Wednesday, September 12, 2007

Nonstandard Analysis: A (Lost) French Perspective

The most famous mathematical mystic was no doubt Blaise Pascal. In answering those of his contemporaries who objected to reasoning with infinitely small quantities, Pascal was fond of saying that the heart intervenes to make the work clear. Pascal looked on the infinitely large and the infinitely small as mysteries, something that nature has proposed to man not for him to understand but for him to admire.

The full flower of infinitesimal reasoning came with the generations after Pascal: Newton, Leibniz, the Bernoulli brothers (Jakob and Johann) and Leonhard Euler. The fundamental theorems of the calculus were found by Newton and Leibniz in the 1660s and 1670s. The first textbook on the calculus was written in 1696 by the Marquis de L’Hospital, a pupil of Leibniz and Johann Bernoulli. Here it is stated at the outset as an axiom that two quantities differing by an infinitesimal can be considered to be equal. In other words, the quantities are at the same time considered to be equal to each other and not equal to each other! A second axiom states that a curve is “the totality of an infinity of straight segments, each infinitely small.” This is an open embracing of methods that Aristotle had outlawed 2,000 years earlier.

Indeed, wrote L’Hospital, “ordinary analysis deals only with finite quantities; this one penetrates as far as infinity itself. It compares the infinitely small differences of finite quantities; it discovered the relations between these differences, and in this way makes known the relationships between finite quantities that are, as it were, infinite compared with the infinitely small quantities. One may even say that this analysis extends beyond infinity, for it does not confine itself to the infinitely small differences but discovers the relationships between the differences of these differences.

Newton and Leibniz did not share L’Hospital’s enthusiasm. Leibniz did not claim that infinitesimals really existed, only that one could reason without error as if they did exist. ... Newton tried to avoid the infinitesimal. In his Principia Mathematica, as in Archimedes’ On the Quadrature of the Parabola, results that were originally found by infinitesimal methods are presented in a purely finite Euclidean fashion.

Phil Davis and Reuben Hersh
The Mathematical Experience,”
Nonstandard Analysis

Abraham Robinson Wiki Page

Non-standard Analysis Wiki Page

No comments: