Monday, March 09, 2009

Augustin Louis Cauchy

Did you know that for a long time, one had to believe in calculus?

Oh yes. Newton and Liebniz invented it, and it was used to grand effect, but it was not really "finished":
...for a long time this higher mathematics lacked inner consistency, and its acceptance rested mainly on the successes it had had in coping with the actual problems of physics. Throughout the eighteenth century the cornerstone of calculus - the concept of limit - under rigorous scrutiny appeared a contradiction in terms, and this gave rise to a long chain of caustic remarks and invective.
In the hands of Bishop Berkeley the difficulty served as a potent weapon to chastise the cocksure confidence of some mathematicians and physicists in the infallible effectiveness of their newly devised tool. ... At times even the most prominent mathematicians joined in the chorus of sarcastic criticism. Rolle, the discoverer of the mean theorem, did not stop pointing out in his lectures that calculus was a collection of ingenious fallacies. Lagrange attributed the success of calculus to the fortuitous effect of errors offsetting one another.
D'Alembert used to advise students of calculus to keep on with their studies, assuring them that faith in it would eventually come to them. The successes of calculus were of course too great to permit any serious doubt about its basic correctness in spite of some equally fundamental difficulties involved in it. In final analysis it was, however, a matter of hope that a solution to those difficulties would come in time, although such a time did not arrive until 1821 when Cauchy finally succeeded in eliminating all inconsistencies from the concept of limit.
[Jaki, The Relevance of Physics 102-3]

Cauchy... Yes, I know this is hard to believe, but it really happened. And Cauchy was not just a brilliant mathematician. He was also a very wise man:
[Cauchy] 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. ... [In the preface of the book which gave that proof] 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 Chance or Reality and Other Essays, 131 quoting Cauchy's Cours d'analyse de l'Ecole Royale Polytechnique, Ire Partie, Analyse algébrique (Paris: de l'Imprimerie Royale, 1821), pp. vi-vii.]

Augustin Louis Cauchy was born in 1789; he died in 1857. Here is some of the details of his life collected by Father Kneller:
At the conclusion of a brilliant course of studies, attached as engineer to the immense operations by which Napoleon undertook to elevate Cherbourg to a first-rate naval harbour confronting England. He was soon compelled by ill-health to resign this position; and he thenceforth devoted himself exclusively to scientific research, and published in rapid succession a number of brilliant works. That which attracted most attention was his demonstration of a Theory of Fermat, at which the most eminent mathematicians, including even Euler and Gauss, had worked in vain. Election to the Academy, and the highest educational posts in France were the reward of these first works.

His subsequent achievements were no less remarkable; they are, indeed, beyond praise. Joseph Bertrand, a learned colleague of Cauchy's, but by no means a sharer of his convictions, in his discourse on Cauchy in the Academy in 1897 expressly declares his inability to render him the praise which is his due.

"For a long time now", he says, "it has been beyond the power of any eulogy to heighten the splendour of his reputation, a reputation which can never die. We come too late to say anything but what all the world knows, and yet our predecessors who spoke immediately after his death were too early. The reputation of Cauchy grew wider as years went on; his most enthusiastic admirers of half a century ago could not have foreseen or foretold it. He had explored new regions: the heights to which he had climbed all the world knew, but no one could then have rightly appreciated the spaciousness, the consistency, the inexhaustible fertility of his researches."

Cauchy's biographer, A. Valson, joint editor of the collection of his works published at the expense of the Academy, disscussing the place which Cauchy occupies in his Science, writes: "I confine myself to the statement that many scientific authorities consider Cauchy the first mathematician of our century: no one at any rate disputes his claim to be ranked among the greatest masters. His methods and conclusions form the point of departure of most contemporary mathematicians. Herein, above all, lies the characteristic quality of his life's work."

According to O. Terquem also, the distinguishing feature of Cauchy is the creative verve of his genius he laid new paths open whereever he went. Our admiration of his extraordinary endowments is increased by the fact that his works are not limited to a single province of Mathematics, but deal with almost every part of this science. His genius for work was so amazing that at nearly every weekly meeting of the Academy he had something new to offer, and the reprint of his collected works will, according to Valson's calculation, run to 11,531 quarto pages.

This rapid sketch of Cauchy's life is sufficient to show that he was not only a man of genius, but also a man of character. And this greatness of character rested wholly and absolutely on Christian conviction and practical piety. He not merely discharged faithfully all the duties of a Catholic, but was ever foremost among those who are ready at the call of events to defend or propagate religion or to set on foot works of charity. He was a zealous member of the Conference of St. Vincent de Paul, and did all that he could do to relieve the needs of others. "Nearly every day", said the Mayor of Sceaux (where Cauchy had a country house), over his grave, "he paid me a visit. He had a poor invalid, or a foundling to recommend, a young person looking for a situation, or a soldier who was the sole support of his family and begged to be allowed to return to it." Many pious and philanthropic societies were established by his efforts, as, for example, a Society to secure the universal observance of the Sunday's rest, and the still existing Society for the maintenance of schools in the East. Of other charitable enterprises he was a leading supporter as e. g. the Society of St. Francis Regis for the legitimation of irregular unions. When in 1846 Ireland was visited by a terrible famine, Cauchy succeeded in inducing the Pope to issue a Rescript on behalf of that country. In Sceaux he secured the settlement of a community of nuns, and founded a union for the protection of youth. On his deathbed he gave a touching manifestation of his deep piety. When the priest informed him that he was about to bring the Consecrated Host to him, he gave instructions that all the most beautiful flowers in the garden should be strewn along the stairs which were to be honoured by the passage of Our Lord. The thought which most troubled his last hours was the future of a community of the Christian Brothers which he had introduced into Sceaux.

He was very intimate with many Jesuit priests, especially with the celebrated pulpit orator P. Ravignan. When shortly before the February Revolution a violent assault was made on the Schools of the Order in France, Cauchy published two pamphlets in defence of them. In one of these we find the following explicit Confession of Faith which may well find a place here.
"I am a Christian, that is to say, I believe in the divinity of Jesus Christ as did Tycho Brahe, Copernicus, Descartes, Newton, Fermat, Leibniz, Pascal, Grimaldi, Euler, Guldin Boscovich, Gerdil; as did all the great astronomers, physicists, and geometricians of past ages: nay more - I am like the greater part of these a Catholic: and were I asked for the reasons of my faith I would willingly give them. I would show that my convictions have their source not in mere prejudice but in reason and a resolute enquiry. I am a sincere Catholic as were Corneille, Racine, La Bruyère, Bossuet, Bourdaloue, Fénelon, as were and still are so many of the most distinguished men of our time, so many of those who have done most for the honour of our science, philosophy, and literature, and have conferred such brilliant lustre on our Academies. I share the deep convictions openly manifested in word, deed, and writings by so many savants of the first rank, by a Ruffini, a Haüy, a Laënnec, an Ampère, a Pelletier, a Freycinet, a Coriolis, and if I avoid naming any of those living, for fear of paining their modesty, I may at least be allowed to say that I loved to recognise all the noble generosity of the Christian faith in my illustrious friends the creator of Crystallography (Haüy), the introducers of quinine and the stethoscope (Pelletier and Laennec), the famous voyager on board the 'Urania', and the immortal founders of the theory of Dynamic Electricity (Freycinet and Ampère)."
"The life of Augustin Cauchy", wrote the celebrated J. B. Biot a, "offers a perfect model of Christian virtue, as well as of supreme intellectual activity. He was one of the most eminent mathematicians that France has produced, and his nobility of character was not less remarkable than his genius for mathematics."
[Kneller, Christianity and the Leaders of Modern Science]
Note: This book by Kneller is amazing. It was reprinted recently with an introduction by Fr. Jaki, and is available through Real View Books.


At 22 August, 2013 15:29, Blogger Harald Sack said...

Cauchy really was an extraordinary mathematician! Thanks for your interesting article about him. Yesterday (August 21) was Cauchy's birthday, and we have also dedicated an article to him in our 'Daily History of Science, Technology, and Arts' Blog. If you are interested in our article on Cauchy, you will find it here:



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