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Why bother about rigor in physics?
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Approximate methods are almost always more efficient than rigorous ones.
You can see this, for example, from the way integrals are calculated in
numerical analysis. No one uses the 'constructive proof' by
Riemann sums or, harder, by measure theory.
Most approximations in engineering and physics are based on rules of
thumb, not on rigor. If we would insist on riger before using
approximations, we couldn't solve any of the fluid mechanics problems
that are very successfully solved today for the design and optimization
of aircraft, say - one of the uses of large-scale scientific
computation important in our culture.
For it is not even known whether the underlying systems of partial
differential equations are solvable, a theoretical task that must be
solved before one can produce rigorous error bounds, and hence a
rigorous justification of the numerical approximations involved in
the current fluid dynamics codes.
But for the logical coherence of a theory, the rigorous approach
is important.
To prove that a long, complicated expression in a single variable is
monotone may be quite hard and exceed the capacity of a typical
mathematician or phycisist, but to evaluate it at a few hundred points
and look at the plot generated is easy.
If you (the reader) are satisfied with the latter, never try to
understand mathematical physics - it will be a waste of your time.
But if you want to have physics in general look like classical
Hamiltonian mechanics - a beautiful piece of mathematically rich
and powerful theory, then you should not be satisfied with the way
current quantum field theory (say) is done, and keep looking for
a better, more solid, foundation.
About the pitfalls of using mathematics ''formally'' (i.e., without
bothering about convergence of the expressions, existence or
interchangability of limits, etc.), I recommend reading
F. Gieres,
Mathematical surprises and Dirac's formalism in quantum mechanics,
Rep. Prog. Phys. 63 (2000) 1893-1931.
quant-ph/9907069
and
G. Bonneau, J. Faraut, G. Valent,
Self-adjoint extensions of operators and the teaching of quantum
mechanics,
Amer. J. Phys. 69 (2001) 322-331.
quant-ph/0103153
See also:
K Davey,
Is Mathematical Rigor Necessary in Physics?
British J. Phil. Science 54 (2003), 39-463.
http://philsci-archive.pitt.edu/archive/00000787/
On the other hand, on the way towards finding out what is true,
nonrigorous first steps are the rule, even for hard die
mathematicians. The role of intuition and nonrigorous thinking in
mathematics is well depicted in the classics
J. Hadamard,
An essay on the psychology of invention in the mathematical field,
Princeton 1945.
and
G. Polya,
Mathematics and plausible reasoning,
2 Vols., 1954.
or
G. Polya,
Mathematical discovery,
John Wiley and Sons, New York, 1962.
More recently, the article
A. Jaffe and F. Quinn,
"Theoretical mathematics": Toward a cultural synthesis of
mathematics and theoretical physics,
Bull. Amer. Math. Soc. (N.S.) 29 (1993) 1-13.
math.HO/9307227
reports on the potential and dangers of nonrigorous approaches
to scientific truth. This paper was commented in contributions
by a number of influential mathematicians and mathematical physcists in
M. Atiyah et al.,
Responses to ``Theoretical Mathematics: Toward a cultural
synthesis of mathematics and theoretical physics'',
by A. Jaffe and F. Quinn,
Bull. Amer. Math. Soc. 30 (1994) 178-207.
math/9404229
and the response of Jaffe and Quinn is given in
A. Jaffe and F. Quinn,
Response to comments on ``Theoretical mathematics'',
Bull. Amer. Math. Soc. 30 (1994) 208-211.
math/9404231
See also
D. Zeilberger,
Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture,
math.CO/9301202,
J. Borwein, P. Borwein, R. Girgensohn and S. Parnes
Experimental Mathematics: A Discussion
(1996?)
http://grace.wharton.upenn.edu/~sok/papers/age/expmath.pdf