Friday, November 23, 2012

Are you a 'rigor mortis' kind of guy? or maybe even a 'rigor post mortem' one?

"Certain generalities seem to have been drawn from this, namely that a concern for rigor comes at the end of a mathematical develop- ment, after the "creative ferment" has subsided, that rigor in fact means rigor mortis. Weierstrass himself provides a good counterexample to this generality, for all his work on the spectral theory of forms was motivated by a concern for rigor, a concern that was vital to his accomplishments.

Weierstrass was dissatisfied with the kind of algebraic proofs that were com- monplace in his time. These proofs proceeded by formal manipulation of the symbols involved, and no attention was given to the singular cases that could arise when the symbols were given actual values. One operated with symbols that were regarded as having "general" values, and hence such proofs were sometimes re- ferred to as treating the "general case", although it would be more appropriate to speak of the generic case. Generic reasoning had led Lagrange and Laplace to the incorrect conclusion that, in their problems, stability of the solutions to the system of linear differential equations required not only reality but the nonexistence of multiple roots. (Hence their problem had seemed all the more formidable !) In fact, Sturm who was the first to study the eigenvalue problem (1) proved among other things the "theorem" that the eigenvalues are not only real but distinct as well. His proof was of course generic, and he himself appears to have been uneasy about it; for at the end of his paper he confessed that some of his theorems might be subject to exceptions if the matrix coefficients are given specific values. Cauchy was much more careful to avoid what he called disparagingly "the generalities of algebra," but multiple roots also proved problematic for him. As he realized, his proof of the existence of an orthogonal substitution which diagonalizes the given quadratic form depended upon the nonexistence of multiple roots. He tried to brush away the cases not covered by his proof with a vague reference to an infinitesimal argument that was anything but satisfactory.

It was to clear up the muddle surrounding multiple roots by replacing generic arguments with truly general ones that Weierstrass was led to create his theory of elementary divisors. Here is a good example in which a concern for rigor proved productive rather than sterile. Another good example is to be found in the work of Frobenius, Weierstrass' student, as I shall shortly indicate.
(The Theory of Matrices in the 19th Century, Hawkins)

Oh matrix you special substitution.

"Thus by the mid-1850's the idea of treating linear substitutions as objects which can be treated algebraically much like ordinary numbers was not very novel" (The Theory of Matrices in the 19th Century, Hawkins)

Wednesday, November 21, 2012

And I was right about the curved light rays.

"He asked his readers to imagine some ex- periment in which a seemingly decisive result had been obtained, for example the construc- tion of a figure with light rays marking out four equal sides meeting at four equal angles for which the sum of the angles was less than 2π . This would seem to suggest that space was non-Euclidean,but,said,Poincare ́,thereis another interpretation, which was that space was Euclidean and light rays were curved. There could be no way of deciding logically between these two interpretations, and all we could do would be to settle for the geometry we found most convenient, which, indeed, he said would be the Euclidean one. His reasons were, however, unexpected, and will be con- sidered shortly." (Poincare ́ and the idea of a group, Gray)

Sunday, November 18, 2012

woohoo, 20 errata.

Thursday, November 1, 2012

The search for maximum simplicity.

"Numerous factors compel the scientist to revise constantly his conceptual construction. Apart from general cultural predisposi- tions, conditioned by specific philosophical, theological, or politi- cal considerations, the three most important methodological fac- tors calling for such revisions seem to be: (1) the outcome of further experimentation and observation, introducing new effects hitherto unaccounted for; (2) possible inconsistencies in the logi- cal network of derived concepts and their interrelations; (3) the search for maximum simplicity and elegance of the conceptual construction. In most cases it is a combination of two of these factors, and often even the simultaneous consideration of all of them, that leads to a readjustment or basic change of the con- ceptual structure." (Max Jammer)

(3) is definitely related to the drive for abstraction when it comes to Mathematics. One could wonder why we have (3) as a goal. But the answer is simple: because it allows our poor limited intellect to make new progress while tackling increasingly complex subjects by ... making them conceptually as simple as they possibly can be.