[FOM] Unreasonable effectiveness
Steven Ericsson-Zenith
steven at iase.us
Mon Nov 4 23:23:26 EST 2013
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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