Das Kalenderblatt 091029


How does it happen that some important facets of the real world can in
fact be accurately analyzed by austere deductions from axioms? In
other words, how does it happen that logic fits the world; how can one
account for the extraordinary and unexpected effectiveness of formal
mathematics?
This issue can also be stated for particular cases:
How is it that the formal calculation by Newtonian mechanics of the
motions of bodies turns out to fit their actual motions?
Why is it that the formal deduction of the possible groups of symmetry
is matched by those groups as they occur in the world?
Why is it that the theoretical properties of boundary value problems
for differential equations describe so well so many aspects of
electricity, optics, mechanics, hydrodynamics, and electrodynamics?
How is it that the differential calculus seems to work both for
physics and for the economists' problems of local maxima?
Such questions of the relations of formal logical deductions to actual
events raise metaphysical problems to which I have no adequate answer.

{{But I have. The world is matched by meaningful mathematics, which
does not need axioms but confirmation by reality. That part of
mathematics, namely set theory, which is rigorously deduced from
axioms does not match anything of the world - but this wisdom has been
hidden and suppressed with great care.}}

In the practice of mathematics this notion of absolute rigor is
certainly present; but a mathematician, in addition to being guided by
his concepts of precision, is guided also by his understanding of the
breadth and depth of his subject. By "breadth" I refer to the other
objects within or without mathematics to which this subject applies,
while the issue of "depth" involves judgment as to the choice of
abstractions which will lead to the heart of the problems at issue. We
can today clearly understand notions of rigor and formulate them in
metamathematical terms, but there is no comparable analysis of breadth
or depth of mathematical research. For example, why are the simple
axioms for group theory so powerful?
One aspect of such an analysis is the choice of the direction for
mathematical research: What topic should be studied next? On this
there can be sharp opinions, for example, with Bourbaki. In the hands
of Dieudonné: this doctrine of chosen directions has become firm, not
to say frozen. It reads: "To see whether you are doing good
mathematics, look to see what Bourbaki notices; if your subject has
not come to their favorable notice, it is not worth doing," Such a
dogma can be stifling {{drum: To see whether you are doing good
mathematics, look to see what Bourbaki notices; if your subject has
come to their favorable notice, then its worth is dubious.}}

[SAUNDERS MAC LANE: "MATHEMATICAL MODELS: A SKETCH FOR THE PHILOSOPHY
OF MATHEMATICS", The American Mathematical Monthly, Vol. 88, No. 7
(1981) 462-472.]
http://home.dei.polimi.it/schiaffo/TFIS/philofmaths.pdf

Gruß, WM
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