[FOM] Unreasonable effectiveness

[Dustin Wehr]

Timothy,
Do we think that Peano Arithmetic is "unreasonably effective" in the study
of the natural numbers?
That is a special case of the question we're talking about, if you accept
the view that Max Tegmark nicely explains in his response to Wigner's
thesis. The Wikipedia entry on Wigner's page has a good summary:
"A different response, advocated by Physicist Max
Tegmark<http://en.wikipedia.org/wiki/Max_Tegmark> in
2007, is that physics is so successfully described by mathematics because the
physical world *is *completely
mathematical<http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis>,
isomorphic to a mathematical structure, and that we are simply uncovering
this bit by bit.[6]<http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences#cite_note-tegmark-6>
In
this interpretation, the various approximations that constitute our current
physics theories are successful because simple mathematical structures can
provide good approximations of certain aspects of more complex mathematical
structures. In other words, our successful theories are not mathematics
approximating physics, but mathematics approximating mathematics."

>From that view, Wigner's thesis boils down to "simple mathematical
structures can provide good approximations of certain aspects of more
complex mathematical structures" (which _is_ remarkable), and I'm wondering
if maybe that is the place you should start in your search for an
interesting definition of "unreasonably effective".

Dustin Wehr


> On Sun, Nov 3, 2013 at 5:21 PM, Timothy Y. Chow <tchow at alum.mit.edu>wrote:
>
>> In light of some of the responses, I think I should state explicitly some
>> things that I was taking for granted but that may not have been clear to
>> everyone.
>>
>> I am not (currently) interested in a general philosophical discussion of
>> Wigner's thesis.  Such a discussion tends to provoke directionless rambling
>> of a kind that is likely to cause the moderator to shut down the thread
>> faster than you can say, "Julia Robinson."
>>
>> Instead, what I am wondering is whether there is an interesting technical
>> question lurking in the vicinity.  Namely, is it possible to write down a
>> precise mathematical definition of "reasonable effectiveness" and then test
>> the hypothesis that mathematics is "unreasonably effective"?  The
>> mathematical model will necessarily have to be very much a spherical cow at
>> first if any non-trivial result is to emerge.  But it might still be
>> interesting.  In this regard, Jacques Carette's response has been the most
>> helpful one so far.  (As of this writing, I think Carette's message is
>> still awaiting moderator attention---Carette copied me separately on his
>> email---but I'm expecting it to be approved.)
>>
>>
>> Tim
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>
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