FOM: Aristotle's critique of Platonism in mathematics

[Stephen G Simpson]

Bill Tait <wtait at ix.netcom.com> 21 Jan 98 00:21:03 writes:

 > Have you actually tried to find in Plato's writings the source of the 
 > `Platonic' doctrines that Aristotle refers to in Books M and N of 
 > _Metaphysics_?

I confess I haven't tried hard to correlate Aristotle's critique in
books M and N with Plato's actual writings.  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.

(One of the many studies of books M and N which I *haven't*
absorbed thoroughly is Julia Annas, "Aristotle's Metaphysics, Books M
and N", Oxford, 1976.)

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.

 > 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?

One way of reading the history is that Plato's Timaeus exercised a
tremendous influence in the Christian world through many centuries
during which Aristotle's writings were lost and unknown.  When
Aristotle again became available in the 12th century (via the William
of Moerbeke translation, from the Arabic if I'm not mistaken) and
influential (via Thomas Aquinas), this broke the log-jam and paved the
way for reality-oriented science and the Renaissance.

When discussing the Middle Ages and the Renaissance, we need to keep
in mind that "Platonism" and "Aristoteleanism" were highly
Christianized and may have drifted very far from Plato and Aristotle.
For example, Aristotle used mathematical demonstrations as examples of
the best kind of scientific proof, but some of the "Aristoteleans"
argued against this and went so far as to say that geometry and
arithmetic are not sciences.  (I'm getting this from Mancosu's
discussion of the Quaestio de Certitudine Mathematicarum, in his 1996
book "Philosophy of Mathematics and Mathematical Practice in the
Seventeenth Century".)

 > For Aristotle, geometry is to be understood in just this way: when
 > we speak of geometrical objects, we are really speaking of sensible
 > substance, but only in the vocabulary of extension.

Well, I wouldn't put it that way, but yes, that will give an idea to
those who are totally unfamiliar with Aristotle.

 > Plato understood that this abstractionist conception of mathematics
 > would not work: ....

  [ many other interesting points omitted -- see Bill's posting ]

These are debatable points.  I'd like to see them debated more on the
FOM list.  I think that the debate over books M and N can be highly
relevant to f.o.m. and lead to specific programs for f.o.m. research.

 > Just as Aristotle's epistemology is the epistemology of
 > classification, the logic which supports it, the syllogistic, is
 > the logic of classification.

This account of Aristotle's epistemology and logic is grossly
oversimplified.

 > For all the hype to the contrary (especially Heath), it had nothing
 > to do with mathematical reasoning.

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.

 > 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).

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?

 > Contemporary analytic philosophers like Aristotle: they see him as
 > an analytic philosopher like themselves, only not so acute.

I'm not very fond of contemporary analytic philosophy.  (See Brand
Blanshard, "Reason and Analysis", Open Court, 1962.)  But we can also
appreciate Aristotle for reasons that have little or nothing to do
with contemporary analytic philosophy.

 > The essential difference, from the point of view of f.o.m., is that
 > Plato believed in pure mathematics---the autonomy of reason---and
 > Aristotle's philosophy can not accommodate it.

I feel that this last point is highly debatable.  I grant you that
contemporary f.o.m. research has not paid much attention to Aristotle,
but that's a mistake.  Aristotelean ideas could be relevant to a
reality-oriented, objectivity-oriented foundational scheme which would
accommodate both pure and applied mathematics.  This could lead to
many interesting f.o.m. research problems.  Perhaps I'll post my ideas
on this later.  In the meantime, those who want to press me could read
the comments on Aristotle in my paper on Hilbert's program,
http://www.math.psu.edu/simpson/papers/hilbert/.

 > Steve, I had decided not to get involved in what is only a
 > historical issue: but you have gone too far!

All right!  That's what I want to see on the FOM list!  Passion!

 > Bill Tait
 > Pensioner  

In case anyone here didn't know, Bill Tait is an extremely
distinguished philosopher and f.o.m. researcher.

-- Steve