[FOM] Did f.o.m. begin in the 19th century & other amazing questions
Martin Davis
martin at eipye.com
Fri Oct 10 15:08:22 EDT 2003
Dean Buckner is under the impression that foundations of mathematics was a
subject invented by Frege and continued by Russell, Wittgenstein (heaven
help us), and that mathematical logians have stolen the word logic as used
by such as Mill (and why didn't he add Hegel)?
This is in haste. (I'm leaving for a week in Mexico tomorrow, and WARNING
TO FOMers: I'll not be able to attend to FOM on a daily basis during that
period.)
With respect to the first point: My teacher when I was an undergraduate
B.P. Gill, remarked that mathematics progresses in two directions, upward
seeking new information ("truths" if you like), and downward seeking
rootedness. Mathematical formalisms have more spread than imagined by their
inventors, and are always encroaching on new domains. Foundational
questions arise in the need to provide rational grounding for these new
areas. Sometimes, mathematicians find themselves doing things that
explicitly contradict previously held beliefs, and there can be talk of a
"crisis". What has always happened in the past is that mathematics has been
enriched by the new conceptual FOUNDATIONAL understandings that come from
resolving the problems.
The discovery by the Pythagoreans of the incommensurability of the diagonal
of a square with its side was a problem from the Greek understanding of
number. The properties of similar triangles leading to a proof of the
Pythagorean Theorem are easily derived under the assumption that any pair
of lengths are commensurable. But this path was unavailable. Euclid's
Elements present two different solutions to this FOUNDATIONAL problem. Book
I develops a proof of the Pythagorean Theorem (its climax) that does not
use similar triangles, and evades the problem of incommensurability. Later
(I don't remember in which book), Eudoxus's brilliant theory of
proportionality (anticipating Dedekind's development of the real number
continuum) provided a beautiful solution to the problem, leading to another
proof of the Pythagorean Theorem.
I could go on and on. Negative numbers, "imaginary" numbers, etc. etc. The
systematic development of methods for dealing with limit processes, the
"calculus", brought in its wake FOUNDATIONAL problems that took two
centuries to solve. Frege (who not only wrote in German - as Buckner
reminds us - but also was a mathematician) developed his foundational
scheme very much as part of that great effort.
The formal study of deductive reasoning was begun by Aristotle, and further
developed by the stoics and much later by the Scholastics. Then the subject
languished until it began to be studied by mathematicians. Modern
mathematical logic is the contemporary continuation of that tradition.
Alonzo Church once remarked to me that it was because formal logic had been
so static for so long that philosophers like Hegel used the word for
studies having nothing to do with formal deduction.
In haste,
Martin
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