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What is the most difficult text on mathematics?

  1. May 8, 2015 #1

    Demystifier

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    In your opinion, what is the most difficult written text (e.g. a book or a paper) on mathematics?

    My candidate: Principia Mathematica by Whitehead and Russell
     
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  3. May 8, 2015 #2

    jedishrfu

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    What about the notebooks of Ramanujan?
     
  4. May 8, 2015 #3

    Demystifier

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    At least I understand the notation in it (which cannot be said for PM). :biggrin:
     
  5. May 8, 2015 #4

    lavinia

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    why?
     
  6. May 8, 2015 #5

    Demystifier

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    Have you tried to read it?
     
  7. May 8, 2015 #6
    Difficult math or difficult writing?
     
  8. May 8, 2015 #7

    Demystifier

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    In the case of PM, it is definitely difficult writing.
    First, they use a rather unfamiliar notation (at least to modern mathematicians, including logicians).
    Second, while other books on logic have a human friendly combination of formal and informal talk, PM is almost entirely formal.
    Third, it's really big, probably much bigger than necessary to explain all what really needs to be explained.
     
  9. May 8, 2015 #8

    micromass

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    And fourth, the kind of logic they are using makes things much more complicated than using a more modern logic. The entire endaveour was to eliminate Russel's paradox by introducing type theory. This goes on to make an extremely complicated kind of mathematics. The more "modern" elimination of Russel's paradox (by the ZFC axioms) is much easier and intuitive.
     
  10. May 8, 2015 #9

    micromass

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    Anyway, to answer your question: I find about any old mathematics text very difficult to read. The older the text is, the more difficult in general. This is mainly because they have a certain kind of philosophy of mathematics that is not common anymore. For example, Newton's Principia or ancient Greek texts are difficult to read because they wanted to do everything geometrically (again: to eliminate certain paradoxes that we have solved much more adequately). The modern texts have a more balanced view of geometry/algebra. Other difficult texts (to me) are the ones written by Galois (which proves that introducing some abstraction is certainly beneficial).

    Just open any mathematical history book and try to read the old statements of mathematical results. You will almost always find the modern statement to be more comprehensible. And this (I think) mainly because we are more used to the modern philosophy of mathematics.
     
  11. May 8, 2015 #10

    wabbit

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  12. May 8, 2015 #11

    lavinia

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    The things that make math hard - other than bad writing - are non-intuitive arguments,formalism, and high levels of abstraction. Every area has some of this.
    Most mathematicians find Spanier's Algebraic Topology unreadable. Try Rational Homotopy theory by Halperin and Felix. Or have fun with grown-up Rudin.

    I sat in on a course in Algebraic Topology given by one of the immortals, and he shunned abstraction and formalism. In fact if a student tried a formal demonstration he would say, "That's not a proof" To him the proof is the idea not the demonstration. I believe that all mathematics can be seen as ideas but sadly many books have neither the time or space for it. The only way to sunlight is to talk with others and to concentrate until the ideas come through.
     
  13. May 8, 2015 #12

    lavinia

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    Another thought.

    Mathematics in 2015 is a larger and more elaborate field than it was in the early 20'th century. Whole new fields have come into existence and older fields have achieved a new sophistication. The sheer volume of theory that arose on the 20'th century dwarfs all mathematical knowledge of prior centuries. Because of this mathematics is much harder today than it was back then. To read a book or a paper nowadays requires knowing a lot of machinery. From this point of view, the hardest books are probably be written now.
     
    Last edited: May 8, 2015
  14. May 8, 2015 #13

    pwsnafu

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    An example from integration theory would be texts written by Henstock.
     
  15. May 11, 2015 #14

    Demystifier

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    The type theory may me complicated when studied in all details, but the idea of type theory (which I read about from other books, not directly from PM) is quite intuitive to me.
     
  16. May 11, 2015 #15

    Fredrik

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    The most difficult math (or mathematical physics) book I own is "Geometry of quantum theory" by Varadarajan. The second most difficult is "A course in functional analysis", by Conway.
     
  17. May 11, 2015 #16

    PAllen

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    While I can't follow the details, I have always found Perelman's papers to seem well written. There is plenty of description of the idea to be established, how it fits with other idea, and how it will be used. On the other hand, experts in the field are pretty unanimous that the logical 'step size' is way above average. This is the aspect that made it so hard, and meant every verification of it was 10 times the size of the original. On the other hand, my first point about a clear game plan led, in my recollection, to experts 'believing the program' way before they completed detailed verification.
     
  18. May 11, 2015 #17

    wabbit

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    I agree there's a lot of motivation - it's really not that the exposition is poor, rather than the content is mathematically very hard - the step size may be a good explanation, its just several sizes above my league :)
     
  19. May 12, 2015 #18
    I don't know if it is actually hard (or am I plain stupid :P), but I find the exercises in Mathematical methods for physicists by Arfken and Weber quite a handful. Rudin and Goldberg's mathematical analysis is also demanding. In case of mathematical physics, I am not good friends with Straumann's GR book.
     
  20. May 12, 2015 #19

    MathematicalPhysicist

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    PM by Russell and Whitehead is indeed quite hard to read. I read the Quine did what they do in shorter amount of pages (something like ~200 pages in comparison to what they do, three volumes of more than 200 pages).

    For me alongside this there's also Schwinger's monograph in Particles, Fields and Sources volume I which is hard to read and follow, but that's a physics book, and physics books are notoriously hard to follow.
     
  21. May 12, 2015 #20
    I think the original question overlooks what may be a valid distinction: some thing may be difficult and yet well-written.
     
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