FOM: The unreasonable effectiveness of mathematics

Julio Gonzalez Cabillon jgc at
Tue Jan 27 13:53:45 EST 1998

On Fri, 19 Dec 1997, 19:51:47 -0200, Julio Gonzalez Cabillon wrote:

|  Now, if one believes that mathematics is "socially constructed" (as I do
|  believe at times), how can we explain Wigner's dilemma of "unreasonable
|  effectiveness of mathematics in the natural sciences"?
|  How does a "socially constructed" discipline possess these science-like
|  qualifications?

Let me rephrase the wording of my previous questions:

Now, if one believes that mathematics is "socially constructed", 

i)  how can we explain Wigner's dilemma of "unreasonable effectiveness
of mathematics in the natural sciences"?

Richard Wesley Hamming in his "The Unreasonable Effectiveness of Mathematics"
[_The American Mathematical Monthly_, vol. 87, pp. 81-90, February 1980]
concludes that

          "we select the mathematics to fit the situation, and it is
          simply not true that the same mathematics works every place".
Perhaps, this statement might give the impression that we invent 'ad hoc'
mathematics, but this is simply not *always* the case. At any rate, it is
not very clear what Hamming had in mind when he said "we select". At the
beginning of his article he writes:

          "In his paper, Wigner gives a large number of examples of the
          effectiveness of mathematics in the physical sciences.
          Let me, therefore, draw on my own experiences that are closer
          to engineering. My first real experience in the use of
          mathematics to predict things in the real world was in
          connection with the design of atomic bombs during the Second
          World War. How was it that the numbers we so patiently
          computed on the primitive relay computers agreed so well with
          what happened on the first test shot at Almagordo? There were,
          and could be, no small-scale experiments to check the
          computations directly. Later experience with guided missiles
          showed me that this was not an isolated phenomenon - constantly
          what we predict from the manipulation of mathematical symbols
          is realized in the real world.

          "Many of you know the story of Maxwell's equations, how to some
          extent for reasons of symmetry he put in a certain term, and in
          time the radio waves that the theory predicted were found by Hertz.
          Many other examples of successfully predicting unknown physical
          effects from a mathematical formulation are well known and need
          not be repeated here. 

ii) how can we explain the unreasonable fitness of large portions of certain
fields mathematics in *other* areas of mathematics"?

I think I do not need to recall examples here, don't I?

Therefore, if one believes that MATHEMATICS is *just* a "social construction"
how can we explain usefulness and unreasonable effectiveness of the same
pieces of mathematics in widely different contexts?

What will be our "definitions" of mathematics when computers have better
"understanding"? [It might be interesting to re-read Kasparov's statements
concerning computer's power ("intelligence") in his controversial book "The
child of change", about ten years ago!]

It may not be irrelevant to recall that Wigner's "The Unreasonable
Effectiveness of Mathematics in the Natural Sciences" [_Communications on
Pure and Applied Mathematics_, vol. 13, pp. 1-14, February 1960] was
reprinted in:

   Mickens, Ronald E (ed.): "Mathematics and science", World Scientific
   Publishing, Inc., Teaneck, NJ, pp. 342, 1990. ISBN: 981-02-0233-4.
    1. Barut, A. O.:
       "On the effectiveness and limits of mathematics in physics" (1-13),
    2. Davies, P. C. W.:
      "Why is the Universe knowable?" (14-33),
    3. Doreian, Patrick:
       "Mathematics in sociology: Cinderella's carriage or pumpkin?" (34-54),

    4. Greenspan, Donald:
       "Fundamental roles of mathematics in science" (55-63),

    5. Hersh, Reuben:
       "Inner vision, outer truth" (67-72),

    6. Holladay, Wendell G.:
       "Mathematics and the natural order" (73-93),

    7. Lin, Yi:
       "A few systems-colored views of the world" (94-114),

    8. Mac Lane, Saunders:
       "The reasonable effectiveness of mathematical reasoning" (115-121),

    9. Narens Louis; Luce R. Duncan:
       "Three aspects of the effectiveness of mathematics in science" (122-135),

   10. Oldershaw, Robert L.:
       "Mathematics and natural philosophy" (136-153),

   11. Peat, F. David:
       "Mathematics and the language of nature" (154-172),

   12. Polkinghorne, John:
       "The reason within and the reason without" (173-182),

   13. Rosen, Robert:
       "The modelling relation and natural law" (183-199),

   14. Shelton, La Verne:
       "Structure and effectiveness" (200-222),

   15. Townsend James T.; Kadlec, Helena:
       "Psychology and mathematics" (223-248),

   16. von Baeyer, Hans C.:
       "Ariadne's thread: the role of mathematics in physics" (249-257),

   17. West, Bruce J.:
       "The disproportionate response" (258-290),

   18. Wigner, Eugene P.:
       "The unreasonable effectiveness of mathematics in the natural sciences"

   19. Zee, A.:
       "The effectiveness of mathematics in fundamental physics" (307-323). 

On Mon, 26 Jan 1998 18:58:19 -0500 (EST) Stephen G Simpson wrote:

     | Basically, I regard Wigner's position as disastrous, because
     | it is so mystical and unscientific.  It's terrible to say this
     | about a physicist of Wigner's stature, but there you are.

I do NOT agree with Stephen. "Mystical and unscientific" is perhaps so
emotional and subjective as Wigner's personal views.
     |   "The miracle of the appropriateness of the language of
     |   mathematics for the formulation of the laws of physics
     |    is a wonderful gift which we neither understand nor deserve."
     | This descent into the world of MacLane (Shirley, not Saunders)
     | makes me heartsick.

No comment!

     What we need is a philosophy of mathematics that will account
     for not only pure mathematics but also applications of
     mathematics to physics and other applied areas.

I wholeheartedly agree with this. But where (which) is that PhOM?

     | This is why
     | I see Aristotle's ideas as so important and inspiring.
     | For Aristotle, mathematics is simply the science of quantity,
     | i.e. of quantitative aspects of the real world.
     | From this point of view, there is no mystery or miracle about
     | the fact that quantitative knowledge is possible, no more than
     | for any other type of human knowledge.

We should try to avoid Whig history!

On Sat, 27 Dec 1997 10:58:19 -0700 (MST), Reuben Hersh wrote:

     | ...
     |    "There are 9 planets travelling round the sun. There were those 9
     |    planets before any human being stirred on Earth. *Ergo*, 9 is
     |    not a human creation.  It is part of the physical universe."
     | We have to analyze the word and concept "9".  In "There are 9 planets,"
     | "9" is an adjective.  It's a "counting number."  Counting numbers
     | applied to things like the solar system are indeed part of physical
     | reality.  The planets shine, they have certain masses, momenta, and
     | orbits, and they are nine.  Their numerosity is a fact of astronomy
     | as much as their luminosity.
     | ...

I do NOT think we need to analyze the word and concept "9". In my opinion
the problem lies elsewhere. If we admit that the physical universe has not
carbon-based brain (nor silicon-based brain), how does it possess any *concept*
of planet? Therefore the physical universe simply cannot count -- I am not
competent in theology, so I leave aside religious considerations. The
*concept* of planet belongs to us -- human beings. I happen to agree with
Hersh that the "planets shine, they have certain masses, momenta, and orbits,
and they are nine". But their numerosity (9) as much as their luminosity
(for instance) are facts of astronomy, and lies in the possibility of making classifications, distinctions, and similarities. 

Enough for now!

Best wishes from Montevideo,
                                  Julio Gonzalez Cabillon

"It is that mathematicians (or at least some of them)
have sold their souls to the devil in return for
advance information about what sort of mathematics
will be of scientific importance." Steven Weinberg

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