The first half of W H Young's lecture The mathematical method and its limitations is given at this link
Here is the text of the second half of the lecture:-
1. The experimentalist, whatever branch of scientific activity he may be engaged on, who disclaims much indebtedness to Mathematics, does not realise how unceasingly he relies on mathematical analogies and on mathematical calculation and reasoning. If he is a Physicist, or an Astronomer, he is not likely to underrate its value, he is now fully aware that when physical hypotheses have turned out to be most erroneous, it has been often due to the fact that Mathematics was still too undeveloped to provide any means of checking them.
2. The successes of the Mathematical Method in Physical Science have been, moreover, of late years, so tremendous and so far-reaching in their effects that these have been felt in the remotest corners of the world, thereby removing distinctions between Civilisation and Barbarism.
And although these have put in the shade for the moment the still more striking progress in Astronomy of an earlier age, we are expecting equally momentous consequences to the human race to emerge from the mathematical discussion of the electro-magnetic field of the sun, based on the joint work of astronomer and physicist.
On the other hand, the physicists have revived in a modified form older physico-astronomical theories of the atom. And they have been so confident in the outcome, that they have told us in their enthusiasm that we really an but hold the Electron in our hands and behold it with our eyes; and yet its existence is but a mathematical one, just as was that of the molecule, even before the new theories of De Broglie and Schrödinger placed the whole theory in a still more mathematical light.
In Chemistry, the epoch-making discovery of Argon in 1894, as I heard from Ramsay himself, had as determining factor the numerical work of Cavendish, more than a century before. While today, large portions of the science have already taken a mathematical form, and have come almost to be regarded as a branch of Mathematical Physics.
1. But, if the Mathematical Method has been most fruitful in the Physical and cognate Sciences, it can point to an increasing influence in other departments, - and we may perhaps learn something of this from other speakers, - in Heredity, in Economics and Art.
In these and in many branches of intellectual research, the Higher Mathematics of our epoch has hitherto proved of little use; it is for the most part what I should call Large-Scale Mathematics which has been employed, primitive concepts and rough-and-ready processes, too humble to have sought to be represented at an International Congress of Mathematicians. And yet, these also deserve their place of honour, if only for services like that rendered by Sir Ronald Ross in utilising the simple idea that it is on the percentage of mosquitoes to the individual, not on their mere presence, or even their number, that depends the epidemic of malaria, thereby creating anew the science of tropical medicine.
2. But a new Fine-Scale Mathematics, the outcome of that mathematical inheritance which we shall pass down and the new environment which these new sciences create, will arise to cope with the finer problems that suggest themselves. One of the chief interests to us in those fresh problems, on which our colleagues are to speak, will lie in the vistas that may be opened for the development of Mathematics itself.
3. Indeed, the idea of degrading Mathematics in Biology, Sociology, Economics, Politics, to mere Statistics, [or even to the Theory of Probability, magnificent as are its promises], can only take form in the minds of men immersed in the details of purely observational work.
The task of Mathematics is above all that of devising abstract concepts, of immaterialising, so to speak, some phase, some event, some reality, in some single idea. Thus did the concept of Attraction emerge in all its power of application from the mind Of Newton. A modern, still somewhat embryonic parallel is the idea of Ophelimity in Social Economics, due, I believe, chiefly to Pareto. On the obtaining of such conceptions as these, the progress of Science must depend. The discovery of a new law is bound up with the framing of a new concept.
1. This question belongs to the second part of my subject: The Limitations of the Mathematical Method. If the Mathematical Method is felt to be far from applying directly to all the problems of today, because Mathematics appears to lack many of the concepts necessary for their adequate discussion, this does not prejudice in any way the possibility of such concepts being devised in the Future, but, on the contrary, acts as an incentive to the Mathematician to enlarge and enrich his own science, and constitutes no final cheek on the scope of the Method.
2. But that is not the only type of limitation which must be recognised in the application of the Mathematical Method. That relates merely to the absence, in any one mathematical scheme, of entities, or facts, corresponding to possible objects, or phenomena, in a given region of observation.
The converse limitation also is inherent in the Mathematical Method, as a form of the Method of Analogy, namely the lack of phenomena corresponding to accepted mathematical concepts, or relations.
3. The two forms of limitation are indeed the precise analogue of those found in the most current arguments from Analogy; whenever a scheme A is compared to a scheme B, the comparison unfailingly breaks down at some point; entities exist in A which have no correlatives in B, and vice versa; properties of A may be adduced which find no parallel in the scheme B, and vice versa.
It is scarcely necessary to point out, however, that there are great differences, in this respect, between the Mathematical Method and that of Analogy in its ordinary and every-day use. A mathematical discussion of limitations often involves serious difficulties, as we shall be reminded if we think of the efforts that have been required to elucidate the Existence Theorems of Mathematics itself. Where, on the contrary, the discussion is carried on in ordinary language, and in terms of every-day life, errors are relatively easily detected and avoided, though even this is only in a partial sense true. It would be more correct perhaps to say that subtle errors slip in at quite different points in different types of the argument from analogy, and that a mind trained to avoid them in one type, is not thereby qualified to discover them in another, even on a detailed examination. If the specific limitations of the Method of Analogy, in the widest sense of the term, are to be defined at all, they must be discussed very profoundly in connection with each application.
This discussion has to bring out what is the Region of Validity of the argument in each case, its Geltungsbereich. At present no systematic attempt of this kind, on a large scale, seems to have been made, and men seem, for the most part, divided between the tacit assumption that the Region of Validity is unlimited, and the overt opinion that it reduces to zero; both these standpoints are certainly untenable.
4. The two forms of the limitation of the Mathematical Method must be looked upon essentially in different lights. While on the one hand each failure of the Scientist to find the tools he requires must be considered carefully by the Mathematician with a view to supplying the want, on the other hand the presence in any mathematical scheme of entities and facts not corresponding to specific objective facts in Nature must be pointed out, with a view to the safeguarding of those who appeal to Mathematics. These last must not be allowed to assume that a demonstrated mathematical result constitutes a proof of the existence of a so-called truth in the world of phenomena.
5. The experimental Physicist hardly falls into this latter error, one that reappears at more or less regular intervals in the History of Thought. Even the theoretical Physicist rarely concedes such absolute power to Mathematics. We have, for instance, been told by the enquirer himself, of the frigid reception accorded to his question: What is the physical analogue of the most general group of conformal transformations of four-dimensional space that leaves unaltered the equations of Maxwell-Lorentz?
6. What can be said with safety is that the existence of mathematical relations, properties, and entities - among which we may particularly mention the notion of Infinity - can only suggest the question of the existence in Nature of corresponding objects, that may, at least approximately, or from a particular point of view, be assimilated to them. It was not as an argument, but as a confession of faith, that Leibniz wrote, with reference to the Infinite in Nature:
"Je suis tellement pour l'infini actuel, qu'au lieu d'admettre que la nature I'abhorre,.... je tiens qu'elle I'affecte partout, pour marquer les perfections de son auteur. Ainsi je crois qu'il n'y a aucune partie de la matière qui ne soit, je ne dis pas divisible, mais actuellement divisée, et, par conséquent, la moindre particule doit être considérée comme un monde plein d'une infinité de créatures, différentes".This subject of the actual infinite has been discussed interminably in philosophical circles. What however has not been sufficiently emphasized is that the infinite is a convention, or, more strictly perhaps, part of a convention, which has enormously simplified mathematics, and rendered possible theories, calculations and consequent practical discoveries, that centuries of work would not have produced, if our mind had magisterially excluded the concept of the Infinite. If we could prove that the universe was in every sense of the word bounded, we should continue to use the notion of infinity with profit as much as before.
Two-dimensional intelligent insects, living on the surface of a great sphere would certainly introduce the notion of parallelism, and construct a Euclidian geometry similar to ours. If in the course of time, and at the cost of tremendous labours, these insects succeeded in compassing the whole of their universe, thereby realising that it was not a flat space, but a curved one, this would not in any way invalidate their mathematics, nor prejudice its utility.
We have ourselves no hope of mapping out our universe by actually penetrating into every part. Only the poet Dante could permit himself to face such a conception, - Dante, who seems to have imagined the Universe, more or less vaguely, as a three-dimensional analogue of the above two-dimensional spherical surface.
7. The necessity for the recognition of the region of validity may be illustrated by the following simple fable:
The barbaric chief of a primitive tribe, who was able to count as far as three or four, determined to arrange his warriors in order of increasing strength. The cataloguing was entrusted to the Chief Priest, an ancient sage, imbued with the study of numbers and their arithmetic, and was carried out on the results of individual combats. The classification finished, the chief asked the advice of the priest as to the arranging of the warriors in parties of two or three, of equal strength, to attack the enemy at various points. The Priest advised him in the first place to send out the first with the fourth, the second with the third. But the second and the third succeeded so much better in the fight, that the Chief, in dudgeon, disgraced the first warrior from his high estate, and renounced the advice of the priest.
8. In this fable we perceive the most primitive form of the application of the Mathematical Method, the seriating in a certain order. In this process, the idea of quantity does not enter. A certain attribute is recognised as common to the objects of a given class, an attribute such as strong which may be said to be more or less, in the pure comparative sense. The idea of there being a corresponding quality, the introduction, perhaps, of a corresponding word, strength, leads to the concept of that quality being possessed in a greater or less degree, and to a numerical classification of more or less effective form. As long as the ordinal character of the numbers is respected, the results are confined to ordinal mathematics. But once the order-numbers are assimilated -to cardinal numbers, whose values represent, and even to some extent measure, the various degrees of the quality considered, processes valid for cardinal numbers will suggest combinations of the seriated objects, which may, or may not, prove legitimate.
9. The combinatory properties of cardinal numbers, which comprise ultimately all formal analysis, provide a practically unlimited series of questions, as to the corresponding properties of the objects under consideration. But they cannot of themselves lead to valid results; and the inquiry may be entirely misleading, if the order-numbers used do not happen to be chosen in the most favourable manner. This is what physicists mean, when, for instance, they insist that the proper choice of the variable is paramount in the investigation of a law.
Is there, however, in all cases, a most favourable manner of choosing the variables ? There seems no reason for supposing this to be true.
A case in which such a choice seems, at any rate, not yet to have been discovered, arises in the Theory of Colour-Vision. In so far as it appeals to Mathematics, this theory borrows a number of its mathematical tools from Physical Optics, and, in particular, seems to have adopted the mode of characterising a monochromatic light-radiation by its Wave-length. One of the fundamental problems, that of Colour Mixture, has, however, no counterpart in Physical Optics, properly so called, since the fusion of "mixed colours", as far as appears to be known at present, is a physiological, and not a physical, phenomenon. The assumption of the wave- length as the characteristic of a monochromatic light, does not seem to have availed much towards the solution of the problem.
The questions are complicated by the fact that the "colour" of a mixed light varies when the intensity of one of its constituents is altered, so that this also must be among the characteristic data.
Thus, for instance, in the natural enquiry into the mutual relation of Complementary Colours, the attempt to discover a relation between their Wave Lengths might with fair probability have been condemned on a priori grounds: the unconcerned retention of the wave length as a working variable, after it had served only as a characterising Order-Number, and the ignoration of the second variable in the case, seem sufficiently crave errors to warrant any failure.
There is one point which I wish to make before concluding. In characterising the Mathematical Method as a form of the Method of Analogy, and in pointing out the only limitations which it can naturally possess as such, we do away with a limitation which a more or less conscious bias has often artificially impressed upon the Method.
It does not matter if, in different portions of the same region of enquiry, comparison-schemes are employed which, when pushed to their extremes, are mathematically incompatible. The fact that Physics, today, is using at one and the same time the Theory of Relativity and the Newtonian Theory, is no indication of a Collapse of Science, preparatory to a Period of Scepticism. It must be taken as an index rather of a New Birth. There is no sign of decrepitude in the extraordinarily prolific output of physical hypotheses during our time, descending into greater and greater detail: the Ether, with its contradictory properties, diaphanous, so that a gram of it would scarcely fill a cubic kilometre, denser than platinum, rigid as steel, and in a state of stress, but possessing a porous structure, so that solid bodies pass freely through it; the Bohr Atom, endowed with a complex planetary system of spinning Electrons, which, Mahatma-like, pop in and out of existence, like the hypothetical motor-car, "moving along a road at thirty miles an hour, not continuously, but so as to appear at each successive mile-stone for two minutes". These ideas may seem to our descendents quaint, but not senile. If we smile at the Myths of our forefathers we admire the prowess of which those myths were the guiding stars.
Indeed it does not matter, as the Theory of Mathematical Correspondences would suggest to us, without our leaving our own realm, how fanciful the mathematical scheme may be, or how distorted. The essential thing is, that we should be able to read, from the picture, properties of Reality. It is not for us to say that such a scheme should be discarded, merely because it does not appeal to us. Let those who can do so, use it! They will be judged by the results they obtain, by their power to predict the Future, by the plausibility and fertility of the guesses they are enabled to make as to the Past and to the Unseen, by the progress they may initiate in the mastery of Nature, and not merely by the measure of success they may have in giving concrete form to their ideas, or in realising them materially.