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People who do foundations of maths?

  1. Yes

    3 vote(s)
    16.7%
  2. No

    15 vote(s)
    83.3%
  1. May 13, 2009 #1

    tgt

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    Conjecture: Most of the people doing research in the foundations of maths are 'not good enough' for maths.

    conjecture here is obviously a joke (but not completely).
    not good enough as in feel that maths is too difficult to understand (i.e not clear enough, too abstract).
    There's obviously also a personal taste as to why some do foundations and some don't.

    I got this idea from Hilbert who thought that abstract mathematics was an elegant way of stating mathematical proofs but that all mathematical proofs could be reduced to a concrete and constructive manner. Godel showed he was wrong but the idea can be applied to wide areas of maths. So in that sense my conjecture seems very true.
     
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  3. May 13, 2009 #2
    Foundations of math seems like a very very difficult and subtle subject from my limited experience with it. I've never heard of anyone saying it was easy.
     
  4. May 13, 2009 #3

    tgt

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    Don't you hear many people when they don't understand a proof complain it's not clear enough. Then the other person explains it in more detail, in other words (inpolitely) dumbing it down until the person understands it.

    Foundations of maths is more then just dumbing down maths but there is an aspect of it to it.
     
  5. May 13, 2009 #4

    dx

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    What exactly do you mean by 'foundations' in this context?
     
  6. May 13, 2009 #5
    I want to make sure we're on the same page here - when I hear "foundations of maths", I think of logic, set theory, category theory, and things like that, and people like Cantor and Godel. Is this what you have in mind?
     
  7. May 14, 2009 #6

    CRGreathouse

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    Foundations is a hard field -- harder than most, perhaps. I dabble in it, but I don't think I could ever do more.
     
  8. May 14, 2009 #7

    tgt

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    Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory.
     
  9. May 14, 2009 #8

    tgt

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    How come?
     
  10. May 14, 2009 #9

    tgt

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    sure.
     
  11. May 14, 2009 #10

    CRGreathouse

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    Have you tried category theory or model theory? It's serious stuff. Also all the the large cardinal stuff falls cleanly into foundations, and that's even more heady: I'm just waiting to see how much crashes down the day someone shows a really strong one turns out to be inconsistent.
     
  12. May 14, 2009 #11

    tgt

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    How much of it have you studied? At what level?
     
  13. May 14, 2009 #12

    dx

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    That's the first sentence from the wikipedia page. Do you know anything at all about these fields? If you do, you will know that none of these are 'easy'.
     
  14. May 14, 2009 #13

    tgt

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    I'm a beginner but if one was to generalize what Hilbert is describing, it doesn't seem so.
     
  15. May 14, 2009 #14

    dx

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    huh?
     
  16. May 14, 2009 #15

    tgt

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    To Hilbert, foundations of maths is making the abstract concrete, which was his programme as well. If we take the foundations of maths as achieving that goal then it would be simpler. Didn't they say that all mathematical proofs can be expanded out to very long if necessary? Is that just one aspect of mathematical logic?
     
  17. May 14, 2009 #16

    dx

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    Where did he say this? Do you have the exact quote?
     
  18. May 14, 2009 #17

    Hurkyl

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    I've heard a joke that 0=1 is the last of the large cardinal axioms. :smile: (ah, wikipedia is where I saw it)
     
  19. May 14, 2009 #18

    HallsofIvy

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    So, basically, you don't really know what "foundations of mathematics" is, and you don't really know what Hilbert was saying, but you still have the audacity to say "Most of the people doing research in the foundations of maths are 'not good enough' for maths."?

    Or are you just hoping that is true so you can do research in the foundations of maths?
     
  20. May 14, 2009 #19
    :rofl:

    Perhaps a weaker axiom would suffice: [tex]0\approx1[/tex] or [tex]0\in0[/tex]
     
  21. May 14, 2009 #20
    I focused on set theory during my undergrad years; it's a very difficult subject that is not by any means populated by people who are "not good enough" for other fields. You are very misinformed about foundations, and Hilbert.
     
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