More on the Poetry of Limits

Over on the ACS blogg I posted something about Chesterton and "the poetry of limits", and I thought I would try to add just a little more.

But, since it was posted on the feast of St. Ignatius, I thought it would be worth saying a little more about the larger matters at hand, than giving the usual explanations one can find in any calculus test. Because I remembered someone had already written something quite important about the matter, so I shall quote it for you:

Engels' knowledge of physics, especially of nineteenth-century physics, was very inexact indeed. Of its many gaps two illustrations should suffice. Engels did not mention Cauchy, who in 1821 succeeded in giving an exact formulation of the theory of the limit, the cornerstone of calculus and the very basis of exact, mathematical physics. Prior to 1821 great physicists, Lagrange for example, kept
telling their students that they must take calculus on faith and wait until the rigorous proof of the limit would come. It is not likely that Engels would have found Cauchy too much to his liking had he known of Cauchy's epoch-making book containing the rigorous proof of the theory of limit. In its Preface Cauchy took pains to emphasize that calculus was not everything and that it would be a grave error to think that all valid proofs should be based on integral and differential equations. Such was a daring statement especially in the France of those times where graduates of the École Polytechnique occupied in large numbers high civil service posts and were busy in introducing the spirit of infinitesimals into politics. But as Cauchy wrote: "Nobody has up to now tried to prove by calculus the existence of Louis XIV; yet all in their right mind agree that his existence is as certain as Pythagoras' theorem.... What I have said of a historical event, can be applied equally well to a great number of questions, in religion, in ethics, in politics. Therefore, let us remain convinced that there are truths other than those of geometry, and realities other than those of sensible objects." His concluding advice was: "Let us therefore cultivate with fervor the mathematical sciences, without wishing to extend them beyond their range; and let us not imagine that one could attack the problems of history with mathematical formulas, or that one could sanction the principles of morality by theorems of algebra and calculus."
[Jaki, "Knowledge in an Age of Science" in Chance or Reality and Other Essays; the Cauchy quote is from his Cours d'analyse de l'Ecole Royale Polytechnique, Ire Partie, Analyse algébrique (Paris: de l'Imprimerie Royale, 1821), pp. vi-vii.]

Engels is the one who collaborated with Marx to bring such evils into the world.

Augustin Louis Cauchy (1789-1857) is the great mathematician who gave us the true theory of "limit" which brought the calculus - the great work of Laplace and Newton - to its stable foundation. He was a Catholic, and had a role in the return of Hermite (another mathematician) to the faith. About Hermite Fr. Jaki recalled a hilarious episode from his own student days: "I still remember the embarrassment of an excellent teacher of quantum mechanics, whom I asked in class as to why certain very useful polynomials are called Hermite polynomials. He answered that probably because of their mysterious character they were found to resemble those elusive figures, called hermits." [Jaki, A Mind's Matter 25]

Joseph-Louis Lagrange ((1736-1813) was another great mathematician who did work in celestial mechanics (that means the motion of planets, not what God has under the hood!)

It may be good to here recall Chesterton's famous line when asked "Well, can you tell me any man of intellect, great in science or philosophy, who accepted the miraculous?" he replied "With pleasure. Descartes, Dr. Johnson, Newton, Faraday, Newman, Gladstone, Pasteur, Browning, Brunetière - as many more as you please." [GKC, ILN May 4, 1907 CW27:456]