W H Young gave his lecture The mathematical method and its limitations on Wednesday 5 September at 9.00. Here is the text of the lecture:-
The Mathematical Method and its Limitations
1. If, in addressing this Meeting of the International Congress of Mathematicians, I have chosen as my subject The mathematical method and its Limitations, it is because I wish to direct the attention of Mathematicians of an countries to the importance of adopting it as a topic for all future Congresses, in view of the wide range of issues it involves, and this all the more that it must be regarded as not belonging properly to any one of the existing sections.
In a certain sense it may be said to require all the resources of the Historian, of the Philosopher, of the Scientist, - using these terms in their most general sense - as well as of the expert Mathematician.
2. It will be well first to point out that I conceive Mathematics as not only that which has been, or shall come to be, embodied in processes and concepts, successively developed from ordering by ordinal, and calculating by cardinal, numbers, through Arithmetic, Analysis, Theory of Numbers, Theory of Groups and the like, disciplines evolved, so to speak, step by step, of themselves, but also as including Geometry, Mechanics, Mathematical Physics, and other such disciplines which have been devised to image the so-called World of Realities.
By the Mathematical Method I understand, not only the utilisation of mathematical tools, and, in particular, of Symbolism, but also the application of mathematical Intuition, from the most abstract to the most nearly concrete. The Mathematical Method is employed in the most familiar operations of daily life, as well as in the laboratory of the scientist and the study of the Pure Mathematician.
In view of the wide sense in which I use the term Mathematical Method, I need hardly say that I shall not attempt in the short time at my disposal to discuss the subject in all its bearings.
1. From what I have said you will understand that the Mathematical Method is not confined to the employment of the theory of Measurement, extraordinarily fertile though this branch of our subject has certainly been.
The work of Pure Mathematicians of the 19th Century was partly applied to convince mathematicians themselves that the idea of measurement did not enter even into Geometry to the extent that was supposed. I need hardly say that the Theory of Groups does not involve measurement, and the same is true of the notion of Function, which, indeed, so little depends in itself on the idea of measurement, that, when the occurrence, or non-occurrence, of an event depends on the occurrence, or non-occurrence, of various combinations of other events, we are patently in presence of a functional relationship of a two-valued function of several variables, each assuming only two values, say zero when the event in question does not occur, and unity when it does. Here the idea of measurement is wholly absent, and this though the final decision as to whether an event has happened, or not, may depend in practice on an act of measurement, such, for instance, as the doctor has to carry out with his clinical thermometer.
Measurement, as such, represents the persistent attempt to replace Quality by Quantity. Yet when we try to express quality by a series of numbers, each number must itself have a Quality-factor, before the series of numbers individually and collectively can be intelligible. This is true, not only in Applied, but even in Pure, Mathematics. It is this alone which renders possible cooperation between Mathematics and other branches of human activity.
Plato compared the utilisation of mathematical notions, in the investigation of the actual, to the fitting of the sandal to the foot. No more picturesque and suggestive image for the Mathematical Method, by which mathematical concepts, whatever these may be, are fastened on to the complex realities, could have been devised. The mathematical concepts must take the imprint of these realities, and only in so far as they are supple and lend themselves to natural deformations, do they add to, and strengthen, our conception of those realities.
2. I must also in this connexion emphasize the fact that the tools which the Mathematical Method borrows from Mathematics are not merely its symbolism. If this is most frequently involved, it is because symbolism in Mathematics is merely a scheme of abbreviations for current terms and phrases in the Mathematical Language.
This Mathematical Language, which, though limited in its power of expression, is par excellence a universal language, bears too every analogy to ordinary speech, while its development has been, and is, far more rapid. For this reason, if for no other, its study might be recommended, even to Philologists, as revealing, with a clearness all its own, the processes at work in the infancy of all language, and as indicating lines on which the growth of a language should be encouraged.
3. I do not propose to speak at length of the Methods of Mathematics itself, and it is unfortunate that a typographical error in the early edition of the Programme might give this impression. To one of these methods, attention must, however, be called, since with it is bound up, to a great extent, the ever increasing efficacy of Mathematics. I mean The Method of Generalisation.
The key-note in this method is the eradication of the exceptional. Some particular law, relation, or property, is regarded as paramount; and to render it supreme, certain concepts are generalised. In saving the one law, it is true, we sacrifice other laws, judged to be of minor import. But the balance is on the side of progress. The result is not merely the conceiving of a broader class of entities, comprising as a particular case those already known; but, in admitting the rebels, we emancipate the loyalists from the trammels of their original definition. In reality, no members of the new class coincide with the entities from which that class was evolved, for the definition of each such member is founded on the prior existence of its forbears. The whole body of Mathematics is thus raised to a higher level; as, for instance, in the passage from rational to real numbers, from real to complex, or from the Cauchy integral through the Riemann, to the Lebesgue, or again, from the concept of a single unique limit to the Theory of Sets of Points.
The active principle, the Entelechie, of the Method of Generalisation is precisely its elimination of inessentials; by this it proceeds to the simplification of its underlying concepts, in the sense of reducing their generic properties. At the same time, the generalisation most frequently reveals properties hitherto unperceived, as by the removal of a veil.
4. These potentialities and the efficacy of the Method of Generalisation within the bounds of Mathematics, may serve, in part, to explain why there is a Mathematical Method in Science, what is its nature, and why the scientific investigator relies so implicitly upon it at every turn. The process of disregarding characteristics in the objects considered, is fundamental in every pursuit of human knowledge; as far as we can see, it leads ultimately to Mathematical Concepts, in esse or in posse, as the only ones that cannot be further simplified: and the advance in the development of the science of Mathematics over that of other sciences is such as to encourage hopes in those who appeal to it, which no other discipline would warrant to an equal degree.
At the same time, this appeal is vital to Mathematics itself. The science of Mathematics is not so self-contained as it tends to ultimately appear. It borrows its vivifying germs from other sciences, which appeal more urgently to the human mind, whose demands are greater incentives to work, than the purely methodological, or even aesthetic, demands of Mathematics left to itself.
It is for this reason that a study of the Mathematical Method in Science is as necessary to the mathematician as to the scientist. He cannot afford either to lose the confidence of the scientific man, or to submit to the contempt which an ill-regulated appeal to mathematical resources inevitably entails.
5. We are however encouraged to hope that Mathematics may rise in time to deal with most, or at least with many, of the essential problems that present themselves. Far be it from me to class myself with those who accept the dictum that the whole universe could be summed up in one Differential Equation, or even in a system of such equations. The solution of many even of the problems of today lies far beyond the Mathematics of our time, or of any definite epoch.
1. What is the nature of the problems confronting the human race for which a method is required? The problems which dire necessity sets before us are not solved by lengthy reflection; for unforeseen accidents we require unforeseen expedients, foresight for apprehended dangers. And the Method has as much to furnish the question as to elicit the answer. Verily in the clear statement of the problem lies more than half its solution. How this familiar phrase confutes the attitude of those who put forward, as the one and only principle of experiment, the putting the ear to Nature, without any idea what to listen for!
2. We may go further, and say that the life of our mind consists, above all, in a germination of new problems, each of which suggests another. There is no actual solving of problems, but a referring back from the more to the less complex, or vice versa, according to the trend of our thoughts. We cannot even claim to judge the unknown by the known, but only the less known by the more familiar.
1. The principle just characterised is in the main that of the Method of Analogy. From this point of view, all methods, and the Mathematical Method among the rest, are particular modes of the appeal to analogy. This is the one argument all men use. On the hustings, in the law-courts, in the pulpit, in Parliament, on the Stock Exchange, everywhere where men meet - here, if we look back to the Past, in Italy, on the Aventine Hill, - men have been impelled by the argument from analogy. It is all-powerful; 'it is the one force of an intellectual type which everyone understands: it acts, as far as we can judge, even in the realm of the lower animals. If we contemplate dispassionately the achievements of Science, nay if we read intelligently the works of our own most famous mathematicians, we shall find Analogy to have been everywhere their guide. In the unscientific world, the potency of the method is due largely to the fact that the questions which it substitutes for other questions appeal to the mind as though they were statements of fact. In Science, those who own its sway claim, at least now-a-days, no more than the fertility of the method, and endeavour to be cautious in its application. In earlier times Kepler might boldly state: plurimum namque amo analogias fidelissimos meos magistros, omnium naturae areanorum conscios (For above all I love analogies, my most faithful teachers, acquainted with all the secrets of Nature),but, since then, imagination in Science has gone somewhat out of fashion: and, in fear of losing caste, the scientist has grown proverbially matter of fact. Some of the most brilliant ideas which mankind has ever conceived have doubtless never been uttered.
2. That is why we may often learn from Poets what we cannot from scientists. Sheltered behind the cult of form, the artist is far freer to encourage the flight of imagination, guided, as it always is, by the sense of analogies.
3. What we want is not, indeed, a belief that the Method of Analogy will unlock all portals to knowledge, but to realise that the argument from analogy is the only one we possess. It seems to guide even Nature herself, as she proceeds from lower to higher forms, from existing species to new varieties. This is an aspect of Nature so suggestive and so impressive, that even one of the most abstract of Pure Mathematicians, Riemann, wrote down his belief in the Soul of the Earth.
Yet, as far as I can judge, no systematic investigation of even the most hackneyed analogies, no investigation, I mean, as to the extent to which they hold good, the nature of their limitations and their origin, has ever been attempted.
4. In its most common form, as applied to the incidents of everyday life, the argument from analogy is scarcely a reasoned process at all, it reduces to a mere habit of thought.
Yet to how many men this is the only kind of cerebral commotion which they care to seek for! The mediocrity loves to do the same thing time after time. He likes to be in situations with which he is familiar, he likes to recognise a circumstance and apply his experience to take the next move.
To welcome a new situation, which sets the mind working in unfamiliar grooves, and awakens fresh trains of associations, requires already an intellectual power with which the common man is not endowed; it would, indeed, unfit him for the monotonous routine in which he revels.
That is why the Intellectual in Plato's parable of the Cave, has only too often been despised and rejected of men. Unfit to be an employé, he has not learnt to be a ruler.
Yet, as this common method of thought, however mechanical, is but a particular and degenerate form of an intellectual process, the intellectual alone can properly analyse it, and check its consequences. Whole classes and nations of men are precipitated into anarchy and war by the evil power of imperfect analogies. Political and philosophical creeds are built up on scarcely any other foundation. Passwords are created which clinch the tacit argument in a way that no true argument ever could, and which contain in themselves a whole chain of analogies. Even the grammatical form of a word suffices to influence the attitude of mind, as students of Philosophy and Sociology are well aware.
Such causes of error also occur in the application of the Mathematical Method; but here, at least, they are reduced to a minimum. This alone would be a reason for the study, before all others, of the Mathematical Method, even were it not the most widely applied.