The Mathematician, Mathematics and Eternity

Hilda P Hudson wrote two papers which tell us much about the way she thought about mathematics. These are

(i) Hilda P Hudson, Mathematics and Eternity, The Mathematical Gazette 12 (174) (1925), 265-270.

(ii) Hilda P Hudson, The Mathematician in Ordinary Intercourse, The Mathematical Gazette 15 (206) (1930), 62-67.

We note that this second paper is the report of a talk by Hilda Hudson to the Mathematical Association. This was the final talk of the Annual Meeting of the Mathematical Association held at the London Day Training College on 6-7 January 1930.

Below we give some extracts from these two papers.

1. Mathematics and Eternity.

To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology. The relationships of men to God and to the universe have each an accurate, and even a numerical, side, capable of scientific treatment, called pure and applied mathematics. It is more than an accident that theology, in the narrower sense, continually uses mathematical terms and illustrations, some of which we shall consider. Since truth is God, and therefore one, its different departments are related and overlapping; each is capable of illuminating the others, and the lower aspects are continually recognised as symbols and sacraments of the higher.

Thus research is a mystical and a passionate quest. All scientists know this, though most have an unnecessary and unscientific shyness of admitting it. The lower passions may blind the eyes and cloud the intellect; but the high passion for truth is enlightening and steadying, and in some form or other is the only power that enables men to scorn delights and live laborious days in the pursuit of knowledge.

It is worth while for once to turn our minds away from the professional and temporal aspects of mathematics, to its relation to the things that are unseen and eternal.

Let us begin by making an act of thanksgiving for the common blessings of our intellectual life, as we give thanks for air and light, food and shelter. Just as the universe corresponds to our physical needs, and makes the life of the body possible and good and glorious, so does it also correspond to our intellectual needs, supplying us with the laws of thought: logic and arithmetic and geometry, which we use and depend on every moment. Try to imagine a world without the multiplication table, in which twice two were three today and five tomorrow. There would be no possibility of common sense or foresight or reason. Intellectual life would cease, we should all be reduced to idiocy, to mental death, just as without air or food we should all be reduced to physical death. Let us give thanks for the common blessings of our mental environment, which makes thought possible and good and glorious.

The phrase mental environment makes the great assumption that truth is something outside of, and independent of, ourselves, the same for you as for me: this underlies all study, all teaching and all religion. An old Greek, a French child and a self-taught Indian, each finds for himself the same theory of geometrical conics. The simplest, and therefore the most scientific way of describing this, is that they have each discovered, not created, a geometry that exists by itself eternally, the same for them all, the same for teacher as for taught, the same for man as for God.
We have His mind, not as in a glass darkly, but accurately. In applied science, and particularly in the sciences of ethics and conduct, we feel after the divine truth, and reach it partially, doubtfully, inaccurately. But the thoughts of pure mathematics are true, not approximate or doubtful; they may not be the most interesting or important of God's thoughts, but they are the only ones that we know exactly.

Therefore analysis and geometry are the most sacred of all sacred studies; and they are indeed profane persons who pursue them only for examination purposes.

Because pure mathematics is thus a direct contact with God, it is possible for a Christian to dedicate his whole time to research in it: the usual gibe about its uselessness has no meaning. Other professions, and the arts and crafts of everyday life, are worth pursuing only because of their useful applications; they are indirect approaches to God, through the service of man. But mathematics is a direct approach, and can bring the sense of divine companionship more intensely than anything else; for it is God's thoughts that we are thinking after Him, and with Him. We can practise the presence of God in an algebra class, better than in Brother Lawrence's kitchen; and in the utter loneliness of an unfashionable corner of research work, better than on a mountain top. If the whole duty of man is to glorify God and enjoy Him for ever, who is in a better way of doing his duty than the mathematician?

The great idea of enjoying God points to the unity of beauty, truth and goodness; the common distinction does not go very deep: these three are one. For scientists especially, who have developed the habit of association of ideas, any suggestion of falsehood prevents pleasure and destroys beauty. It was a perverted old cynic who said that "a mixture of a lie doth ever add pleasure."

And there is a kind of beauty found in intellectual truth and nowhere else: austere perhaps, but all the more beautiful for that. If a flower or a rainbow moves us to wonder and worship, how much more some exquisite piece of deduction, or a great theorem spanning whole ranges of applications, like Laplace's equation, or the law of the inverse square? God shines through His works as clearly in logic as in matter.
It is significant that theologians use so many mathematical terms, and they would gain enormously if they accepted, or at least understood, the mathematical definitions. For example, in theology, infinite has no one accepted meaning and this vagueness brings the danger of falsehood, which is blasphemy. All mathematicians have accepted one meaning for the word, which would serve the theologians admirably, if only they knew it. An infinite aggregate is one that is equal to one of its parts, and by equal we mean here the usual 1, 1 relationship. No finite whole is equal to any of its parts: that is the correct form of Euclid's axiom, which is the definition of finiteness. Juliet says of her love to Romeo:
The more I give to thee
The more I have, for both are infinite.
That is not poetic licence, but pure mathematics.

If this use of finite and infinite were accepted in theology we should hear no more of the "finite human mind." A finite mind could never form the idea of infinity. That the range of human thought is infinite follows directly from the definition, coupled with the fact that we can think about each of our thoughts; for thoughts about thoughts are a part of the whole set of thoughts, in 1, 1 relation to the whole set.

When a part is equal to the whole, it can enter into the same relationships with other aggregates: the whole is mapped on, or represented by, or revealed by the part. The assumption that God is mathematically infinite implies a self-revelation through a part of Himself. If God is infinite, the incarnation is possible: if we know that He is infinite, the incarnation has happened. This is expressed in the Athanasian Creed, which calls our Lord
Equal to the Father, as touching His Godhead,
And inferior to the Father, as touching His Manhood.
A theory is more or less true according as it is a more or less satisfactory description of certain observed facts. The satisfactoriness is aesthetic as well as logical; its chief element is simplicity, which is a form of beauty.

The heliocentric theory of the solar system did not account for more facts than the "Eccentricks and Epicycles, and such Engines of Orbs" of the earlier astronomers, which were carried to the degree of complication needed to describe the observed facts of the time. But this caused such a loss of simplicity, compared with the earlier theory of seven celestial spheres, so intense a dissatisfaction and strain, that the new idea was able to burst through into the mind of Copernicus: or rather an idea, old as the heavens themselves, came to its right place at last. It prevailed by being, not a fuller account than the epicycles, but a simpler, not so much through its truth as through its beauty, proved in the only way possible for a physical theory, by satisfying men's souls as well as their minds.

We owe a great debt to the old school, cranks and faddists though they were, Their courage and persistence, in stretching the current theories to all the facts, alone gave the needed background and stimulus for the next advance. It is often only the cranks who have just this kind of courage.

As in astronomy, so in politics. For centuries we have fitted the facts of everyday intercourse, between nations and neighbours, to a theory based on the old geocentric idea of division - race, class, sex, denomination, education. This has been carried through to the intolerable complication and strain of the war and the peace. On every side there is abroad the new idea, old as God Himself, based on unity instead of division. The two sets of people who ought to be the first to accept the new basis of human relationships are on the one hand scientists, whose work is already based on the unity of truth, and on the other hand Christians, who believe in the fatherhood of God; and before all others, Christians who are scientists. We can pass naturally from abstract mathematical research to practical work for European Student Relief or the League of Nations, with a full sense of continuity and appropriateness. The tragedy of our times is that so many people, partly from fear of being thought cranks, are wavering between the two contradictory principles, between sacrifice and security, love and fear. There is neither simplicity nor satisfaction for such, till they take a more scientific attitude. But wherever the principle of love is applied consistently, the simplification which it brings to all the relationships of life carries its own conviction.

2. The Mathematician in Ordinary Intercourse.

I received this morning a pamphlet in which the master of Stowe School, describing the Acropolis, says: "The physical beauty of the place alone would have moved even a mathematician's heart," and that appears to be what the world thinks of us, or, at any rate, a different part of the scholastic world. I am, of course, speaking to mathematicians and not to teachers, and I suppose we have all at one time or another suffered from the misguided hostess who introduces one as a mathematician. When that terrible word is mentioned the other person instantly curls up and says to the so-called mathematician, "How terribly clever you must be," or else, "I never could do maths at school." Neither is a good opening in conversation, and it takes some time to beat down that barrier and get back to ordinary friendly terms. A mathematical reputation is rather like a clerical collar; it is a hindrance to intercourse and condemns one to a very great deal of loneliness. This may, of course, be partly due to jealousy of those who are academically superior on one particular point, although they are inferior on a great many other points, because there is a certain amount of truth in the gibe that a specialist is a person who has failed in every subject except one, having had neither time nor energy to succeed in the others.

Perhaps it is worth while to look for a few minutes into one or two things in mathematical training that are either handicaps or advantages when it comes to intercourse in other fields. First of all, there is the mathematical language, which I think has rightly been described as the most unintelligible on earth. I remember at college, when three of us were doing research, that the historian could say she was dealing with charters of the reign of King John, which meant something, the bio-chemist that she was seeing how and why geraniums were scarlet, which also meant something, and I could say I was dealing with the fundamental points of Cremona transformations, which meant nothing to anybody. It is a thing one simply cannot share. When people ask you what your last book is about, the best answer you can give, and the one I always give, is: "It took me forty years to find out; if you will give me another twenty, I will explain it." And it is not only in the strict realm of mathematics that our language is unintelligible. There are many quite common objects of general interest for which we mathematicians have beautiful and elegant expressions which, on the score of politeness, we have no right to use. If we want to describe the shape of a hill as a truncated cone we must not say so, because it is just as rude to say that to people who do not know what it means as to talk French or German in the presence of those who do not know either language. It may be true that we do sometimes overstep the bounds of politeness. Besides this great temptation, which is always ours, to use our own language rather than that of other people, we have other habits which are very likely annoying. There are, however, some habits we would perhaps prefer to see more widespread, especially the habit of mental arithmetic and being able to multiply small quantities by large multipliers rapidly and approximately. To most people a penny a day is just a penny a day, but, as a matter of fact, it is something like £100 in a lifetime. A person who comes quickly to a fact such as that is apt to be treated as a magician. I think that is a habit which one may hope eventually to see more widespread. Five minutes is not five minutes but a whole hour wasted if the number on the committee which is kept waiting is twelve, but the average person who turns up late does not think of that beforehand. Mathematicians have no excuse for ever being late.

Take another of our bad habits, which is really a foolish one and is annoying in ordinary intercourse. Certain logical methods that are important and valuable in mathematics are foolish and misleading in topics of general inter- course. For instance, if one point of a mathematical argument is suspected of fallacy, one pays attention to that point and ignores all the rest till satisfied that that particular link holds good; we are drawn to the weak spot and stress the weakest link. But if the argument is one of politics or economics, and we make any amount of not only weak but false links, so long as some links are good and strong, however weak another link may be it is not worth while paying attention to it. What is important there is to go for the strongest and not the weakest; to go for the weakest part first is waste of time, and hence people think us the fools that we are.

Another habit which is sensible in mathematics and foolish in other regions is that of dealing with the extreme case. The old Latin tag extremum probat regulam means "the extreme case tests the rule," though it is usually mistranslated. One tests a proposition by taking half the coefficients to be zero, and the other half, unity; and if it fails then, one does not consider the general case at all.

If you try to do that in a political argument or in any argument in another region, you are always certain to be applying it to some case for which it was not intended, because, the enunciation not being mathematically expressed, with all the provisos put in, the extremely simple case one takes to test it by is one which would have been definitely ruled out if the enunciation had been accurate. If we use that kind of test, people think we are quibbling. Perhaps we are.

Another bad habit is that of reductio ad absurdum, because your hearer thinks it is himself and not a particular proposition which is being made to look absurd, and nobody likes that.

In all these different ways our real mistake is a lack of proportion. It is very difficult, I think, for those who are trained to put first what has logical primacy when going over to other spheres of intercourse to put first what is of importance in other than logical ways; we find it more difficult to come by a true sense of proportion than people with different training.

What first set me off on the somewhat thin subject of this paper was the experience I had of running a series of conferences on cooperative lines with very mixed participants, and it fell to me to have to select certain people for helpers. I found myself picking out mathematicians for certain jobs. It is perfectly true that there are certain services that people with mathematical training can render easily and therefore ought to render, because they do them better and with much less effort than those with different training. There are other jobs that others can do better than mathematicians. No one would ask a mathematician to do the flowers or play for dancing. I think there are limits. There was one society whose annual meeting I gave up attending because for three years running I was asked to scrutinise the ballot and missed the whole of the proceedings!

I think one element in the dislike that mathematicians arouse is that there is supposed to be a certain moral as well as intellectual superiority, which arises from the curious fact, which has no moral value really, that in the sphere of mathematical research one is automatically safeguarded from certain common temptations. There is no sphere in existence in which lying or swank is so unattractive and unprofitable. You are found out directly - you find yourself out - and it does not get you on, in the way it may be supposed to in other spheres. It may be that we poor people do get out of our training, for which we pay a high price, a certain tendency to honesty that comes to us more easily than - well ... (Laughter).

Another thing from which we are entirely sheltered is avarice, because there is no money in mathematics and therefore that particular root of evil does not come into our universe of discourse. It does not mean we are any stronger morally or any better characters, but simply that for those particular hours that we give to our own work, we have been automatically sheltered from those particular temptations, though of course liable to certain others which I leave to your imagination. And when we go into the world and come up against those sheltered from other temptations and whose standards are higher than ours in other matters, ought we to expect of them our standards where we are strong? Ought we not to be prepared to bow to their standards where we are weak? Do we ask too much of the world when we get annoyed with other people's inferiority, ourselves being beyond reproach? I think it must be admitted that the world at large ought to take its standards from where they are most easily kept high, and that each profession and each craft has in trust for the universe the safeguarding of certain standards that come easily its way. Perhaps the most important thing that we have in trust for the world is our standard of what you may call truth, if you want to make the most of it, or what you may call mere accuracy, if you want to be scornful. There is an amount of truth in mathematics, from the multiplication table upwards, which does not occur in any other science or art, and I think that is one of the things we have got to guard and, as far as possible, make clear in all its beauty to the rest of the world.

The loneliness of the mathematician or the man with scientific training is generally due, in the long run, to the great gulf fixed between clear-minded and woolly-minded. There are those who see the world black and white and miss all the beautiful colour, and those who see the world through a rainbow fog and doubtless miss all the beauty of form. Like every other gulf in creation, that can be bridged. It takes patience and affection on both sides but it can be done. And perhaps by and by it may be filled up, when we get education worth calling education, but that will hardly happen in our time. Meanwhile, the great majority of mankind are on the soft side of the gulf and those on the rocks - well, it is of all the greater importance to have such meetings as these where we can get together and talk our own language amongst ourselves with ease.

Last Updated December 2021