[FOM] Unreasonable effectiveness

[John Kadvany]

The historian of mathematics Ivor Grattan-Guinness published a critique of Wigner's  claim in The Mathematical Intelligencer Summer 2008, Volume 30, Issue 3, pp 7-17: Solving Wigner’s mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences. 
JK

Antonino Drago <drago at unina.it> wrote:

>I seem that Chow's premise (i.e. the effective existence in the history of 
>science, of a persisting matching of mathematics with natural science) is 
>ill-stated.
>
>Wigner supported the idea of the unreasonable effectiveness of mathematics 
>in theoretical physics by referring to the mathematics of differential 
>equations.
>First. No whatsoever mathematical theory is suitable for a specific natural 
>science. Along centuries physics developed differential equations, but (pace 
>Rashevsky) last century theoretical biology started by means of an at all 
>different kinds mathematics.
>Second. Wigner's consideration was anticipated by M. Born, who was surprised 
>that thermodynamics resisted to be formulated by means of differential 
>equations. He asked to the friend C. Carathéodory to provide a suitable 
>system. The latter suggested a new formulation of thermodynamics, but 1) it 
>starts from abstract, unoperative axioms, as Carathéodory himself 
>recognised; 2) the differential equation is rather the resolution of the 
>problem of the exact differential (so, a 1st order equation, like Fourier's 
>equation of heat transport; very different from the reversible equations).
>Third. The time in which Born suggested the problem, was just before that 
>theoretical physics changed abruptly its mathematics from the continuous one 
>to the discrete one (se the introduction of 1905 Enistein's paper on 
>quanta); which was the the mathematics of chemistry since two centuries.
>Fourth. The mathematics in physics changed again abruptly in '60s, after the 
>discovery of the non parity by Yang and Lee; rather than differential 
>equations, symmetries became the basic mathematical technique (for 
>elementary particles, e.g.).
>Fifth. Also the effectiveness of the mathematics of the symmeties  is not 
>universal. There is no formulation of quantum mechanics entirely based on 
>symmetries (although Weyl tried it since the beginnings of the theory). By 
>reflecting about all past physical theories, A.O. Barut wrote a fine 
>reflection on the complementarity of the two mathematical techniques in 
>theoretical physics.
>In conclusion, in order to develop a scientific theory of nature, one has to 
>try to apply or even to invent (e.g. the infinitesimal analysis for Neton 
>mechanics) one among several kinds mathematical theories, without any 
>persisteance if not along some periods of time (and so contributing to some 
>paradigms). Hence, there is no constant historical phenomenon to be 
>parametrized.
>Best greetings
>Antonino Drago
>
>----- Original Message ----- 
>From: "Timothy Y. Chow" <tchow at alum.mit.edu>
>To: <fom at cs.nyu.edu>
>Sent: Saturday, November 02, 2013 4:40 PM
>Subject: [FOM] Unreasonable effectiveness
>
>
>> 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
>> _______________________________________________
>> FOM mailing list
>> FOM at cs.nyu.edu
>> http://www.cs.nyu.edu/mailman/listinfo/fom
>> 
>
>_______________________________________________
>FOM mailing list
>FOM at cs.nyu.edu
>http://www.cs.nyu.edu/mailman/listinfo/fom