[FOM] Unreasonable effectiveness

[joeshipman at aol.com]

Here is another way of organizing a response to Wigner's question of whether mathematics is unreasonably effective in the natural sciences.


First, separate the metaphysical from the metamathematical by dividing it into two questions:


1) Is the Universe fundamentally mathematical, and if so why?


2) What is the relationship between 


(a) the mathematics we can effectively do,


(b) the mathematics relevant to the natural sciences, and 


(c) all mathematics, 


and in particular, is (a) more closely related to (b) than we might otherwise expect?


The first question can be bypassed, because on this forum we are competent in metamathematics but not metaphysics (is anyone?). Let us take as given that the Universe is fundamentally mathematical, and that the natural sciences (meaning "the process of obtaining knowledge about and understanding of the universe") are based in mathematics.

To address the second question, I start by noting that I have no reason not to suppose that (a) and (b) are each contained in an initial segment of all mathematics.


The next step is to note that, by anthropic arguments, we should expect that (b) must be extensive enough to allow for the Universe to be inhabited by minds like ours. If we accept a form of the Strong AI thesis that systems that work like Turing machines can be minds like ours, then  (b) seems much more extensive, because much simpler models (such as cellular automata) suffice to produce minds.


Without the strong AI thesis, I do not know how we would model "mathematicians" (creative ones, as distinct from automated theorem-provers), and I can't answer how effective mathematics "ought to be", thus failing to shed much light on Wigner's question.


With the strong AI thesis, we can talk about 


(a') the mathematics capable of being discovered, developed, and applied to natural sciences, by beings that can be modeled as Turing machines interacting with an environment related to our physics, 


which is contained in


(a'')  the mathematics capable of being discovered and developed, by beings that can be modeled as Turing machines. 


(a') is roughly speaking the intersection of (a'') and (b), and lies between (a) and (b), and the question has now become "how close is (a') to (b), and should we have expected that?


The experience of mathematicians and physicists over the centuries has been that both (b) and (a') were seen to be much more extensive than had previously been understood. 


On the one hand,  there is a common feeling that (b) is not unimaginably vast, while (a'') is (since Godel) considered unlimited in principle, so that we should not be surprised to see (a') approach (b).


On the other hand, even if (b) is only an initial segment of mathematics, it may contain questions that are unanswerable according to now-standard metamathematical investigation. In particular, if physically measurable sequences are uncomputable, human mathematics (a'') fails to be effective in the natural sciences in the standard meaning of the word "effective" and there is a gulf between (a') and (b). In this case the effectiveness we see is unreasonable because we are at an unreasonably early stage in our understanding, and once we understand the importance of the uncomputable in physics, we will no longer regard (human) mathematics as "effective"


-- JS


-----Original Message-----
From: Steven Ericsson-Zenith <steven at iase.us>
To: tchow <tchow at alum.mit.edu>; Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Tue, Nov 5, 2013 11:38 am
Subject: Re: [FOM] Unreasonable effectiveness


This naturally begs the question of what it is to say, in exact terms, that mathematics is "unreasonably effective." What would an answer to the question look like? It is a general answer or specific to one or more mathematical domains?



My own view is that we do not have the means currently, in any form of mathematical logic (computerized or not), to address the question. But you might begin with the end cases and they may point the way. These are not philosophical questions but necessary existential questions in relation to the physical sciences: I would not take the axiomatic method for granted, for example, without asking what it is that binds (or selects) the axioms. The other end case to consider is how you demonstrate "unreasonable effectiveness" in the face of necessary fallibility in the physical sciences. This pretty much brings you back to reasoning in terms of known properties of mathematical systems.  


The physical sciences has no clearly articulated consensus on epistemology right now, so I do not know where you would begin, except to address the epistemic issues.


I will be proposing the following in a Stanford University lecture next week: if you chose Einstein's advocacy of general covariance as the basis of all physical law (that, I argue, is to be preferred to conventional axiomatic systems since it provides the lost binding between premises) and a form of structuralism as the basis of the physical sciences, and you take structuralism to be the subject of pure mathematics, then you will begin to get traction on your question - at least then pure mathematics and the physical sciences have a common subject, structure. But I do not expect that such an approach will initially be welcomed by contemporary theoretical physics.


More controversial, in terms of mathematical logic, I believe that the way you build the bridge between pure mathematics and the physical sciences is by building on the above; that is, the common subject for mathematical logic must also be a structuralism. In historical terms, I think that Benjamin Peirce's intuition, mentioned earlier, concerning the potential of complex analysis was good.


Regards,
Steven







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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