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What is the point of studying maths at a very high level?

  1. Mar 10, 2013 #1
    Please do not misunderstand the purpose of my question! I love maths and I respect mathematics and mathematicians very much but I want to know the point of doing maths at a very high level. To me it seems like maths is only useful when used as a tool in engineering or sciences like chemistry and physics.
    Last edited: Mar 10, 2013
  2. jcsd
  3. Mar 10, 2013 #2
    And how would you develop those tools without doing research in mathematics?
  4. Mar 10, 2013 #3
    they use very advanced math in physics. group theory, topology and more
  5. Mar 10, 2013 #4


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    This pretty much sums it all up. I don't think there is anything left to say.
  6. Mar 10, 2013 #5
    I agree with the above posters. In some cases, though... people just do it because it's cool. "What's the point of art?" is in a similar vein.
  7. Mar 13, 2013 #6
    Those people who do very high level math are obsessed. It isn't terribly rational.

    It has no application today, but tomorrow never knows. My guess is that some day it will come in handy.
  8. Mar 13, 2013 #7


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    I disagree. All of my math professors have been pretty normal people who just also happen to like math.

    If you judge rationality based on perceived utility, then perhaps. But I think that sort of view is shortsighted. For most of the mathematicians I know, in one way or another, their interest in math comes down to its aesthetic quality. Doing math because you find it beautiful or because you like the way it makes you think seems like a perfectly rational decision to me.

    This depends on what you mean by application. Much of current mathematics research will not be directly useful for any sort of concrete product. On the other hand, a lot of current research is directly (and immediately) applicable to problems in theoretical physics, cryptography, etc.
  9. Mar 13, 2013 #8
    This is quite an ignorant point-of-view. But I guess many people think this way.

    Mathematicians are not obsessed and they are not irrational. Most mathematicians are very normal people. If they are not talking about mathematics, then many people would have a difficult time pointing out who is a mathematician and who is not. You would be very surprised.

    And although mathematicians don't tend to care about applications, there really are many applications of pure mathematics. Just because you don't know them, doesn't mean that they don't exist!!
    However, there are some parts of mathematics that don't have applications at all. But there are also parts of engineering or physics without applications. I don't think it's fair to single out pure mathematics here.
  10. Mar 13, 2013 #9
    I only learnt advanced Math to help me think in a certain way (I like physics and I program applications on a computer). Training your brain to think in a logical manner is very useful application in my opinion and is a highly transferable skill.
  11. Mar 13, 2013 #10
    The IAS organized a special year on Quantum Field Theory in 1996-97. Below is the cover from the proceedings :


    In the 1970s, physicists were celebrating the incredible successes of the standard model of particle physics (which culminated recently with the discovery of the Higgs boson). Meanwhile, mathematicians had developed powerful algebraic tools to solve wide classes of complex geometrical problems. At the end of the 1990s, the physicists found themselves working on problems for 20 year ago mathematicians, and vice-versa.

    Two very specific examples, showing how alive and well the interface between mathematics and physics is, how we could be on the verge of a new revolution in our conception of space-time, can be found in
    Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics
    Scattering Amplitudes and the Positive Grassmannian

    There is nothing more than circumstantial in my choice of examples above. The point is : by the very nature of research, you can not presume what will be important in the future. I illustrated that point with applications in physics because I am familiar with that, but please look up the history behind the jpg format, or as others have alluded to, modern cryptography technics for instance. Mathematics is a universal langage underlying all of science. Mathematicians mostly do it with the same motivations as musicians, painters or philosophers, and we ought to support and even cherish them only for that. Yet, if you are looking for practical reasons, there are plenty.
  12. Mar 13, 2013 #11
    The same can be said of math.
  13. Mar 13, 2013 #12
    You don't work for Luminosity.com, do you Daminc?

    Paul Dirac was notable for his advocay of mathematical puritism in theoretical physics research. It was more important for him that a theory be mathematically beautiful than it conform to experimental data, and he felt that major advances in physics could be achieved through pure mathematical insight. This philosophy was anathema to the hard core experimentalists of the day like Rutherford. However, Dirac "faced" them by predicting spin and antimatter through simply using pure mathematical reasoning. These were not properties of matter he could have conceived of a priori to using these techniques, they were "found" through the maths.

    Other than that, there are many other examples of research in maths that were either pure or originally intended for something else LATER being adopted by physicists to attack some newly emerging issue. Of course, Einstein's use of differental geometry to formulate GR is one famous example.
  14. Mar 13, 2013 #13
    Counting cards in Vegas. I've seen that on TV.
  15. Mar 13, 2013 #14
    False. I have never been successful in trying to express a Mathematician in the form of a fraction.
    Last edited: Mar 13, 2013
  16. Mar 14, 2013 #15
    Just because you can't do it, doesn't mean that it can't be done!
  17. Mar 14, 2013 #16
    What do you get when you divide a mathemetician by a physicist, and then raise that sum to the power of a cosmologist?
  18. Mar 14, 2013 #17


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    You can call me insane, but I don't see a big difference between doing pure maths or applied maths, and I enjoy doing both.

    Though algebra maybe boring at times.
  19. Mar 14, 2013 #18
    You get a philosopher.
  20. Mar 14, 2013 #19
    Bingo! You got it Micromass, good job. First try. Can you see by this result how things have come full circle? Doesn't it all make sense now?
  21. Mar 14, 2013 #20
    The way I see it, only about fifty people in this world are doing "very high level" math. The Andrew Wiles's and Grigori Perelmans and Alexander Grothendiecks and Ed Wittens of this world strike me as being very math-focused to the point of obsession. But I don't know them personally, so what do I know.
  22. Mar 14, 2013 #21
    and Terry Tao*
  23. Mar 14, 2013 #22
    That opinion is indeed enlightening... I suppose you should contact the authors on
    to let them know your piece of mind on their work.
  24. Mar 14, 2013 #23
    Why does 'very high level maths' look very simple?
    When I watch videos or look at pictures of a top mathematician all I see is something simple like: [tex]\tilde{g}_{\alpha \beta }=\iota g_{\alpha \beta }[/tex] or [tex]x^{n}+y^{n}=z^{n}[/tex]
  25. Mar 14, 2013 #24


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    See if this is simple: prove that [itex]\mathbb{R}/\mathbb{Z}\cong \vee _{n = 1}^{\infty }S^{1}_{n}[/itex] where [itex]\cong [/itex] denotes homeomorphic, [itex]\vee [/itex] is the wedge sum, and [itex]\mathbb{R}/\mathbb{Z}[/itex] is the quotient space obtained by collapsing all the integers to a point :wink:.
  26. Mar 14, 2013 #25

    By the way, can you please help me with this: https://www.physicsforums.com/showthread.php?t=678363

    My attempt is terrible I know :(
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