FOM: Aristotle; critique of Platonism in mathematics

[wtait@ix.netcom.com]

Steve,

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_? There is no source there. There is of course the 
traditional view of Plato's unwritten doctrines; but the difficulty with 
that is that Plato was writing up to the time of his death, and it is 
unreasonable to suppose that what he was saying in lectures was in flat 
contradiction to what he was writing. The so-called intermediates of 
mathematicals are a case in point. Where in Plato's writings does one 
find such things? Again, the difficulties that Aristotle finds with the 
notion of the `idea` Two, for example,  being itself a paradigm of a 
two-element set (a set of pure units), whether valid or not, do not apply 
to Plato's doctrine as we can read it. (As a matter of fact, they are not 
valid at all, any more than were Frege's objections to that idea in late 
19th century.) Aristotle's reference to the  `indefinite dyad', which is 
generally refered to Plato's `lecture on the Good`, indeed has its source 
in Plato's writings, namely in the _Philebus_ (which is on the Good). It 
refers to a wonderful account of magnitude (in the passages on peras and 
a peiron), including intensive magnitudes, which Aristotle completely 
misunderstood and replaced in his own writings by his piddling `mixture 
of opposites'. Due to Aristotle's influence among the ignorant churchmen 
in the middle ages, it wasn't until the 14th century that the 
mathematical treatment of intensive magnitudes reached the level implicit 
already in the _Philebus_. 

In past postings, you have praised the _Posterior Analytics_. To my mind, 
that work is transformation of the central view of Plato on exact 
science, to be found in the passage on the `second best method' in the 
_Phaedo_ and in the discussion of noesis in the passage on the Divided 
Line in Book VI of the _Republic_, adopting it, unsuccessively, to 
Aristotle's empiricism. The essential difference between Plato's and 
Aristotle's view is that, for the latter, all knowledge not only begins 
in perception of sensibles, but is essentially tied to it. In the 
perception of Socrates we see by abstraction man, in that we see animal, 
etc. It is an epistemology of classification---suitable to Aristotle's 
biology. But more to the point, all truth is ultimately empirical truth: 
I can abstract from my shirt its color; but that only means that when I 
speak of the color, I am really only speaking of the shirt, but in the 
restricted vocabulary of color. 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.

Plato understood that this abstractionist conception of mathematics would 
not work: you can't abstract what isn't there. For example, no empirical 
truth will support, via abstraction, the existence of incommensurable 
line segments. (My own hypothesis is that it was this particular fact 
which most guided Plato's conception of exact science---though there were 
other influences, such as Hereclitus and Parmenides.) The most important 
feature of Plato's view of science (so obvious to us that we miss it) is 
the autonomy of reason. We do not abstract from the empirical; but from 
the phenomenon we may (causally) come to entertain a certain kind of 
structure. We look for the first principles (axioms) for this kind of 
structure (by a dialectical process); and from that point on, all 
reasoning is purely logical.

For Plato, these axioms are not literally true of the phenomena, they are 
idealizations. (He says that they are true of the forms; but one should 
understand this as meaning nothing more than that the axioms aren't 
arbitrary; there is an idea of a certain kind of structure behind them.)

This distinction between abstraction and idealization is the critical 
distinction between  Plato and Aristotle on exact science. Plato clearly 
wins. The problem with Aristotle's Post. Analy., discussed over and over 
again, is that there is a tension between two notions of truth: derivable 
syllogistically from the first principles and empirically true. 
Ultimately, the problem is: how can we be certain that the first 
principles are true (emopirically). For Plato, the two questions are 
separate: truth in exact science means derivable from the axioms. The 
question of why the exact science (mathematics) should have empirical 
significance is treated seperately---as it should be. 

Just as Aristotle's epistemology is the epistemology of classification, 
the logic which supports it, the syllogistic, is the logic of 
classification. For all the hype to the contrary (especially Heath), it 
had nothing to do with mathematical reasoning. 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). 

Contemporary analytic philosophers like Aristotle: they see him as an 
analytic philosopher like themselves, only not so accute. (They overllook 
the fact that he actually knew something, such as the functions of some 
biological systems.) They like to think of Plato as a primitive 
Aristotle, but who didn't understand the real nature of universals. 
(MIchael Hardy (1/14/98) is absolutely right in his criticism of this.) 
This is entirely wrong. 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 accomodate it.

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

Bill Tait
Pensioner