[FOM] Unreasonable effectiveness
Steven Ericsson-Zenith
steven at iase.us
Sat Nov 2 20:00:49 EDT 2013
Benjamin Peirce, who in 1870 said: "In every form of material
manifestation, there is a corresponding form of human thought, so that the
human mind is as wide in its range of thought as the physical universe in
which it thinks." (Linear Associative Algebra) argued that it is for logic,
as a natural science, to build the bridge between pure mathematics and the
physical sciences. I argue that he set his son Charles, upon this task -
and this led to Charles' development of pragmaticism and logic as semeiotic
theory.
Benjamin, however, had in mind that Hamilton's, newly devised, quaternions
promised a way ahead. This was a path that Charles tried but abandoned but
it was also pursued by his elder brother James, who took over Benjamin's
chair in mathematics at Harvard, and, like his father served that
institution for 50 years. James was the world's leading expert on
quaternions when he died in 1909 (although the subject was out of favor at
the time). Benjamin would point out, I think, that both quantitative and
qualitative approaches are necessary. His 19th century reference to the
latter, as an example, being the work of Boole.
The problem has been to find a better way of dealing with the qualitative.
I am hopeful in this case that the physical sciences can aid us through
developments in biophysics.
Regards,
Steven
On Sat, Nov 2, 2013 at 8:40 AM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> 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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