Everything and More: A Compact History of Infinity by David Foster Wallace

Author:David Foster Wallace
Language: eng
Format: mobi, pdf
Tags: Infinity, Science, Science & Technology, Mathematics, Biography & Autobiography, Mathematical & Computational, Physics
ISBN: 9780393326291
Publisher: Norton
Published: 2004-12-15T18:42:39.771000+00:00

Weierstrass is not directly in the Fourier-Cauchy-Dirichlet-Riemann academic line—up until his 40s, he’s obscure in the same way Bolzano was. His early career is spent teaching high school in West Prussia (not exactly a hub),61 and he’s said to have been literally too poor to afford the postage for submitting work to journals. He finally starts publishing in the late 1850s, and sets math on its collective ear, and gets hired by prestigious U. Berlin as a prof—it’s all a long and kind of romantic story. (IYI Weierstrass is also conspicuous among mathematicians for being physically large, a gifted athlete, an inveterate partier and blowoff in college, indifferent to music (most mathematicians are fiends for music), and a cheery, non-neurotic, gregarious, wholly good and much-loved fellow. He’s also widely regarded as the greatest math teacher of the century, even though he never published his lectures or even let his students take notes.62)

The specific reason we’re now talking about Weierstrass is that it is mostly his discoveries that enable math to attack the questions that Dirichlet and Riemann’s work on the G.C.P.F.S. had raised. So much so that q.v., from math-historian I. Grattan-Guinness, “[T]he history of mathematical analysis during the last third of the 19th century is in notable measure the story of mathematicians applying Weierstrassian techniques to Riemannian problems.” The real inspiration behind these techniques is not Fourier or Riemann but the tangentially aforementioned N. H. Abel (Weierstrass being a huge Abel fan), specifically an innovation called elliptic functions that Abel had derived c. 1825 from elliptic integrals—which latter, to make a long story short, emerge in calculating the arc-length of an ellipse and are a big deal in both pure and applied math.63 Weierstrass’s first significant work (back in W. Prussia, by candlelight, in between grading quizzes) involves the power-series expansions of elliptic functions, which leads him into problems regarding the convergence of power series,64 and thence to convergence, continuity, and functions in general.

The reason Russell lauds him w/r/t the problem of infinitesimals is the same reason Weierstrass gets top billing in the Arithmetization of Analysis. He is the first to give a wholly rigorous and metaphysically untainted theory of limits. Because it’s important, and underlies the way most of us are now taught calc in school, let’s at least quickly observe that Weierstrass’s definition of limits replaces Abel/Bolzano/Cauchy’s natural-language terms like ‘approaches a limit’ and ‘becomes less than any given quantity’ with the little epsilon and delta and the ‘| |’ brackets of absolute value. A great fringe benefit of Weierstrass’s theory is that it characterizes limits and continuity in such a way that either can be defined in terms of the other. See for example his definition of continuous function, which is still the industry standard 150 years later65: f(x) is continuous at some point xn if and only if, for any positive number ε, there exists a positive δ such that for any x in the interval |x − xn| < δ, |f(x) − f(xn)| < ε.

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