Address by GH Darwin

It is interesting that this noted astronomer and gifted mathematician of the Victorian era proved that the weather is not controlled by the moon. …

Address by GH Darwin (noted astronomer and gifted mathematician of the Victorian era) delivered before the International Congress of Mathematicians at Cambridge in 1912)

(I converted this text from a PDF with the unfortunate habit of munging words with the letter F in them. If I missed any corrections out, you will know how to mind wipe them.)

Four years ago at our Conference at Rome the Cambridge Philosophical Society did itself the honour of inviting the International Congress of Mathematicians to hold its next meeting at Cambridge. And now I, as President of the Society, have the pleasure of making you welcome here. I shall leave it to the Vice-Chancellor, who will speak after me, to express the feeling of the University as a whole on this occasion and I shall confine myself to my proper duty as the representative of our Scientific Society.

The Science of Mathematics is now so wide and is already so much specialised that it may be doubted whether there exists to-day any man fully competent to understand mathematical research in all its many diverse branches. I, at least, feel how profoundly ill-equipped I am to represent our Society as regards all that vast field of knowledge which we classify as pure mathematics. I must tell you frankly that when I gaze on some of the papers written by men in this room I feel myself much in the same position as if they were written in Sanskrit.

But if there is any place in the world in which so one-sided a President of the body which has the honour to bid you welcome is not wholly out of place it is perhaps Cambridge. It is true that there have been in the past at Cambridge great pure mathematicians such as Cayley and Sylvester, but we surely may claim without undue boasting that our University has played a conspicuous part in the advance of applied mathematics.

Newton was a glory to all mankind, yet we Cambridge men are proud that fate ordained that he should have been Lucasian Professor here. But as regards the part played by Cambridge I refer rather to the men of the last hundred years, such as Airy, Adams, Maxwell, Stokes, Kelvin and other lesser lights, who have marked out the lines of research in applied mathematics as studied in this University. Then too there are others such as our Chancellor, Lord Rayleigh, who are happily still with us.

Up to a few weeks ago there was one man who alone of all mathematicians might have occupied the place which I hold without misgivings as to his fitness; I mean Henri Poincare. It was at Rome just four years ago that the first dark shadow fell on us of that illness which has now terminated so fatally. You all remember the dismay which fell on us when the word passed from man to man "Poincare is ill."

We had hoped that we might again have heard from his mouth some such luminous address as that which he gave at Rome; but it was not to be and the loss of France in his death affects the whole world.

It was in 1900 that, as president of the Royal Astronomical Society, I had the privilege of handing to Poincare the medal of the Society and I then attempted to give an appreciation of his work on the theory of the tides, on figures of equilibrium of rotating fluid and on the problem of the three bodies.

Again in the preface to the third volume of my collected papers I ventured to describe him as my patron Saint as regards the papers contained in that volume. It brings vividly home to me how great a man he was when I reflect that to one incompetent to appreciate fully one half of his work yet he appears as a star of the first magnitude.

It affords an interesting study to attempt to analyze the difference in the textures of the minds of pure and applied mathematicians. I think that I shall not be doing wrong to the reputation of the psychologists of half a century ago when I say that they thought that when they had successfully analyzed the way in which their own minds work they had solved the problem before them.

But it was Sir Francis Galton who shewed that such a view is erroneous. He pointed out that for many men, visual images form the most potent apparatus of thought, but that for others this is not the case. Such visual images are often quaint and illogical, being probably often founded on infantile impressions, but they form the wheels of the clockwork of many minds.

The pure geometrician must be a man who is endowed with great powers of visualisation and this view is confirmed by my recollection of the difficulty of attaining to clear conceptions of the geometry of space until practice in the art of visualisation had enabled one to picture clearly the relationship of lines and surfaces to one another.

The pure analyst probably relies far less on visual images, or at least his pictures are not of a geometrical character. I suspect that the mathematician will drift naturally to one branch or another of our science according to the texture of his mind and the nature of the mechanism by which he works.

I wish Galton, who died but recently, could have been here to collect from the great mathematicians now assembled, an introspective account of the way in which their minds work. One would like to know whether students of the theory of groups picture to themselves little groups of dots; or are they sheep grazing in a field?

Do those who work at the theory of numbers associate colour, or good or bad characters with the lower ordinal numbers and what are the shapes of the curves in which the successive numbers are arranged?

What I have just said will appear pure nonsense to some in this room, others will be recalling what they see and perhaps some will now for the first time be conscious of their own visual images.

The minds of pure and applied mathematicians probably also tend to differ from one another in the sense of aesthetic beauty. Poincare has well remarked in his Science et Methode (p. 57):

"On peut s'etonner de voir invoquer la sensibilite apropos de demonstrations
mathematiques qui, semble-t-il, ne peuvent interesser que
l'intelligence. Ce serait oublier le sentiment de la beaute mathematique, de
l'harmonie des nombres et des formes, del' elegance geometrique. C'est un
vrai sentiment esthetique que tous les vrais mathematiciens connaissent.
Et c'est bien la de la sensibilite."

And again he writes:

"Les combinaisons utiles, ce sont precisement les plus belles, je veux
dire celles qui peuvent le mieux charmer cette sensibilite speciale que tous
les mathematiciens connaissent, mais que les profanes ignorent au point
qu'ils sont souvent tentes d'en sourire."

Of course there is every gradation from one class of mind to the other and in some, the aesthetic sense is dominant and in others subordinate.

In this connection I would remark on the extraordinary psychological interest of Poincare's account, in the chapter from which I have already quoted, of the manner in which he proceeded in attacking a mathematical problem. He describes the unconscious working of the mind, so that his conclusions appeared to his conscious self as revelations from another world.

I suspect that we have all been aware of something of the same sort and like Poincare have also found that the revelations were not always to be trusted.

Both the pure and the applied mathematician are in search of truth but the former seeks truth in itself and the latter truths about the universe in which we live. To some men abstract truth has the greater charm, to others the interest in our universe is dominant. In both fields there is room for indefinite advance; but while in pure mathematics every new discovery is a gain, in applied mathematics it is not always easy to find the direction in which progress can be made, because the selection of the conditions essential to the problem presents a preliminary task and afterwards there arise the purely mathematical difficulties.

Thus it appears to me at least, that it is easier to find a field for advantageous research in pure [rather] than in applied mathematics. Of course if we regard an investigation in applied mathematics as an exercise in analysis, the correct selection of the essential conditions is immaterial; but if the choice has been wrong, the results lose almost all their interest.

I may illustrate what I mean by reference to Lord Kelvin's celebrated investigation as to the cooling of the earth. He was not and could not be aware of the radio-activity of the materials of which the earth is formed and I think it is now generally acknowledged that the conclusions which he deduced as to the age of the earth cannot be maintained; yet the mathematical investigation remains intact.

The appropriate formulation of the problem to be solved is one of the greatest difficulties which beset the applied mathematician and when he has [he has] attained to a true insight but too often there remains the fact that his problem is beyond the reach of mathematical solution.

To the layman the problem of the three bodies seems so simple that he is surprised to learn that it cannot be solved completely and yet we know what prodigies of mathematical skill have been bestowed on it. My own work on the subject cannot be said to involve any such skill at all, unless indeed you describe as skill the procedure of a housebreaker who blows in a safe door with dynamite instead of picking the lock. It is thus by brute force that this tantalising problem has been compelled to give up some few of its secrets and great as has been the labour involved.

I think it has been worth while. Perhaps this work too has done something to encourage others such as St ormer to similar tasks as in the computation of the orbits of electrons in the neighbourhood of the earth, thus affording an explanation of some of the phenomena of the aurora borealis. To put at their lowest the claims of this clumsy method, which may almost excite the derision of the pure mathematician, it has served to throw light on the celebrated generalisations of Hill and Poincare.

I appeal then for mercy to the applied mathematician and would ask you to consider in a kindly spirit the dificulties under which he labours. If our methods are often wanting in elegance and do but little to satisfy that aesthetic sense of which I spoke before, yet they are honest attempts to unravel the secrets of the universe in which we live.

We are met here to consider mathematical science in all its branches. Specialisation has become a necessity of modern work and the intercourse which will take place between us in the course of this week will serve to promote some measure of comprehension of the work which is being carried on in other fields than our own. The papers and lectures which you will hear will serve towards this end, but perhaps the personal conversations outside the regular meetings may prove even more useful.

The Project Gutenberg ebook #35588

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