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Book about mathematical sloppiness by physicists?

  1. Jan 28, 2008 #1
    Does anyone know of a book that discusses the ways that physicists are
    sloppy with mathematics?
     
  2. jcsd
  3. Jan 29, 2008 #2
    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?


    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
     
  4. Jan 29, 2008 #3
    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?

    >
    > 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
    >
     
  5. Jan 30, 2008 #4
    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?

    >>
    >> 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
    >>

    >
    >


    --
     
  6. Jan 31, 2008 #5
    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?
    >


    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.
     
  7. Jan 31, 2008 #6
    Edward C. Jones schrieb:

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


    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
     
  8. Feb 1, 2008 #7
    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?
    >>

    >
    > 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 ...


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

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

    Arnold Neumaier
     
  9. Feb 1, 2008 #8
    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

    "Edward C. Jones" <edcjones@comcast.net> wrote in message
    news:HYWdnQbuFqPsDgbanZ2dnUVZ_vqpnZ2d@comcast.com...
    > Does anyone know of a book that discusses the ways that physicists are
    > sloppy with mathematics?
    >
     
  10. Feb 1, 2008 #9
    Edward C. Jones <edcjones@comcast.net> wrote:

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


    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
     
  11. Feb 2, 2008 #10
    Thus spake Edward C. Jones <edcjones@comcast.net>
    >Does anyone know of a book that discusses the ways that physicists are
    >sloppy with mathematics?
    >

    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
     
  12. Feb 2, 2008 #11
    "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 ...

    >
    > .. but at least your use of dates is very sloppy.
    >
    > Schroedinger invented his equation in 1926, but his cat only in 1935.


    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
    >
     
  13. Feb 3, 2008 #12
    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.


    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
     
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