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Eugene Wigner's take on math's role in the natural sciences #1

  1. Mar 30, 2012 #1
    I had this posted under the wrong field earlier until I realized it's an applied mathematics question

    An excerpt from The Unreasonable Effectiveness of Mathematics in the Natural Sciences
    by Eugene Wigner
    "There is a story about two friends, who were classmates in high school, talking about their
    jobs. One of them became a statistician and was working on population trends. He showed a
    reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution
    and the statistician explained to his former classmate the meaning of the symbols for the actual
    population, for the average population, and so on. His classmate was a bit incredulous and was
    not quite sure whether the statistician was pulling his leg. “How can you know that?” was
    his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What
    is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are
    pushing your joke too far,” said the classmate, “surely the population has nothing to do with
    the circumference of the circle.” Naturally, we are inclined to smile about the simplicity of
    the classmate’s approach. Nevertheless, when I heard this story, I had to admit to an eerie
    feeling because, surely, the reaction of the classmate betrayed only plain common sense. I
    was even more confused when, not many days later, someone came to me and expressed his
    bewilderment2 with the fact that we make a rather narrow selection when choosing the data
    on which we test our theories. “How do we know that, if we made a theory which focuses its
    attention on phenomena we disregard and disregards some of the phenomena now commanding
    our attention, that we could not build another theory which has little in common with the
    present one but which, nevertheless, explains just as many phenomena as the present theory?”
    It has to be admitted that we have no definite evidence that there is no such theory.
    The preceding two stories illustrate the two main points which are the subjects of the present
    discourse. The first point is that mathematical concepts turn up in entirely unexpected connections.
    Moreover, they often permit an unexpectedly close and accurate description of the
    phenomena in these connections. Secondly, just because of this circumstance, and because we
    do not understand the reasons of their usefulness, we cannot know whether a theory formulated
    in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that
    of a man who was provided with a bunch of keys and who, having to open several doors in succession,
    always hit on the right key on the first or second trial. He became skeptical concerning
    the uniqueness of the coordination between keys and doors."

    So we dont understand why math works?

    Also just curious what exactly does pi have to do with the population (distribution)?

    LINK: http://www.dartmouth.edu/~matc/MathD...ng/Wigner.html [Broken]
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Mar 30, 2012 #2

    mathman

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    There are essentially two parts to the answer. First the population distribution is being approximated by the normal distribution. Second the normal distribution (like any probability distribution) has a requirement that the total probability be one. In order to get the total equal one for the normal distribution, π appears in the normalization constant.

    Aside: your link didn't work for me.
     
  4. Mar 30, 2012 #3
    Sorry about that. I should've checked:
    http://www.ipod.org.uk/reality/reality_wigner.pdf [Broken]

    QUOTE=mathman;3842087]There are essentially two parts to the answer. First the population distribution is being approximated by the normal distribution. Second the normal distribution (like any probability distribution) has a requirement that the total probability be one. In order to get the total equal one for the normal distribution, π appears in the normalization constant.

    How did the mathematician analytically come to the constant pi (apart from what you mentioned about the normal distribution assumptions)? Sorry if my question is too rudimentary.
     
    Last edited by a moderator: May 5, 2017
  5. Mar 30, 2012 #4
    Almost all ancient mathematicians who concerned themselves with pi (I say almost because I'm not an expert on the subject and there might be exceptions) used the circle in some extent to approximate pi. Ratio of circumference to diameter. I believe Archimedes first approximated the value significantly by enclosing the circle in n-gons on the inside and outside and calculating those perimeters to give a range for pi.
     
  6. Mar 31, 2012 #5
    I see so its the ratio, not a numerical value put in there, got it.
     
  7. Mar 31, 2012 #6

    mathman

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    The integrand for the standard normal distribution is of the form exp(-x2/2). The integral over the whole real line = √(2π). Since the integral has to be 1 (total probability), we need to divide the integrand by √(2π).
     
    Last edited by a moderator: May 5, 2017
  8. Mar 31, 2012 #7
    math started from totally common sense concepts, e.g., natural numbers, ,straight lines, etc. and gradually developed into totally unexpected concepts through logical reasoning. I guess the question is what is logic, and is there another set of logic that works as well as ours, and come up with different theories that explain as many different phenomena as ours.
     
  9. Mar 31, 2012 #8
    http://www.infoocean.info/avatar2.jpg [Broken]There are essentially two parts to the answer.
     
    Last edited by a moderator: May 5, 2017
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