So far as I know, and I do know something, the concept of "symmetry", insofar as it is useful in mathematics, does *not* constitute in any sense a mere prop - in practice the usages are entirely mathematical; though there are some who utter various catchphrases, espouse paradigms, support the odd yoga, but we are all very well aware that such things do not constitute real mathematics (however useful/beautiful such formulations might be).
Let us contribute a few words in response to Mr. Patio. We were thinking of symmetry in regard not so much to its use in proofs, as to the part it plays in the construction and invention of new mathematical entities. One very simple example is the way a mirror is held up to the whole numbers and lo! the negative integers are born.
I thank you for your contribution. Indeed, your statement just above carries much profound truth in it.
We were thinking also of Dirac the little Bristol man when he said "It is more important to have beauty in one's equations than to have them fit the experiment." Observe there if you will that fundamental driver of us all importance.
Of course, he was a physicist. But yes, theoretical physicists have something of a similar sense of the "importance" of "beauty" as do mathematicians.
Poincaré too tells us that the ęsthetic rather than the logical is the dominant element in mathematical creativity. And Hardy himself wrote "The mathematician's patterns, like the painter's or the poet's, must be beautiful. A mathematician is a master of pattern."
Order, as opposed to chaos, makes life comprehensible. Pattern or symmetry - one type of order - is defined and analyzed in terms of the invariants of transformation groups. Thus a plane figure possesses axial symmetry around the line
y = 0 if it is unchanged by the transformation
x' = -x; y' = y.
Yes; indeed, almost everything in geometry (and, indeed, all algebra and most of analysis) is studied in relationship to symmetry transformations.
Davis and Hersch - two American professors - give us an amusing description - much too long to reproduce here in its entirety - of what the average mathematician actually does. They point out that in his work he never bothers to define the terms "exist" "rigour" or "completeness." The "objects" which he studies were unknown until thirty years ago, and even now are known to only a few dozen persons.
This is what one does in any specialist field of science! Or certainly in technical articles contained therein. All the background material should be accessible in common text books or referenced in the bibliography, of course, and one might be able to force one's self to write the occasional expository article.
"His writing," they go on, "follows an unbreakable convention: to conceal any sign that the author or the intended reader is a human being. It gives the impression that, from the stated definitions, the desired results follow infallibly by a purely mechanical procedure. To read his proofs, one must be privy to a whole subculture of motivations, standard arguments and examples, habits of thought and agreed-upon modes of reasoning."
This is all conventional; mathematicians usually understand mathematical articles...they are not written for the non-mathematician. I do not see any particular problem here here. (of course, and this is a welcome development, recently there have been many informal blogs set up by accomplished mathematicians who are willing to ramble on about their work in a rather more accessible register; but for a professional in their field, one must have access to rigorous documentation).
But yes, in what sense does such concept constitute a "prop"? I do understand that mathematics is very-much an aesthetics-driven profession, and that one of these buzzwords that connotes beauty is "symmetry". But yes, I do not fully understand.
