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I hate axiomatics

  1. Aug 23, 2014 #1
    I have an incredible distaste for the axiomatic approach ,it's a very bad method,I think ,for teaching or learning about mathematics.I don't understand why I feel this way, I always thought inductive reasoning in mathematics ,the sort you find with physicist,is better than the deductive reasoning you find with the Bourbaki group.
    What do math people on this forum prefer ? Why ? How is it better?
    Any comment?
    P.S here's a very interesting and somewhat related quote from one of George Polya's books:
    ''Induction often begins with observation. A naturalist may observe bird life, a crystallographer the shapes of crystals. A mathematician, interested in the Theory of Numbers, observes the
    properties of the integers 1, 2, 3, 4, 5, . . . . If you wish to observe bird life with some chance of obtaining interesting results, you should be somewhat familiar with birds, interested in birds perhaps you should even like birds. Similarly, if you wish to observe the numbers, you should be interested in, and somewhat familiar with, them. You should distinguish even and odd numbers, you should know the squares 1,4,9,16,25, . . . and the primes 2,3,5,7, 11, 13, 17, 19,23,
    29, . . .. Even with so modest a knowledge you may be able to observe something interesting.''
     
  2. jcsd
  3. Aug 23, 2014 #2
    Inductive reasoning has absolutely no place in mathematics in my opinion.
     
  4. Aug 23, 2014 #3
    Some very good mathematicians like Newton,Euler,Poincaré,Minkowski, Weyl, Kolmogorov were prominent users of it,according to Vladimir Arnold who's also one of them,on other side you may find Leibniz and Descatres,the Bourbakists,Artin,Noether.
    I think the best discoveries in math were made this way(induction).
     
  5. Aug 23, 2014 #4
    Inductive reasoning cannot prove anything by its very nature. I won't argue that it isn't useful for making conjectures, but it can never be used in proof.
     
  6. Aug 23, 2014 #5
    I thought Axiomatics were a hand held calculators sold on late night infomercials.:devil:
     
  7. Aug 23, 2014 #6
    A related joke: http://abstrusegoose.com/504

    The thread opening reminded me about my attempts to study Galois theory. All books on it are perfect examples of where rigor definitions are merely piled up on each other with no apparent motivation, and they are extremely difficult to understand.

    Have you attempted to understand Galois theory? Have you succeeded or failed?
     
  8. Aug 24, 2014 #7

    Matterwave

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    Are you saying proofs by induction are invalid?
     
  9. Aug 24, 2014 #8
    These are supposed to be different things:

    http://en.wikipedia.org/wiki/Inductive_reasoning

    http://en.wikipedia.org/wiki/Mathematical_induction

    1MileCrash already contradicted himself by first stating

    and then

    so obviously he isn't choosing his words carefully.

    Anyway, I'm sure we know that whyevengothere is talking about a real thing. Sometimes mathematics is pure axiomatic definitions piled on each other, and it can get incomprehensible.
     
  10. Aug 24, 2014 #9

    disregardthat

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    The axiomatic approach is entirely necessary, especially in modern mathematics, to even do the slightest actual work of mathematics.
     
  11. Aug 24, 2014 #10
    Sounds simple when your are talking about this on the general level, but has anyone here totally enjoyed studying Galois theory? In Galois theory you are given properties of groups and fields in such way that you have no clue how the pieces are supposed to work in the end. I haven't liked it, I can admit on my part.
     
  12. Aug 24, 2014 #11

    micromass

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    I hope you know that all of these people used axioms and deductive reasoning.
     
  13. Aug 24, 2014 #12

    disregardthat

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    Not trying to assume too much, but it seems to me that you are equating the "axiomatic approach" with a way of presenting mathematics for the student, and not a way of presenting mathematics as a subject.

    If a book teaches Galois theory intuitively, with lots of examples and outside reasoning for every new piece, that would not undermine what we understand as the "axiomatic approach". I personally found the introduction to galois theory when I first encountered it in the book on Abstract algebra by Fraleigh as very difficult to put together in my mind. I do not however blame the general approach, just that particular presentation or set-up.
     
  14. Aug 24, 2014 #13
    Hello

    In my analysis

    Inductive reasoning leads from observations ( (and what we imagine to accommodate the observations in a conceptual framework) to the model (Physics).

    Once the formal model in hand, we proceed by deduction to make predictions (mathematics, calculus).

    Patrick
     
  15. Aug 24, 2014 #14
    Yes ,maybe, but I think that most of their mathematics was done ''experimentally''( there's a video you can google by V.I Arnold titled ''mathématique expérimentale'' online,you should see it).
     
  16. Aug 24, 2014 #15
    Yes ,that's exactly what I meant.
     
  17. Aug 24, 2014 #16

    micromass

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    Yes, of course. Mathematics is an experimental science in the sense that every mathematician first does some experiments on known objects and known theories in order to obtain something new. However, after this, they proceed with proving their theories rigorously using deductive reasoning. This is simply how mathematical research works.
     
  18. Aug 24, 2014 #17
    A Professor of Mathematics fan of George Polya’s classic Mathematics and Plausible Reasoning.

    Patrick
     
  19. Aug 24, 2014 #18
    Proofs by mathematical induction are not inductive reasoning. It's deductive reasoning. If I prove something via mathematical induction, it's over, the theorem is true. Yes, a mathematical "proof" that uses inductive reasoning is not a proof, completely invalid, and means nothing.

    A "proof" using inductive reasoning would be something like "well, for the first hundred natural n, the sum from 1 to n is n(n+1)/2. C'mon man, it works for the first hundred, that makes me feel a lot of feelings. Therefore, for all natural numbers, the sum from 1 to n is n(n+1)/2."

    What I mean is that inductive reasoning can be used for us to come up with "inklings" or conjectures that we think may be true (IE, they give us the idea of a problem to solve), but in actually doing mathematics (ie, proving theorems or solving problems) inductive reasoning has no place. Show me one theorem proven with inductive reasoning, and you win. Inductive reasoning by definition does not guarantee its consequent, it doesn't prove anything, ever.
     
    Last edited: Aug 24, 2014
  20. Aug 24, 2014 #19
    Read Polya 's book ''Mathematics and Plausible Reasoning''.
     
  21. Aug 24, 2014 #20

    WWGD

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    I can summarize my perspective: I know "they" (concepts, theorems) are abstractions .
    Abstractions of _what_? Information can be lost in the process of abstracting. What was the
    author thinking, aiming for when s/he coined the term?
     
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