Mathematica Book about mathematical sloppiness by physicists?

[Edward C. Jones]

Does anyone know of a book that discusses the ways that physicists are
sloppy with mathematics?

 

[pellis]

On Jan 28, 7:16=A0am, "Edward C. Jones" <edcjo...@comcast.net> wrote:
> Does anyone know of a book that discusses the ways that physicists are
> sloppy with mathematics?[/color]

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]

I had in mind the use of mathematical theorems without checking out all
their hypotheses. Also useful but mathematically dubious concepts like
the Feynman path integral.

pellis wrote:
> On Jan 28, 7:16=A0am, "Edward C. Jones" <edcjo...@comcast.net> wrote:
>> Does anyone know of a book that discusses the ways that physicists are
>> sloppy with mathematics?[/color]
>
> 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
>[/color]

 

[Artie Prendergast Smith]

Not a book, I know, but if I remember correctly the following article of
Arthur Jaffe and Frank Quinn discusses exactly that phenomenon:

Arthur Jaffe, Frank Quinn. "_Theoretical Mathematics_: Toward a Cultural
Synthesis of Mathematics and Theoretical Physics". 13 pages. Bull. Amer.
Math. Soc. (N.S.) 29 (1993) 1-13.

Or on the arXiv: math.HO/9307227.

I hope that's useful.

Artie

On Tue, 29 Jan 2008, Edward C. Jones wrote:

> I had in mind the use of mathematical theorems without checking out all their
> hypotheses. Also useful but mathematically dubious concepts like the Feynman
> path integral.
>
> pellis wrote:
>> On Jan 28, 7:16=A0am, "Edward C. Jones" <edcjo...@comcast.net> wrote:
>> > Does anyone know of a book that discusses the ways that physicists are
>> > sloppy with mathematics?[/color]
>>
>> 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
>>[/color]
>
>[/color]

--

 

[Salviati]

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

> Does anyone know of a book that discusses the ways that physicists are
> sloppy with mathematics?
>[/color]

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.

 

[Arnold Neumaier]

Edward C. Jones schrieb:

> Does anyone know of a book that discusses the ways that physicists are
> sloppy with mathematics?[/color]

My theoretical physics FAQ at
http://arnold-neumaier.at/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

 

[Arnold Neumaier]

Salviati schrieb:
> What is your intention? Perhaps there are not even serious papers on
> that topic. You asked:
>
>> Does anyone know of a book that discusses the ways that physicists are
>> sloppy with mathematics?
>>[/color]
>
> 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 ...[/color]

... but at least your use of dates is very sloppy.

Schroedinger invented his equation in 1926, but his cat only in 1935.

Arnold Neumaier

 

[Mike]

Check out:

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

https://www.amazon.com/dp/0198515723/?tag=pfamazon01-20

"Edward C. Jones" <edcjones@comcast.net> wrote in message
newsYWdnQbuFqPsDgbanZ2dnUVZ_vqpnZ2d@comcast.com...
> Does anyone know of a book that discusses the ways that physicists are
> sloppy with mathematics?
>[/color]

 

[J. J. Lodder]

Edward C. Jones <edcjones@comcast.net> wrote:

> Does anyone know of a book that discusses the ways that physicists are
> sloppy with mathematics?[/color]

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.
Jan

 

[Oh No]

Thus spake Edward C. Jones <edcjones@comcast.net>
>Does anyone know of a book that discusses the ways that physicists are
>sloppy with mathematics?
>[/color]
I have not read it as yet, but I believe Peter Woit's Not Even Wrong
contains a certain amount on this topic.

Regards

--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)

http://www.teleconnection.info/rqg/MainIndex

 

[Salviati]

"Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> wrote
Salviati wrote
>>
>> 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 ...[/color]
>
> .. but at least your use of dates is very sloppy.
>
> Schroedinger invented his equation in 1926, but his cat only in 1935.[/color]

In case of 1926 I referred to Quantisation as Eigenwertproblem, 4th
Communication, Ann. Phys. 81(4), 109-139 where Schroedinger wrote

"... one may consider the real part of psi as the real wave function, if
necessary." He returned from complex domain into the real one just by
multiplying psi with its complex conjugate psi*. "

This quite common method was formally flawless from the perspective of
those who sloppily concluded from their belief and from the symmetry of
differential equations that the difference between past and "future has
merely the meaning of an albeit obstinate illusion". However, it is
insufficient if one is ready to accept the argument by Ritz that future
events cannot influence the past.

At the end of the same paper Schroedinger wrote:

"If the use of a complex wave function was in principle inevitable and
not just a mere advantage in calculation, then this would imply that
there are in principle two wave functions which only together give
information about the state of the system."

Such sloppy illusions gave rise to the EPR paper, the cat, and v.
Neumann's letter to Garret Birkhoff, dated Nov. 13: "I would like to
make a confession, which may seem immoral. I do not believe absolutely
in Hilbert-space any more." They also gave rise to naive hope for
quantum computing in the far far future. Do already announced quantum
computers work as the sellers are claiming? I was told the opposite.

The same man who was mocking about the battle between frogs and mice in
the ongoing fundamental crisis of mathematics but nonetheless reproached
Hilbert's behaviour towards Brouwer, the same man agreed with
Schroedinger on: "... two physical quantities described by non-commuting
operators, the knowledge of one precludes the knowledge of the other"
and "The psi-function must not describe a sort of blend of not yet
exploded and exploded systems."

Salviati

>
> Arnold Neumaier
>[/color]

 

[Salviati]

Pronounced mathematical rigorosity goes back to efforts of
mathematicians like Gauss who was also a physicist to some extent. Was
he rigorous when he wrote to Bessel in 1830, April 9 the following?
'..we have to admit ... that even if the number is merely manmade, space
has a reality outside our ideas, a reality to which we cannot a priori
completely dictate the physical laws'? I wonder if we can dictate any
natural law to reality.

"Edward C. Jones" <edcjones@comcast.net> wrote in
news:PuudnWNf5YJ3rgPanZ2dnUVZ_uOmnZ2d@comcast.com...
>I had in mind the use of mathematical theorems without checking out all
>their hypotheses. Also useful but mathematically dubious concepts like the
>Feynman path integral.[/color]

Gauss wrote in 'Theorie der biquadratischen Reste' 1831: 'Positive and
negative numbers can only find an application where the counted object
has an opposite, which is to be considered equal with cancellation if
united with it. Strictly speaking, this precondition only happens where
the object of counting it not concrete items but relations between each
two of them.'

While Gauss this time was correct, his complex plane was and frequently
is still applied in a sloppy manner. I consider the spectrogram at least
as mathematically dubious as the Feynman path integral. I argue one
should always know what one is doing. Mathematical rigorosity does not
always ensure physical correctness. Progress has been largely based on
superficially seen sloppy seeming thinking by Leibniz, Euler, Fourier,
Heaviside, etc.

Salviati