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522Re: Eternal Universe With New Cosmological Model

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  • Ian
    Oct 3, 2010
      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
      > > > > > > > > > >
      > > > > > > > > >
      > > > > > > > >
      > > > > > > >
      > > > > > >
      > > > > >
      > > > >
      > > >
      > >
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