[FOM] Unreasonable effectiveness

[Don Palmer]

I think you are going to have problems with a definition of 'reasonable effectiveness', as this is heavily dependent upon how well we know the final specification of nature.  How 'reasonable' is our current mathematical model of nature?  Is it really that close to the 'real thing'?

This is a major issue with the general concept of Wigner's assertion as well as any attempt at making a quantitative measure.  We must assume we have a 'reasonably close' qualitative understanding of what nature really is so that we can produce some quantitative measure of 'reasonableness' of our current models.   However, our qualitative understanding of nature is based upon the (quantitative) accuracy of our current models, so we are going to end up in a tautological loop - with our current models defining how close those same models are to nature.

If, in the next 50 years, there is a major re-structuring of our (qualitative) understanding of nature (maybe 'scale' becomes the 4th dimension?), then what will any measure of 'reasonableness' mean then?  As others have mentioned, taking an historical view suggests a less than strong correlation between how well scientific models match nature over time.  The mathematical models used in the mid 1800's were nearly entirely replaced by the mid 1900's.  Further, what constituted 'nature' was qualitatively very different at these two times.  What makes us think this same situation will not repeat - the accuracy of measurement of our current models?

So I think you are going to need an appropriate concept of the 'qualitative accuracy' of a model - how well does our current model map onto (describe) nature?  If we define 'model accuracy' in terms of measurement accuracy, then we presume the qualitative concept of nature is essentially correct.  Historically this has not been a good indicator of our qualitative understanding.

Take care,
-Don

Sent from my iPad

> On Nov 3, 2013, at 8:21 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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