FOM: Aristotle's critique of Platonism in mathematics

[wtait@ix.netcom.com]

In Steve Simpson, FOM: Aristotle's critique of Platonism in mathematics (1/22/98):
> Many Plato scholars
>complain that Aristotle's exposition of Platonism is a caricature .
>Many Aristotle scholars disagree and say that Aristotle's exposition
>is accurate.  I'm not prepared to judge this; I don't understand
>Plato's doctrines well enough.
and then later on
>In any case, I believe that some points of Aristotle's critique apply
>interestingly to the "Platonism" of John Steel and other contemporary
>set theorists, even if their ideas diverge significantly from Plato's.
Fair enough. But then you (Steve) are refering, not to P's doctrines, but to others that you call `Platonic', and think that Books M and N refute those other doctrines. But then spell out the views of Steel, et al, and the refutation to be found in M and N.

I am not trying to be contentious. But there have been many instances of arguments of the form: that is Platonism and therefore Š. [Filled in by some inditement of what the author is pleased to call Platonism but which may bear no relation to the text in question] In your case it is filled in by a reference to _Metaphysics_ M and N. I want to suggest that the label `Platonism' is lazy. I don't believe that Plato's doctrines are really involved (or rather: to the extent that they are, they form a common background of agreement among most of us). As the term is frequently used on FOM (e.g. by you or by Sol Feferman) I don't believe that it has a clear and univocal sense. The `Platonism' of Steel, for example, at least as manifested in his postings, is not the `Platonism' of Godel. (Incidently, I've never been sure of the origin of its contemporary usage in f.o.m. Poincare, Bernays ?) 

> > Due to Aristotle's influence among the ignorant churchmen in the
> > middle ages, ...
>
>Bill, what are you saying here?  Were these churchmen ignorant because
>that was a long time ago and they don't know as much as we do now?  Or
>were they willfully ignorant, in the sense that they didn't even want
>to absorb the scientific knowledge that was available at the time?  Or
>were they ignorant because they were influenced by Aristotle?

None of the above. I was simply referring to the bearers of the scientific knowledge of their time. I meant that Greek exact science had disappeared and, when science began to be rediscovered and incorporated into scholastic natural philosophy, it was Aristotelian. You are right that it was a transformation of A's doctrines. But in the relevant respects it was Aristotelian as opposed to Platonic. The exact science of, for example,  Euclid's _Optics_ or Archimedes' _On the Equilibrium of Planes_ or _ON Floating Bodies_, with its characteristic idealization of the phenomena, was lost: everthing had to be fitted in to the grand system of Aristotelian science. The beginnings of the mathematical treatment of intensive magnitudes and of motion in the 14th Century involved rejecting Aristotle's (and not just Aristotelian) doctrines in the physics and in particular his theory of change (`motion').

> Plato's Timaeus exercised a
>tremendous influence in the Christian world through many centuries
>during which Aristotle's writings were lost and unknown.


You are right that the _Timaeus_ was known; but this was Plato's ``likely story'' of cosmology, containing little hint of his view of mathematics (or better, exact science).  Actually, some of Aristotle's logic (the _Categories_, Topics_, _Prior Analytics_ and _On Interpretation_) was known very early via translations from the Greek and commentaries by Boethius. 
> When
>Aristotle again became available in the 12th century (via the William
>of Moerbeke translation, from the Arabic if I'm not mistaken)
 
Moerbeke is 13th century and from the Greek, but some of A's work did get translated in the middle 12th C, including the _Physics_, _Post Analytics_ and some part of the _Metaphysics_, by James of Venice---also from Greek editions. (I'm cheating: I just looked up all this stuff in _Science in the Middle Ages_ (ed. David C. Linberg, Univ. of Chicago Press, 1978) Chapter 2 (written by Lindberg).)

There is no point arguing about your evaluations of some of my statements about Aristotle as summaries for beginners or as oversimplified. I think there you are begging the question; but it is also true that my remarks were obiter dicta. But

> > For all the hype to the contrary (especially Heath), it had nothing
> > to do with mathematical reasoning. (Tait)
>
>If it had nothing to do with mathematical reasoning, how do you
>explain the geometrical and arithmetical examples in the "juicy
>quotes" in my FOM posting of 22 Dec 1997 16:56:53?  Have a look at
>"Aristotle's Philosophy of Mathematics", by H. G. Apostle, University
>of Chicago Press, 1952. (Simpson)

My statement was the mathematical reasoning does not fit the mold of syllogism. (Aristotle claims in at least 2 places---in _Post. Analy._ and in _Prior Analy._ that there is a syllogistic proof that the interior angles of a triangle equal 2 right angles. Can you make sense of that? (I tried once, but had to give up.) I didn't know about Apostle's book, though I know and like his translations of Aristotle. Thanks. I might mention that Heath's _Mathematics in Aristotle_, a source of more juicy quotes, has been reprinted (Thoemmes Press, 1996).

As for my comment on Aristotle's philosophy not accomodating mathematics, the question is not whether Aristotle appreciated mathematics or knew some. It is whether his view accomodates the fact that e.g. geometric truth cannot be regarded as simply abstracted from empirical shapes. To me, the central determinant of Plato's view of exact science was just this fact. There is no evidence in Aristotle of the need to spell out, as Plato called for in _Republic_ Book VII (and as Euclid did in his Postulates) what is to count as a construction, so as to separate the geometric notion from empirical construction. (Aristotle's own notion of `postulate', incidently, is something else.) This does not mean that Aristotle did not appreciate mathematical proof. In the sense of proving the less obvious from the more obvious, proof had been around for a very long time and in other cultures. It is the idea of being able to entertain the idea of a structure not lierally exempified in the empirical world and of finding first principles---axioms---that define that structure that is the springboard to mathematics in our sense of pure mathematics. 

>> For centuries after people spoke of the syllogistic method versus
> > the geometric method. (As late as Kant, this was the distinction
> > between demonstration (mathematical reasoning) and discursive
> > reasoning (syllogistic).(Tait)
>
>I don't think Aristotle made this distinction (syllogistic versus
>geometric).  In fact, Aristotle used geometric syllogisms to
>illustrate some of his points.  Aren't you talking about some kind of
>"Aristoteleanism" that is rather foreign to Aristotle? (Simpson)

I think we are crossing paths. My point was that a consequence of Aristotle's view of scientific demonstration in the _Post. Analy_ was that logic got split off from mathematical reasoning. 

> > Steve, I had decided not to get involved in what is only a
> > historical issue: but you have gone too far! (Tait)
>
>All right!  That's what I want to see on the FOM list!  Passion! (Simpson)

For ``passion'' read light-headedness. It was late at night when I read your posting and I had been celebrating. If it weren't for that, I would have stayed out of this on FOM. (I am very interested, but worry whether extensive discussion of history on the list is appropriate (?))

Bill Tait