[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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