Saturday, March 21, 2009

Charles Hermite

Whenever I hear someone speak of "transcendental" - usually in modification of the word "meditation", which for me as a Catholic is always the extreme presence of thought, and not its absence, as in the "eastern" form...

Ahem. Let me start over.

Whenever I hear someone speak of "transcendental" I think of Hermite.

Charles Hermite (1822-1901) is the great French Catholic mathematician who established that the number e is transcendental. This number e, just like the much more famous number p, is represented by a letter - it's approximately 2.71828, but it never repeats. It is transcendental because it is not the root of any algebraic equation with integer coefficients. This was not Hermite's only work; other results are far harder to state, but play important roles in quantum physics, which comes up in - oh, semiconductors. (That means COMPUTERS, you know. And cell phones, and video games and all that. Oh!)

Let us hear Kneller's entire report, both informative and moving:
On Jan. 14th 1901 there died at Paris "the last of the great mathematicians of the second half of the nineteenth century", a gray-haired veteran of seventy-eight, whose eminence in the history of science was proclaimed in the highest terms of praise by the highest authorities. The first of these eulogists was Fouqué, President of the Academy of Sciences of Paris.

"Hermite", he writes, "doyen of our Geometrical section, and member of the Academy since 1856, was one of our special glories. All who sit here as geometricians think it their chief honour to have been pupils of his, all are penetrated with gratitude for the generous aid which he has constantly given them. Wherever science is cultivated, the name of Hermite is spoken with veneration."

To the praise of his scientific genius there was added recognition of his lovable character.

"With Hermite", wrote La Marc, "there disappears one of the unsullied glories of French science.
Hermite not only stood among the masters of mathematics of the last century; his private life, also, was a model. No one ever pushed unselfish devotion to science farther than he did.

"He leaves to history an imperishable name, and to all those who had the happiness to know him, the memory of a man as great of heart as of intellect. A convinced spiritualist he believed that the soul would one day be crowned with a complete revelation of those mathematical harmonies of which only the reflex is accessible to human nature."

Not very long before, on Dec. 24th, 1892, France had celebrated with the greatest splendour Hermite's seventieth birthday. The Minister of Education presided at the Jubilee Assembly, the most eminent French mathematicians were present, and nearly every learned society offered an address. The King of Sweden decorated Hermite with an order, hitherto conferred on no one in France except the President of the Republic and Pasteur. And all these honours were well deserved, for, as Poincaré said in his discourses, Hermite had for fifty years laboured incessantly in all the most difficult departments of mathematics and had enriched mathematical analysis, Algebra, and the Theory of Numbers with "inestimable conquests". When only twenty years of age he had addressed to Jacobi an essay of a very few pages which placed him at one bound on a level with the first mathematicians in Europe. After the death of Cauchy, Gauss, Jacobi and Dirichlet, he was universally regarded as the leader of his science. At the mathematical Congresses at Zurich in 1897, and at Paris in 1900, he was elected, with acclamation, Honorary President.

Of his discoveries we may mention one, which will be understood even by those who have no technical knowledge of mathematics. It was Hermite who in 1873 demonstrated for the first time that the quantity e is transcendent i. e. it cannot be the root of an algebraic equation with integral co-efficients. Hermite's line of proof was in 1881 extended by Lindemann to the quantity p, and by means of it an interesting fact was established. Inasmuch as p cannot be the root of a quadratic equation it follows that it cannot be determined with rule and compass, in other words, that the "squaring of the circle" is impossible.

That p does not admit of determination with other instruments does not of course follow; in point of fact it can be determined with the help of the integraph, invented in 1880 by the Russian Abakanowicz. An old problem was thus at last solved: it had taken two thousand years, and the expenditure of a vast deal of pains and penetration to discover that the attempt so often renewed was a priori impossible. Professor F. Klein may justly say that this discovery, suggested and made possible by Termite, marks an epoch in the history of mathematical science.

Painlevé's reference to Hermite's "spiritualism" will be more clearly understood if it is stated that the great mathematician was simply a member of the Catholic Church. In his younger days his religious views had undergone certain variations. But "thanks to the charity of the Sisters of Mercy who nursed him through a severe illness, thanks also without doubt to the influence of Cauchy" he returned to the Faith, and, "from the day in i856 on which he found his road to Damascus, the fervour of his religion never diminished". "From 1877 on", writes the Catholic Review from which we borrow these notes, "he took a lively interest in our review, and constantly congratulated our late-lamented general secretary and his collaborators on their articles, which he found so solid and so appropriate to the intellectual needs of the day".... Fifteen years later he expressed himself to the same effect, and added that he had the happiness to share the Faith professed by the writers of the Review. To the Congress of Catholic Scientists, held at Brussels in 1894, he contributed a paper.

"Hermite", says the celebrated mathematician Émile Borel, "was deeply attached to the Catholic religion; it was the stay and the centre of his life.... His opinions and his works were in perfect harmony with Catholic ideas, and this is certainly no ordinary merit."
[Kneller, Christianity and the Leaders of Modern Science]

It seems a bit odd that I've posted all these little reports with hardly any humour - and that is sad. So, in recognition of tomorrow being Laetare Sunday, I shall give you a curious little vignette from Fr. Jaki's "Intellectual Autobiography":
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. Perhaps he was joking. But that was the last time I raised such questions in class. As I came to see later on, only one out of ten textbooks of quantum mechanics would reveal that Hermite was a 19th-century French mathematician. None of those textbooks contained a single word on a historically far more intriguing, and epistemologically far more revealing, fact of modern physics. Both in relativity theory and in quantum mechanics crucial role was played by mathematical functions that had been worked out for decades, at times a century earlier before physicists found them useful, indeed, indispensable.
[Jaki, A Mind's Matter 25]

Also see here.


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