## Moderatus on Number

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• One fragment of Moderatus found in Stob. Anth. I p. 21 claims that number is a system of units, or a progression of multiplicity (propodismos plethous)
Message 1 of 19 , Oct 26 3:59 PM
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One fragment of Moderatus found in Stob. Anth. I p. 21 claims that number is "a system of units, or a progression of multiplicity (propodismos plethous) beginning from unity, and a regression (anapodismos) ending in unity" (trans. John Dillon).

This formulation has always seemed a bit odd, at least as regards thinking of number as a regression as much as a progression. Progression in this context of course seems quite natural, as we all think of counting up from one to whatever number. And there it ends, one would think at least in an everyday, intuitive way. But Moderatus pairs the progression from one, with a regression ending in one.

So where, I think it is natural to ask then, if regression is also defining, where does that regression begin? It seems to me that he implies it is from plethos, multiplicity, itself. Multiplicity means of course more than one, simply put, as we all of course know. Is two then enough to be the same as multiplicity? Or three? No, I think it means rather the principle of more than one and so must then encompass somehow all numbers without however being defined as any one number in particular, and as a principle cannot contain what it engenders - at last that is the usual Platonic requirement for a principle such as this one. I won't however equate it with infinity, at least for this discussion.

So how does one define any given number? On the one hand a number is limited to be a certain set of monads, in Moderatus' view, coming out of the one, and as a group has itself a sort of limit or unity imposed upon it, to be just that group and not another group, so that for instance 5 is a group of 5 monads as opposed to 6 which is rather a group of 6 monads. But there also is I think a sort of tension, a balance, that has to attained in order for that progression not to spin off, as it were, into ultimate multiplicity, or infinity if you will; because both principles, unity and multiplicity are equally required in order to create a number, or number. There then does really have to be an equally reactive force, reining in from the other side, from multiplicity back to unity, but of course not receding entirely back into the one, in order to reach the particular number.

So we must also regress from that multiplicity which is all numbers, back to some specific point, if one wants to think perhaps of the number as some point on an infinite line. If there are two principles at work, and Moderatus appears here definitely if tersely to make such a claim, or he is being summarized in such a way, and he does not also make the claim that one is greater than the other, so that we must take the position then without more detailed evidence that they must also be equal partners in definition, then they must both contribute equally as principles, and that then progression is not enough, that it only effects the contribution of the one, and that also we must allow the contribution of the many, which is characterized then as a regression, not somehow the other way around, with one as the regression and multiplicity as the progression, because of course in such Pythagorean and Platonic thinking the one is superior in all things.

But where does Moderatus derive these two principles? The answer is most likely Speusippus, who posited these two principles as a part of his own system, we well know. And so we have here another example of what Moderatus himself complains of the Platonists, as Porphyry tells us in his Life of Pythagoras, that they stole what was best from the Pythagoreans (cf John Dillon, "Plotinus, Speusippus, and the Platonic Parmenides" in Plotin Ekei Entautha, ed. Daniel Montet, Kairos, pp68-69).

As to whether my reasoning is echt Moderatan, or even resasonable in general I am not sure, but this is the way I try to understand Moderatus' definition of number, in I supose I have to say a rather Speusippan way. Whether Moderatus did as well, who can say? Have I then made Moderatus a follower of Speusippus, at least in this definition? Or perhaps is he rather echoing, or even, may I be so bold as to suggest, reproducing Speusippus here?

Dennis Clark

"Alles, was tief ist, liebt die Maske."
Friedrich Nietzsche, Jenseits von Gut und Böse
• ... (apparent) time of Moderatus the words *propodismos* and *anapodismos* are technical terms of astronomy: they signify two of the three movements adopted
Message 2 of 19 , Oct 26 11:30 PM
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> One fragment of Moderatus found in Stob. Anth. I p. 21 claims that number
> is "a system of units, or a progression of multiplicity (propodismos
> plethous) beginning from unity, and a regression (anapodismos) ending in
> unity" (trans. John Dillon).
>
> M.C. I'd be inclined to adopt another, complementary approach. By the
(apparent) time of Moderatus the words *propodismos* and *anapodismos*
are technical terms of astronomy: they signify two of the three
movements adopted by the planets in their irregular motion : going
backwards (*anapodismos*), standing still (*stêrigmos*), going forward
(*propodismos*). That the same terminology was used in astronomy and
number is attested by Nicomachus (Introd. 1,5,2; Theolog. p. 19). Which
came first, the astronomical chicken or the numerological egg ? I don't
known, and I don't know whether it's possible to know for sure.

But some of the ancients certainly saw a connection. What might that be?
Well, in the case of the motion of the planets, one can perhaps imagine
the triad *anapodismos* - *stêrigmos* - *propodismos* as a fluctuation
around a central point, not unlike the way state spaces cycle around an
attractor in dynamics (I will leave it up to Bob Wallace to draw the
obvious parallels with Hegelian dialectics!).

What might this mean in number theory ? Well, negative numbers were
already known to Diophantus, probably a near-contemporary of Moderatus.
Could we not imagine that what's alluded to is the oscillation of the
number line from negative numbers to zero to positive numbers?

All this is just off the top of my head, and perhaps *real* scholars of
mathematics have figured all this out long ago. But thanks for bringing
this up, Dennis!

Michael Chase
CNRS UPR 76
Paris-Villejuif
France
• Re below, I think I m going to withdraw my suggestion below that there may be a reference in this matter to the number line and to negative numbers. Having had
Message 3 of 19 , Oct 27 3:28 AM
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Re below,

I think I'm going to withdraw my suggestion below that there may be a
reference in this matter to the number line and to negative numbers.
Having had a look at the sources in a decent library, I can find no
confirmation that Diophantus, or any other Greek, really had a complete
understanding of negative integers.

Oh well, never mind, as Gilda Radner used to say.

Best, Mike

>
>> One fragment of Moderatus found in Stob. Anth. I p. 21 claims that
>> number
>> is "a system of units, or a progression of multiplicity (propodismos
>> plethous) beginning from unity, and a regression (anapodismos) ending in
>> unity" (trans. John Dillon).
>>
>> M.C. I'd be inclined to adopt another, complementary approach. By the
> (apparent) time of Moderatus the words *propodismos* and *anapodismos*
> are technical terms of astronomy: they signify two of the three
> movements adopted by the planets in their irregular motion : going
> backwards (*anapodismos*), standing still (*stêrigmos*), going forward
> (*propodismos*). That the same terminology was used in astronomy and
> number is attested by Nicomachus (Introd. 1,5,2; Theolog. p. 19). Which
> came first, the astronomical chicken or the numerological egg ? I don't
> known, and I don't know whether it's possible to know for sure.
>
> But some of the ancients certainly saw a connection. What might that be?
> Well, in the case of the motion of the planets, one can perhaps imagine
> the triad *anapodismos* - *stêrigmos* - *propodismos* as a fluctuation
> around a central point, not unlike the way state spaces cycle around an
> attractor in dynamics (I will leave it up to Bob Wallace to draw the
> obvious parallels with Hegelian dialectics!).
>
> What might this mean in number theory ? Well, negative numbers were
> already known to Diophantus, probably a near-contemporary of Moderatus.
> Could we not imagine that what's alluded to is the oscillation of the
> number line from negative numbers to zero to positive numbers?
>
> All this is just off the top of my head, and perhaps *real* scholars of
> mathematics have figured all this out long ago. But thanks for bringing
> this up, Dennis!
>
> Michael Chase
> CNRS UPR 76
> Paris-Villejuif
> France
>
>
>

Michael Chase
CNRS UPR 76
Paris-Villejuif
France
• Dennis, I believe you raise a--if not *the*--fundamental ontological question. For me personally, it can induce a mystical state. It s sublimely
Message 4 of 19 , Oct 27 8:37 AM
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Dennis,

I believe you raise a--if not *the*--fundamental ontological question.
For me personally, it can induce a mystical state. It's sublimely
ontological, in my opinion, because it goes to the heart of the constitution of the
substance of Being. This brief comment is a sort of immediate
giving-in-to-emotion; a kind of gut reaction that must settle down in order for rational
intellect to apply. Hopefully I'll be able to get to the latter in due
course.

Suffice for the moment to say that, to me, the question brings us back to
the earlier question whether 1 and 2 are numbers. What's involved is the
dyad and the genera of being per Plotinus and Proclus. In order to do both
questions justice, I need to comment point-by-point on your post. I'm very
motivated toward doing just that to the end of honing my thinking on the
subject. And I'd again mention Prof. Slaveva-Griffin's exegesis of
Speusippus and Moderatus which implicates Nichomachus and Numenius as well. What
I'll have to share will not address your Moderatus question *per se*, but
rather the implicate issue concerning number.

More as time enables. I've also been deferring a response to Prof.
Kallio's post on astrological influences. Too many distractions of the
earn-a-living kind.

Grateful for this list,

David Gallagher

In a message dated 10/26/2010 7:00:03 P.M. Eastern Daylight Time,
vaeringjar@... writes:

One fragment of Moderatus found in Stob. Anth. I p. 21 claims that number
is "a system of units, or a progression of multiplicity (propodismos
plethous) beginning from unity, and a regression (anapodismos) ending in unity"
(trans. John Dillon).

This formulation has always seemed a bit odd, at least as regards thinking
of number as a regression as much as a progression. Progression in this
context of course seems quite natural, as we all think of counting up from
one to whatever number. And there it ends, one would think at least in an
everyday, intuitive way. But Moderatus pairs the progression from one, with a
regression ending in one.

So where, I think it is natural to ask then, if regression is also
defining, where does that regression begin? It seems to me that he implies it is
from plethos, multiplicity, itself. Multiplicity means of course more than
one, simply put, as we all of course know. Is two then enough to be the same
as multiplicity? Or three? No, I think it means rather the principle of
more than one and so must then encompass somehow all numbers without however
being defined as any one number in particular, and as a principle cannot
contain what it engenders - at last that is the usual Platonic requirement
for a principle such as this one. I won't however equate it with infinity, at
least for this discussion.

So how does one define any given number? On the one hand a number is
limited to be a certain set of monads, in Moderatus' view, coming out of the
one, and as a group has itself a sort of limit or unity imposed upon it, to be
just that group and not another group, so that for instance 5 is a group
of 5 monads as opposed to 6 which is rather a group of 6 monads. But there
also is I think a sort of tension, a balance, that has to attained in order
for that progression not to spin off, as it were, into ultimate
multiplicity, or infinity if you will; because both principles, unity and multiplicity
are equally required in order to create a number, or number. There then
does really have to be an equally reactive force, reining in from the other
side, from multiplicity back to unity, but of course not receding entirely
back into the one, in order to reach the particular number.

So we must also regress from that multiplicity which is all numbers, back
to some specific point, if one wants to think perhaps of the number as some
point on an infinite line. If there are two principles at work, and
Moderatus appears here definitely if tersely to make such a claim, or he is being
summarized in such a way, and he does not also make the claim that one is
greater than the other, so that we must take the position then without more
detailed evidence that they must also be equal partners in definition,
then they must both contribute equally as principles, and that then
progression is not enough, that it only effects the contribution of the one, and that
also we must allow the contribution of the many, which is characterized
then as a regression, not somehow the other way around, with one as the
regression and multiplicity as the progression, because of course in such
Pythagorean and Platonic thinking the one is superior in all things.

But where does Moderatus derive these two principles? The answer is most
likely Speusippus, who posited these two principles as a part of his own
system, we well know. And so we have here another example of what Moderatus
himself complains of the Platonists, as Porphyry tells us in his Life of
Pythagoras, that they stole what was best from the Pythagoreans (cf John
Dillon, "Plotinus, Speusippus, and the Platonic Parmenides" in Plotin Ekei
Entautha, ed. Daniel Montet, Kairos, pp68-69).

As to whether my reasoning is echt Moderatan, or even resasonable in
general I am not sure, but this is the way I try to understand Moderatus'
definition of number, in I supose I have to say a rather Speusippan way. Whether
Moderatus did as well, who can say? Have I then made Moderatus a follower
of Speusippus, at least in this definition? Or perhaps is he rather echoing,
or even, may I be so bold as to suggest, reproducing Speusippus here?

Dennis Clark

"Alles, was tief ist, liebt die Maske."
Friedrich Nietzsche, Jenseits von Gut und Böse

[Non-text portions of this message have been removed]
• ... Nice, Mike - I was not aware at all of the other uses of the terms, and so I should be mindful of them, especially with an extract like this handed down to
Message 5 of 19 , Oct 27 2:54 PM
• 0 Attachment
--- In neoplatonism@yahoogroups.com, "Goya" <goya@...> wrote:
>
>
> > One fragment of Moderatus found in Stob. Anth. I p. 21 claims that number
> > is "a system of units, or a progression of multiplicity (propodismos
> > plethous) beginning from unity, and a regression (anapodismos) ending in
> > unity" (trans. John Dillon).
> >
> > M.C. I'd be inclined to adopt another, complementary approach. By the
> (apparent) time of Moderatus the words *propodismos* and *anapodismos*
> are technical terms of astronomy: they signify two of the three
> movements adopted by the planets in their irregular motion : going
> backwards (*anapodismos*), standing still (*stêrigmos*), going forward
> (*propodismos*). That the same terminology was used in astronomy and
> number is attested by Nicomachus (Introd. 1,5,2; Theolog. p. 19). Which
> came first, the astronomical chicken or the numerological egg ? I don't
> known, and I don't know whether it's possible to know for sure.
>
> But some of the ancients certainly saw a connection. What might that be?
> Well, in the case of the motion of the planets, one can perhaps imagine
> the triad *anapodismos* - *stêrigmos* - *propodismos* as a fluctuation
> around a central point, not unlike the way state spaces cycle around an
> attractor in dynamics (I will leave it up to Bob Wallace to draw the
> obvious parallels with Hegelian dialectics!).
>
> What might this mean in number theory ? Well, negative numbers were
> already known to Diophantus, probably a near-contemporary of Moderatus.
> Could we not imagine that what's alluded to is the oscillation of the
> number line from negative numbers to zero to positive numbers?
>
> All this is just off the top of my head, and perhaps *real* scholars of
> mathematics have figured all this out long ago. But thanks for bringing
> this up, Dennis!
>
> Michael Chase
> CNRS UPR 76
> Paris-Villejuif
> France
>

Nice, Mike - I was not aware at all of the other uses of the terms, and so I should be mindful of them, especially with an extract like this handed down to us baldly without any of the original context within which it appeared.

Regardless of the negative number aspect you think is not fruitful now, I think the astronomical context of the terms you bring up still fits in with the notion of location of a given number along, let's say, a spectrum, and you noticed you described this notion as a complementary approach. If I am right, we maybe also should allow the possibility that there might not have been at that time exact philosophical terms for this sort of "positioning" - and I use that term with its spatial reference then rather metaphorically in the frame of my main argument for the interaction of two absolute principles - so a somewhat allusory approach, using astronomical terminology might have been the best Moderatus could do if there was nothing in the philosophical vocabulary that served already by main definition to describe progression and recession. (though we still have the issue of why not use 'prohodos', for instance, but as I recall it are not attested in philosophical context til later, with Plotinus, of course). Though I suppose allowing this astronomical origin - and as you say it's hard to know which came first - rather puts a damper on my Speusippan provenance, unless of course Moderatus is again importing newer language from another discipline and just repackaging truely Speusippan concepts in those newer terms.

Who is to say, speculating here now, but I do think my idea offers at least one reasonable explanation for his giving weight to both progression and regression, and I wasn't originally even thinking of Speusippus when it came to me several weeks ago, even though he is otherwise much on my tiny mind these days. But as I finished the posting it just occurred to me the whole passage may go back as well, if we can of course credit my argument.

Another aspect of this, how numbers relate in this hierarchy to the Forms, is something else I have been trying to ponder, looking of course at the old Academy chiefly to try to work that one out and understand their own differing views. I agree however with Prof. Dillon that Speusippus did not actually totally annihilate the Forms, regardless of the fragment claiming that, rather that the situation is a bit more complicated, but that he did give precedence to number I think is certain.

Dennis Clark
• ... Dennis ­ I had toyed with the idea that Moderatus¹ anapodismos of number was just the turning back upon itself of each number, which made it what it was,
Message 6 of 19 , Oct 27 3:55 PM
• 0 Attachment
>
>
>
>
>
>
>
> --- In neoplatonism@yahoogroups.com <mailto:neoplatonism%40yahoogroups.com> ,
> "Goya" <goya@...> wrote:
>> >
>> >
>>> > > One fragment of Moderatus found in Stob. Anth. I p. 21 claims that
>>> number
>>> > > is "a system of units, or a progression of multiplicity (propodismos
>>> > > plethous) beginning from unity, and a regression (anapodismos) ending in
>>> > > unity" (trans. John Dillon).
>>> > >
>>> > > M.C. I'd be inclined to adopt another, complementary approach. By the
>> > (apparent) time of Moderatus the words *propodismos* and *anapodismos*
>> > are technical terms of astronomy: they signify two of the three
>> > movements adopted by the planets in their irregular motion : going
>> > backwards (*anapodismos*), standing still (*stêrigmos*), going forward
>> > (*propodismos*). That the same terminology was used in astronomy and
>> > number is attested by Nicomachus (Introd. 1,5,2; Theolog. p. 19). Which
>> > came first, the astronomical chicken or the numerological egg ? I don't
>> > known, and I don't know whether it's possible to know for sure.
>> >
>> > But some of the ancients certainly saw a connection. What might that be?
>> > Well, in the case of the motion of the planets, one can perhaps imagine
>> > the triad *anapodismos* - *stêrigmos* - *propodismos* as a fluctuation
>> > around a central point, not unlike the way state spaces cycle around an
>> > attractor in dynamics (I will leave it up to Bob Wallace to draw the
>> > obvious parallels with Hegelian dialectics!).
>> >
>> > What might this mean in number theory ? Well, negative numbers were
>> > already known to Diophantus, probably a near-contemporary of Moderatus.
>> > Could we not imagine that what's alluded to is the oscillation of the
>> > number line from negative numbers to zero to positive numbers?
>> >
>> > All this is just off the top of my head, and perhaps *real* scholars of
>> > mathematics have figured all this out long ago. But thanks for bringing
>> > this up, Dennis!
>> >
>> > Michael Chase
>> > CNRS UPR 76
>> > Paris-Villejuif
>> > France
>> >
>
> Nice, Mike - I was not aware at all of the other uses of the terms, and so I
> should be mindful of them, especially with an extract like this handed down to
> us baldly without any of the original context within which it appeared.
>
> Regardless of the negative number aspect you think is not fruitful now, I
> think the astronomical context of the terms you bring up still fits in with
> the notion of location of a given number along, let's say, a spectrum, and you
> noticed you described this notion as a complementary approach. If I am right,
> we maybe also should allow the possibility that there might not have been at
> that time exact philosophical terms for this sort of "positioning" - and I use
> that term with its spatial reference then rather metaphorically in the frame
> of my main argument for the interaction of two absolute principles - so a
> somewhat allusory approach, using astronomical terminology might have been the
> best Moderatus could do if there was nothing in the philosophical vocabulary
> that served already by main definition to describe progression and recession.
> (though we still have the issue of why not use 'prohodos', for instance, but
> as I recall it are not attested in philosophical context til later, with
> Plotinus, of course). Though I suppose allowing this astronomical origin - and
> as you say it's hard to know which came first - rather puts a damper on my
> Speusippan provenance, unless of course Moderatus is again importing newer
> language from another discipline and just repackaging truely Speusippan
> concepts in those newer terms.
>
> Who is to say, speculating here now, but I do think my idea offers at least
> one reasonable explanation for his giving weight to both progression and
> regression, and I wasn't originally even thinking of Speusippus when it came
> to me several weeks ago, even though he is otherwise much on my tiny mind
> these days. But as I finished the posting it just occurred to me the whole
> passage may go back as well, if we can of course credit my argument.
>
> Another aspect of this, how numbers relate in this hierarchy to the Forms, is
> something else I have been trying to ponder, looking of course at the old
> Academy chiefly to try to work that one out and understand their own differing
> views. I agree however with Prof. Dillon that Speusippus did not actually
> totally annihilate the Forms, regardless of the fragment claiming that, rather
> that the situation is a bit more complicated, but that he did give precedence
> to number I think is certain.
>
> Dennis Clark
>
>
>
>
>

Dennis ­ I had toyed with the idea that Moderatus¹ anapodismos of number was
just the turning back upon itself of each number, which made it what it was,
a sort of jelling¹. This would then be an ancestor of Plotinus¹ idea that
the turning back of each lower hypostasis -- Intellect, Soul  upon its
higher served to constitute itself as such. But I can¹t claim to be very

[Non-text portions of this message have been removed]
• ... Yahoo mangled my posting yet again, but not to be too ungracious to our host here: and you noticed you described this notion as a should be and I
Message 7 of 19 , Oct 27 4:21 PM
• 0 Attachment
--- In neoplatonism@yahoogroups.com, "vaeringjar" <vaeringjar@...> wrote:
>
>
>
> --- In neoplatonism@yahoogroups.com, "Goya" <goya@> wrote:
> >
> >
> > > One fragment of Moderatus found in Stob. Anth. I p. 21 claims that number
> > > is "a system of units, or a progression of multiplicity (propodismos
> > > plethous) beginning from unity, and a regression (anapodismos) ending in
> > > unity" (trans. John Dillon).
> > >
> > > M.C. I'd be inclined to adopt another, complementary approach. By the
> > (apparent) time of Moderatus the words *propodismos* and *anapodismos*
> > are technical terms of astronomy: they signify two of the three
> > movements adopted by the planets in their irregular motion : going
> > backwards (*anapodismos*), standing still (*stêrigmos*), going forward
> > (*propodismos*). That the same terminology was used in astronomy and
> > number is attested by Nicomachus (Introd. 1,5,2; Theolog. p. 19). Which
> > came first, the astronomical chicken or the numerological egg ? I don't
> > known, and I don't know whether it's possible to know for sure.
> >
> > But some of the ancients certainly saw a connection. What might that be?
> > Well, in the case of the motion of the planets, one can perhaps imagine
> > the triad *anapodismos* - *stêrigmos* - *propodismos* as a fluctuation
> > around a central point, not unlike the way state spaces cycle around an
> > attractor in dynamics (I will leave it up to Bob Wallace to draw the
> > obvious parallels with Hegelian dialectics!).
> >
> > What might this mean in number theory ? Well, negative numbers were
> > already known to Diophantus, probably a near-contemporary of Moderatus.
> > Could we not imagine that what's alluded to is the oscillation of the
> > number line from negative numbers to zero to positive numbers?
> >
> > All this is just off the top of my head, and perhaps *real* scholars of
> > mathematics have figured all this out long ago. But thanks for bringing
> > this up, Dennis!
> >
> > Michael Chase
> > CNRS UPR 76
> > Paris-Villejuif
> > France
> >
>
> Nice, Mike - I was not aware at all of the other uses of the terms, and so I should be mindful of them, especially with an extract like this handed down to us baldly without any of the original context within which it appeared.
>
> Regardless of the negative number aspect you think is not fruitful now, I think the astronomical context of the terms you bring up still fits in with the notion of location of a given number along, let's say, a spectrum, and you noticed you described this notion as a complementary approach. If I am right, we maybe also should allow the possibility that there might not have been at that time exact philosophical terms for this sort of "positioning" - and I use that term with its spatial reference then rather metaphorically in the frame of my main argument for the interaction of two absolute principles - so a somewhat allusory approach, using astronomical terminology might have been the best Moderatus could do if there was nothing in the philosophical vocabulary that served already by main definition to describe progression and recession. (though we still have the issue of why not use 'prohodos', for instance, but as I recall it are not attested in philosophical context til later, with Plotinus, of course). Though I suppose allowing this astronomical origin - and as you say it's hard to know which came first - rather puts a damper on my Speusippan provenance, unless of course Moderatus is again importing newer language from another discipline and just repackaging truely Speusippan concepts in those newer terms.
>
> Who is to say, speculating here now, but I do think my idea offers at least one reasonable explanation for his giving weight to both progression and regression, and I wasn't originally even thinking of Speusippus when it came to me several weeks ago, even though he is otherwise much on my tiny mind these days. But as I finished the posting it just occurred to me the whole passage may go back as well, if we can of course credit my argument.
>
> Another aspect of this, how numbers relate in this hierarchy to the Forms, is something else I have been trying to ponder, looking of course at the old Academy chiefly to try to work that one out and understand their own differing views. I agree however with Prof. Dillon that Speusippus did not actually totally annihilate the Forms, regardless of the fragment claiming that, rather that the situation is a bit more complicated, but that he did give precedence to number I think is certain.
>
> Dennis Clark
>

Yahoo mangled my posting yet again, but not to be too ungracious to our host here:

"and you noticed you described this notion as a"

should be "and I noticed you..."

"but as I recall it are not attested"

should be "...is not..."

"whole passage may go back as well"

should be "back to Speusippus as well"

Dennis Clark

"Alles, was tief ist, liebt die Maske."
Friedrich Nietzsche, Jenseits von Gut und Böse
• ... That is much simpler for sure, so that number like any other entity finds itself again on that spectrum and at the point of its existence starts its
Message 8 of 19 , Oct 27 4:33 PM
• 0 Attachment
>
> Dennis ­ I had toyed with the idea that Moderatus¹ anapodismos of number was
> just the turning back upon itself of each number, which made it what it was,
> a sort of jelling¹. This would then be an ancestor of Plotinus¹ idea that
> the turning back of each lower hypostasis -- Intellect, Soul  upon its
> higher served to constitute itself as such. But I can¹t claim to be very
>
>

That is much simpler for sure, so that number like any other entity finds itself again on that 'spectrum' and at the point of its existence starts its epistrophe, and both forces are applied in balance at that point. Though I suppose that still doesn't mean we have to toss out multiplicity, that it's push and pull between the two, progression pushing to multiplicity from the one and regression pulling back from multiplicity to the one, and the epistrophe is the turning point, the exact location on the spectrum of the number or entity, with again a prejudice to the one as superior in a Platonic system, hence the directional coloring of epistrophe back to the one.

But numbers aren't really entities, are they? Yet they seem more to me in this context than just another Aristotelian category or aspect of an entity? Or am I making this too complicated? Somehow number intuitively, or what I hope I am taking correctly as in a sort of Platonic intuition, is more basic or superior.

I suppose though that Heraclitus would just laugh at all my manoeuverings here - "he kata he ana"!

Dennis Clark
• Dear All, There are some good ideas going around here, but might not the answer be a bit simpler? The definition reads a system of units, or a progression of
Message 9 of 19 , Oct 27 6:25 PM
• 0 Attachment
Dear All,

There are some good ideas going around here, but might not the answer be
a bit simpler? The definition reads "a system of units, or a progression
of multiplicity (propodismos plethous) beginning from unity, and a
regression (anapodismos) ending in unity". The 'system of units' sounds
pretty unimaginative, and it is the colourful terms that suggest there
might be something rather unusual about the latter. Clearly the terms
were carefully chosen, but they seem to me to suggest a *step-by-step*
advance or retreat. Damascius uses the verb propodizo of numbers I
believe, but I've not been and looked it up.

But we should also bear in mind the possibility that arithmos is being
used in two slightly different senses (which if one thinks about it has
to be so, since a system cannot surely be identical with a movement).
One meaning given for arithmos by LSJ is simply 'counting' (= to
arithmein), which to my mind is well defined as a step-by-step movement
towards a greater or lesser plurality. I don't affirm that this is the
right solution, but just for once I'm inclined to keep both Speusippus
and Neoplatonism out of the equation.

Harold

Prof. Harold Tarrant,
School of Humanities and Social Science,
University of Newcastle,
NSW 2308 Australia
Ph: (+61) 2 49215230
Fax: (+61) 2 49216933
*Eu Prattein*
>>> vaeringjar <vaeringjar@...> 28/10/10 10:34 AM >>>

>
> Dennis - I had toyed with the idea that Moderatus¹ anapodismos of
number was
> just the turning back upon itself of each number, which made it what
it was,
> a sort of Œjelling¹. This would then be an ancestor of Plotinus¹ idea
that
> the turning back of each lower hypostasis -- Intellect, Soul ‹ upon
its
> higher served to constitute itself as such. But I can¹t claim to be
very
>
>

That is much simpler for sure, so that number like any other entity
finds itself again on that 'spectrum' and at the point of its existence
starts its epistrophe, and both forces are applied in balance at that
point. Though I suppose that still doesn't mean we have to toss out
multiplicity, that it's push and pull between the two, progression
pushing to multiplicity from the one and regression pulling back from
multiplicity to the one, and the epistrophe is the turning point, the
exact location on the spectrum of the number or entity, with again a
prejudice to the one as superior in a Platonic system, hence the
directional coloring of epistrophe back to the one.

But numbers aren't really entities, are they? Yet they seem more to me
in this context than just another Aristotelian category or aspect of an
entity? Or am I making this too complicated? Somehow number intuitively,
or what I hope I am taking correctly as in a sort of Platonic intuition,
is more basic or superior.

I suppose though that Heraclitus would just laugh at all my
manoeuverings here - "he kata he ana"!

Dennis Clark
• I m afraid Professor Tarrant s rationalizing explanation seems hard to swallow. System of units may not sound - *to us* - particularly imaginative, but the
Message 10 of 19 , Oct 28 12:59 AM
• 0 Attachment
I'm afraid Professor Tarrant's rationalizing explanation seems hard to
swallow. "System of units" may not sound - *to us* - particularly
imaginative, but the fact remains that the expression is used *only* by
authors of Pythagorean allegiance (Moderatus, Nicomachus, Theon of Smyrna,
Iamblichus, Syrianus, Damascius) and *never* by less philosophically
engaged mathematicians such as Ptolemy, Archimedes, Diophantus, Euclid,
etc. etc. This cannot be a coincidence, in my view.

Nor can it be a coincidence, I think, that words derived from
*propodizein* occur a couple of times in Homer, then *not at all for six
or seven centuries*, then suddenly re-emerge in two, and only two, quite
specific domains: number theory and astronomy. Top it all of by the fact
that Nicomachus points out and explains these parallels, and I'm afraid I
don't see how there can be much room for doubt that a connection exists,
whatever its precise nature may be.

Best, Mike

> Dear All,
>
> There are some good ideas going around here, but might not the answer be
> a bit simpler? The definition reads "a system of units, or a progression
> of multiplicity (propodismos plethous) beginning from unity, and a
> regression (anapodismos) ending in unity". The 'system of units' sounds
> pretty unimaginative, and it is the colourful terms that suggest there
> might be something rather unusual about the latter. Clearly the terms
> were carefully chosen, but they seem to me to suggest a *step-by-step*
> advance or retreat. Damascius uses the verb propodizo of numbers I
> believe, but I've not been and looked it up.
>
> But we should also bear in mind the possibility that arithmos is being
> used in two slightly different senses (which if one thinks about it has
> to be so, since a system cannot surely be identical with a movement).
> One meaning given for arithmos by LSJ is simply 'counting' (= to
> arithmein), which to my mind is well defined as a step-by-step movement
> towards a greater or lesser plurality. I don't affirm that this is the
> right solution, but just for once I'm inclined to keep both Speusippus
> and Neoplatonism out of the equation.
>
> Harold
>
> Prof. Harold Tarrant,
> School of Humanities and Social Science,
> University of Newcastle,
> NSW 2308 Australia
> Ph: (+61) 2 49215230
> Fax: (+61) 2 49216933
> *Eu Prattein*
>>>> vaeringjar <vaeringjar@...> 28/10/10 10:34 AM >>>
>
>
>>
>> Dennis - I had toyed with the idea that ModeratusÂ¹ anapodismos of
> number was
>> just the turning back upon itself of each number, which made it what
> it was,
>> a sort of ÅjellingÂ¹. This would then be an ancestor of PlotinusÂ¹ idea
> that
>> the turning back of each lower hypostasis -- Intellect, Soul â¹ upon
> its
>> higher served to constitute itself as such. But I canÂ¹t claim to be
> very
>>
>>
>
>
> That is much simpler for sure, so that number like any other entity
> finds itself again on that 'spectrum' and at the point of its existence
> starts its epistrophe, and both forces are applied in balance at that
> point. Though I suppose that still doesn't mean we have to toss out
> multiplicity, that it's push and pull between the two, progression
> pushing to multiplicity from the one and regression pulling back from
> multiplicity to the one, and the epistrophe is the turning point, the
> exact location on the spectrum of the number or entity, with again a
> prejudice to the one as superior in a Platonic system, hence the
> directional coloring of epistrophe back to the one.
>
> But numbers aren't really entities, are they? Yet they seem more to me
> in this context than just another Aristotelian category or aspect of an
> entity? Or am I making this too complicated? Somehow number intuitively,
> or what I hope I am taking correctly as in a sort of Platonic intuition,
> is more basic or superior.
>
> I suppose though that Heraclitus would just laugh at all my
> manoeuverings here - "he kata he ana"!
>
> Dennis Clark
>
>
>

Michael Chase
CNRS UPR 76
Paris-Villejuif
France
• ... was ... Yes, Œfive¹ or Œseven¹ would be in effect in either case an anapodismos of Unity in its progress towards infinite Multiplicity. Thus would be
Message 11 of 19 , Oct 28 1:54 AM
• 0 Attachment
>
>
>
>
>
>
>
>> >
>> > Dennis ­ I had toyed with the idea that Moderatus¹ anapodismos of number >>
was
>> > just the turning back upon itself of each number, which made it what it
>> was,
>> > a sort of jelling¹. This would then be an ancestor of Plotinus¹ idea that
>> > the turning back of each lower hypostasis -- Intellect, Soul  upon its
>> > higher served to constitute itself as such. But I can¹t claim to be very
>> > confident about that. John
>> >
>> >
>
> That is much simpler for sure, so that number like any other entity finds
> itself again on that 'spectrum' and at the point of its existence starts its
> epistrophe, and both forces are applied in balance at that point. Though I
> suppose that still doesn't mean we have to toss out multiplicity, that it's
> push and pull between the two, progression pushing to multiplicity from the
> one and regression pulling back from multiplicity to the one, and the
> epistrophe is the turning point, the exact location on the spectrum of the
> number or entity, with again a prejudice to the one as superior in a Platonic
> system, hence the directional coloring of epistrophe back to the one.
>
> But numbers aren't really entities, are they? Yet they seem more to me in this
> context than just another Aristotelian category or aspect of an entity? Or am
> I making this too complicated? Somehow number intuitively, or what I hope I am
> taking correctly as in a sort of Platonic intuition, is more basic or
> superior.
>
> I suppose though that Heraclitus would just laugh at all my manoeuverings here
> - "he kata he ana"!
>
> Dennis Clark
>
>
>
>
>

Yes, five¹ or seven¹ would be in effect in either case an anapodismos of
Unity in its progress towards infinite Multiplicity. Thus would be formed
the numbers of the Decad, which seem to be regarded as the basic numbers ­
all others being combinations of these. Or is that quite crazy?

[Non-text portions of this message have been removed]
• ... Possibly I m miles off, but am wondering how faithful this translation is? Mainly as the word units entered the English language as a formulation of
Message 12 of 19 , Oct 28 4:11 AM
• 0 Attachment
On 26 October 2010 23:59, vaeringjar <vaeringjar@...> wrote:
>
>
>
> One fragment of Moderatus found in Stob. Anth. I p. 21 claims that number is "a system of units, or a progression of multiplicity (propodismos plethous) beginning from unity, and a regression (anapodismos) ending in unity" (trans. John Dillon).

Possibly I'm miles off, but am wondering how faithful this translation
is? Mainly as the word 'units' entered the English language as a
formulation of Elizabethan 'mathemagician' John Dee. As I understand
it, the previous idea of 'Unity' virtually excluded the concept of
units as 'things in themselves'. As said, might be miles off but
thought it worth a query.

Jake

http://www.underworld-apothecary.com/
• ... Well, monades, as I recall. One can make what one likes of that. JMD [Non-text portions of this message have been removed]
Message 13 of 19 , Oct 28 6:43 AM
• 0 Attachment
>
>
>
>
>
> On 26 October 2010 23:59, vaeringjar <vaeringjar@...
> <mailto:vaeringjar%40yahoo.com> > wrote:
>> >
>> >
>> >
>> > One fragment of Moderatus found in Stob. Anth. I p. 21 claims that number
>> is "a system of units, or a progression of multiplicity (propodismos
>> plethous) beginning from unity, and a regression (anapodismos) ending in
>> unity" (trans. John Dillon).
>
> Possibly I'm miles off, but am wondering how faithful this translation
> is? Mainly as the word 'units' entered the English language as a
> formulation of Elizabethan 'mathemagician' John Dee. As I understand
> it, the previous idea of 'Unity' virtually excluded the concept of
> units as 'things in themselves'. As said, might be miles off but
> thought it worth a query.
>
> Jake
>
> http://www.underworld-apothecary.com/
>
>
>
>

Well, monades, as I recall. One can make what one likes of that. JMD

[Non-text portions of this message have been removed]
• In a message dated 10/27/2010 5:54:42 P.M. Eastern Daylight Time, ... number ... in ... Nice, Mike - I was not aware at all of the other uses of the terms,
Message 14 of 19 , Oct 28 7:35 AM
• 0 Attachment
In a message dated 10/27/2010 5:54:42 P.M. Eastern Daylight Time,
vaeringjar@... writes:

--- In _neoplatonism@yahoogroups.com_ (mailto:neoplatonism@yahoogroups.com)
, "Goya" <goya@...> wrote:
>
>
> > One fragment of Moderatus found in Stob. Anth. I p. 21 claims that
number
> > is "a system of units, or a progression of multiplicity (propodismos
> > plethous) beginning from unity, and a regression (anapodismos) ending
in
> > unity" (trans. John Dillon).
> >
> > M.C. I'd be inclined to adopt another, complementary approach. By the
> (apparent) time of Moderatus the words *propodismos* and *anapodismos*
> are technical terms of astronomy: they signify two of the three
> movements adopted by the planets in their irregular motion : going
> backwards (*anapodismos*), standing still (*stêrigmos*), going forward
> (*propodismos*). That the same terminology was used in astronomy and
> number is attested by Nicomachus (Introd. 1,5,2; Theolog. p. 19). Which
> came first, the astronomical chicken or the numerological egg ? I don't
> known, and I don't know whether it's possible to know for sure.
>
> But some of the ancients certainly saw a connection. What might that be?
> Well, in the case of the motion of the planets, one can perhaps imagine
> the triad *anapodismos* - *stêrigmos* - *propodismos* as a fluctuation
> around a central point, not unlike the way state spaces cycle around an
> attractor in dynamics (I will leave it up to Bob Wallace to draw the
> obvious parallels with Hegelian dialectics!).
>
> What might this mean in number theory ? Well, negative numbers were
> already known to Diophantus, probably a near-contemporary of Moderatus.
> Could we not imagine that what's alluded to is the oscillation of the
> number line from negative numbers to zero to positive numbers?
>
> All this is just off the top of my head, and perhaps *real* scholars of
> mathematics have figured all this out long ago. But thanks for bringing
> this up, Dennis!
>
> Michael Chase
> CNRS UPR 76
> Paris-Villejuif
> France
>

Nice, Mike - I was not aware at all of the other uses of the terms, and so
I should be mindful of them, especially with an extract like this handed
down to us baldly without any of the original context within which it
appeared.

Regardless of the negative number aspect you think is not fruitful now, I
think the astronomical context of the terms you bring up still fits in with
the notion of location of a given number along, let's say, a spectrum, and
you noticed you described this notion as a complementary approach. If I am
right, we maybe also should allow the possibility that there might not
have been at that time exact philosophical terms for this sort of
"positioning" - and I use that term with its spatial reference then rather
metaphorically in the frame of my main argument for the interaction of two absolute
principles - so a somewhat allusory approach, using astronomical terminology
might have been the best Moderatus could do if there was nothing in the
philosophical vocabulary that served already by main definition to describe
progression and recession. (though we still have the issue of why not use
'prohodos', for instance, but as I recall it are not attested in philosophical
context til later, with Plotinus, of course). Though I suppose allowing
this astronomical origin - and as you say it's hard to know which came first -
rather puts a damper on my Speusippan provenance, unless of course
Moderatus is again importing newer language from another discipline and just
repackaging truely Speusippan concepts in those newer terms.

Who is to say, speculating here now, but I do think my idea offers at
least one reasonable explanation for his giving weight to both progression and
regression, and I wasn't originally even thinking of Speusippus when it
came to me several weeks ago, even though he is otherwise much on my tiny mind
these days. But as I finished the posting it just occurred to me the whole
passage may go back as well, if we can of course credit my argument.

Another aspect of this, how numbers relate in this hierarchy to the Forms,
is something else I have been trying to ponder, looking of course at the
old Academy chiefly to try to work that one out and understand their own
differing views. I agree however with Prof. Dillon that Speusippus did not
actually totally annihilate the Forms, regardless of the fragment claiming
that, rather that the situation is a bit more complicated, but that he did
give precedence to number I think is certain.

Dennis Clark

DG: It seems logical that number would be given precedence, because I
believe the issue at hand, given progression and recession, is the dyad and the
genera. Complementarity is implicate as well. Might the use of
astronomical terms suggest or indicate possible astrological intent?

David Gallagher

[Non-text portions of this message have been removed]
• In a message dated 10/27/2010 6:56:08 P.M. Eastern Daylight Time, ... (mailto:neoplatonism@yahoogroups.com) , ...
Message 15 of 19 , Oct 28 7:48 AM
• 0 Attachment
In a message dated 10/27/2010 6:56:08 P.M. Eastern Daylight Time,
jmdillon@... writes:

> --- In _neoplatonism@yahoogroups.com_
(mailto:neoplatonism@yahoogroups.com) <mailto:neoplatonism%40yahoogroups.com> ,
> "Goya" <goya@...> wrote:
>> >
>> >
>>> > > One fragment of Moderatus found in Stob. Anth. I p. 21 claims that
>>> number
>>> > > is "a system of units, or a progression of multiplicity
(propodismos
>>> > > plethous) beginning from unity, and a regression (anapodismos)
ending in
>>> > > unity" (trans. John Dillon).
>>> > >
>>> > > M.C. I'd be inclined to adopt another, complementary approach. By
the
>> > (apparent) time of Moderatus the words *propodismos* and *anapodismos*
>> > are technical terms of astronomy: they signify two of the three
>> > movements adopted by the planets in their irregular motion : going
>> > backwards (*anapodismos*), standing still (*stêrigmos*), going forward
>> > (*propodismos*). That the same terminology was used in astronomy and
>> > number is attested by Nicomachus (Introd. 1,5,2; Theolog. p. 19).
Which
>> > came first, the astronomical chicken or the numerological egg ? I
don't
>> > known, and I don't know whether it's possible to know for sure.
>> >
>> > But some of the ancients certainly saw a connection. What might that
be?
>> > Well, in the case of the motion of the planets, one can perhaps
imagine
>> > the triad *anapodismos* - *stêrigmos* - *propodismos* as a fluctuation
>> > around a central point, not unlike the way state spaces cycle around
an
>> > attractor in dynamics (I will leave it up to Bob Wallace to draw the
>> > obvious parallels with Hegelian dialectics!).
>> >
>> > What might this mean in number theory ? Well, negative numbers were
>> > already known to Diophantus, probably a near-contemporary of
Moderatus.
>> > Could we not imagine that what's alluded to is the oscillation of the
>> > number line from negative numbers to zero to positive numbers?
>> >
>> > All this is just off the top of my head, and perhaps *real* scholars
of
>> > mathematics have figured all this out long ago. But thanks for
bringing
>> > this up, Dennis!
>> >
>> > Michael Chase
>> > CNRS UPR 76
>> > Paris-Villejuif
>> > France
>> >
>
> Nice, Mike - I was not aware at all of the other uses of the terms, and
so I
> should be mindful of them, especially with an extract like this handed
down to
> us baldly without any of the original context within which it appeared.
>
> Regardless of the negative number aspect you think is not fruitful now, I
> think the astronomical context of the terms you bring up still fits in
with
> the notion of location of a given number along, let's say, a spectrum,
and you
> noticed you described this notion as a complementary approach. If I am
right,
> we maybe also should allow the possibility that there might not have
been at
> that time exact philosophical terms for this sort of "positioning" - and
I use
> that term with its spatial reference then rather metaphorically in the
frame
> of my main argument for the interaction of two absolute principles - so a
> somewhat allusory approach, using astronomical terminology might have
been the
> best Moderatus could do if there was nothing in the philosophical
vocabulary
> that served already by main definition to describe progression and
recession.
> (though we still have the issue of why not use 'prohodos', for instance,
but
> as I recall it are not attested in philosophical context til later, with
> Plotinus, of course). Though I suppose allowing this astronomical origin
- and
> as you say it's hard to know which came first - rather puts a damper on
my
> Speusippan provenance, unless of course Moderatus is again importing
> language from another discipline and just repackaging truely Speusippan
> concepts in those newer terms.
>
> Who is to say, speculating here now, but I do think my idea offers at
least
> one reasonable explanation for his giving weight to both progression and
> regression, and I wasn't originally even thinking of Speusippus when it
came
> to me several weeks ago, even though he is otherwise much on my tiny mind
> these days. But as I finished the posting it just occurred to me the
whole
> passage may go back as well, if we can of course credit my argument.
>
> Another aspect of this, how numbers relate in this hierarchy to the
Forms, is
> something else I have been trying to ponder, looking of course at the old
> Academy chiefly to try to work that one out and understand their own
differing
> views. I agree however with Prof. Dillon that Speusippus did not actually
> totally annihilate the Forms, regardless of the fragment claiming that,
rather
> that the situation is a bit more complicated, but that he did give
precedence
> to number I think is certain.
>
> Dennis Clark
>
>
>
>
>

Dennis ­ I had toyed with the idea that Moderatus¹ anapodismos of
number was
just the turning back upon itself of each number, which made it what it
was,
a sort of Œjelling¹. This would then be an ancestor of Plotinus¹ idea that
the turning back of each lower hypostasis -- Intellect, Soul ‹ upon its
higher served to constitute itself as such. But I can¹t claim to be very

I think you're onto it, John. Is it interesting that the customary mode
of expression is same-different and motion-rest? That's asymmetrical as
otherwise it ought to be same-different, rest-motion. But if we think
progression-regression and complementarity from the symbolism of the circle, same
is center with progression (outgoing to different) circumference. The
complementary regression then motion (circumference) returning to rest (center).
And in that one can see dyad, and, therefore, number. Number, therefore,
must be precedent.

David Gallagher

[Non-text portions of this message have been removed]
• In a message dated 10/27/2010 7:33:36 P.M. Eastern Daylight Time, ... number was ... was, ... that ... That is much simpler for sure, so that number like any
Message 16 of 19 , Oct 28 8:06 AM
• 0 Attachment
In a message dated 10/27/2010 7:33:36 P.M. Eastern Daylight Time,
vaeringjar@... writes:

>
> Dennis ­ I had toyed with the idea that Moderatus¹ anapodismos of
number was
> just the turning back upon itself of each number, which made it what it
was,
> a sort of Œjelling¹. This would then be an ancestor of Plotinus¹ idea
that
> the turning back of each lower hypostasis -- Intellect, Soul ‹ upon its
> higher served to constitute itself as such. But I can¹t claim to be very
>
>

That is much simpler for sure, so that number like any other entity finds
itself again on that 'spectrum' and at the point of its existence starts
its epistrophe, and both forces are applied in balance at that point. Though
I suppose that still doesn't mean we have to toss out multiplicity, that
it's push and pull between the two, progression pushing to multiplicity from
the one and regression pulling back from multiplicity to the one, and the
epistrophe is the turning point, the exact location on the spectrum of the
number or entity, with again a prejudice to the one as superior in a
Platonic system, hence the directional coloring of epistrophe back to the one.

But numbers aren't really entities, are they? Yet they seem more to me in
this context than just another Aristotelian category or aspect of an
entity? Or am I making this too complicated? Somehow number intuitively, or what
I hope I am taking correctly as in a sort of Platonic intuition, is more
basic or superior.

I suppose though that Heraclitus would just laugh at all my manoeuverings
here - "he kata he ana"!

Dennis Clark

Might finding itself = self-knowing; that is, thinking itself?

Numbers, in my current opinion, are not entities. Moreover, we need to
distinguish between Number and numbers. Number, as it seems to me, is the
power of the One lending itself to distinction; its presence to Intellect.
If the One is the Principle of Being, it must also, in a manner of speaking,
be the Principle of Number. The the idea of number as principle, however,
is problematic because number itself does not produce, yet seems
indispensable to the substance of Being in order for principles to be existents. The
idea of some sort of reciprocal progression-recession complementarity
seems essentially essential, so to speak.

David Gallagher

David Gallagher

[Non-text portions of this message have been removed]
• In the interest of simplicity, might we bear in mind the triplicity of number: cardinal, ordinal and nominal. Just wondering if there s an apples and
Message 17 of 19 , Oct 28 8:13 AM
• 0 Attachment
In the interest of simplicity, might we bear in mind the "triplicity" of
number: cardinal, ordinal and nominal. Just wondering if there's an apples
and oranges element in the discussion.

David Gallagher

In a message dated 10/27/2010 9:26:08 P.M. Eastern Daylight Time,
Harold.Tarrant@... writes:

Dear All,

There are some good ideas going around here, but might not the answer be
a bit simpler? The definition reads "a system of units, or a progression
of multiplicity (propodismos plethous) beginning from unity, and a
regression (anapodismos) ending in unity". The 'system of units' sounds
pretty unimaginative, and it is the colourful terms that suggest there
might be something rather unusual about the latter. Clearly the terms
were carefully chosen, but they seem to me to suggest a *step-by-step*
advance or retreat. Damascius uses the verb propodizo of numbers I
believe, but I've not been and looked it up.

But we should also bear in mind the possibility that arithmos is being
used in two slightly different senses (which if one thinks about it has
to be so, since a system cannot surely be identical with a movement).
One meaning given for arithmos by LSJ is simply 'counting' (= to
arithmein), which to my mind is well defined as a step-by-step movement
towards a greater or lesser plurality. I don't affirm that this is the
right solution, but just for once I'm inclined to keep both Speusippus
and Neoplatonism out of the equation.

Harold

Prof. Harold Tarrant,
School of Humanities and Social Science,
University of Newcastle,
NSW 2308 Australia
Ph: (+61) 2 49215230
Fax: (+61) 2 49216933
*Eu Prattein*
>>> vaeringjar <_vaeringjar@..._ (mailto:vaeringjar@...) >
28/10/10 10:34 AM >>>

>
> Dennis - I had toyed with the idea that Moderatus¹ anapodismos of
number was
> just the turning back upon itself of each number, which made it what
it was,
> a sort of Œjelling¹. This would then be an ancestor of Plotinus¹ idea
that
> the turning back of each lower hypostasis -- Intellect, Soul ‹ upon
its
> higher served to constitute itself as such. But I can¹t claim to be
very
>
>

That is much simpler for sure, so that number like any other entity
finds itself again on that 'spectrum' and at the point of its existence
starts its epistrophe, and both forces are applied in balance at that
point. Though I suppose that still doesn't mean we have to toss out
multiplicity, that it's push and pull between the two, progression
pushing to multiplicity from the one and regression pulling back from
multiplicity to the one, and the epistrophe is the turning point, the
exact location on the spectrum of the number or entity, with again a
prejudice to the one as superior in a Platonic system, hence the
directional coloring of epistrophe back to the one.

But numbers aren't really entities, are they? Yet they seem more to me
in this context than just another Aristotelian category or aspect of an
entity? Or am I making this too complicated? Somehow number intuitively,
or what I hope I am taking correctly as in a sort of Platonic intuition,
is more basic or superior.

I suppose though that Heraclitus would just laugh at all my
manoeuverings here - "he kata he ana"!

Dennis Clark

[Non-text portions of this message have been removed]
• ... To both Mike and Prof. Tarrant - Yes, well, I am liking seeing too much through the Speusippan lens, if there even is such a thing, since I am
Message 18 of 19 , Oct 28 11:08 AM
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--- In neoplatonism@yahoogroups.com, "Goya" <goya@...> wrote:
>
>
> I'm afraid Professor Tarrant's rationalizing explanation seems hard to
> swallow. "System of units" may not sound - *to us* - particularly
> imaginative, but the fact remains that the expression is used *only* by
> authors of Pythagorean allegiance (Moderatus, Nicomachus, Theon of Smyrna,
> Iamblichus, Syrianus, Damascius) and *never* by less philosophically
> engaged mathematicians such as Ptolemy, Archimedes, Diophantus, Euclid,
> etc. etc. This cannot be a coincidence, in my view.
>
> Nor can it be a coincidence, I think, that words derived from
> *propodizein* occur a couple of times in Homer, then *not at all for six
> or seven centuries*, then suddenly re-emerge in two, and only two, quite
> specific domains: number theory and astronomy. Top it all of by the fact
> that Nicomachus points out and explains these parallels, and I'm afraid I
> don't see how there can be much room for doubt that a connection exists,
> whatever its precise nature may be.
>
> Best, Mike
>
> > Dear All,
> >
> > There are some good ideas going around here, but might not the answer be
> > a bit simpler? The definition reads "a system of units, or a progression
> > of multiplicity (propodismos plethous) beginning from unity, and a
> > regression (anapodismos) ending in unity". The 'system of units' sounds
> > pretty unimaginative, and it is the colourful terms that suggest there
> > might be something rather unusual about the latter. Clearly the terms
> > were carefully chosen, but they seem to me to suggest a *step-by-step*
> > advance or retreat. Damascius uses the verb propodizo of numbers I
> > believe, but I've not been and looked it up.
> >
> > But we should also bear in mind the possibility that arithmos is being
> > used in two slightly different senses (which if one thinks about it has
> > to be so, since a system cannot surely be identical with a movement).
> > One meaning given for arithmos by LSJ is simply 'counting' (= to
> > arithmein), which to my mind is well defined as a step-by-step movement
> > towards a greater or lesser plurality. I don't affirm that this is the
> > right solution, but just for once I'm inclined to keep both Speusippus
> > and Neoplatonism out of the equation.
> >
> > Harold
> >
> > Prof. Harold Tarrant,
> > School of Humanities and Social Science,
> > University of Newcastle,
> > NSW 2308 Australia
> > Ph: (+61) 2 49215230
> > Fax: (+61) 2 49216933
> > *Eu Prattein*
> >>>> vaeringjar <vaeringjar@...> 28/10/10 10:34 AM >>>
> >
> >
> >>
> >> Dennis - I had toyed with the idea that ModeratusÂ¹ anapodismos of
> > number was
> >> just the turning back upon itself of each number, which made it what
> > it was,
> >> a sort of Å'jellingÂ¹. This would then be an ancestor of PlotinusÂ¹ idea
> > that
> >> the turning back of each lower hypostasis -- Intellect, Soul â¹ upon
> > its
> >> higher served to constitute itself as such. But I canÂ¹t claim to be
> > very
> >> confident about that. John
> >>
> >>
> >
> >
> > That is much simpler for sure, so that number like any other entity
> > finds itself again on that 'spectrum' and at the point of its existence
> > starts its epistrophe, and both forces are applied in balance at that
> > point. Though I suppose that still doesn't mean we have to toss out
> > multiplicity, that it's push and pull between the two, progression
> > pushing to multiplicity from the one and regression pulling back from
> > multiplicity to the one, and the epistrophe is the turning point, the
> > exact location on the spectrum of the number or entity, with again a
> > prejudice to the one as superior in a Platonic system, hence the
> > directional coloring of epistrophe back to the one.
> >
> > But numbers aren't really entities, are they? Yet they seem more to me
> > in this context than just another Aristotelian category or aspect of an
> > entity? Or am I making this too complicated? Somehow number intuitively,
> > or what I hope I am taking correctly as in a sort of Platonic intuition,
> > is more basic or superior.
> >
> > I suppose though that Heraclitus would just laugh at all my
> > manoeuverings here - "he kata he ana"!
> >
> > Dennis Clark
> >
> >
> >
>
>
> Michael Chase
> CNRS UPR 76
> Paris-Villejuif
> France
>

To both Mike and Prof. Tarrant -

Yes, well, I am liking seeing too much through the Speusippan lens, if there even is such a thing, since I am concentrating on him or trying to at least at the moment, and if all you have is a hammer you then see nails everywhere, or so they say. Turn a Platonic rock, and out pops old Speusippus, but no one later will admit it, and not just Iamblichus? He is used him more than cited - ? Perhaps not - it's easy to get carried away, especially when so much of the true origins of this Pythagorean/Platonic doctrine is completely unknowable with the present state of the textual evidence, but one wants to try at least to get some handle on these strands crossing almost 1000 years of ancient philosophy and apparently enduring.

But that lack is one thing that really attracts me personally, as I like working with sparse evidence and trying to tease - ever reasonably, I also try - something useful out of it. And I suppose one advantage to being an amateur is being able to stick my neck out at least a bit, but I would never want to go off into unsupported speculation either. And airing ideas, which I think we do here in a good environment, is one way of testing them.

I also always think if we just knew more about the simple availability of earlier texts themselves in the Middle Platonic period that it would help us a lot just to form a sort of basis to start from for such influences. I wouldn't think much of Speusippus had survived intact at all to that period, but then there is that extract in the Theog. Arith. and all the things in Iamblichus, which I do personally accept as echt, though those could be just doxologically maintained too and not original. I suspect his indirect influence however may be greater than we know, or certainly greater than we can prove conclusively, particularly on Iamblichus and so those also who followed his doctrine.

Dennis Clark
• ... OK, Yahoo is on my nerves now officially - I do check what I enter here and make corrections, and it LOSES them. above, liking should be likely . DC
Message 19 of 19 , Oct 28 12:20 PM
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>
> To both Mike and Prof. Tarrant -
>
> Yes, well, I am liking seeing too much through the Speusippan lens,

OK, Yahoo is on my nerves now officially - I do check what I enter here and make corrections, and it LOSES them.

above, "liking" should be "likely".

DC
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