[FOM] The lure of the infinite

Martin Davis martin at eipye.com
Sat Feb 18 17:39:58 EST 2006

Arnon Avron wrote:

 >Obviously, Berkeley's devastating critique did NOT fall
 >on deaf ears. Otherwise we would still teach and make proofs
 >in mathematics using infinitesimals (the fact that some justification
 >for using them was found many years after they have been BANNED
 >from official mathematical proofs, though not from less
 >rigorous applications, is irrelevant). The fact is that
 >nowadays we totally reject "proofs" that were given in
 >the 17th and 18th century, and also some of the "theorems"
 >proved then. So although mathematicians to worthwhile results using
 >the dubious methods, a quest for more reliable methods
 >did start - and even succeeded. So based on this experience
 >one can confidently predict that sooner or later mathematicians
 >will pay attention to justified criticisms and replace
 >dubious set-theoretic methods by more reliable one.

This is bad history and dubious prophecy. Deaf ears are exactly what 
Berkeley's work fell on. Certainly Euler paid no attention and did 
beautiful work. It was only when the foundational issues forced their way 
into mathematical practice. that the reforms were developed. It's too long 
a history to go into detail here, but a few hints:

1. Euler was content to work with the idea of a function as a mathematical 
expression. The discovery that Fourier series could represent something 
whose graph looked like bits and pieces led Dirichlet to propose the 
current idea of a function as an arbitrary map.
2. Cauchy's "theorem" that the convergent sum of a series of continuous 
functions is continuous and its counterexamples led to the notion of 
uniform convergence.
3. Amusingly, nonstandard analysis has managed the resurrection of some of 
Euler's most outrageous proofs. My favorite example is his derivation of 
the Weierstrass infinite product for the sine by factoring its McClaurin 
series as though it were a polynomial.

I agree that further insight will likely eventually clear up the anomalies 
that remain at the outer edge of set-theoretic reasoning, but my guess is 
that this will serve to justify set-theoretic foundations rather than to 
replace them. Also (and that was the main point of my original post), as in 
the past, the needed insights will  come from mathematical practice rather 
from a critique proposing retrenchment.

Martin Davis 

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