[FOM] Unreasonable effectiveness
Timothy Y. Chow
tchow at alum.mit.edu
Sat Nov 2 11:40:51 EDT 2013
In 1960, Wigner argued for the unreasonable effectiveness of mathematics
in the natural sciences, and his thesis has been enthusiastically accepted
by many others.
Occasionally, someone will express a contrarian view. The two main
contrarian arguments I am aware of are:
1. The effectiveness of mathematics is about what one would expect at
random, but humans have a notorious tendency to pick patterns out of
random data and insist on an "explanation" for them when no such
explanation exists.
2. The effectiveness of mathematics is higher than one would expect from a
completely random process, but there is a form of natural selection going
on. Ideas are generated randomly, and ineffective ideas are silently
weeded out, leaving only the most effective ideas as survivors. The
combination of random generation and natural selection suffices to explain
the observed effectiveness of mathematics.
Unfortunately, the application of mathematics to the natural sciences is
such a complex and poorly understood process that I see no way of modeling
it in a way that would allow us to investigate the above controversy in a
quantitative manner. I am wondering, however, if recent progress in
computerized formal proofs might enable one to investigate the analogous
question of the (alleged) "unreasonable effectiveness of mathematics in
mathematics."
I am not sure exactly how this might go, but here is a vague outline.
Theorems are built on lemmas. We want to construct some kind of model of
the probability that Lemma X will be "useful" for proving Theorem Y. This
model would be time-dependent; that is, at any given time t, we would have
a probabilistic model, trained on the corpus of mathematics known up to
time t, that could be used to predict future uses of lemmas in theorems.
This model would represent "reasonable effectiveness." Then the thesis of
"unreasonable effectiveness" would be that this model really does evolve
noticeably over time---that the model at time t systematically
underestimates uses of Lemma X in Theorem Y at times t' > t.
I am wondering if anyone else has thought along these lines. Also I am
wondering if there is any plausible way of using the growing body of
computerized proofs to make the above outline more precise. There is of
course the problem that the "ontogeny" of computerized proofs does not
exactly recapitulate the "phylogeny" of how the theorems were arrived at
historically, but nevertheless maybe something can still be done.
Tim
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