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The trouble with mathematics?

  1. Nov 9, 2007 #1
    Just wondering have there been articles/books written by mathematicians critising current trends in research mathematics. Whether it be technical troubles or people trouble or any other trouble that is fundamental.

    i.e In string theory there is a book written by Lee Smolin critising the current trend in reserach in physics.
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  3. Nov 9, 2007 #2


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    What particular current trends?
  4. Nov 9, 2007 #3
    Even though I'm not a professional mathematician, I'm quite sure that there is not such big problems like there is in physics with the string theory. But there seems to be some disagreements still. For example, the Wikipedia mentions, that not everybody agrees that the set theory should be taken as the basis of all mathematics.

  5. Nov 9, 2007 #4
    I don't know any trends. That is also part of the question but mainly focusing on trends that is problematic, if any.
  6. Nov 11, 2007 #5
    The only such "criticism" that I've heard of is that long, tedious proofs are becoming the norm rather than the exception nowadays. Of course there's no guarantee that every problem has a "book" proof.
  7. Nov 11, 2007 #6
    Institutions, universities in particular expect their mathematicians to dish out research papers in ever increasing frequency. This has led to a myriad of trivial and plainly uninteresting 'facts' to be published in journals that very obviously have had little motivation apart from the mathematician doing it to survive in the increasingly cut-throat world of academia.
    This sort of thing gives rise to other issues such as accusations of plagarism and the sort. A prime example is the Perelman case as of late. These problems however, I believe, have not really put a dent in mathematical advancement to any comparable degree with other disciplines.
  8. Nov 11, 2007 #7


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    You just said that profs. will spend time mindless trivial journals with nothing truly important. That's a lot of time wasted pleasing academia and those who supply grants.

    Sounds to me it is putting a dent on mathematical advancement. All that time wasted could be put to better use.
  9. Nov 12, 2007 #8
    Mathematics is so vast a discipline that they are bound to stumble upon something truly extraordinary once in a while. Besides we often have problem lists in mathematics that spark attention most of the time in a way that it should be!
  10. Nov 13, 2007 #9
    This seems like a correctable problem. I'm sure if it gets serious enough, somebody will try to fix it. I think it's the case that the alternatives are worse options.
  11. Nov 13, 2007 #10
    That problem may not be very serious as it will not damage mathematics per se.

    What I was looking for are problems that gets down and breaks the heart of mathematics. i.e mathematical activity that contradicts the essense of mathematics.

    i.e string theory is a serious probem as a physics theory because it does not make experimentally testable predictions or is not falsifable.

    Mathematics is a broad discipline but if we restrict our attention to pure mathematics only then we see that pure mathematics is about proving implications in the form of A=>B. To sum it up in one sentence. This implication once proved should be undisputable.
    However are there proofs or even fields of mathematics (i.e when they became too abstract) that have been disputed?

    There was a quote by a mathematician critising Hilbert saying 'This is not mathematics, it's theology.'

    Other problems may be a discipline getting overly technical.
    ie. 'Serre subsequently changed his research focus; he apparently thought that homotopy theory, where he had started, was already overly technical.'
    Last edited: Nov 13, 2007
  12. Nov 13, 2007 #11
    That is not a fundamental problem. It just means the discipline is getting more mature with most simple problems with short solutions being solved.

    I can think of a physicist complaining that the maths in physics seems to be getting more complicated but that is (possibly) a good thing as the discipline is growing and advancing.
  13. Nov 14, 2007 #12
    You'll be hard pressed to find any established field or proof therein within mathematics where disputes linger. It boils down to the fact that a proof can either be right, conditionally right(whereupon this shall be put to light) or simply wrong.

    The style of mathematics we now strive for is based upon definition, proposition and proof, any disputes that may occur(which often do not) are cleared most definitely along one of these lines.
  14. Nov 14, 2007 #13
    As for the axiomatisation of mathematics itself, we need not bother ourselves quite so much prematurely because of Godel's incompleteness result.
  15. Nov 16, 2007 #14
    As far as folks publishing lame results to generate employment, that has always been the case and is harmless as far as mathematics is concerned - as Hilbert said the subject would remain unchanged if 99% of mathematicians had never lived.

    But there are clearly problems when a Mathematician like Alexander Grothendick, considered by some to be the greatest living mathematician (if he is indeed alive), wanders alone into the woods after expressing sentiments like this:


    This speaks more to the plague of plagiarism and anti-camaraderie among mathematicians.
  16. Nov 16, 2007 #15
    Hardy also said something along those lines, saying that third rate mathematicians which is more mathematicians were useless or something.

    I wonder if the problem is worse now, 20 years later. It could be the same reason why Perelmen refused the fields medal.

    This problem seems more apparent in physics.
  17. Nov 17, 2007 #16
    Perelman was discouraged by the fact that people who wrote 'about' his results proceeded to give themselves more credit than was due as I read. Regardless though, we do not do what we do in mathematics(I do not know about the landscape in physics that well) for politics and fame, we do it for the passion for its results. The problem is infact worse now though not as pronounced(if that makes sense?!) as before. People seem to have gotten used to the risk of plagiarism and the sort and seek only to encourage the 'good habits' at college(where the policies are stronger than ever in stark contrast to the professional world right around them that never quite manages to penetrate the barrier that prevents the good as well as the bad from coming through)
  18. Nov 17, 2007 #17
    True. The best policy is to be true to yourself and others which is also the best preparation for creative mathematics.

    A materialistic attitude I find, destroys the passion and creativity for mathematics so don't bother. ALthough modern (pure) mathematics is tough and abstract so for the mortals, the only way to survive in an environment where everyone else is plagarising is to join them. Otherwise they may get kicked out of academia so for most it's not a matter of choice. THat's only my guess.
    Last edited: Nov 17, 2007
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