > > > > > >
                                  > > > > > > > Granted, GR doesn't work on small scales, and doesn't
                                  > explain how spacetime is bent by forces other than gravity (and attempts
                                  > thus far to explain this, e.g. supergravity, have failed). However, the
                                  > theory is uncannily accurate on large scales (which *is* what we're
                                  > talking about when we're talking about scales on which things like
                                  > eternal recurrence become important), since gravity is the only force of
                                  > any importance on this scale (since the nuclear forces only work on
                                  > small scales, and electric charges nearly cancel out perfectly on this
                                  > scale).
                                  > > > > > > >
                                  > > > > > > > ~Ian
                                  > > > > > > >
                                  > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                  > <sauwelios@> wrote:
                                  > > > > > > > >
                                  > > > > > > > > I agree, but it *is* necessarily true if space/time is all
                                  > there is (which is the case according to Nietzsche's theory of the
                                  > eternal recurrence).
                                  > > > > > > > >
                                  > > > > > > > >
                                  > > > > > > > > --- In human_superhuman@yahoogroups.com, "Ian"
                                  > <ianmathwiz7@> wrote:
                                  > > > > > > > > >
                                  > > > > > > > > > By the way, it is not necessarily true that there is
                                  > nothingness in the middle and the outside of curved, even circular,
                                  > space (and, by Minkowski's extension, time). I believe we have Bernhard
                                  > Riemann, one of the greatest mathematicians of all time, to thank for
                                  > proving that theorem.
                                  > > > > > > > > >
                                  > > > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                  > <sauwelios@> wrote:
                                  > > > > > > > > > >
                                  > > > > > > > > > > I think we will always keep running into the
                                  > infinity/nothingness problem.
                                  > > > > > > > > > >
                                  > > > > > > > > > > If time or causality forms a circle, there must be
                                  > nothingness around it and in the middle. But to say that there is
                                  > nothingness in the middle is to say that there is *nothing* in the
                                  > middle, which makes it something different from a circle. And to say
                                  > that there is nothingness around it is to say that it is bounded by
                                  > *nothing*, which means it is infinite. And infinity is unthinkable.
                                  > > > > > > > > > >
                                  > > > > > > > > > > My contemplations on this problem, which I have here
                                  > reported very briefly, drive me toward the conclusion that, contrary to
                                  > the will to power, the eternal recurrence is not meant to be regarded as
                                  > a fact. This seems to be supported by the fact that, as far as I know,
                                  > Nietzsche never presented an argument for it in his published works,
                                  > except perhaps in *TSZ*, which also stands out in every other way. There
                                  > he provides an argument for it, but only in order to vanquish the dwarf
                                  > ('The Vision and the Enigma'). And that only works because the dwarf has
                                  > already asserted that time be a circle. Zarathustra is merely pointing
                                  > out the *ramifications* of that idea to him, which is what then defeats
                                  > him. But Zarathustra omits one of the two premises of his argument,
                                  > which he only provides later:
                                  > > > > > > > > > >
                                  > > > > > > > > > > "[My laughing, wide-awake day-wisdom, which mocketh at
                                  > all 'infinite worlds',] saith: "Where force is, there becometh number
                                  > the master: it hath more force.""
                                  > > > > > > > > > > ['The Three Evil Things', 1.]
                                  > > > > > > > > > >
                                  > > > > > > > > > > This premise, that force be finite, leads inevitably
                                  > to the infinity/nothingness problem:
                                  > > > > > > > > > >
                                  > > > > > > > > > > "This world: [...] enclosed by 'nothingness' as by a
                                  > boundary; not something blurry or wasted, not something endlessly
                                  > extended, but set in a definite space as a definite force, and not a
                                  > space that might be 'empty' here or there, but rather as force
                                  > throughout[.]"
                                  > > > > > > > > > > [*WP* 1067.]
                                  > > > > > > > > > >
                                  > > > > > > > > > > The phrase here translated as "'nothingness'" is *'das
                                  > Nichts'*, "'the Nothing'". But what is the difference between 'the
                                  > Nothing' and *nothing*? How does this not amount to saying that this
                                  > world is bounded by *nothing*, i.e., not bounded at all? Unless it
                                  > amounts to saying that, though there is no empty space *within* this
                                  > world, there is an infinite empty space *outside* it. But if this were
                                  > the case, its finite force would inevitably disperse into that infinite
                                  > emptiness. In any case, neither emptiness nor infinity nor nothingness
                                  > is thinkable, so we still arrive at the same problem Nietzsche tried to
                                  > solve:
                                  > > > > > > > > > >
                                  > > > > > > > > > > "[T]he world, as force, may not be thought of as
                                  > unlimited, for it *cannot* be so thought of[.]"
                                  > > > > > > > > > > [*WP* 1062.]
                                  > > > > > > > > > >
                                  > > > > > > > > > > Likewise, it cannot be thought of as being enclosed by
                                  > 'nothingness' as by a boundary. Hence I conclude that the eternal
                                  > recurrence is not a fact but a *value*.
                                  > > > > > > > > > >
                                  > > > > > > > > > >
                                  > > > > > > > > > > --- In human_superhuman@yahoogroups.com, "Fred"
                                  > <nietzschefred@> wrote:
                                  > > > > > > > > > > >
                                  > > > > > > > > > > >
                                  > > > > > > > > > > > New model of the universe does away with the
                                  > singularity, beginning and end and describes physical laws as constantly
                                  > changing and evolving...there are no cosmological constants.
                                  > > > > > > > > > > >
                                  > > > > > > > > > > >
                                  > > > > > > > > > > > • The speed of light and the gravitational
                                  > "constant" are not constant, but vary with the evolution of the
                                  > universe.
                                  > > > > > > > > > > > • Time has no beginning and no end; i.e., there
                                  > is neither a big bang nor a big crunch singularity.
                                  > > > > > > > > > > > • The spatial section of the universe is a
                                  > 3-sphere [a higher-dimensional analogue of a sphere], ruling out the
                                  > possibility of a flat or hyperboloid geometry.
                                  > > > > > > > > > > > • The universe experiences phases of both
                                  > acceleration and deceleration.
                                  > > > > > > > > > > >
                                  > > > > > > > > > > > New life for a scientific consideration of the
                                  > eternal recurrence?
                                  > > > > > > > > > > >
                                  > > > > > > > > > > > http://www.physorg.com/news199591806.html
                                  > > > > > > > > > > >
                                  > > > > > > > > > >
                                  > > > > > > > > >
                                  > > > > > > > >
                                  > > > > > > >
                                  > > > > > >
                                  > > > > >
                                  > > > >
                                  > > >
                                  > >
                                  >
                                • Ian
                                  Well, for one thing, I d say a concept *is* a thing. I don t mean that all things can be formed or had in the mind. Well, we can dispute that. Ultimately,
                                  Message 16 of 24 , Oct 4, 2010
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                                    "Well, for one thing, I'd say a concept *is* a thing. I don't mean that all things can be formed or had in the mind."

                                    Well, we can dispute that.  Ultimately, it's just a metaphysical argument, and, as it does not take away from the soundness of your argument anyway, I think that everyone would appreciate it if we stuck to things that matter.  Also, I didn't mean to imply that you said that all things can exist in the mind.


                                    "A rational argument might be that the mind is a function of the brain, and that the brain is finite: therefore, the mind is finite."

                                    Perhaps, but, of course, it would then be that first premise that those who believe in an immaterial soul or similar idea would attack, but I shan't press further.

                                    "I think properties, like velocity, are completely abstract concepts. Like Nietzsche, I don't think there are things that 'have' properties; I think that if one would strip a thing of all its properties, nothing would remain.
                                    >
                                    > A property, then, is not a thing. This seems in line with what I said, as one cannot think of a property, e.g. velocity, in itself; one can only think of an object moving at a certain velocity, for instance."

                                    I fully agree with that statement.  So, my question is, then, if we imagine a region with neither space nor time, would it be that quality that gives it its "nothingness," or the fact that it has no objects in it?

                                    "This is already a bit too abstract language for my liking: "contains a predicate measuring". Anyway, I guess to be finite is to be bounded, and to be infinite is to not be bounded. Does that answer your question somewhat?"

                                    For future reference, a predicate is a statement that applies to an object x, such as "x is finite," "x is unthinkable," etc, but I'll try to stay away from abstractions like this from now on.  You've already somewhat answered my question somewhat, by defining "existence" as, basically, the set of all things that exists.  My understanding is now that to say that "existence is finite" is to say that this set is finite (i.e. there are only finitely many objects that exist), and to say that "existence is infinite" is to say that this is an infinite set.  Is this accurate?

                                    "I don't know how to define something by truth tables."

                                    Well, basically you would define an operation by taking an input value (true or false) and showing the output corresponding to that input.  For example, the NOT operation ~ is defined by:

                                    A|~A

                                    T|F
                                    F|T

                                    However, I've already done this for you, by assuming that syllogism 6 is valid and working backwards.  The B operator (x is bounded by y) would have to work exactly like the conditional operator ==> (IF x THEN y) in order for this syllogism to work; the syllogism then reduces to a Modus Ponens argument.  The crux now rests on how this operator does exactly that.  However, if the B operator did work that way, syllogisms like the one I presented would work.

                                    If you want to get technical about the atmosphere being part of the Earth, then, since the Earth itself (including its atmosphere) is surrounded by almost nothing but hydrogen gas, we would still have syllogisms like this:

                                    -1.  The Earth is enclosed by hydrogen gas [Experience].
                                    -2.  Hydrogen gas contains 100% hydrogen [Definition]
                                     C.  The Earth contains 100% hydrogen [Given (1) and (2)].

                                    This hydrogen gas is not really part of the Earth, and we know that the Earth is not 100% hydrogen.   Therefore, we would have to make the argument that syllogism (6) is only valid if we include both the B operator and the U(x) predicate.  In other words, you have to show that anything enclosed by something unthinkable is necessarily unthinkable.  Indeed, you try to do just that:

                                    "If existence *is* enclosed by 'nothing', then existence as it *is* cannot be conceived without conceiving it as enclosed by 'nothing'; and as 'nothing' cannot be conceived, existence as it is cannot be conceived."

                                    How is this true?  I can conceive the Earth without conceiving it as being enclosed by hydrogen gas.  Why does unthinkability get a free pass?  We seem to have a special pleading fallacy here.

                                    While you're waiting to respond to my definition of "curvature," I would like to elaborate a little more.  It seems that I've defined all of the terms that you did not understand, so if you don't understand any of those definitions, please let me know.

                                    To recap, my revised definition was:

                                    "A space X is curved insofar as the distance between two arbitrarily close points in X cannot be related by a one-to-one function to the distance between two arbitrarily close points in a Euclidean space Y"

                                    although, that definition is not quite correct.  I should have said "A space X is curved insofar as two arbitrarily close points in X cannot be related by a one-to-one function to two arbitrarily close points in a Euclidean space Y such that the distance between those points is equal in both spaces."

                                    However, let's look at a more intuitive, albeit less precise, definition.  Let's imagine an extrinsically curved object such as a globe, and drawing an evenly-spaced grid on its surface.  Then, let's imagine doing the same thing to a flat sheet of paper, which represents zero curvature.  Can we reversibly map the points on the globe to those on the paper in such a way that the distances between those points are equal?  No, and here's why:  Our so-called "evenly-spaced grid" on the globe is *not* evenly spaced after all.  This (rather crude) analogy brings me to my third (and hopefully final) definition:

                                    "A space is curved if and only if an evenly-spaced grid cannot be constructed within it.  The curvature metric, then, is a measurement of how this grid deviates from being evenly-spaced."

                                    We know, for example, that space-time is curved, because, while the curvature is too small to be noticed with the naked eye, we've measured our "grid" at the quantum level and found small deviations.  Space-time is also curved more than usual in the presence of a force field, and we interpret this "excess curvature" as force.

                                    ~Ian

                                    P.S.  Wow, have I completely lost my ability to write short posts?  Is this what it's come to?

                                    --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@...> wrote:
                                    >
                                    > --- In human_superhuman@yahoogroups.com, "Ian" ianmathwiz7@ wrote:
                                    > >
                                    > > Thank you for taking the time to write those syllogisms. Some problems
                                    > > with it are:
                                    > >
                                    > > 1
                                    > >
                                    > > (a): "To think is to form or have in the mind"
                                    > >
                                    > > I agree with this definition, as long as we remember that when we say
                                    > > that something is "in the mind," we don't mean that we actually have
                                    > > that thing in the mind; rather, we have a concept or representation of
                                    > > that thing in the mind. I would thus amend that definition to
                                    > >
                                    > > To think is to form or have a concept in the mind
                                    > >
                                    > > simply to make this clear. However, you can still logically derive
                                    > > "Whatever is thinkable is a thing" from that definition, as long as you
                                    > > use a precise and appropriate definition of "concept," so the soundness
                                    > > of your argument is unaffected here.
                                    > >
                                    >
                                    > Well, for one thing, I'd say a concept *is* a thing. I don't mean that all things can be formed or had in the mind.
                                    >
                                    >
                                    > > (b): "The mind is finite [...]."
                                    > >
                                    > > I'm not going to dispute this premise myself, but it should be made
                                    > > clear that some would dispute this claim on several bases, such as a
                                    > > cosmic consciousness or some other assumption for which there is no
                                    > > evidence. Because there is no evidence for these claims at present,
                                    > > however, I don't dispute this premise at all (Incidentally, this premise
                                    > > also destroys the famous Ontological argument for God's existence, but I
                                    > > won't go there).
                                    > >
                                    >
                                    > Thank you. I would indeed not mind offending religious minds with that statement.
                                    >
                                    > A rational argument might be that the mind is a function of the brain, and that the brain is finite: therefore, the mind is finite.
                                    >
                                    >
                                    > > 5 and 6, meanwhile, are invalid syllogisms.
                                    > >
                                    > > In the case of 5, we suppose that existence is finite and derive the
                                    > > conclusion that space-time is enclosed by "nothingness." I'm disputing
                                    > > the validity of this syllogism on the grounds of Riemannian geometry,
                                    > > which I will get to in a moment.
                                    > >
                                    > > However, it depends on the precise definition of nothingness; or rather,
                                    > > of a "thing." Would you consider space-time a "thing?" For purposes
                                    > > such as GR, physicists consider it to be a property rather than a thing,
                                    > > rather like spin, velocity, etc. On what grounds do you dispute this
                                    > > (if at all)? If you do not dispute this, on the other hand, then the
                                    > > fact that space-time is finite does not mean that it is enclosed by
                                    > > "nothingness" at all.
                                    > >
                                    >
                                    > I did not say "space-time", but "existence". And I did not mean existence as a property, as in "the existence of that existent", but as the sum of all existents. At first, I wrote "the All".
                                    >
                                    > I think properties, like velocity, are completely abstract concepts. Like Nietzsche, I don't think there are things that 'have' properties; I think that if one would strip a thing of all its properties, nothing would remain.
                                    >
                                    > A property, then, is not a thing. This seems in line with what I said, as one cannot think of a property, e.g. velocity, in itself; one can only think of an object moving at a certain velocity, for instance.
                                    >
                                    >
                                    > > I have a more fundamental objection, however: What do you mean when you
                                    > > say that "existence is finite/infinite?" What is it about existence
                                    > > that contains a predicate measuring whether it is finite or infinite?
                                    > >
                                    >
                                    > This is already a bit too abstract language for my liking: "contains a predicate measuring". Anyway, I guess to be finite is to be bounded, and to be infinite is to not be bounded. Does that answer your question somewhat?
                                    >
                                    >
                                    > > In the case of 6, this syllogism seems rather intuitively true, until we
                                    > > write it in predicate form:
                                    > >
                                    > > LET e="existence" AS String
                                    > > n="nothingness" AS String
                                    > > U(x)="x is unthinkable" AS Proposition
                                    > > x B y="x is enclosed by y as by a boundary" AS Operator
                                    > >
                                    > > -1. e B n [Given 5 (c)]
                                    > > -2. U(n) [Given 4 (c)]
                                    > > C. U(e) [From (1) and (2)]
                                    > >
                                    > > The problem here is, B is not a well-defined operator. Predicates like
                                    > > U(x) don't just transfer between two variables just because you can put
                                    > > both of them in a syllogism together. Define the operator B for me,
                                    > > preferably by truth tables, and show how this corresponds to something
                                    > > "being enclosed" by something else, and I'll be able to actually
                                    > > evaluate the validity of this syllogism.
                                    > >
                                    > > I might add, however, that if B *did* work that way, the following
                                    > > syllogism wouldn't just be valid, but sound.
                                    > >
                                    > > -1. The Earth is enclosed by the atmosphere as by a boundary
                                    > > [Experience].
                                    > > -2. The atmosphere is transparent [Experience].
                                    > > C. The Earth is transparent [From (1) and (2)].
                                    > >
                                    >
                                    > Well, the atmosphere is not *completely* transparent: which is why we see it as blue (due to the oxygen). In the time of the dinosaurs, we would have seen it as orange (due to methane).
                                    >
                                    > Anyway, that doesn't harm your argument. And I don't know how to define something by truth tables.
                                    >
                                    > The Earth is indeed like the atmosphere insofar as the atmosphere is part of the Earth. That part of the Earth is then 'transparent', for instance, and so the statement "the Earth is transparent" is true (though "the *whole* Earth is transparent" would be false). This seems to already come somewhat close to what I meant.
                                    >
                                    > If existence *is* enclosed by 'nothing', then existence as it *is* cannot be conceived without conceiving it as enclosed by 'nothing'; and as 'nothing' cannot be conceived, existence as it is cannot be conceived.
                                    >
                                    >
                                    > >
                                    > >
                                    > > "With this definition, you [have] withdrawn too far into jargon for me
                                    > > to follow you."
                                    > >
                                    > > It honestly would have really helped if you pointed out precisely which
                                    > > terms you did not understand here. However, I will assume a basic high
                                    > > school literacy of mathematics, and particularly geometry, and try to
                                    > > break this definition down. I'm going to attempt to do so in a way that
                                    > > is both precise and understandably, but does not over-simplify it.
                                    > >
                                    > > (Intrinsic) curvature is a measure of the extent to which the metric
                                    > > tensor of a Riemannian manifold is locally non-isometric to a Euclidean
                                    > > manifold.
                                    > >
                                    >
                                    > "metric"
                                    > "tensor"
                                    > "Riemannian manifold"
                                    > "(non-)isometric"
                                    > "locally (non-)isometric"
                                    > "Euclidean manifold"
                                    >
                                    > I may go into the rest of your post later. I knew what the Greek word "metrikos" would mean, by the way.
                                    >
                                    >
                                    > > Metric tensor: actually fairly simple: it is a tensor that measures
                                    > > metrics.
                                    > >
                                    > > To understand what a tensor is, I'll draw an analogy to scalars,
                                    > > vectors, matrices, and the empty set. You, presumably, know that a
                                    > > matrix is a two dimensional array of numbers, such as:
                                    > >
                                    > > [ 3 5 9]
                                    > > [-2 0 1]
                                    > > [ 7 9 0]
                                    > >
                                    > > A vector, or a quantity with both magnitude and direction, can be
                                    > > represented by a one-dimensional array of numbers, such as (-3, 8, 0).
                                    > >
                                    > > A scalar (a term for just a number) can be represented by a zero
                                    > > dimensional array, as it only contains one number.
                                    > >
                                    > > The empty set, on the other hand, is -1 dimensional; it is represented
                                    > > as {}.
                                    > >
                                    > > You may have already guessed that a tensor is the general name for these
                                    > > things. The empty set is a -1 tensor, a scalar is a 0-tensor, a vector
                                    > > is a 1-tensor, and a matrix is a 2-tensor. You can go beyond two, by
                                    > > the way, and talk about 3-tensors, 4-tensors, or even (hypothetically)
                                    > > 1,789,032,853-tensors, though I can't think of a possible use for that
                                    > > last one.
                                    > >
                                    > > Metric comes from the greek word for "distance-measurement." A metric
                                    > > function is one that defines a distance between elements in a set (and
                                    > > not just distance in space; it is also used to measure angle and time).
                                    > >
                                    > > A metric tensor, then, is one that, given two elements, produces a
                                    > > quantity which defines the distance (in space, time, or angle) between
                                    > > them. This tensor is always a 2-tensor (a matrix).
                                    > >
                                    > > Riemannian manifold: For our purposes, space-time.
                                    > >
                                    > > Local: The region just barely around a point (or set of points). Put
                                    > > another way, given a point x0 and another point x1, which is a distance
                                    > > of dx away from x0, the local region around x0 is the region as dx gets
                                    > > arbitrarily small.
                                    > >
                                    > > Isometric: This one is the hardest for a layman to understand, and,
                                    > > unfortunately, the crux of the whole definition. For now, this
                                    > > definition will have to do. If you can't understand it, I'll try to use
                                    > > a better one, but this is a lot harder than it looks.
                                    > >
                                    > > Suppose we have two metric spaces (sets of points in which the notion of
                                    > > distance exists; e.g. space-time) X and Y. These spaces are said to be
                                    > > isomorphic iff (if and only if) there exists a one-to-one function f
                                    > > from X to Y such that for any a,b in X
                                    > >
                                    > > d(f(a),f(b))=d(a,b).
                                    > >
                                    > > In other words, the distance between a and b in space X is equal to the
                                    > > distance between corresponding points f(a) and f(b) in space Y.
                                    > >
                                    > > Non-isometric spaces, then, are spaces that do not meet this criteria,
                                    > > i.e. there is no such function.
                                    > >
                                    > > Euclidean manifold: Basically, a space-time in which Euclidean geometry
                                    > > holds exactly. This is basically the "zero-point" of my curvature
                                    > > measurement; just like temperature and mass have to have a point defined
                                    > > as "zero," so does curvature.
                                    > >
                                    > > The definition thus becomes:
                                    > >
                                    > > A space X is curved insofar as the distance between two arbitrarily
                                    > > close points in X cannot be related by a one-to-one function to the
                                    > > distance between two arbitrarily close points in a Euclidean space Y.
                                    > >
                                    > > Hope that definition helps,
                                    > >
                                    > > ~Ian
                                    > >
                                    > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@>
                                    > > wrote:
                                    > > >
                                    > > >
                                    > > >
                                    > > > --- In human_superhuman@yahoogroups.com, "Ian" ianmathwiz7@ wrote:
                                    > > > >
                                    > > > > "Well, as you have not grounded any of your assertions, there is
                                    > > nothing for me to go into."
                                    > > > >
                                    > > > > Do the words "Riemannian geometry" ring a bell here?
                                    > > > >
                                    > > >
                                    > > > I did some reading on it. That didn't help to ground your assertions.
                                    > > >
                                    > > >
                                    > > > > "I don't know what you mean by 'common sense'. The
                                    > > infinity/nothingness problem is based on *logic* [emphasis original]."
                                    > > > >
                                    > > > > So is Riemannian geometry.
                                    > > > >
                                    > > >
                                    > > > Of course.
                                    > > >
                                    > > >
                                    > > >
                                    > > > > Besides, if that were really true, you should be able to present
                                    > > your argument in a sound syllogism, or at least a paragraph proof. If
                                    > > you were to do so, I would then be able to tell exactly what, if
                                    > > anything, is wrong with your logic.
                                    > > > >
                                    > > >
                                    > > > Below are syllogisms. In each case, "a" is the first premise, "b" is
                                    > > the second premise, and "c" is the conclusion.
                                    > > >
                                    > > > 1
                                    > > > a. To think is to form or have in the mind:
                                    > > http://www.merriam-webster.com/dictionary/think
                                    > > > b. The mind is finite: this is a fact of experience.
                                    > > > c. Whatever is thinkable is finite.
                                    > > >
                                    > > > 2
                                    > > > a. Whatever is thinkable is finite: see 1.
                                    > > > b. Infinity is not finite: by definition.
                                    > > > c. Infinity is not thinkable.
                                    > > >
                                    > > > 3
                                    > > > a. To think is to form or have in the mind: see 1.
                                    > > > b. Only *things* (i.e., existents) can be formed or had: forming and
                                    > > having both require an object.
                                    > > > c. Whatever is thinkable is a thing.
                                    > > >
                                    > > > 4
                                    > > > a. Whatever is thinkable is a thing: see 3.
                                    > > > b. 'Nothing' is no thing: by definition.
                                    > > > c. 'Nothing' is not thinkable.
                                    > > >
                                    > > > 5
                                    > > > a. Existence is all that exists: by definition.
                                    > > > b. Existence is finite: suppose.
                                    > > > c. Existence is enclosed by 'nothing' as by a boundary.
                                    > > >
                                    > > > 6
                                    > > > a. Existence is enclosed by 'nothing' as by a boundary: see 5.
                                    > > > b. 'Nothing' is unthinkable: see 4.
                                    > > > c. Existence is unthinkable.
                                    > > >
                                    > > > 7
                                    > > > a. Existence is infinite: suppose.
                                    > > > b. Infinity is unthinkable: see 2.
                                    > > > c. Existence is unthinkable.
                                    > > >
                                    > > > 8
                                    > > > a. Existence is unthinkable as finite: see 6.
                                    > > > b. Existence is unthinkable as infinite: see 7.
                                    > > > c. Existence is unthinkable.
                                    > > >
                                    > > >
                                    > > > > If you can define "curve" without doing so in terms of Euclidean
                                    > > geometry (e.g., straight lines or plane surfaces), please do so. If you
                                    > > can't, a space-time surface apparently cannot be thought of as "curved
                                    > > without having to be curved within anything else"; it can then at most
                                    > > be thought of as "'curved' without having to be curved"---whatever that
                                    > > means.
                                    > > > >
                                    > > > >
                                    > > > > I don't understand why it cannot make reference to Euclidean
                                    > > geometry. If you mean in reference to a linear shape (a straight line
                                    > > or plane), then I can certainly do so. If you mean in reference to the
                                    > > geometry itself, that's impossible, as curvature has no meaning without
                                    > > straightness. I certainly don't understand how defining it in relation
                                    > > to the geometry itself means that "it can then at most be thought of as
                                    > > 'curved' without having to be 'curved' [...]."
                                    > > > >
                                    > > >
                                    > > > You haven't reproduced my quotation marks right in your last sentence.
                                    > > Anyway, that doesn't matter anymore:
                                    > > >
                                    > > >
                                    > > > > Before I give my definition, however, I will make the distinction,
                                    > > which I probably should have done from the outset (as you and I seem to
                                    > > be talking apples and oranges here, and I blame myself for this),
                                    > > between *intrinsic* and *extrinsic* curvature.
                                    > > > >
                                    > > > > You seem to be talking as though space-time could be extrinsically
                                    > > curved. This is the definition that your dictionary is using, as well
                                    > > as any layman. Be assured that from now on, when I use the word
                                    > > "curvature" without any label, I am not using this definition. The
                                    > > reason, in fact, that I brought up "common sense" in my last post is
                                    > > because our intuitive definition of curvature is the *extrinsic*
                                    > > definition, not the one that I will be using here.
                                    > > > >
                                    > > > > When we are talking about the "curvature of space-time," however, we
                                    > > are necessarily talking about *intrinsic* curvature. This definition is
                                    > > *not* the same as our intuitive definition. Indeed, the only reason we
                                    > > use the same word is because we can apply it to space and time being
                                    > > cyclical, hyperbolic, etc.
                                    > > > >
                                    > > > > With that in mind, my definition is:
                                    > > > >
                                    > > > > (Intrinsic) curvature is a measure of the extent to which the metric
                                    > > tensor of a Riemannian manifold is locally non-isometric to a Euclidean
                                    > > manifold.
                                    > > > >
                                    > > >
                                    > > > With this definition, you withdrawn too far into jargon for me to
                                    > > follow you.
                                    > > >
                                    > > >
                                    > > > > Finally, I again grace you with the suggestion that you actually do
                                    > > some reading on Riemannian geometry before you post again. This time, I
                                    > > will even be nice enough to provide some links:
                                    > > > >
                                    > > > > https://secure.wikimedia.org/wikipedia/en/wiki/Riemannian_geometry
                                    > > > > http://comet.lehman.cuny.edu/sormani/research/riemgeom.html
                                    > > > > http://www.mathpages.com/rr/s5-07/5-07.htm
                                    > > > >
                                    > > > > These pages give fairly accurate descriptions for the layman, and
                                    > > they are not that long.
                                    > > > >
                                    > > > > ~Ian
                                    > > > >
                                    > > > >
                                    > > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@>
                                    > > wrote:
                                    > > > > >
                                    > > > > > Well, as you have not grounded any of your assertions, there is
                                    > > nothing for me to go into.
                                    > > > > >
                                    > > > > > I don't know what you mean by "common sense". The
                                    > > infinity/nothingness problem is based on *logic*.
                                    > > > > >
                                    > > > > > You say: "a space-time surface can be curved, without having to be
                                    > > curved 'within' anything else". I think a curve must *by definition*
                                    > > exist within something else. Merriam-Webster's Online Dictionary, for
                                    > > instance, defines "curve" (noun) in terms of "curve" (verb), which it in
                                    > > turn defines as:
                                    > > > > >
                                    > > > > > "to have or take a turn, change, or deviation from a straight line
                                    > > or plane surface without sharp breaks or angularity"
                                    > > > > > http://www.merriam-webster.com/dictionary/curve
                                    > > > > >
                                    > > > > > If you can define "curve" without doing so in terms of Euclidean
                                    > > geometry (e.g., straight lines or plane surfaces), please do so. If you
                                    > > can't, a space-time surface apparently cannot be thought of as "curved
                                    > > without having to be curved within anything else"; it can then at most
                                    > > be thought of as "'curved' without having to be curved"---whatever that
                                    > > means.
                                    > > > > >
                                    > > > > >
                                    > > > > >
                                    > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@>
                                    > > wrote:
                                    > > > > > >
                                    > > > > > > "I don't think you've thought through the infinity/nothingness
                                    > > problem."
                                    > > > > > >
                                    > > > > > > Actually, I have thought this through, and I knew what, in
                                    > > essence, your argument was. GR predicts many phenomena that seem
                                    > > counter-intuitive to us humans, simply because we are used to energy
                                    > > scales where these phenomena aren't a major part of our lives. Quantum
                                    > > mechanics is even "stranger" according to our common sense.
                                    > > > > > >
                                    > > > > > > On the large energy/time scales that these theories deal with,
                                    > > our common sense fails. Massively. We conclude, therefore, that our
                                    > > common sense is not a good tool to use on these scales. For example,
                                    > > some quantum gravity theories (e.g. loop quantum gravity) predict that
                                    > > time is discrete, not continuous. This contradicts everything we
                                    > > intuitively know about time, because the smallest time unit is too damn
                                    > > small for us to notice.
                                    > > > > > >
                                    > > > > > > GR, which works almost perfectly on the timescales we're talking
                                    > > about here, is based on a non-Euclidean geometry (Euclidean geometry is
                                    > > the geometry that intuitively makes sense to us), which means that a
                                    > > space-time surface can be curved, without having to be curved "within"
                                    > > anything else; curvature is a metric property of space-time. I would
                                    > > recommend that you do some reading on Riemannian geometry, so that you
                                    > > can see this for yourself (although the mathematics are pretty
                                    > > advanced). With this in mind, questions like
                                    > > > > > >
                                    > > > > > > "If the space of the universe is the surface of a 4D
                                    > > hypersphere, what encloses the hypersphere [...]?"
                                    > > > > > >
                                    > > > > > > become as utterly meaningless as "What is the absolute velocity
                                    > > of the Earth?"
                                    > > > > > >
                                    > > > > > > ~Ian
                                    > > > > > >
                                    > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                    > > <sauwelios@> wrote:
                                    > > > > > > >
                                    > > > > > > > I don't think you've thought through the infinity/nothingness
                                    > > problem. Thus you say:
                                    > > > > > > >
                                    > > > > > > > "According to Einstein's theory of General Relativity (which
                                    > > is based heavily on Riemannian geometry), the space of the universe is
                                    > > cyclical (or, more accurately, is the surface of a 4D hypersphere),
                                    > > without there having to be anything other than the universe".
                                    > > > > > > >
                                    > > > > > > > If the space of the universe is the surface of a 4D
                                    > > hypersphere, what encloses the hypersphere -- 'nothingness'?
                                    > > > > > > >
                                    > > > > > > >
                                    > > > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@>
                                    > > wrote:
                                    > > > > > > > >
                                    > > > > > > > > "[I]t *is* necessarily true if space/time is all there is."
                                    > > > > > > > >
                                    > > > > > > > > Says who? According to Einstein's theory of General
                                    > > Relativity (which is based heavily on Riemannian geometry), the space of
                                    > > the universe is cyclical (or, more accurately, is the surface of a 4D
                                    > > hypersphere), without there having to be anything other than the
                                    > > universe, and without there having to be "nothingness" inside the
                                    > > sphere. According to Minkowskian geometry, which forms the other main
                                    > > geometric basis of GR, the same arguments can be applied to time. The
                                    > > *only reason* that this seems counter-intuitive to us is that, in our
                                    > > everyday lives, we don't encounter things that behave like spacetime
                                    > > does.
                                    > > > > > > > >
                                    > > > > > > > > Granted, GR doesn't work on small scales, and doesn't
                                    > > explain how spacetime is bent by forces other than gravity (and attempts
                                    > > thus far to explain this, e.g. supergravity, have failed). However, the
                                    > > theory is uncannily accurate on large scales (which *is* what we're
                                    > > talking about when we're talking about scales on which things like
                                    > > eternal recurrence become important), since gravity is the only force of
                                    > > any importance on this scale (since the nuclear forces only work on
                                    > > small scales, and electric charges nearly cancel out perfectly on this
                                    > > scale).
                                    > > > > > > > >
                                    > > > > > > > > ~Ian
                                    > > > > > > > >
                                    > > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                    > > <sauwelios@> wrote:
                                    > > > > > > > > >
                                    > > > > > > > > > I agree, but it *is* necessarily true if space/time is all
                                    > > there is (which is the case according to Nietzsche's theory of the
                                    > > eternal recurrence).
                                    > > > > > > > > >
                                    > > > > > > > > >
                                    > > > > > > > > > --- In human_superhuman@yahoogroups.com, "Ian"
                                    > > <ianmathwiz7@> wrote:
                                    > > > > > > > > > >
                                    > > > > > > > > > > By the way, it is not necessarily true that there is
                                    > > nothingness in the middle and the outside of curved, even circular,
                                    > > space (and, by Minkowski's extension, time). I believe we have Bernhard
                                    > > Riemann, one of the greatest mathematicians of all time, to thank for
                                    > > proving that theorem.
                                    > > > > > > > > > >
                                    > > > > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                    > > <sauwelios@> wrote:
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > I think we will always keep running into the
                                    > > infinity/nothingness problem.
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > If time or causality forms a circle, there must be
                                    > > nothingness around it and in the middle. But to say that there is
                                    > > nothingness in the middle is to say that there is *nothing* in the
                                    > > middle, which makes it something different from a circle. And to say
                                    > > that there is nothingness around it is to say that it is bounded by
                                    > > *nothing*, which means it is infinite. And infinity is unthinkable.
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > My contemplations on this problem, which I have here
                                    > > reported very briefly, drive me toward the conclusion that, contrary to
                                    > > the will to power, the eternal recurrence is not meant to be regarded as
                                    > > a fact. This seems to be supported by the fact that, as far as I know,
                                    > > Nietzsche never presented an argument for it in his published works,
                                    > > except perhaps in *TSZ*, which also stands out in every other way. There
                                    > > he provides an argument for it, but only in order to vanquish the dwarf
                                    > > ('The Vision and the Enigma'). And that only works because the dwarf has
                                    > > already asserted that time be a circle. Zarathustra is merely pointing
                                    > > out the *ramifications* of that idea to him, which is what then defeats
                                    > > him. But Zarathustra omits one of the two premises of his argument,
                                    > > which he only provides later:
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > "[My laughing, wide-awake day-wisdom, which mocketh at
                                    > > all 'infinite worlds',] saith: "Where force is, there becometh number
                                    > > the master: it hath more force.""
                                    > > > > > > > > > > > ['The Three Evil Things', 1.]
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > This premise, that force be finite, leads inevitably
                                    > > to the infinity/nothingness problem:
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > "This world: [...] enclosed by 'nothingness' as by a
                                    > > boundary; not something blurry or wasted, not something endlessly
                                    > > extended, but set in a definite space as a definite force, and not a
                                    > > space that might be 'empty' here or there, but rather as force
                                    > > throughout[.]"
                                    > > > > > > > > > > > [*WP* 1067.]
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > The phrase here translated as "'nothingness'" is *'das
                                    > > Nichts'*, "'the Nothing'". But what is the difference between 'the
                                    > > Nothing' and *nothing*? How does this not amount to saying that this
                                    > > world is bounded by *nothing*, i.e., not bounded at all? Unless it
                                    > > amounts to saying that, though there is no empty space *within* this
                                    > > world, there is an infinite empty space *outside* it. But if this were
                                    > > the case, its finite force would inevitably disperse into that infinite
                                    > > emptiness. In any case, neither emptiness nor infinity nor nothingness
                                    > > is thinkable, so we still arrive at the same problem Nietzsche tried to
                                    > > solve:
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > "[T]he world, as force, may not be thought of as
                                    > > unlimited, for it *cannot* be so thought of[.]"
                                    > > > > > > > > > > > [*WP* 1062.]
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > Likewise, it cannot be thought of as being enclosed by
                                    > > 'nothingness' as by a boundary. Hence I conclude that the eternal
                                    > > recurrence is not a fact but a *value*.
                                    > > > > > > > > > > >
                                    > > > > > > > > > > >
                                    > > > > > > > > > > > --- In human_superhuman@yahoogroups.com, "Fred"
                                    > > <nietzschefred@> wrote:
                                    > > > > > > > > > > > >
                                    > > > > > > > > > > > >
                                    > > > > > > > > > > > > New model of the universe does away with the
                                    > > singularity, beginning and end and describes physical laws as constantly
                                    > > changing and evolving...there are no cosmological constants.
                                    > > > > > > > > > > > >
                                    > > > > > > > > > > > >
                                    > > > > > > > > > > > > • The speed of light and the gravitational
                                    > > "constant" are not constant, but vary with the evolution of the
                                    > > universe.
                                    > > > > > > > > > > > > • Time has no beginning and no end; i.e., there
                                    > > is neither a big bang nor a big crunch singularity.
                                    > > > > > > > > > > > > • The spatial section of the universe is a
                                    > > 3-sphere [a higher-dimensional analogue of a sphere], ruling out the
                                    > > possibility of a flat or hyperboloid geometry.
                                    > > > > > > > > > > > > • The universe experiences phases of both
                                    > > acceleration and deceleration.
                                    > > > > > > > > > > > >
                                    > > > > > > > > > > > > New life for a scientific consideration of the
                                    > > eternal recurrence?
                                    > > > > > > > > > > > >
                                    > > > > > > > > > > > > http://www.physorg.com/news199591806.html
                                    > > > > > > > > > > > >
                                    > > > > > > > > > > >
                                    > > > > > > > > > >
                                    > > > > > > > > >
                                    > > > > > > > >
                                    > > > > > > >
                                    > > > > > >
                                    > > > > >
                                    > > > >
                                    > > >
                                    > >
                                    >
                                  • sauwelios
                                    ... And by saying that a concept is a thing, I did not mean to imply that it had substance . So it s not a metaphysical, but a definitional argument. I define
                                    Message 17 of 24 , Oct 5, 2010
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                                      --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@...> wrote:
                                      >
                                      > "Well, for one thing, I'd say a concept *is* a thing. I don't mean that
                                      > all things can be formed or had in the mind."
                                      >
                                      > Well, we can dispute that. Ultimately, it's just a metaphysical
                                      > argument, and, as it does not take away from the soundness of your
                                      > argument anyway, I think that everyone would appreciate it if we stuck
                                      > to things that matter. Also, I didn't mean to imply that you said that
                                      > all things can exist in the mind.
                                      >

                                      And by saying that a concept is a thing, I did not mean to imply that it had 'substance'. So it's not a metaphysical, but a definitional argument. I define "thing" so, that it can refer to concepts. Compare again Merriam-Webster:

                                      "3 a : a separate and distinct individual quality, fact, idea, or usually entity"
                                      http://www.merriam-webster.com/dictionary/thing

                                      Though it usually refers to entities, it may also refer to ideas.

                                      Is entities what you meant by "things that matter"?


                                      >
                                      > "A rational argument might be that the mind is a function of the brain,
                                      > and that the brain is finite: therefore, the mind is finite."
                                      >
                                      > Perhaps, but, of course, it would then be that first premise that those
                                      > who believe in an immaterial soul or similar idea would attack, but I
                                      > shan't press further.
                                      >

                                      Surely, but we need not take that seriously---intellectually, at least (religion remains of course a force to be reckoned with).


                                      > "I think properties, like velocity, are completely abstract concepts.
                                      > Like Nietzsche, I don't think there are things that 'have' properties; I
                                      > think that if one would strip a thing of all its properties, nothing
                                      > would remain.
                                      > >
                                      > > A property, then, is not a thing. This seems in line with what I said,
                                      > as one cannot think of a property, e.g. velocity, in itself; one can
                                      > only think of an object moving at a certain velocity, for instance."
                                      >
                                      > I fully agree with that statement. So, my question is, then, if we
                                      > imagine a region with neither space nor time, would it be that quality
                                      > that gives it its "nothingness," or the fact that it has no objects in
                                      > it?
                                      >

                                      To me, this is an utterly meaningless question. "A region with neither space nor time" does not mean anything to me. Therefore, neither does "imagining" it. I think "region" already refers to space---or, by extension, to time.

                                      Your question also seems to suppose that nothingness is a *quality*.

                                      In my experience, talking about nothingness always leads to logical contradictions. I think this is further evidence for my claim that nothingness be unthinkable for us.


                                      > "This is already a bit too abstract language for my liking: "contains a
                                      > predicate measuring". Anyway, I guess to be finite is to be bounded, and
                                      > to be infinite is to not be bounded. Does that answer your question
                                      > somewhat?"
                                      >
                                      > For future reference, a predicate is a statement that applies to an
                                      > object x, such as "x is finite," "x is unthinkable," etc, but I'll try
                                      > to stay away from abstractions like this from now on. You've already
                                      > somewhat answered my question somewhat, by defining "existence" as,
                                      > basically, the set of all things that exists. My understanding is now
                                      > that to say that "existence is finite" is to say that this set is finite
                                      > (i.e. there are only finitely many objects that exist), and to say that
                                      > "existence is infinite" is to say that this is an infinite set. Is this
                                      > accurate?
                                      >

                                      Not quite, as objects tend to consist of parts that are objects in themselves. So instead of "objects", I would say "fundamental particles". If fundamental particles exist, then yes, "existence is finite" means the set of all fundamental particles is finite, whereas "existence is infinity" means that set is infinite.


                                      > "I don't know how to define something by truth tables."
                                      >
                                      > Well, basically you would define an operation by taking an input value
                                      > (true or false) and showing the output corresponding to that input. For
                                      > example, the NOT operation ~ is defined by:
                                      >
                                      > A|~A
                                      > T|F
                                      > F|T
                                      >
                                      > However, I've already done this for you, by assuming that syllogism 6 is
                                      > valid and working backwards. The B operator (x is bounded by y) would
                                      > have to work exactly like the conditional operator ==> (IF x THEN y) in
                                      > order for this syllogism to work; the syllogism then reduces to a Modus
                                      > Ponens argument. The crux now rests on how this operator does exactly
                                      > that. However, if the B operator did work that way, syllogisms like the
                                      > one I presented would work.
                                      >
                                      > If you want to get technical about the atmosphere being part of the
                                      > Earth, then, since the Earth itself (including its atmosphere) is
                                      > surrounded by almost nothing but hydrogen gas, we would still have
                                      > syllogisms like this:
                                      >
                                      > -1. The Earth is enclosed by hydrogen gas [Experience].
                                      > -2. Hydrogen gas contains 100% hydrogen [Definition]
                                      > C. The Earth contains 100% hydrogen [Given (1) and (2)].
                                      >
                                      > This hydrogen gas is not really part of the Earth, and we know that the
                                      > Earth is not 100% hydrogen. Therefore, we would have to make the
                                      > argument that syllogism (6) is only valid if we include both the B
                                      > operator and the U(x) predicate. In other words, you have to show that
                                      > anything enclosed by something unthinkable is necessarily unthinkable.
                                      > Indeed, you try to do just that:
                                      >
                                      > "If existence *is* enclosed by 'nothing', then existence as it *is*
                                      > cannot be conceived without conceiving it as enclosed by 'nothing'; and
                                      > as 'nothing' cannot be conceived, existence as it is cannot be
                                      > conceived."
                                      >
                                      > How is this true? I can conceive the Earth without conceiving it as
                                      > being enclosed by hydrogen gas. Why does unthinkability get a free
                                      > pass? We seem to have a special pleading fallacy here.
                                      >

                                      Well, can you conceive the Earth as not being enclosed by *anything*?

                                      By the way, even if you can conceive the Earth as being enclosed by nitrogen gas, for instance, you would then not be conceiving the Earth as it *is*---that is, you would technically not be conceiving the Earth at all. In your terms, being enclosed by hydrogen gas is a *property* of the Earth.

                                      I do appreciate your endeavour to formulate my argument in terms of formal logic. I had some more thoughts which it seems in order to express now. I think that what you've been saying about the universe is that it's neither infinite nor enclosed by nothingness, but enclosed by *itself*. And Nietzsche seems to say much the same thing:

                                      "The world [...] lives on itself: its excrements are its food."
                                      (WP 1066.)

                                      This, however, is further evidence for my claim that Nietzsche's philosophy be the philosophy of the eternal recurrence: "philosophy" understood in the sense of BGE 9:

                                      "Philosophy is [the] tyrannical drive [to create^ the world in its image], the most spiritual will to power, to 'creation^ of the world', to the causa prima ["first cause"]."

                                      For the idea of something being enclosed by itself is like the idea of a self-cause: it is absurd.

                                      ^ The word translated as "create" is *schaffen*, cognate with "shape". Compare:

                                      ""Will to Truth" do ye call it, ye wisest ones [i.e., genuine philosophers], that which impelleth you and maketh you ardent?
                                      Will for the thinkableness of all being: thus do I call your will!
                                      All being would ye make thinkable: for ye doubt with good reason whether it be already thinkable.
                                      But it shall accommodate and bend itself to you! So willeth your will. Smooth shall it become and subject to the spirit, as its mirror and reflection."
                                      (Nietzsche, TSZ, "Self-Surpassing".)

                                      "In the new wisdom of [Nietzsche's] Zarathustra, the wise man, the spirited knower, is a lover who transforms the beloved [the world as will to power] into something still more beautiful than she is [namely, into the world as *eternally recurring* will to power], and she is beautiful as she is."
                                      [Lampert, *Nietzsche's Teaching*, page 120.]


                                      > While you're waiting to respond to my definition of "curvature," I would
                                      > like to elaborate a little more. It seems that I've defined all of the
                                      > terms that you did not understand, so if you don't understand any of
                                      > those definitions, please let me know.
                                      >
                                      > To recap, my revised definition was:
                                      >
                                      > "A space X is curved insofar as the distance between two arbitrarily
                                      > close points in X cannot be related by a one-to-one function to the
                                      > distance between two arbitrarily close points in a Euclidean space Y"
                                      >
                                      > although, that definition is not quite correct. I should have said "A
                                      > space X is curved insofar as two arbitrarily close points in X cannot
                                      > be related by a one-to-one function to two arbitrarily close points in a
                                      > Euclidean space Y such that the distance between those points is equal
                                      > in both spaces."
                                      >

                                      Ah, that actually makes sense to me. Thanks!


                                      > However, let's look at a more intuitive, albeit less precise,
                                      > definition. Let's imagine an extrinsically curved object such as a
                                      > globe, and drawing an evenly-spaced grid on its surface. Then, let's
                                      > imagine doing the same thing to a flat sheet of paper, which represents
                                      > zero curvature. Can we reversibly map the points on the globe to those
                                      > on the paper in such a way that the distances between those points are
                                      > equal? No, and here's why: Our so-called "evenly-spaced grid" on the
                                      > globe is *not* evenly spaced after all. This (rather crude) analogy
                                      > brings me to my third (and hopefully final) definition:
                                      >
                                      > "A space is curved if and only if an evenly-spaced grid cannot be
                                      > constructed within it. The curvature metric, then, is a measurement of
                                      > how this grid deviates from being evenly-spaced."
                                      >
                                      > We know, for example, that space-time is curved, because, while the
                                      > curvature is too small to be noticed with the naked eye, we've measured
                                      > our "grid" at the quantum level and found small deviations. Space-time
                                      > is also curved more than usual in the presence of a force field, and we
                                      > interpret this "excess curvature" as force.
                                      >
                                      > ~Ian
                                      >
                                      > P.S. Wow, have I completely lost my ability to write short posts? Is
                                      > this what it's come to?
                                      >
                                      > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@>
                                      > wrote:
                                      > >
                                      > > --- In human_superhuman@yahoogroups.com, "Ian" ianmathwiz7@ wrote:
                                      > > >
                                      > > > Thank you for taking the time to write those syllogisms. Some
                                      > problems
                                      > > > with it are:
                                      > > >
                                      > > > 1
                                      > > >
                                      > > > (a): "To think is to form or have in the mind"
                                      > > >
                                      > > > I agree with this definition, as long as we remember that when we
                                      > say
                                      > > > that something is "in the mind," we don't mean that we actually have
                                      > > > that thing in the mind; rather, we have a concept or representation
                                      > of
                                      > > > that thing in the mind. I would thus amend that definition to
                                      > > >
                                      > > > To think is to form or have a concept in the mind
                                      > > >
                                      > > > simply to make this clear. However, you can still logically derive
                                      > > > "Whatever is thinkable is a thing" from that definition, as long as
                                      > you
                                      > > > use a precise and appropriate definition of "concept," so the
                                      > soundness
                                      > > > of your argument is unaffected here.
                                      > > >
                                      > >
                                      > > Well, for one thing, I'd say a concept *is* a thing. I don't mean that
                                      > all things can be formed or had in the mind.
                                      > >
                                      > >
                                      > > > (b): "The mind is finite [...]."
                                      > > >
                                      > > > I'm not going to dispute this premise myself, but it should be made
                                      > > > clear that some would dispute this claim on several bases, such as a
                                      > > > cosmic consciousness or some other assumption for which there is no
                                      > > > evidence. Because there is no evidence for these claims at present,
                                      > > > however, I don't dispute this premise at all (Incidentally, this
                                      > premise
                                      > > > also destroys the famous Ontological argument for God's existence,
                                      > but I
                                      > > > won't go there).
                                      > > >
                                      > >
                                      > > Thank you. I would indeed not mind offending religious minds with that
                                      > statement.
                                      > >
                                      > > A rational argument might be that the mind is a function of the brain,
                                      > and that the brain is finite: therefore, the mind is finite.
                                      > >
                                      > >
                                      > > > 5 and 6, meanwhile, are invalid syllogisms.
                                      > > >
                                      > > > In the case of 5, we suppose that existence is finite and derive the
                                      > > > conclusion that space-time is enclosed by "nothingness." I'm
                                      > disputing
                                      > > > the validity of this syllogism on the grounds of Riemannian
                                      > geometry,
                                      > > > which I will get to in a moment.
                                      > > >
                                      > > > However, it depends on the precise definition of nothingness; or
                                      > rather,
                                      > > > of a "thing." Would you consider space-time a "thing?" For
                                      > purposes
                                      > > > such as GR, physicists consider it to be a property rather than a
                                      > thing,
                                      > > > rather like spin, velocity, etc. On what grounds do you dispute
                                      > this
                                      > > > (if at all)? If you do not dispute this, on the other hand, then
                                      > the
                                      > > > fact that space-time is finite does not mean that it is enclosed by
                                      > > > "nothingness" at all.
                                      > > >
                                      > >
                                      > > I did not say "space-time", but "existence". And I did not mean
                                      > existence as a property, as in "the existence of that existent", but as
                                      > the sum of all existents. At first, I wrote "the All".
                                      > >
                                      > > I think properties, like velocity, are completely abstract concepts.
                                      > Like Nietzsche, I don't think there are things that 'have' properties; I
                                      > think that if one would strip a thing of all its properties, nothing
                                      > would remain.
                                      > >
                                      > > A property, then, is not a thing. This seems in line with what I said,
                                      > as one cannot think of a property, e.g. velocity, in itself; one can
                                      > only think of an object moving at a certain velocity, for instance.
                                      > >
                                      > >
                                      > > > I have a more fundamental objection, however: What do you mean when
                                      > you
                                      > > > say that "existence is finite/infinite?" What is it about existence
                                      > > > that contains a predicate measuring whether it is finite or
                                      > infinite?
                                      > > >
                                      > >
                                      > > This is already a bit too abstract language for my liking: "contains a
                                      > predicate measuring". Anyway, I guess to be finite is to be bounded, and
                                      > to be infinite is to not be bounded. Does that answer your question
                                      > somewhat?
                                      > >
                                      > >
                                      > > > In the case of 6, this syllogism seems rather intuitively true,
                                      > until we
                                      > > > write it in predicate form:
                                      > > >
                                      > > > LET e="existence" AS String
                                      > > > n="nothingness" AS String
                                      > > > U(x)="x is unthinkable" AS Proposition
                                      > > > x B y="x is enclosed by y as by a boundary" AS Operator
                                      > > >
                                      > > > -1. e B n [Given 5 (c)]
                                      > > > -2. U(n) [Given 4 (c)]
                                      > > > C. U(e) [From (1) and (2)]
                                      > > >
                                      > > > The problem here is, B is not a well-defined operator. Predicates
                                      > like
                                      > > > U(x) don't just transfer between two variables just because you can
                                      > put
                                      > > > both of them in a syllogism together. Define the operator B for me,
                                      > > > preferably by truth tables, and show how this corresponds to
                                      > something
                                      > > > "being enclosed" by something else, and I'll be able to actually
                                      > > > evaluate the validity of this syllogism.
                                      > > >
                                      > > > I might add, however, that if B *did* work that way, the following
                                      > > > syllogism wouldn't just be valid, but sound.
                                      > > >
                                      > > > -1. The Earth is enclosed by the atmosphere as by a boundary
                                      > > > [Experience].
                                      > > > -2. The atmosphere is transparent [Experience].
                                      > > > C. The Earth is transparent [From (1) and (2)].
                                      > > >
                                      > >
                                      > > Well, the atmosphere is not *completely* transparent: which is why we
                                      > see it as blue (due to the oxygen). In the time of the dinosaurs, we
                                      > would have seen it as orange (due to methane).
                                      > >
                                      > > Anyway, that doesn't harm your argument. And I don't know how to
                                      > define something by truth tables.
                                      > >
                                      > > The Earth is indeed like the atmosphere insofar as the atmosphere is
                                      > part of the Earth. That part of the Earth is then 'transparent', for
                                      > instance, and so the statement "the Earth is transparent" is true
                                      > (though "the *whole* Earth is transparent" would be false). This seems
                                      > to already come somewhat close to what I meant.
                                      > >
                                      > > If existence *is* enclosed by 'nothing', then existence as it *is*
                                      > cannot be conceived without conceiving it as enclosed by 'nothing'; and
                                      > as 'nothing' cannot be conceived, existence as it is cannot be
                                      > conceived.
                                      > >
                                      > >
                                      > > >
                                      > > >
                                      > > > "With this definition, you [have] withdrawn too far into jargon for
                                      > me
                                      > > > to follow you."
                                      > > >
                                      > > > It honestly would have really helped if you pointed out precisely
                                      > which
                                      > > > terms you did not understand here. However, I will assume a basic
                                      > high
                                      > > > school literacy of mathematics, and particularly geometry, and try
                                      > to
                                      > > > break this definition down. I'm going to attempt to do so in a way
                                      > that
                                      > > > is both precise and understandably, but does not over-simplify it.
                                      > > >
                                      > > > (Intrinsic) curvature is a measure of the extent to which the metric
                                      > > > tensor of a Riemannian manifold is locally non-isometric to a
                                      > Euclidean
                                      > > > manifold.
                                      > > >
                                      > >
                                      > > "metric"
                                      > > "tensor"
                                      > > "Riemannian manifold"
                                      > > "(non-)isometric"
                                      > > "locally (non-)isometric"
                                      > > "Euclidean manifold"
                                      > >
                                      > > I may go into the rest of your post later. I knew what the Greek word
                                      > "metrikos" would mean, by the way.
                                      > >
                                      > >
                                      > > > Metric tensor: actually fairly simple: it is a tensor that
                                      > measures
                                      > > > metrics.
                                      > > >
                                      > > > To understand what a tensor is, I'll draw an analogy to scalars,
                                      > > > vectors, matrices, and the empty set. You, presumably, know that a
                                      > > > matrix is a two dimensional array of numbers, such as:
                                      > > >
                                      > > > [ 3 5 9]
                                      > > > [-2 0 1]
                                      > > > [ 7 9 0]
                                      > > >
                                      > > > A vector, or a quantity with both magnitude and direction, can be
                                      > > > represented by a one-dimensional array of numbers, such as (-3, 8,
                                      > 0).
                                      > > >
                                      > > > A scalar (a term for just a number) can be represented by a zero
                                      > > > dimensional array, as it only contains one number.
                                      > > >
                                      > > > The empty set, on the other hand, is -1 dimensional; it is
                                      > represented
                                      > > > as {}.
                                      > > >
                                      > > > You may have already guessed that a tensor is the general name for
                                      > these
                                      > > > things. The empty set is a -1 tensor, a scalar is a 0-tensor, a
                                      > vector
                                      > > > is a 1-tensor, and a matrix is a 2-tensor. You can go beyond two,
                                      > by
                                      > > > the way, and talk about 3-tensors, 4-tensors, or even
                                      > (hypothetically)
                                      > > > 1,789,032,853-tensors, though I can't think of a possible use for
                                      > that
                                      > > > last one.
                                      > > >
                                      > > > Metric comes from the greek word for "distance-measurement." A
                                      > metric
                                      > > > function is one that defines a distance between elements in a set
                                      > (and
                                      > > > not just distance in space; it is also used to measure angle and
                                      > time).
                                      > > >
                                      > > > A metric tensor, then, is one that, given two elements, produces a
                                      > > > quantity which defines the distance (in space, time, or angle)
                                      > between
                                      > > > them. This tensor is always a 2-tensor (a matrix).
                                      > > >
                                      > > > Riemannian manifold: For our purposes, space-time.
                                      > > >
                                      > > > Local: The region just barely around a point (or set of points).
                                      > Put
                                      > > > another way, given a point x0 and another point x1, which is a
                                      > distance
                                      > > > of dx away from x0, the local region around x0 is the region as dx
                                      > gets
                                      > > > arbitrarily small.
                                      > > >
                                      > > > Isometric: This one is the hardest for a layman to understand, and,
                                      > > > unfortunately, the crux of the whole definition. For now, this
                                      > > > definition will have to do. If you can't understand it, I'll try to
                                      > use
                                      > > > a better one, but this is a lot harder than it looks.
                                      > > >
                                      > > > Suppose we have two metric spaces (sets of points in which the
                                      > notion of
                                      > > > distance exists; e.g. space-time) X and Y. These spaces are said to
                                      > be
                                      > > > isomorphic iff (if and only if) there exists a one-to-one function
                                      > f
                                      > > > from X to Y such that for any a,b in X
                                      > > >
                                      > > > d(f(a),f(b))=d(a,b).
                                      > > >
                                      > > > In other words, the distance between a and b in space X is equal to
                                      > the
                                      > > > distance between corresponding points f(a) and f(b) in space Y.
                                      > > >
                                      > > > Non-isometric spaces, then, are spaces that do not meet this
                                      > criteria,
                                      > > > i.e. there is no such function.
                                      > > >
                                      > > > Euclidean manifold: Basically, a space-time in which Euclidean
                                      > geometry
                                      > > > holds exactly. This is basically the "zero-point" of my curvature
                                      > > > measurement; just like temperature and mass have to have a point
                                      > defined
                                      > > > as "zero," so does curvature.
                                      > > >
                                      > > > The definition thus becomes:
                                      > > >
                                      > > > A space X is curved insofar as the distance between two arbitrarily
                                      > > > close points in X cannot be related by a one-to-one function to the
                                      > > > distance between two arbitrarily close points in a Euclidean space
                                      > Y.
                                      > > >
                                      > > > Hope that definition helps,
                                      > > >
                                      > > > ~Ian
                                      > > >
                                      > > > --- In human_superhuman@yahoogroups.com, "sauwelios" <sauwelios@>
                                      > > > wrote:
                                      > > > >
                                      > > > >
                                      > > > >
                                      > > > > --- In human_superhuman@yahoogroups.com, "Ian" ianmathwiz7@ wrote:
                                      > > > > >
                                      > > > > > "Well, as you have not grounded any of your assertions, there is
                                      > > > nothing for me to go into."
                                      > > > > >
                                      > > > > > Do the words "Riemannian geometry" ring a bell here?
                                      > > > > >
                                      > > > >
                                      > > > > I did some reading on it. That didn't help to ground your
                                      > assertions.
                                      > > > >
                                      > > > >
                                      > > > > > "I don't know what you mean by 'common sense'. The
                                      > > > infinity/nothingness problem is based on *logic* [emphasis
                                      > original]."
                                      > > > > >
                                      > > > > > So is Riemannian geometry.
                                      > > > > >
                                      > > > >
                                      > > > > Of course.
                                      > > > >
                                      > > > >
                                      > > > >
                                      > > > > > Besides, if that were really true, you should be able to present
                                      > > > your argument in a sound syllogism, or at least a paragraph proof.
                                      > If
                                      > > > you were to do so, I would then be able to tell exactly what, if
                                      > > > anything, is wrong with your logic.
                                      > > > > >
                                      > > > >
                                      > > > > Below are syllogisms. In each case, "a" is the first premise, "b"
                                      > is
                                      > > > the second premise, and "c" is the conclusion.
                                      > > > >
                                      > > > > 1
                                      > > > > a. To think is to form or have in the mind:
                                      > > > http://www.merriam-webster.com/dictionary/think
                                      > > > > b. The mind is finite: this is a fact of experience.
                                      > > > > c. Whatever is thinkable is finite.
                                      > > > >
                                      > > > > 2
                                      > > > > a. Whatever is thinkable is finite: see 1.
                                      > > > > b. Infinity is not finite: by definition.
                                      > > > > c. Infinity is not thinkable.
                                      > > > >
                                      > > > > 3
                                      > > > > a. To think is to form or have in the mind: see 1.
                                      > > > > b. Only *things* (i.e., existents) can be formed or had: forming
                                      > and
                                      > > > having both require an object.
                                      > > > > c. Whatever is thinkable is a thing.
                                      > > > >
                                      > > > > 4
                                      > > > > a. Whatever is thinkable is a thing: see 3.
                                      > > > > b. 'Nothing' is no thing: by definition.
                                      > > > > c. 'Nothing' is not thinkable.
                                      > > > >
                                      > > > > 5
                                      > > > > a. Existence is all that exists: by definition.
                                      > > > > b. Existence is finite: suppose.
                                      > > > > c. Existence is enclosed by 'nothing' as by a boundary.
                                      > > > >
                                      > > > > 6
                                      > > > > a. Existence is enclosed by 'nothing' as by a boundary: see 5.
                                      > > > > b. 'Nothing' is unthinkable: see 4.
                                      > > > > c. Existence is unthinkable.
                                      > > > >
                                      > > > > 7
                                      > > > > a. Existence is infinite: suppose.
                                      > > > > b. Infinity is unthinkable: see 2.
                                      > > > > c. Existence is unthinkable.
                                      > > > >
                                      > > > > 8
                                      > > > > a. Existence is unthinkable as finite: see 6.
                                      > > > > b. Existence is unthinkable as infinite: see 7.
                                      > > > > c. Existence is unthinkable.
                                      > > > >
                                      > > > >
                                      > > > > > If you can define "curve" without doing so in terms of Euclidean
                                      > > > geometry (e.g., straight lines or plane surfaces), please do so. If
                                      > you
                                      > > > can't, a space-time surface apparently cannot be thought of as
                                      > "curved
                                      > > > without having to be curved within anything else"; it can then at
                                      > most
                                      > > > be thought of as "'curved' without having to be curved"---whatever
                                      > that
                                      > > > means.
                                      > > > > >
                                      > > > > >
                                      > > > > > I don't understand why it cannot make reference to Euclidean
                                      > > > geometry. If you mean in reference to a linear shape (a straight
                                      > line
                                      > > > or plane), then I can certainly do so. If you mean in reference to
                                      > the
                                      > > > geometry itself, that's impossible, as curvature has no meaning
                                      > without
                                      > > > straightness. I certainly don't understand how defining it in
                                      > relation
                                      > > > to the geometry itself means that "it can then at most be thought of
                                      > as
                                      > > > 'curved' without having to be 'curved' [...]."
                                      > > > > >
                                      > > > >
                                      > > > > You haven't reproduced my quotation marks right in your last
                                      > sentence.
                                      > > > Anyway, that doesn't matter anymore:
                                      > > > >
                                      > > > >
                                      > > > > > Before I give my definition, however, I will make the
                                      > distinction,
                                      > > > which I probably should have done from the outset (as you and I seem
                                      > to
                                      > > > be talking apples and oranges here, and I blame myself for this),
                                      > > > between *intrinsic* and *extrinsic* curvature.
                                      > > > > >
                                      > > > > > You seem to be talking as though space-time could be
                                      > extrinsically
                                      > > > curved. This is the definition that your dictionary is using, as
                                      > well
                                      > > > as any layman. Be assured that from now on, when I use the word
                                      > > > "curvature" without any label, I am not using this definition. The
                                      > > > reason, in fact, that I brought up "common sense" in my last post is
                                      > > > because our intuitive definition of curvature is the *extrinsic*
                                      > > > definition, not the one that I will be using here.
                                      > > > > >
                                      > > > > > When we are talking about the "curvature of space-time,"
                                      > however, we
                                      > > > are necessarily talking about *intrinsic* curvature. This
                                      > definition is
                                      > > > *not* the same as our intuitive definition. Indeed, the only reason
                                      > we
                                      > > > use the same word is because we can apply it to space and time being
                                      > > > cyclical, hyperbolic, etc.
                                      > > > > >
                                      > > > > > With that in mind, my definition is:
                                      > > > > >
                                      > > > > > (Intrinsic) curvature is a measure of the extent to which the
                                      > metric
                                      > > > tensor of a Riemannian manifold is locally non-isometric to a
                                      > Euclidean
                                      > > > manifold.
                                      > > > > >
                                      > > > >
                                      > > > > With this definition, you withdrawn too far into jargon for me to
                                      > > > follow you.
                                      > > > >
                                      > > > >
                                      > > > > > Finally, I again grace you with the suggestion that you actually
                                      > do
                                      > > > some reading on Riemannian geometry before you post again. This
                                      > time, I
                                      > > > will even be nice enough to provide some links:
                                      > > > > >
                                      > > > > >
                                      > https://secure.wikimedia.org/wikipedia/en/wiki/Riemannian_geometry
                                      > > > > > http://comet.lehman.cuny.edu/sormani/research/riemgeom.html
                                      > > > > > http://www.mathpages.com/rr/s5-07/5-07.htm
                                      > > > > >
                                      > > > > > These pages give fairly accurate descriptions for the layman,
                                      > and
                                      > > > they are not that long.
                                      > > > > >
                                      > > > > > ~Ian
                                      > > > > >
                                      > > > > >
                                      > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                      > <sauwelios@>
                                      > > > wrote:
                                      > > > > > >
                                      > > > > > > Well, as you have not grounded any of your assertions, there
                                      > is
                                      > > > nothing for me to go into.
                                      > > > > > >
                                      > > > > > > I don't know what you mean by "common sense". The
                                      > > > infinity/nothingness problem is based on *logic*.
                                      > > > > > >
                                      > > > > > > You say: "a space-time surface can be curved, without having
                                      > to be
                                      > > > curved 'within' anything else". I think a curve must *by definition*
                                      > > > exist within something else. Merriam-Webster's Online Dictionary,
                                      > for
                                      > > > instance, defines "curve" (noun) in terms of "curve" (verb), which
                                      > it in
                                      > > > turn defines as:
                                      > > > > > >
                                      > > > > > > "to have or take a turn, change, or deviation from a straight
                                      > line
                                      > > > or plane surface without sharp breaks or angularity"
                                      > > > > > > http://www.merriam-webster.com/dictionary/curve
                                      > > > > > >
                                      > > > > > > If you can define "curve" without doing so in terms of
                                      > Euclidean
                                      > > > geometry (e.g., straight lines or plane surfaces), please do so. If
                                      > you
                                      > > > can't, a space-time surface apparently cannot be thought of as
                                      > "curved
                                      > > > without having to be curved within anything else"; it can then at
                                      > most
                                      > > > be thought of as "'curved' without having to be curved"---whatever
                                      > that
                                      > > > means.
                                      > > > > > >
                                      > > > > > >
                                      > > > > > >
                                      > > > > > > --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@>
                                      > > > wrote:
                                      > > > > > > >
                                      > > > > > > > "I don't think you've thought through the
                                      > infinity/nothingness
                                      > > > problem."
                                      > > > > > > >
                                      > > > > > > > Actually, I have thought this through, and I knew what, in
                                      > > > essence, your argument was. GR predicts many phenomena that seem
                                      > > > counter-intuitive to us humans, simply because we are used to energy
                                      > > > scales where these phenomena aren't a major part of our lives.
                                      > Quantum
                                      > > > mechanics is even "stranger" according to our common sense.
                                      > > > > > > >
                                      > > > > > > > On the large energy/time scales that these theories deal
                                      > with,
                                      > > > our common sense fails. Massively. We conclude, therefore, that
                                      > our
                                      > > > common sense is not a good tool to use on these scales. For
                                      > example,
                                      > > > some quantum gravity theories (e.g. loop quantum gravity) predict
                                      > that
                                      > > > time is discrete, not continuous. This contradicts everything we
                                      > > > intuitively know about time, because the smallest time unit is too
                                      > damn
                                      > > > small for us to notice.
                                      > > > > > > >
                                      > > > > > > > GR, which works almost perfectly on the timescales we're
                                      > talking
                                      > > > about here, is based on a non-Euclidean geometry (Euclidean geometry
                                      > is
                                      > > > the geometry that intuitively makes sense to us), which means that a
                                      > > > space-time surface can be curved, without having to be curved
                                      > "within"
                                      > > > anything else; curvature is a metric property of space-time. I
                                      > would
                                      > > > recommend that you do some reading on Riemannian geometry, so that
                                      > you
                                      > > > can see this for yourself (although the mathematics are pretty
                                      > > > advanced). With this in mind, questions like
                                      > > > > > > >
                                      > > > > > > > "If the space of the universe is the surface of a 4D
                                      > > > hypersphere, what encloses the hypersphere [...]?"
                                      > > > > > > >
                                      > > > > > > > become as utterly meaningless as "What is the absolute
                                      > velocity
                                      > > > of the Earth?"
                                      > > > > > > >
                                      > > > > > > > ~Ian
                                      > > > > > > >
                                      > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                      > > > <sauwelios@> wrote:
                                      > > > > > > > >
                                      > > > > > > > > I don't think you've thought through the
                                      > infinity/nothingness
                                      > > > problem. Thus you say:
                                      > > > > > > > >
                                      > > > > > > > > "According to Einstein's theory of General Relativity
                                      > (which
                                      > > > is based heavily on Riemannian geometry), the space of the universe
                                      > is
                                      > > > cyclical (or, more accurately, is the surface of a 4D hypersphere),
                                      > > > without there having to be anything other than the universe".
                                      > > > > > > > >
                                      > > > > > > > > If the space of the universe is the surface of a 4D
                                      > > > hypersphere, what encloses the hypersphere -- 'nothingness'?
                                      > > > > > > > >
                                      > > > > > > > >
                                      > > > > > > > > --- In human_superhuman@yahoogroups.com, "Ian"
                                      > <ianmathwiz7@>
                                      > > > wrote:
                                      > > > > > > > > >
                                      > > > > > > > > > "[I]t *is* necessarily true if space/time is all there
                                      > is."
                                      > > > > > > > > >
                                      > > > > > > > > > Says who? According to Einstein's theory of General
                                      > > > Relativity (which is based heavily on Riemannian geometry), the
                                      > space of
                                      > > > the universe is cyclical (or, more accurately, is the surface of a
                                      > 4D
                                      > > > hypersphere), without there having to be anything other than the
                                      > > > universe, and without there having to be "nothingness" inside the
                                      > > > sphere. According to Minkowskian geometry, which forms the other
                                      > main
                                      > > > geometric basis of GR, the same arguments can be applied to time.
                                      > The
                                      > > > *only reason* that this seems counter-intuitive to us is that, in
                                      > our
                                      > > > everyday lives, we don't encounter things that behave like spacetime
                                      > > > does.
                                      > > > > > > > > >
                                      > > > > > > > > > Granted, GR doesn't work on small scales, and doesn't
                                      > > > explain how spacetime is bent by forces other than gravity (and
                                      > attempts
                                      > > > thus far to explain this, e.g. supergravity, have failed). However,
                                      > the
                                      > > > theory is uncannily accurate on large scales (which *is* what we're
                                      > > > talking about when we're talking about scales on which things like
                                      > > > eternal recurrence become important), since gravity is the only
                                      > force of
                                      > > > any importance on this scale (since the nuclear forces only work on
                                      > > > small scales, and electric charges nearly cancel out perfectly on
                                      > this
                                      > > > scale).
                                      > > > > > > > > >
                                      > > > > > > > > > ~Ian
                                      > > > > > > > > >
                                      > > > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                      > > > <sauwelios@> wrote:
                                      > > > > > > > > > >
                                      > > > > > > > > > > I agree, but it *is* necessarily true if space/time is
                                      > all
                                      > > > there is (which is the case according to Nietzsche's theory of the
                                      > > > eternal recurrence).
                                      > > > > > > > > > >
                                      > > > > > > > > > >
                                      > > > > > > > > > > --- In human_superhuman@yahoogroups.com, "Ian"
                                      > > > <ianmathwiz7@> wrote:
                                      > > > > > > > > > > >
                                      > > > > > > > > > > > By the way, it is not necessarily true that there is
                                      > > > nothingness in the middle and the outside of curved, even circular,
                                      > > > space (and, by Minkowski's extension, time). I believe we have
                                      > Bernhard
                                      > > > Riemann, one of the greatest mathematicians of all time, to thank
                                      > for
                                      > > > proving that theorem.
                                      > > > > > > > > > > >
                                      > > > > > > > > > > > --- In human_superhuman@yahoogroups.com, "sauwelios"
                                      > > > <sauwelios@> wrote:
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > I think we will always keep running into the
                                      > > > infinity/nothingness problem.
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > If time or causality forms a circle, there must be
                                      > > > nothingness around it and in the middle. But to say that there is
                                      > > > nothingness in the middle is to say that there is *nothing* in the
                                      > > > middle, which makes it something different from a circle. And to say
                                      > > > that there is nothingness around it is to say that it is bounded by
                                      > > > *nothing*, which means it is infinite. And infinity is unthinkable.
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > My contemplations on this problem, which I have
                                      > here
                                      > > > reported very briefly, drive me toward the conclusion that, contrary
                                      > to
                                      > > > the will to power, the eternal recurrence is not meant to be
                                      > regarded as
                                      > > > a fact. This seems to be supported by the fact that, as far as I
                                      > know,
                                      > > > Nietzsche never presented an argument for it in his published works,
                                      > > > except perhaps in *TSZ*, which also stands out in every other way.
                                      > There
                                      > > > he provides an argument for it, but only in order to vanquish the
                                      > dwarf
                                      > > > ('The Vision and the Enigma'). And that only works because the dwarf
                                      > has
                                      > > > already asserted that time be a circle. Zarathustra is merely
                                      > pointing
                                      > > > out the *ramifications* of that idea to him, which is what then
                                      > defeats
                                      > > > him. But Zarathustra omits one of the two premises of his argument,
                                      > > > which he only provides later:
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > "[My laughing, wide-awake day-wisdom, which
                                      > mocketh at
                                      > > > all 'infinite worlds',] saith: "Where force is, there becometh
                                      > number
                                      > > > the master: it hath more force.""
                                      > > > > > > > > > > > > ['The Three Evil Things', 1.]
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > This premise, that force be finite, leads
                                      > inevitably
                                      > > > to the infinity/nothingness problem:
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > "This world: [...] enclosed by 'nothingness' as by
                                      > a
                                      > > > boundary; not something blurry or wasted, not something endlessly
                                      > > > extended, but set in a definite space as a definite force, and not a
                                      > > > space that might be 'empty' here or there, but rather as force
                                      > > > throughout[.]"
                                      > > > > > > > > > > > > [*WP* 1067.]
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > The phrase here translated as "'nothingness'" is
                                      > *'das
                                      > > > Nichts'*, "'the Nothing'". But what is the difference between 'the
                                      > > > Nothing' and *nothing*? How does this not amount to saying that this
                                      > > > world is bounded by *nothing*, i.e., not bounded at all? Unless it
                                      > > > amounts to saying that, though there is no empty space *within* this
                                      > > > world, there is an infinite empty space *outside* it. But if this
                                      > were
                                      > > > the case, its finite force would inevitably disperse into that
                                      > infinite
                                      > > > emptiness. In any case, neither emptiness nor infinity nor
                                      > nothingness
                                      > > > is thinkable, so we still arrive at the same problem Nietzsche tried
                                      > to
                                      > > > solve:
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > "[T]he world, as force, may not be thought of as
                                      > > > unlimited, for it *cannot* be so thought of[.]"
                                      > > > > > > > > > > > > [*WP* 1062.]
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > Likewise, it cannot be thought of as being
                                      > enclosed by
                                      > > > 'nothingness' as by a boundary. Hence I conclude that the eternal
                                      > > > recurrence is not a fact but a *value*.
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > > > --- In human_superhuman@yahoogroups.com, "Fred"
                                      > > > <nietzschefred@> wrote:
                                      > > > > > > > > > > > > >
                                      > > > > > > > > > > > > >
                                      > > > > > > > > > > > > > New model of the universe does away with the
                                      > > > singularity, beginning and end and describes physical laws as
                                      > constantly
                                      > > > changing and evolving...there are no cosmological constants.
                                      > > > > > > > > > > > > >
                                      > > > > > > > > > > > > >
                                      > > > > > > > > > > > > > • The speed of light and the gravitational
                                      > > > "constant" are not constant, but vary with the evolution of the
                                      > > > universe.
                                      > > > > > > > > > > > > > • Time has no beginning and no end; i.e.,
                                      > there
                                      > > > is neither a big bang nor a big crunch singularity.
                                      > > > > > > > > > > > > > • The spatial section of the universe is a
                                      > > > 3-sphere [a higher-dimensional analogue of a sphere], ruling out the
                                      > > > possibility of a flat or hyperboloid geometry.
                                      > > > > > > > > > > > > > • The universe experiences phases of both
                                      > > > acceleration and deceleration.
                                      > > > > > > > > > > > > >
                                      > > > > > > > > > > > > > New life for a scientific consideration of the
                                      > > > eternal recurrence?
                                      > > > > > > > > > > > > >
                                      > > > > > > > > > > > > > http://www.physorg.com/news199591806.html
                                      > > > > > > > > > > > > >
                                      > > > > > > > > > > > >
                                      > > > > > > > > > > >
                                      > > > > > > > > > >
                                      > > > > > > > > >
                                      > > > > > > > >
                                      > > > > > > >
                                      > > > > > >
                                      > > > > >
                                      > > > >
                                      > > >
                                      > >
                                      >
                                    • Ian
                                      Um, hi. It s been quite a while since the last post of this discussion. I have no excuse; I just completely forgot about it. That, of course, isn t
                                      Message 18 of 24 , Aug 26, 2011
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                                        Um, hi. It's been quite a while since the last post of this discussion. I have no excuse; I just completely forgot about it.

                                        That, of course, isn't necessarily a bad thing; it's given me some time to give my arguments a little more solid ground; I think I can articulate myself much better than I could a year ago. I just hope that interest in this discussion hasn't been completely lost in the time since.

                                        So, let's get started again; rather than directly responding to anything in this post, let me just bullet-point where we seem to stand at the moment:

                                        1) Sauwelios has presented me with an argument (reproduced at the bottom), consisting of a progression of eight syllogisms, which represents the infinity/nothingness problem.

                                        2) After having resolved several issues of definition, my objections to this argument reduce to:

                                        (2.1) Syllogism (5) is invalid. While it seems intuitively true (as I've stressed before) that a finite existence must be surrounded by something (or nothing), what we know about Riemannian geometry and General Relativity calls it into question, as a universe can be curved and distorted to the point that it, in effect, can surround itself, without having to be curved "within" anything. I gave three definitions of curvature, which are given in the order of most precise to least precise (and of least clear to most clear):

                                        I: Curvature is a measure of the extent to which the metric tensor of a Riemannian manifold is locally non-isometric to a Euclidean manifold.

                                        II: A space X is curved if and only if the distance between two arbitrarily close points in X cannot be related, by a one-to-one function, to the distance between two corresponding points in a Euclidean space.

                                        III: If a "straight" and evenly-spaced grid can be constructed in a space, that space is Euclidean, and has zero curvature by definition. A curved space, then, does not allow this to be done.

                                        Note that I've amended some of my language to make it somewhat more intelligible and relevant; for example, the first versions of II and III included a definition of the curvature metric as well, but I've dropped that now; it's not necessary to be able to measure curvature, we just need to be able to recognize it.

                                        The point of providing these definitions is to provide a jumping-off point; these definitions can be used to prove that curvature implies extrinsic curvature (i.e. curvature "within" something), and forbids intrinsic curvature (i.e. curvature as an intrinsic property of the space-time), if it indeed does so.

                                        (2.2) Syllogism (6) is also invalid. Nothing about being enclosed by something unthinkable implies that the object in question is unthinkable. For example, I'm currently surrounded (or enclosed) by a gas containing argon gas; that doesn't imply that my body contains argon gas in the same concentration (my body does contain trace amounts of argon, but that cannot be inferred merely from the fact that the atmosphere contains it). Of course, it could just be the property of being unthinkable that allows the inference to be made, but then it would have to be demonstrated that being enclosed by something unthinkable (such as not being enclosed by anything) implies that the object itself is unthinkable.

                                        Sauwelios posed the question of whether I can conceive (e.g.) the Earth as not being enclosed by anything; my answer is that I can certainly conceive the Earth, by itself, without having to conceive of any boundary. In my imagined representation of the Earth, then, the boundary is not anything; it therefore is nothing, by definition.

                                        3) It seems, then, that, starting here, the discussion should focus on two questions:

                                        (3.1) Does the theory of general relativity (GR) and, in particular, the Riemannian geometry used by GR, imply that existence can be enclosed (intrinsically) by itself, given the definitions of "curvature" above, and given that GR implies that it can curve severely enough to enclose itself?

                                        (3.2) Is an object enclosed by something unthinkable itself unthinkable?


                                        I hope to revive this discussion here, especially since there hasn't been *any* activity on this group since.

                                        ~Ian


                                        The infinity/nothingness argument:

                                        1
                                        a. To think is to form or have in the mind: http://www.merriam-webster.com/dictionary/think
                                        b. The mind is finite: this is a fact of experience.
                                        c. Whatever is thinkable is finite.

                                        2
                                        a. Whatever is thinkable is finite: see 1.
                                        b. Infinity is not finite: by definition.
                                        c. Infinity is not thinkable.

                                        3
                                        a. To think is to form or have in the mind: see 1.
                                        b. Only *things* (i.e., existents) can be formed or had: forming and
                                        having both require an object.
                                        c. Whatever is thinkable is a thing.

                                        4
                                        a. Whatever is thinkable is a thing: see 3.
                                        b. 'Nothing' is no thing: by definition.
                                        c. 'Nothing' is not thinkable.

                                        5
                                        a. Existence is all that exists: by definition.
                                        b. Existence is finite: suppose.
                                        c. Existence is enclosed by 'nothing' as by a boundary.

                                        6
                                        a. Existence is enclosed by 'nothing' as by a boundary: see 5.
                                        b. 'Nothing' is unthinkable: see 4.
                                        c. Existence is unthinkable.

                                        7
                                        a. Existence is infinite: suppose.
                                        b. Infinity is unthinkable: see 2.
                                        c. Existence is unthinkable.

                                        8
                                        a. Existence is unthinkable as finite: see 6.
                                        b. Existence is unthinkable as infinite: see 7.
                                        c. Existence is unthinkable.
                                      • sauwelios
                                        Dear Ian, I think I can counter your objection to syllogism 6 very easily. You say: Nothing about being enclosed by something unthinkable implies that the
                                        Message 19 of 24 , Aug 26, 2011
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                                          Dear Ian,

                                          I think I can counter your objection to syllogism 6 very easily. You say:

                                          "Nothing about being enclosed by something unthinkable implies that the object in question is unthinkable."

                                          To say that an object is enclosed by something unthinkable (henceforth "unthinkable-enclosed", for clarity's sake) is to say that an unthinkable-enclosed object exists: see http://www.ilovephilosophy.com/viewtopic.php?f=1&t=168064

                                          Can you conceive of an unthinkable-enclosed object? Note that it's of the _essence_ of an unthinkable-enclosed object that it's enclosed by something unthinkable. I contend thas it's by _definition_ impossible for you to do so, because it would have to include conceiving something inconceivable, thinking something unthinkable.

                                          You say you can conceive the Earth, by itself, without having to conceive of any boundary. I ask you, then: In your imagined representation of it, what encloses the Earth? What do you see, with your "mind's eye", to the left of it, say? Surely at the very least blackness, and not nothing?

                                          As for your objection to syllogism 5: your definitions of curvature are still over my head. But I would not consider the unthinkability of the eternal recurrence an objection to it; indeed, I think proof of it would defeat Nietzsche's end! For if Nietzsche's description of existence as prescription (i.e., will to power) and nothing besides is accurate, then that description must _itself_ be a prescription; but one cannot prescribe to something that it be what it is, as that would be not prescribing _anything_ to it; so one must not prescribe to it that it _be_ what it is, but that it _remain_ what it is or that it _not_ remain what it is; and with regard to individual things, Nietzsche did the latter (as the former would be a negation of the fleeting character of existence), and complemented it by the prescription that they _recur_ as they are (as _not_ doing so would amount to a negation of existence's particular manifestations); to one who knows for a _fact_ that everything eternally recurs, however, such a prescription would be impossible (consider Life's answer to Zarathustra's whisper: "No one knows that." (Nietzsche, _Thus Spake Zarathustra, "The Second Dance Song", section 2)).

                                          Sauwelios


                                          --- In human_superhuman@yahoogroups.com, "Ian" <ianmathwiz7@...> wrote:
                                          >
                                          > Um, hi. It's been quite a while since the last post of this discussion. I have no excuse; I just completely forgot about it.
                                          >
                                          > That, of course, isn't necessarily a bad thing; it's given me some time to give my arguments a little more solid ground; I think I can articulate myself much better than I could a year ago. I just hope that interest in this discussion hasn't been completely lost in the time since.
                                          >
                                          > So, let's get started again; rather than directly responding to anything in this post, let me just bullet-point where we seem to stand at the moment:
                                          >
                                          > 1) Sauwelios has presented me with an argument (reproduced at the bottom), consisting of a progression of eight syllogisms, which represents the infinity/nothingness problem.
                                          >
                                          > 2) After having resolved several issues of definition, my objections to this argument reduce to:
                                          >
                                          > (2.1) Syllogism (5) is invalid. While it seems intuitively true (as I've stressed before) that a finite existence must be surrounded by something (or nothing), what we know about Riemannian geometry and General Relativity calls it into question, as a universe can be curved and distorted to the point that it, in effect, can surround itself, without having to be curved "within" anything. I gave three definitions of curvature, which are given in the order of most precise to least precise (and of least clear to most clear):
                                          >
                                          > I: Curvature is a measure of the extent to which the metric tensor of a Riemannian manifold is locally non-isometric to a Euclidean manifold.
                                          >
                                          > II: A space X is curved if and only if the distance between two arbitrarily close points in X cannot be related, by a one-to-one function, to the distance between two corresponding points in a Euclidean space.
                                          >
                                          > III: If a "straight" and evenly-spaced grid can be constructed in a space, that space is Euclidean, and has zero curvature by definition. A curved space, then, does not allow this to be done.
                                          >
                                          > Note that I've amended some of my language to make it somewhat more intelligible and relevant; for example, the first versions of II and III included a definition of the curvature metric as well, but I've dropped that now; it's not necessary to be able to measure curvature, we just need to be able to recognize it.
                                          >
                                          > The point of providing these definitions is to provide a jumping-off point; these definitions can be used to prove that curvature implies extrinsic curvature (i.e. curvature "within" something), and forbids intrinsic curvature (i.e. curvature as an intrinsic property of the space-time), if it indeed does so.
                                          >
                                          > (2.2) Syllogism (6) is also invalid. Nothing about being enclosed by something unthinkable implies that the object in question is unthinkable. For example, I'm currently surrounded (or enclosed) by a gas containing argon gas; that doesn't imply that my body contains argon gas in the same concentration (my body does contain trace amounts of argon, but that cannot be inferred merely from the fact that the atmosphere contains it). Of course, it could just be the property of being unthinkable that allows the inference to be made, but then it would have to be demonstrated that being enclosed by something unthinkable (such as not being enclosed by anything) implies that the object itself is unthinkable.
                                          >
                                          > Sauwelios posed the question of whether I can conceive (e.g.) the Earth as not being enclosed by anything; my answer is that I can certainly conceive the Earth, by itself, without having to conceive of any boundary. In my imagined representation of the Earth, then, the boundary is not anything; it therefore is nothing, by definition.
                                          >
                                          > 3) It seems, then, that, starting here, the discussion should focus on two questions:
                                          >
                                          > (3.1) Does the theory of general relativity (GR) and, in particular, the Riemannian geometry used by GR, imply that existence can be enclosed (intrinsically) by itself, given the definitions of "curvature" above, and given that GR implies that it can curve severely enough to enclose itself?
                                          >
                                          > (3.2) Is an object enclosed by something unthinkable itself unthinkable?
                                          >
                                          >
                                          > I hope to revive this discussion here, especially since there hasn't been *any* activity on this group since.
                                          >
                                          > ~Ian
                                          >
                                          >
                                          > The infinity/nothingness argument:
                                          >
                                          > 1
                                          > a. To think is to form or have in the mind: http://www.merriam-webster.com/dictionary/think
                                          > b. The mind is finite: this is a fact of experience.
                                          > c. Whatever is thinkable is finite.
                                          >
                                          > 2
                                          > a. Whatever is thinkable is finite: see 1.
                                          > b. Infinity is not finite: by definition.
                                          > c. Infinity is not thinkable.
                                          >
                                          > 3
                                          > a. To think is to form or have in the mind: see 1.
                                          > b. Only *things* (i.e., existents) can be formed or had: forming and
                                          > having both require an object.
                                          > c. Whatever is thinkable is a thing.
                                          >
                                          > 4
                                          > a. Whatever is thinkable is a thing: see 3.
                                          > b. 'Nothing' is no thing: by definition.
                                          > c. 'Nothing' is not thinkable.
                                          >
                                          > 5
                                          > a. Existence is all that exists: by definition.
                                          > b. Existence is finite: suppose.
                                          > c. Existence is enclosed by 'nothing' as by a boundary.
                                          >
                                          > 6
                                          > a. Existence is enclosed by 'nothing' as by a boundary: see 5.
                                          > b. 'Nothing' is unthinkable: see 4.
                                          > c. Existence is unthinkable.
                                          >
                                          > 7
                                          > a. Existence is infinite: suppose.
                                          > b. Infinity is unthinkable: see 2.
                                          > c. Existence is unthinkable.
                                          >
                                          > 8
                                          > a. Existence is unthinkable as finite: see 6.
                                          > b. Existence is unthinkable as infinite: see 7.
                                          > c. Existence is unthinkable.
                                          >
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