The Growth of Mathematics

In January 1957 George Temple gave his Presidential Address to the Mathematical Association. A version of the address was published in The Mathematical Gazette 41 (337) (1957), 161-168. We give a version of George Temple's talk below.

The Growth of Mathematics, by G Temple.

Mathematics is commonly regarded as a subject in which deductive reasoning reigns supreme, but the history of the growth of mathematics is far from being a record of systematic logical development. In fact the main features of the history of mathematics are the impact of the natural sciences, the transmission of the mathematical tradition, and its enrichment by the imaginative and intuitive efforts of individual mathematicians. The purpose of this lecture is to exemplify this statement by selective illustration from different branches of pure and applied mathematics, and to examine some of its practical consequences in the art of teaching.

Mathematics is an abstract study, and the history of mathematics is a history of abstraction. But the nature of mathematical abstraction is frequently misunderstood, and hence flow many misconceptions and difficulties for the student. To say that mathematics is abstract does not mean that it is remote and isolated from the real world of concrete individual things. The mathematical world is not a different world from the physical universe. It is the same world but viewed in a different light - an intellectual light which reveals much that is hidden in the light of common day.

That multivalent word, "abstraction", so rich in its psychological and philosophical associations, still remains charged with an earthy flavour. It suggests a mental activity which extracts secrets from Nature like a man drawing buckets of water from a well, or which purifies our concepts like a chemist distilling a quintessence. Mathematical abstraction has its own special characteristics, but one feature it does share with philosophical abstraction. This vitally important feature is that the concepts obtained by mathematical abstraction are apprehended only when considered with reference to the material from which they are abstracted. Abstraction separates but does not isolate.

It is therefore appropriate to begin our survey of the intrinsic pattern of the growth of mathematics by a consideration of the kind of material available at any epoch to the mathematician engaged in active research. And we shall single out for especial mention two kinds of material - the existing traditions of the craft and the challenge presented by the natural sciences. In the second place we shall consider how this material is utilised by two species of abstraction of particular significance in mathematics, which I will call abstraction by exclusion and abstraction by extension. The study of mathematics in the school or university is a kind of recapitulation of the past history of mathematics. The elucidation of the principles operative in the historical development should therefore have a significance, either immediate or remote, for the teacher and lecturer in mathematics.

Let us then turn first to the material factors which I have mentioned - mathematical tradition and physical problems. It is especially appropriate that here we should begin with what must be the primary influence on the development of mathematics - the records of past achievements as enshrined in books and expounded by teachers. It is well known how the development of analysis in this country has been decisively influenced by G H Hardy's epoch-making textbook entitled A Course of Pure Mathematics. And it seems probable that our whole approach to algebra, analysis and topology will be transformed by the astonishing series of monographs by that group of French mathematicians who write under the collective synonym of "Bourbaki". It would not be difficult to find counter-examples of branches of mathematics which have in the past become almost moribund for the lack of good textbooks. But I will say no more than to contrast the puerile trivialities of some mathematical textbooks on statics with the vital and inspiring engineering texts on the theory of structures.

The great classics of mathematics are naturally beyond the capacity of the schoolboy, but even his elementary lessons can be enriched and enlivened by appropriate references to the history of the subject; and teachers of mathematics are becoming increasingly aware of the inspiration which they can derive from some knowledge of the historical development of mathematical concepts and methods. One of the lamentable features of contemporary university life is the decline in the discipline of the study of the mathematical classics, and the substitution of ephemeral lecture notes for the abiding companionship of a personally-acquired library, however small it may be.

It is not only by their writings that the great mathematicians have influenced the development of the subject, but also in their personal influence on their students and colleagues. I need mention only the famous school of analysis established in Cambridge by Hardy and Littlewood, and the great number of mathematical physicists who owe their training and inspiration to Sir Geoffrey Taylor.

To mention the name of this great mathematician, meteorologist, physicist and engineer brings me to the second material influence on the growth of mathematics - the challenge of the problems presented by the world of nature and art.

It is well known how the contemplation of the geometry of a tiled floor led Pythagoras to discover his famous theorem on the right-angled triangle, and how the study of the vibrating violin string led to the theory of Fourier series. Every branch of applied mathematics has sprung from the endeavour to express in an organised philosophy the complex events of the real external world - although there are certain textbooks which would lead one to believe that applied mathematics is a department of logic! During the last few years a striking example of the growth of a purely mathematical discipline, stimulated - in fact provoked - by the demands of the theoretical physicist, is the invention of the Theory of Distributions, which has given a simple, coherent and consistent expression to the vague, self-contradictory but eminently useful "improper functions" and illicit inferences of quantum theory and wave theory.

The practical requirements of the engineer and his insistent demands for methods of rapid numerical calculations have led to the invention of new techniques for the solution of mathematical problems, such as the method of relaxation, and have sometimes given new significance to the abstract existence theorems of the pure mathematician.

Modern methods of teaching mathematics are well aware of the importance of developing the power of abstraction from concrete material objects and events. Geometry is no longer imposed as a rigid dogmatic system, but is developed by the co-operation of student and teacher in progressive inferences, inductions and abstractions from the study of simple geometrical patterns. The student of applied mathematics is to be found in the laboratory, and the distinction between applied mathematics and theoretical physics is hard to maintain.

From this cursory sketch of some of the external factors which influence the growth of mathematics we must pass to a rapid review of some of the internal mental processes by which the individual mathematician advances his subject. I shall refer in particular to two kinds of abstraction which are of special significance in the development of mathematics - abstraction by exclusion and abstraction by extension, and first I will speak of abstraction by exclusion.

Confronted by the bewildering complexity of the real world the scientist begins by restricting his studies to those objects which can be known, at least in part, without explicit reference to the rest of the universe. Thus, on different scales of operation, we speak of the solar system, the galactic system, the system of extra galactic nebulae; we have theories of the structure of a single atom, a molecule, or of radiation in an isolating enclosure. The concept of a "system" implies that we are prescinding from the effect of all external influences. Such an intellectual exercise may yield results which are fruitful or illusory. Only subsequent experience can be our guide and critic. The effect of the rotation of the Earth can be ignored on a firing range or in aerodynamics, but not in the study of Foucault's pendulum or the gyro compass.

This simplification of an object of study by the neglect of its environment is an obvious example of what I have called abstraction by exclusion. In mathematical studies this method is still further developed by the systematic suppression of certain features of an object of study in the mental pictures which we form. This process is commonly called "idealisation", and its methodical use is a notable characteristic of applied mathematics.

An outstanding example of the success of idealisation, or abstraction, by exclusion, is provided by the modern development of aerodynamics. The subject of study in aerodynamics is indeed a fluid, but a mathematical fluid and not the air in which we move, breathe and fly. The idealised mathematical fluid has been deprived by abstraction of its compressibility, of its viscosity, of its thermal conductivity, of its molecular structure - indeed of every physical property save its density, its fluidity and capacity for exerting pressure. And yet this attenuated ghost of the real air provides a most suitable and appropriate mental model. Its properties are much more readily investigated than those of the physical air with its complex structure and characteristics. And, in a wide range of conditions, the types of motion produced in the perfect mathematical fluid closely simulate the motion of the real air. The success of the abstract theory of the perfect fluid is as complete as it is unmerited.

The explanation of the success of this idealised fluid dynamics is one of the triumphs of the German aerodynamicist, Prandtl - and I refer to it now because it provides another example of successful idealisation. Prandtl showed that, in the motion of a solid body such as an arrow, a missile, or an aircraft, through the real atmosphere, the physical effects of viscosity and thermal conductivity are confined to a thin boundary layer on the surface of the moving body, and to the wake behind the body. Elsewhere the behaviour of the real air conforms to that of the perfect fluid. In the real air there is no sharp transition from the regime of a boundary layer to the regime outside, but in the current idealisation the concept of a boundary layer with a definite thickness forms a most valuable instrument of thought.

In the realm of pedagogics the doctrine of abstraction by exclusion provides a philosophic justification for the kind of examples and exercises which mathematics students in this country are encouraged to attempt. (I say advisedly "in this country" for the Continental tradition in this matter was different.) The principles of dynamics are few and simple, but it is impossible for a student to appreciate their power until he has applied them in numerous problems. But the real problems of physics and engineering are usually far too complex and difficult for this purpose. And therefore idealised problems must be devised in order to provide a suitable training for the novice. This is the justification of the artificialities in our textbooks which arouse the amusement, contempt or disgust of the casual reader. Surfaces which may be "perfectly smooth" or "perfectly rough", bodies of "negligible mass", "inextensible and perfectly flexible" strings - all these are idealisations which serve a most useful purpose in exercising the powers of the student.

Any criticism of the use of these abstractions, precisely because they are abstractions, is quite mistaken. But there are two real criticisms which can be made of these traditional idealised examples. The first is that the student is never, or most seldom, encouraged to make the idealisations himself. It would be of the greatest value to the best students if they were called upon to construct their own idealised versions of practical physical problems. Perhaps the dynamical problems which arise in sport and athletics would provide suitable material for this purpose. The second criticism is that the student is almost invariably told what he is to prove, whereas in real research it is the formulation of the right question which is difficult as compared with its subsequent solution. From time to time exercises might be set in the most general terms so as to stimulate the faculties of abstraction and curiosity. For example, "Discuss the spinning of a penny," "Discuss the rise of bubbles in water," "Discuss the vibrations of railway carriages."

So far our illustrations of abstraction by exclusion have been drawn from applied mathematics, but there is an analogous mental process of proved value in pure mathematics, I mean the abstraction by axiomatisation. There comes a stage in the development of a new subject when our intuition fails, our logical powers seem ineffective and our resources are exhausted. This is the time for a radical stocktaking of our equipment by an explicit cataloguing of all our undefined notions and unproved assumptions and a systematic and logical organisation of all the propositions which we have been able to deduce. This is the process of axiomatisation. In this process it is often advantageous to consider the terms and symbols in which the axioms are expressed as mere counters in a game, and to divest them temporarily of all significance except such as is conveyed by the axioms themselves. The mind of a mathematician seems sometimes to acquire new strength from such a diversion and distraction, and to perceive hitherto unnoticed paths of advance.

Here are two examples drawn from the theory of probability and from topology. The basic concepts and methods of the theory of probability are well known in an implicit and confused manner to all scientists, together with some acquaintance of the outstanding paradoxes of this theory, which naturally create suspicion of its value. One characteristic of the developments in statistics and probability during the last thirty years has been the explicit use of axiomatisation as an instrument of self-criticism and research. Perhaps the most obvious and useful achievement of this method has been the replacement of the old and ill-defined concept of "probability" by the perfectly explicit concept of "relative probability".

To illustrate the need for axiomatisation in topology consider the simple statement that a plane closed curve, which does not intersect itself, has an inside and an outside. As soon as we begin the process of axiomatisation we realise the need for a definition of the term "curve" - and we soon discover that with some definitions there are curves which can fill an area. In order to provide a rigorous proof of the theorem just enunciated it is necessary to discard all our intuitive knowledge of curves and to start afresh with open minds to construct an axiomatic basis for topology.

I need scarcely say that axiomatisation is primarily an instrument of study and research for the advanced student and that it is quite out of place in elementary work, where its abstract character inevitably conduces to boredom and repulsion. The ancient methods of teaching geometry began by learning by rote the Euclidean axioms and postulates, for which the student was quite unable to see the need or significance. Happily modern methods postpone the study of the axioms of geometry to a later stage when the student begins to feel their necessity in order to coordinate his knowledge.

I now approach the most difficult part of my lecture and attempt to give some account of what is beyond doubt the distinctive mental operation by which mathematics is advanced - I mean abstraction by extension, invention or creation. The psychology of invention in the mathematical field has formed the title of an interesting essay by Hadamard (Princeton, 1949), and Henri Poincare has given us a number of personal experiences which form precious material for the psychologist. No better introduction to the subject of abstraction by extension could be found than direct quotation describing the famous lecture which Poincaré gave to the Société de Psychologie in Paris on "Mathematical Creation."

Poincaré's example is taken from one of his greatest discoveries, the first which has consecrated his glory, the theory of fuchsian groups and fuchsian functions. 'In the first place' says Hadamard, 'I must take Poincaré's own precaution and state that we shall have to use technical terms without the reader's needing to understand them. I shall say, for example,' he says, 'that I have found the demonstration of such a theorem under such circumstances. This theorem will have a barbarous name, unfamiliar to many, but that is unimportant; what is of interest for the psychologist is not the theorem, but the circumstances.'

So, we are going to speak of fuchsian functions. At first, Poincaré attacked the subject vainly for a fortnight, attempting to prove that there could not be any such functions: an idea which was going to prove to be a false one.

Indeed, during a night of sleeplessness and under conditions to which we shall come back, he builds up one first class of those functions. Then he wishes to find an expression for them. Poincare tells us:
I wanted to represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate; the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and succeeded without difficulty in forming the series I have called thetafuchsian.

Just at this time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience' sake, I verified the result at my leisure.

Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty, that the arithmetic transformations of indefinite ternary quadratic forms were identical with those of non-Euclidean geometry.
I should like to set side by side with Poincaré's lecture an extract from a celebrated letter by Mozart the musician in which he describes the suddenness and spontaneity of creative work in music:
When I feel well and in a good humour, or when I am taking a drive or walking after a good meal, or in the night when I cannot sleep, thoughts crowd into my mind as easily as you could wish. Whence and how do they come? I do not know and I have nothing to do with it. Those which please me, I keep in my head and hum them; at least others have told me that I do so. Once I have my theme, another melody comes, linking itself to the first one, in accordance with the needs of the composition as a whole: the counterpoint, the part of each instrument, and all these melodic fragments at last produce the entire work. Then my soul is on fire with inspiration, if however nothing occurs to distract my attention. The work grows; I keep expanding it, conceiving it more and more clearly until I have the entire composition finished in my head though it may be long. Then my mind seizes it as a glance of my eye a beautiful picture or a handsome youth. It does not come to me successively, with its various parts worked out in detail, as they will be later on, but it is in its entirety that my imagination lets me hear it.
This lecture on the growth of mathematics would, I feel, have been incomplete without this citation of personal experience of a great mathematician. However, I shall say no more about the nature of the internal mental processes which lead to the higher forms of mathematical abstraction, but turn to a description of their external manifestations.

It is almost true to say that the history of mathematics is the record of successive generalisations. For example, we have the successive elaboration of the concepts of the signless integers, the positive and negative integers, rational fractions, real numbers, complex numbers. Each of these concepts is a generalisation of the preceding. This does not mean that each of these concepts is a particular subspecies of the succeeding concept. That would be a complete misunderstanding of the nature of abstraction by extension. The relation generated by extension is more subtle and more flexible, and it is worth while to attempt a brief description in order to bring out the creative character of mathematical work.

Consider the relation between the integers and fractions. What is a fraction? It is an ordered pair of integers $(p, q)$ called respectively the numerator, $p$, and the denominator, $q$. It is therefore manifest that no fraction can be an integer. A fraction is a different species of mathematical entity. Nevertheless there is a certain class of fractions which behave like integers, to wit, fractions of the form $(p, 1)$ when the denominator is unity. Let us call these integral fractions. Then it is clear that there is a one-to-one correspondence between integers and integral fractions, and between any operations executed on integers and the analogous operation executed on integral fractions. In the terminology of modern algebra, this is expressed by saying that the class of integers and the class of integral fractions are isomorphic, and that the class of fractions is an extension of the class of integers.

This simple example shows up all the characteristics of abstraction by extension. Firstly, the relation between a concept and its extension is susceptible of strict logical analysis and definition, after the extension has been invented by the mathematician. But until the generalisation or extension has been formulated its relation to the primitive concept is unknown, and the way in which the generalisation is gradually given expression can only be compared with the analogous processes of poetic experience.

Secondly, the relation of abstractive extension is certainly not elementary in character, and quite unsuitable for elementary teaching. The problem confronting the teacher is to elicit in the pupil the act of extensive abstraction - without any analysis of the nature of this act. He must teach fractions without any reference to the formal description of fractions which we have sketched above. This is a formidable task which must excite the admiration of those who are occupied with the far easier work of lecturing on differential equations or analytical dynamics.

Thirdly, it is worth noting that the invention of fractions enables us to answer questions which were unanswerable in the restricted language of integers. And this is typical of extensive abstraction in the history of mathematics. The pattern of mathematical progress frequently exhibits the two stages of frustration and invention-frustration at the discovery that certain problems are insoluble with existing concepts and methods, and then the invention of new and more general concepts, new and more powerful methods to resolve these problems. The whole theory of limiting processes is an excellent example of this pattern of abstraction.

Fourthly, there is a feature of the mental processes of abstraction which is of special interest for the teacher and lecturer - i.e. their avowedly tentative and experimental character in their initial stages. Such phrases as "imaginary numbers", "improper functions", "symbolic elements", are relics of past periods of mathematical history when the processes of the creation or invention of complex numbers, distribution functions and nilpotent Aronhold Clebsch algebras were still at a stage of free experimentation unshackled by the factors of a formalised system.

The last characteristic of abstraction by creation which I will mention is the experience of wordless and imageless thought. It is certainly the testimony of many mathematicians, which I can confirm from my own experiences that the great problem is to find words, concepts and images which can embody the free thoughts which we are struggling to express. Perhaps a quotation from Francis Galton, the great geneticist, will elucidate this mysterious activity.
It is a serious drawback to me in writing, and still more in explaining myself, that I do not so easily think in words as otherwise. It often happens that after being hard at work, and having arrived at results that are perfectly clear and satisfactory to myself, when I try to express them in language I feel that I must begin by putting myself upon quite another intellectual plane. I have to translate my thoughts into a language that does not run very evenly with them. I therefore waste a vast deal of time in seeking for appropriate words and phrases, and am conscious, when required to speak on a sudden, of being often very obscure through mere verbal maladroitness, and not through want of clearness of perception. That is one of the small annoyances of my life.
It is now time for me to draw to an end. I have been acutely conscious, during the preparation and delivery of this lecture, of the highly selective nature of the material which I have presented; but I have endeavoured to describe some of the main factors which influence the growth of mathematics, and where possible to refer to their significance in the teaching of this subject. The treatment has been far from exhaustive and many themes have been left untouched. There is in particular one topic, which I have so far neglected, but to which I must refer by way of conclusion - the influence of aesthetic judgements not only in the final presentation of a new theory, but also in the inception and formation of the theory in our minds. The formal elegance of a mathematical proof is not an adventitious decoration superadded to a gaunt logical exposition. It is of the very essence of the work; and a sense of beauty, of the innate fitness of things, is an indispensable guide in that work of choice and selection which forms such a large part of mathematical activity. Moreover, aesthetic considerations can determine the very choice of the problem which a mathematician selects for study. Perhaps this is the clue to those difficult questions, how is mathematical abstraction to be acquired, how is mathematical technique to be learnt? Perhaps also it is the clue to a far more searching question - What is mathematics?

G. T.

Last Updated January 2021