Natural Science Forum / Physics / Research / February 2008

Tho' I'm not aware of any book addressing your question specifically,
a couple of places I'd look are:

1) D'Abro "The Rise of the New Physics" 2 Vols p/back, (1951),
available 2/H on Amazon at remarkably low prices; Vol 2 has more about
physicists and goes into a lot of the background to the development of
C20th physics. (Vol 1 is more on the classical mathematicians who laid
the groundwork for theoretical physics). It might cover some of your
area of interest.

2) Hadamard "(On) The Psychology of Invention in the Mathematical
Field".  Hadamard surveyed mathematicians and theoretical physicists
in the 1930s, asking how they thought about mathematics. It's still in
print and well worth the read (if memory serves, I think I recall
reading that Einstein claimed to think "with his muscles", which when
you think about trying mentally to model curved spaces, might not be a
bad approach).It's a while since I've looked at it, but it might
offfer some hints.

Otherwise, look out for any 'snippy' biographies by physicists who
didn't get on with other physicists. Apparently Fritz Zwicky didn't
get on well with colleagues, and Pauli could be a bit acerbic (as
could Dirac in a less disenchanting way, tho' I can't conceive of
Dirac ever being sloppy.)

Hope this helps

Paul

Edward C. Jones schrieb:

My theoretical physics FAQ at
    http://www.mat.univie.ac.at/~neum/physics-faq.txt
is almost a book, and has a section containing a number of
relevant remarks and references, to which I added some from
the present thread.

The current version of this section reads as follows:

----------------------------------------
S15b. Why bother about rigor in physics?
----------------------------------------

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.

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.

    G. Polya,
    Mathematical Discovery,
    John Wiley and Sons, New York, 1962.
and, more recently, in 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
who also report on the potential and dangers of nonrigorous approaches
to scientific truth. The latter paper was discussed by various
contributions 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

Arnold Neumaier

What is your intention? Perhaps there are not even serious papers on
that topic. You asked:

While 'mathematical sloppiness by physicists' can be considered to
include basic mistakes concerning the appropriateness of particular
tools, 'the way that physicists are sloppy with mathematics' suggests
that you are considering just less qualified or lazy physicists which
tend to perform mathematics incorrectly. As polar lander showed,
experimental physics and technology do not tolerate much sloppiness.

Was John v. Neumann sloppy when he introduced Hilbert space just a few
years before he in 1935 admitted that he did no longer believe in it?

Some months ago I posted here where Schroedinger was sloppy in 1926. His
dramatised cat was not based on sloppy use of mathematics but rather on
formally 'correct' use of a mathematics that was possibly too rigorous
in the sense that usual interpretation of it possibly has been dealing a
bit sloppy with the subtle relationship between set-theoretical
fundamentals of mathematics and more comprehensive aspects of the real
world, as we may experience it.

Check out:

"The Feynman Integral and Feynman's Operational Calculus", by Gerald W.
Johnson and Michel L. Lapidus from the Oxford Science Publications.

http://www.amazon.com/dp/0198515723

Mathematical sloppiness by physicists does not exist.
Physicists do physics, not mathematics.
If the physics is OK mathematicians will find a justification
for whatever misdeeds they may construct ... in retrospect,

And even then: what may be a misdeed at one stage of development
(infinitesimals for example) may find justification
when mathematicians are at last up to it.

Best,

Jan

Thus spake Edward C. Jones <edcjones@comcast.net>

I have not read it as yet, but I believe Peter Woit's Not Even Wrong
contains a certain amount on this topic.

Regards