FOM: "The unreasonable effectiveness of mathematics"

[Solomon Feferman]

Julio Gonzalez Cabillon reminded me of a promise I made way back on
Dec.21, 1997, to say something re Eugene Wigner's oft referred to article
"The unreasonable effectiveness of mathematics" (Comm. Pure & Applied
Math. 13 (1960) 1-14).  The context was a group of postings about the
"social construction" of mathematics, better, the intersubjective basis
for the objectivity of humanly developed mathematical concepts. Julio
wondered, though, how to deal with Wigner's posit.  

For many years before I actually read Wigner's article, I took his article
to represent a neo-Pythogorean position according to which our mathematics
is somehow just a reflection of the underlying structure of the
universe, that it is somehow embodied in nature, for otherwise how
could we account for its effectiveness.  In other words, it would be
unreasonable if that were not the case.  But I was not left with such a
clear-cut view of Wigner's position after reading and re-reading it.  I'll
be interested to hear what other people think about it.  Here are a few
quotes from Wigner (my comments in square brackets):

"The first point is that the enormous usefulness of mathematics in the
natural sciences is something bordering on the mysterious and that there
is no rational explanation for it." (p.2) [If mathematics is embodied in
the world, that would explain it, wouldn't it?]

"...I would say that mathematics is the science of skillful operations
with concepts and rules invented just for this purpose.  The principal
emphasis is on the invention of concepts." (p.2) [This seems to point to
the subjective origin of mathematical concepts and rules.]

"Much more advanced mathematical concepts, such as complex numbers,
algebras, linear operators, Borel sets...were so devised that they are apt
subjects on which the mathematician can demonstrate his ingenuity and
sense of formal beauty." (p.3) [Ditto.]

"The complex numbers provide a particularly striking example for the
foregoing.  Certainly, nothing in our experience suggests the introduction
of these quantities." (p.3) [So does their essential role in various
physical theories demonstrate that they are embodied in nature?]

"The miracle of the appropriateness of the language of mathematics for the
formulation of the laws of physics is a wonderful gift which we neither
understand nor deserve.  We should be grateful for it and hope that it
will remain valid in future research ..." (p.14).

Well, what are we to make of this?  Of course the sample is not enough to
form an opinion of what Wigner's true stance is.  Please judge for
yourself.

At any rate, I want to bring to bear on this two articles, one by Penelope
Maddy, "Indispensability and practice" (J. Philosophy 59 (1992), 275-289)
and myself, "Why a little bit goes a long way: Logical foundations of
scientifically applicable mathematics" (PSA 1992, Vol.II (1993), 442-455).

Maddy points out (op.cit.) that "fundamental mathematized science is
'idealized' (that is, literally false)", for example "the analysis of
water waves by assuming the water to be infinitely deep or the treatment
of matter as continuous in fluid dynamics or the treatment of energy as
continuously varying quantity" (p.281), hence the mathematics involved
cannot be regarded as true other than true in the model.  

In the conclusion to my article, I agreed completely with Maddy about the
extent to which mathematized science depends on highly idealized models.
"What is remarkable [I said, op. cit. p.325], then, is not the
unreasonable effectiveness of mathematics so much as the _unreasonable
effectiveness of (mathematized) natural science_."  

Well, shifting the mystery to the unreasonable effectiveness of such
idealized mathematical models (think also of Newton's treatment of
celestial mechanics) doesn't help explain it, but it should give one pause
about what is really claimed under a Wignerian slogan.

Incidentally, f.o.m. enters the picture in my above-mentioned article.  As
I explained in an earlier posting, that reported the result of work
showing that a certain formal system of variable types conservative over
Peano Arithmetic serves apparently to formalize all (thus far accepted)
scientifically applicable mathematics (18 Nov, 23:51).  So the
indispensability of that mathematics to science can't be counted as an
argument for the real numbers somehow being embedded in the world.  [And
pity the poor engineer who tries to build his bridges on the real numbers
in ZFC starting with the empty set, or worse yet in a topos containing an
NN object.]

Sol Feferman