[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 

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,

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