Re: Proofs of theorems and Symmetries

From: Noam D.Elkies (elkies_AT_m_at_h.harvard.edu)
Date: 09/26/04

  • Next message: Gareth McCaughan: "Re: Winternitz Theorem" 
    Date: Sun, 26 Sep 2004 02:00:02 +0000 (UTC)

    In article <cil7e6$4f1$1@dizzy.math.ohio-state.edu>,
    <tchow@lsa.umich.edu> wrote:

    >In article <cieo0l$oo$1@dizzy.math.ohio-state.edu>,
    >Eric Chopin <eric_chopin@hotmail.com> wrote:

    >> My "conjecture" is that, if T is a proovable theorem such that there
    >> exists one proof which does not require the axiom of choice, and S a
    >> symmetry property such that both the hypothesis and the conclusion are
    >> symmetric with regards to S, then there exists a proof P of T such that
    >> T is explicitely symmetric with regards to S at any stage.

    > [...]
    > I think your conjecture is a useful heuristic that may help you create
    > beautiful mathematics.
    > [...]

    I agree, but feel compelled to point out a flip side:
    once one recognizes the ubiquity and power of symmetry,
    one can become so attached to it that it becomes difficult
    to pursue any attempt at a solution that does not respect
    the full group of known symmetries of a problem. Sometimes
    one does have to break symmetry to make progress -- even
    as the symmetry is kept in mind as a check on the final answer.
    One example that comes to mind is the theory of finite-dimensional
    representations of a Lie algebra, where we begin by choosing a maximal torus.

    NDE

  • Next message: Gareth McCaughan: "Re: Winternitz Theorem"

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