John Theodore Combridge - Presidential address
John Theodore Combridge delivered his Presidential Address Mathematics - Slave, Servant or Sovereign? to the Mathematical Association at the Annual General Meeting at King's College London, on 17 April 1962. It was published as J T Combridge, Presidential Address: Mathematics: Slave, Servant or Sovereign?, The Mathematical Gazette 46 (357) (1962), 179-196. We give a version of this address below.
Mathematics - Slave, Servant or Sovereign?, by J T Combridge.
It is more than 91 years since, on 17 January 1871, a group of twenty-six men met in another place and formed the Association for the Improvement of Geometrical Teaching, which twenty-five years later became the Mathematical Association. Most were schoolmasters; some were clergymen; the account of the proceedings in the first annual report adds: "There were also a few gentlemen present who were not members of the Association." The ladies were allowed to join later.
We may therefore regard our Association as reasonably mature and of an age which, for such a body, does not necessarily imply impending demise. But the age of maturity brings its own dangers: the danger of getting set in one's ways; the danger that the head will fail to understand what the rest of the body is feeling; the danger that the rest of the body will be slow to respond to what the head is thinking.
One of the first characteristics of our organisation that forced itself on my attention when I took up my duties as President a year ago was the difficulty - in spite of the excellent representation afforded on our Council - of ascertaining what might be called the mind of our Association on specific problems of current interest, such as the introduction of a decimal coinage or the adoption of American methods of the early teaching of so-called "modern" mathematics.
There was the difficulty in attempting the reverse operation - that of keeping Council, the Branches, and members generally, informed of matters with which your Officers had had to deal, of action taken and of reasons for it.
Sometimes that action needed to be supplemented by discussion and action in the Branches, and, with ideas about the teaching of mathematics constantly developing at their present pace, the need to wait for a meeting of Council or of the Branches Committee, or alternatively to incur the expense and inconvenience of convening special meetings of such busy people, seemed likely to prove a barrier to effective action and a danger to the health of our Association.
It has been customary for presidential addresses to this Association to deal with mathematics generally or with some mathematical topic, or with problems of teaching, of syllabuses or of examinations. It is not unusual for presidential addresses to other bodies to address the outside world with admonitions or exhortations.
Because of my concern with the possible dangers which I have just put before you it is my intention today to examine our own situation, to indicate reasons for such action as the Council, as the elected head of our body, has felt it necessary to take, and to suggest one or two principles which may guide them and all of us in our activities for some time to come.
It is interesting incidentally to notice that one of the decisions reached at the inaugural meeting of the Association for the Improvement of Geometrical Teaching was that the members, on dispersing, should each concoct a syllabus; should put on paper their ideas of the ideal syllabus in geometry. A comment on our alleged advance in the amenities of civilisation may be found in the fact that the meeting was able to require such members to send twenty printed copies of each syllabus to the Secretary by May 1871. Of course, "printed" may include "jellied on the school duplicating machine," but I doubt it. I doubt still more whether any committee of this Association could lay such a behest on its members today with any hope of success!
I would first, then, notice what seems to me to be one of the most noteworthy features of our present situation, other than that difficulty in communication with which I began. For many years the discussions at our annual meetings have ranged over an extensive area in the realm of teaching: improvements in technique, in approach, in presentation; fresh applications or new illumination. The Reports issued by our Association have brought clarity and definition into our teaching; they have laid down principles which will remain valid long after matters of detail have been changed. But always, in the background, behind the discussions, behind the Reports, there has been, by and large, in our minds and in our works of reference, the same corpus of syllabuses, the same body of subject matter to be taught in the schools, allowance being made for the introduction of the Jeffery Syllabus after the last war.
This being so, the temptation is upon us to continue as a kind of comfortable club, fighting old battles which are losing their relevance, and dwelling on the memories and portraits of the heroes of the past, while outside two great changes have been taking place which we have never really faced up to consciously, though some of our members (all credit to them) are only too keenly aware of them.
What are these changes? Or rather, for our better information and understanding, what has caused them?
The first change has been brought about by the realisation by mathematicians that much of the mathematics we have been teaching for so long is a collection of particular cases of a more general form of mathematics.
Now the human mind tends to begin with particulars in acquiring new knowledge; nevertheless a stage is reached in which its comprehension is enlarged and made firmer if these particulars can be apprehended as parts of a greater and a unified whole. It is given, I think, only to the few to grasp the whole without coming to it through the particular, but when the whole is grasped a collection of disconnected particulars becomes meaningful.
It was, of course, this consideration which motivated the founders of our Association. They were indignant at the waste of effort involved in trying to teach undeveloped schoolchildren the deductive logic of Euclid before they were familiar with the physical counterparts of the concepts to which his logic was to be applied. I sometimes wonder whether the pressure on us to teach "modern" deductive mathematics at an early stage is not a pressure which might pervert the whole of our school mathematics to the condition from which geometry was rescued ninety years ago, and from which we are now trying to rescue statics and dynamics. Professor Newman gave us a valuable warning about this in his presidential address in 1959. However, my object in mentioning now this new understanding of the nature of mathematics is to suggest that we shall be wasting our time if we go on tinkering with details or working with oddments without having been vouchsafed the insight to see the meaning of the whole.
The second change to be noted is the one that is due to the construction and manufacture of machines that can deal with sets of equations and other mathematical processes which, though comparatively elementary in their formulation, have hitherto been beyond the reach of solution within reasonable time. With this I would couple the necessity in almost every occupation for an ability to describe a situation in terms of statistics and to understand critically such descriptions when formulated by others. The new possibilities created by the introduction of these machines have led to a great increase in the practice of stating in mathematical terms a number of problems of operational research, management and other fields into which mathematics had not previously entered. These possibilities have naturally increased the demand for mathematicians which had been already started by the proliferation of scientific research generally; they have also led to a call for more young people able to deal with a new, but not necessarily very advanced, kind of mathematics.
It seems to me unfortunate that the mathematics involved in both the developments I have just indicated tends to be spoken of in brief as "modern mathematics." Although some of the mathematics concerned is indeed common to both developments, this abbreviated label leads to confusion; it also implies (through a common misuse of the word "modern") that such mathematics is in some way better than something which is presumably "ancient mathematics" and that therefore it should at all costs displace much of what is already taught in our schools. I hope we may long share the happy condition of the hymnologists, and be able to retain the mathematical analogies both of Hymns Ancient and Modern and of an English Hymnal.
The increase in the demand for mathematicians of all kinds and grades has resulted in pressure on our Association - a pressure that we share with schools and universities and with the mathematical world throughout this country.
We have been asked to make in the services we offer more provision for mathematicians who are employed in commerce and industry or in government service, and for those who teach in technical and technological institutions. We are asked to persuade teachers in schools to gear their mathematical teaching to the needs of industrial and scientific research. We are asked to make it possible for more men and women to teach mathematics even though they have not studied it themselves since they left school.
Here the first change I have mentioned comes to our aid. For some of the so-called "modern" mathematics that is entering into the university courses, or being met in them at an earlier stage, illuminates much of the elementary mathematics needed in commerce and industry. This fact, however, brings more questions to our attention. Can vacation or term-time courses be arranged at which these developments can be communicated to serving teachers of mathematics? Can courses be devised for serving teachers who have learned no mathematics since their school days so that they may add some knowledge of this subject to their repertory? Can courses in mathematics in training colleges be altered to take advantage of the possibilities of both the developments I have been discussing?
When the first wave of this attack has passed over, a second comes up behind it. Can university courses be so amplified or modified that they may cater not only for the future research worker in mathematics but also for the much larger number who will either teach it in schools and technical colleges or else use it in applications in physics and engineering? Can university teachers of mathematics be induced manifestly to regard all these potential students of mathematics as of equal importance to the community? Can university departments of mathematics be persuaded to accept more entrants and to make it plain that would-be applicants need have far less fear that they will be unable to stay the course for a degree?
A third wave of questions follows. Can mathematics syllabuses in schools be altered to take account of these new developments? If so, how far back do we go with the alterations? It is alleged that considerable success has been achieved in the United States; should the methods adopted in that country be imported here? If the new mathematics is highly abstract, is it suitable for teaching to children before they have reached the stage which in geometry we call stage C? How does a child's mind work at various mental ages? Can it grasp abstractions before it has learned number bonds and practised techniques?
As these pressure waves came upon us another feature of our organisation required urgent attention. This was the need for improvement in means of communication, to which I alluded at the beginning of this address. We were unable easily to call on our members to share these stresses with us and thus diminish the pressure; we were equally unable to let those who felt the stresses know that they were being shared by others. We had practically no means of telling either our members or the outside world what our Council was thinking and doing in response to the calls coming to it.
The editor of the Gazette was inundated with articles, notes and book-reviews, all of first-class quality, and the state of the printing trade today prohibits quick publication of news in a journal such as the Gazette.
The Teaching Committee was constituted to produce well-digested Reports on teaching matters based on long-standing experience; it was not organised for considering and answering questions of such import as I have been giving to you in some detail.
Meanwhile, lack of coordination, and obstacles to corporate thinking and communication, were affecting the entire mathematical world in this country.
As I shall have occasion to use this term "mathematical world" several times in this address, may I say here that what I envisage by it is the combination of all the mathematicians in the region concerned, and all their mathematics. How two quantities so different in their nature may be added, and what the result is likely to be, I hope to discuss later.
Let me briefly enumerate some of these uncoordinated activities. The Southampton Conference of April 1961 had stimulated much interest in some of the questions I have mentioned; Professor Thwaites's subsequent and widely-publicised inaugural lecture had boosted this interest farther still; there followed his pilot scheme with six or eight selected public or grammar schools for modifying school and examination syllabuses.
A conference in Birmingham financed by the Gulbenkian Foundation carried out a survey of G.C.E. Advanced level syllabuses. Isolated teachers in particular schools, known only to H.M. Inspectors, were conducting some successful experiments in teaching some of the new ideas to unusually young pupils.
The Association for Teaching Aids in Mathematics was doing valuable work in a specialised field, chiefly in schools and among teachers for whom the Mathematical Gazette was too high-brow in its limitations, and this work was being done largely by members who are also members of the Mathematical Association.
The London Mathematical Society, the British Mathematical Colloquium, and other interested bodies, were carrying on their normal activities. But there was no overall strategy whereby all these activities could be coordinated and brought to bear on the problems which were pressing on our mathematical world.
A first attempt to remedy this situation - an attempt for which we are not solely responsible but in which we are materially assisting - has been a letter last month over the joint signatures of Sir William Hodge and the Presidents of the London Mathematical Society and the Mathematical Association proposing the formation of a Federal Council to meet the need for coordination and general strategy of which I have just spoken.
The desirability of such a Council was made tragically clear earlier this month by the small proportion of the section devoted to mathematics, and still more by the omissions from that section, in the statement about the plans for utilising the £250,000 which the Nuffield Foundation is munificently making available, in consultation with the Ministry of Education, for improving the teaching of science and mathematics in schools. It is true that we are not quite as much out in the cold and the dark as a first reading of that statement might suggest, but no outside reader would have inferred from it that mathematics might expect to get very much help in the name of science.
I will now consider more explicitly how the pressures and forces to which I have referred impinged on our Association in particular.
The first call for help was the one from teachers of our subject in technical schools and colleges and from users of mathematics in commerce and industry and government service. Next the conference of university professors, and its continuation committee, and a group working with representatives of the schools and of the Ministry of Education, invoked our aid. Conferences of teachers with industrialists looked to us for guidance and action. Organisations with money to spend in overcoming the shortage of mathematicians turned to us for a central authority to advise them in their spending. And in a background of diffuse noise numerous voices were heard: some crying "What is the Mathematical Association doing?" others saying "It is not, of its nature, constituted to deal with such matters;" others, more embarrassingly, asking "What is the Mathematical Association thinking?"
What were we to think?Before I attempt to answer these not altogether rhetorical questions, let me interpolate one observation which carries an implicit warning.
How were we to think?
What were we to do?
Is it our concern?
What are our terms of reference?
How were we to think?
What were we to do?
Is it our concern?
What are our terms of reference?
These pressures, understandably enough, generated fears. There were fears that the teaching of new ideas might be attempted by inadequately prepared people; fears lest "we in the schools" might not be giving to our pupils what the universities or industry require; fears that we may succumb to pressure to train only technologists and thereby bring about the extinction of the race of true mathematicians in this country; fears that all mathematics will be so geared to applications that it will cease to be mathematics; fears that mathematics will become so abstract that it will bear no relation to the "real" world.
Let us remind ourselves that we are not without precedent here. Pendlebury's Arithmetic is a classic example of mathematics geared to commerce. Before we condemn it out of hand, let us ask whether it did not do as much good in its early days as it might do harm if it were adopted as a standard textbook now.
Well, amid these calls, these questions and these fears, we had already begun to do what we could. We had instituted the Diploma in Mathematics to help teachers. (Incidentally, the papers of that examination have perforce had to begin with the old fashioned kind of mathematics that the first candidates might be expected to know. They will have to be considerably modified in the future to take account of recent developments and trends.) When that Diploma was made public there was an outcry because we were not making it available for those interested in mathematics in technology. We felt we must find our feet with one examination at a time, but when it became clear that this first Diploma was going to be a success, we made preparations for a second one. Our proposals for this and for responding to the other calls from mathematicians in technology to which I have already referred will be the subject of this afternoon's discussion, and this morning would be the wrong occasion for me to attempt to deal with them in more detail. I will therefore now begin the search for anything that might guide our Council and our Association amid the clamour and the claims that come upon us.
Our first guidance was found in our constitution. We exist as an Association "to effect improvements in the teaching of elementary mathematics and its applications, to provide means of inter-communication between mathematical students and teachers for this purpose, and to take such measures as may appear expedient to advance the views of the Association on any question affecting the study and teaching of elementary mathematics and its applications." It is at present no part of our terms of reference to set up a professional institute, to create a closed shop, or to interest ourselves as a body in matters of professional status, grades and salaries.
But it soon became clear that better teaching in schools might result in fewer children being frightened off mathematics at an early age; that different attitudes to the subject in the universities might result in fewer pupils deciding to abandon mathematics for science or engineering on leaving school; that a better awareness of the meaning of mathematics, to be gained from some knowledge of recent thought on the subject, might make for more and better teachers; and that an increased awareness of use and applications in commerce, industry and technology might be of considerable value in schools to teachers and pupils alike.
Here, then, was a way in which we could make some contribution towards the efforts to reduce the shortage of mathematicians. There seemed to be a strong case for going forward, and that on as wide a front as possible. This is not the place or the time in which to discuss the practical problems of such an advance. I must devote the rest of my address to developing the thesis that in making such an advance we are acting in accordance with principles that underlie the development of mathematics; that these principles are intrinsic not only in the subject but in nature itself, so that to effect anything of value we must work in harmony with them just as a carver cooperates with the grain of the wood on which he is working, or a sculptor takes account of the nature of his medium. For this thesis I must take the sole responsibility. I cannot say that it has been consciously in the mind of our Council, and I myself have only consciously formulated it in the practice of acting upon it. But I now believe it to be a reliable guide to future policy.
Consider first the reaction of any individual to pressures such as I have indicated. It will vary with the views that he holds of the nature and purpose of mathematics. For some the subject exists simply as a collection of tools, valued only in so far as it is useful; for them the schools have the duty of "training" their pupils to "do" mathematics; and so presumably the universities are here to train further the better trained students from the schools. On this view mathematics has the role of the slave; the trained pupils become fellow-slaves who "make it work;" their teachers are involved in the same slavery; and when the golden age dawns all this unpleasant work will be done entirely by machines.
At the other end of the line are those who exercise a talent for mathematics, partly because they cannot help but do it, and partly because of their love for the subject. For such people mathematics is sovereign; they do not understand why others do not yield to her the same allegiance, or at least recognise her supremacy and her claims; and they do not realise that society will no more support them in what it regards as day-dreaming than it will support artists who do not give it what it wants.
We have then these two classes of extremists, and it is perhaps rather obvious, and not very helpful, to suggest that the truth lies somewhere between the two extremes. The matter is, I think, not so simple as that, and must be discussed in terms which are much less abstract and more human. These two classes of extremists are only a number of individuals in a much larger community which is the human element in what I am calling the mathematical world. Such extremists have this in common, that each ignores the proper relationship between the individual and the community; one makes the community alone the controlling factor; the other allots all rights to the individual.
I suggest that one right function of the community is to promote the proper development of each individual in it, and that the individual finds his true development in service to the community, and I believe that the development of mathematics exhibits this truth in action.
You are all familiar, no doubt, with the concept of mathematics as queen and servant of the sciences. The idea has been worked out in detail by E T Bell in the well-known book with that title; the phrase, as he acknowledges, comes from Gauss. But when a concept has been embedded in a classic phrase, and developed in a standard volume, there is still scope for someone at a later date to indicate its relevance to a contemporary situation. I hope now to do that, and to show that the concept has a deeper intrinsic significance than appears on the surface.
It is advisable at this stage to point out that when we think of mathematics as queen and servant of the sciences we are mistaken if we think of pure mathematics as the queen and of applied mathematics as the servant. The terms "queen" and "servant," as I see them, refer to two interdependent aspects of mathematical development which are often to be seen in action simultaneously.
In fact, I find it difficult to continue with my argument if I am obliged constantly to make a distinction between pure mathematics and applied mathematics. I do not know where to draw the border line. Or perhaps it would be more accurate to say that I am unable to speak of pure mathematicians and applied mathematicians. What we call "Euclid" was first thought of as pure mathematics. Later it was found to be dependent on real objects, and so presumably it became applied mathematics. Then we began to believe that physical space was non-Euclidean, and "Euclid" became a piece of pure mathematics once more.
Professor Stephan Koerner, in his book The Philosophy of Mathematics, says that the statement "1 + 1 = 2" belongs to pure mathematics, while the statement "one apple and one apple make two apples" belongs to applied mathematics.
I shall return to this later, but at the moment I would rather consider the story (possibly apocryphal) about the late W H Berwick. He was a geometer, and he was appointed to a chair at Bangor. It is related that he went to inspect some lodgings. He was a tall, narrow, rather stiff man, and the landlady looked a little doubtfully, first at the bed and then at the professor, and doubted whether he would find it long enough. Berwick is said to have taken his walking-stick and laid it once across the bed and twice along it, paused for a moment and then said "That will be all right; I shall sleep diagonally."
Did Professor Berwick forget himself and become for a moment an applied mathematician? I suppose he did, but the incident seems to me to make nonsense of the distinction. No pure mathematician can be sure that his mathematics will never be put to practical use, or at least used as a possible model for an idealised physical situation in the way that Einstein and Eddington played with various non-Euclidean geometries in the hope of finding a model of the universe.
Norbert Wiener, in his first volume of autobiography Ex-prodigy accuses G H Hardy of escapism on the ground that he valued number theory "precisely for its lack of practical applications." Hardy, in his Apology, denied holding such views.
I have no objection to anyone's being an escapist. I only wish to emphasise that we may go astray if we think of our subject as being anything but an organic union of what are too often regarded as two separate subjects. If I have interpreted Professor Sondheimer correctly, he was saying much the same thing in his inaugural lecture at Westfield College.
I do not wish to deny the possible convenience of this separation administratively for examination purposes, though I have never understood why a certain famous school-certificate question was regarded as a problem in applied mathematics when it related to trams, and became fit for the algebra paper when it was about buying eggs, the algebraic equations involved being identical in the two cases. (For Professor Koerner, of course, both would be problems in applied mathematics.)
But to suggest that the role of mathematics as queen is confined entirely to pure mathematics, and that in some way we are degrading our high calling if we soil our hands with applied mathematics, and with our fellows who make it their chief business, seems to me like refusing to court matrimonial felicity for fear of having to carry coals or help with the washing up.
Mathematics may well be the salt of the scientific earth, but if it is we must remember that there are only two alternatives before it: to be used for the good of the rest or to be cast out and trodden under foot.
In fact, applications of mathematics are not just carrying coals and washing up. They constitute a major means of advance in mathematics as a whole.
Perhaps the real distinction is the one that Hardy made between what he called "real mathematics" and "trivial mathematics." I think that by the latter he meant applied mathematics that had been standardised, turned into rule of thumb, and in fact enslaved. Professor Sondheimer equally refuses to hold a brief for such mathematics. While I hold no brief for slavery I can hardly, with the record of the Exodus before us, write off the most enslaved community as lost to the world.
As some of the modes of development of mathematics are vital to my argument, let us now consider some specific examples, keeping in our minds all the time that the terms "pure mathematics" and "applied mathematics" - if they slip in - must not be thought of as referring to two mutually exclusive activities of the human mind. Professor Sondheimer has carried out this next stage of my task so admirably in the inaugural lecture which I have just mentioned that I cannot do better than refer you to last December's number of the Gazette in which it was printed. To make my argument clear this morning, however, I must step lightly on some of the stepping-stones which he has so effectively established for us.
Let us look first at some geometrical developments.
We have, I think, no historical evidence about the beginnings of Greek mathematics. We may never know whether the need for land measurement stimulated the study of geometry, and the need for counting led to the study of number, or whether the more abstract thought came first; the later history of mathematics suggests that they may well have developed side by side.
And may I here issue a kindly warning to all who like to lace their mathematical teaching with a little history whether in schools or training colleges? Do remember how easily "perhaps" turns into "possibly," and "possibly" into "probably," and how easily then the qualifying word drops out altogether. You gain nothing by asserting as proven facts what are only guesses at history. The most you can do in the majority of cases is to say that Egyptian writings or Sumerian tablets or Professor X's sojourns among the Polynesian Primitives suggest that the course of development may have been such-and-such.
Beware too of the controversy between those who assert that in the development of the child mind geometry comes before number because when a baby puts its toes into its mouth it is carrying out an exercise in spatial relationships, and those who see in this activity involving ten toes the first steps in arithmetic; the probability seems to me to be that what the baby is really interested in is the taste of its toes.
Euclidean geometry, then, however it began, became a highly abstract system of axioms and postulates and deductions therefrom. At the same time, (as the legend of Professor Berwick shows us) it remained directly applicable to the world around us. It continued as the chief instrument for mathematical applications until two events jolted the mathematical world into deeper thought.
Firstly, Bolyai and Lobachevsky queried Euclid's fifth postulate: that through any point one and only one straight line can be drawn parallel to a given straight line not passing through that point. Secondly, Einstein conceived the idea of thinking of the paths of particles in a gravitational field as geodesics in a curved space-time continuum.
The first jolt enabled mathematicians to live with the possibility of more than one kind of geometry; the second put an end to the idea of one geometry as in some way "true" while others were "fictitious." The first event represented a break-through advance in mathematics made by pure mathematicians querying one of the previously unquestioned postulates. The second event represented an advance initiated by a mathematical physicist seeking a mathematical model to represent physical events. To quote a free translation of some words of the late Tullio Levi-Civita:
Einstein's gravitational theory ... considers the geometrical structure of space as ... depending on the physical phenomena which take place there; unlike the classical theories, which all assume the physical space as given a priori. The mathematical development of the magnificent concept of Einstein (which finds in the Absolute Differential Calculus of Ricci its natural analytical instrument) brings in as an essential element the curvature of a certain four-dimensional variety and the related Riemann symbols.This concept of Einstein's then led to investigations into the idea of parallelism on a curved surface and in a curved space - the comparison of directions at neighbouring points - and resulted further in a tremendous development of affine geometry. It has been well remarked somewhere that later generations may wonder how we ever managed to think of space as being flat or Euclidean, when all the time we knew we were living on a curved surface and using parallels of latitude and meridians of longitude.
Einstein himself has made a comment on the genesis of his concept which is not without interest for teachers generally:-
When I asked myself how it happened that I in particular discovered the relativity theory, it seemed to lie in the following circumstance. The normal adult never bothers his head about space-time problems. Everything there is to be thought about it, in his opinion, has already been done in early childhood. I, on the contrary, developed so slowly that I only began to wonder about space and time when I was already grown up. In consequence I probed deeper into the problems than an ordinary child would have done.Turning from geometry, we can think of cases in which a significant contribution to pure mathematics has been made for the express purpose of dealing with a physical problem. As G T Guilbaud has put it:-
It was in order to establish a rational theory of the propagation of heat that Fourier was led (in 1807-12) to revise completely the very basis of the notion of a mathematical function.Sometimes an advance comes through a mathematician giving mathematical expression to some profound and fruitful ideas of a physicist. In this very college, one hundred years ago, Clerk Maxwell was doing just that for the electromagnetic concepts of Faraday, and formulating what have become known as Maxwell's equations - the basis of electromagnetic theory.
Sometimes the mathematical physicist invents the mathematical theory, and years later a mathematician (whom I dare not classify as pure or applied) undertakes perforce or from choice the task of tidying up his forerunner's mathematical apparatus. Thus Norbert Wiener, in his second autobiographical volume I am a mathematician relates this experience:-
What the communication engineers actually did was to use a formal calculus of communication theory which had been developed some twenty years before by Oliver Heaviside. This Heaviside calculus had not as yet been given a thoroughly rigorous justification, but it had worked for Heaviside and for those of his followers who had absorbed the spirit of his theory sufficiently to use it intelligently. For several years the chief demand made on me at Massachusetts Institute of Technology by the electrical engineering department was to put the Heaviside calculus on a proper logical foundation. ... In performing this task I had to study harmonic analysis on an extremely general basis, and I found out that Heaviside's work could be translated word for word into the language of this generalised harmonic analysis.So also Dirac introduces his delta function into Fourier theory for mathematical-physical purposes, Schwartz produces the rigorous generalised theory, Temple simplifies it, and Lighthill produces a textbook which might have satisfied Hardy and might even have been classified by him as "real" mathematics.
These instances may help to illustrate the way in which mathematics develops; the way in which at one and the same time she serves and reigns. Summon her to your aid, treat her not so much with deference as with an acceptance of her nature and of her laws of development, be prepared to work with her and not against her, and she will open her treasure-houses and admit you freely. She will give you, almost as a by-product, instruments for treating all kinds of physical problems, and when you tackle these problems you will be rewarded with new advances in mathematics. And so the chain reaction continues.
At this point some questions and words of caution are needful. Am I personifying mathematics and being sentimental? Am I making too big a jump by passing from this recognition of the interrelation of service and supremacy in the development of mathematics to the adoption of a guiding principle of conduct? What has all this to do with the obligations of the Mathematical Association in its present situation?
My answer to these questions is that I find myself unable to separate mathematics from mathematicians.
The conquest of man over nature - in so far as it is achieved - is won by finding out how nature works and cooperating with her in her ways. We used to call it "discovering the laws of nature" and then using them. And this is successful not because we are deities creating and standing apart from nature but because we ourselves are part of nature and are made with the same warp and woof. Many of the world's troubles today arise from our forgetting our oneness with nature and giving too much attention to its non-human aspects and too little to its human constituents.
These considerations raise the question not of how mathematics develops but of the nature of the mathematical world. Here I must go carefully, because I am no philosopher. But I think it is correct to say that there is a continuing discussion in the philosophy of mathematics between those who hold that mathematics is entirely an expression of abstract human thought - a kind of emanation from the human brain - and those who think of it as existing objectively and awaiting investigation and discovery.
My guess is that here, as so often, we have not an either-or, but that the truth contains both elements. I see an analogy in what happens when the physicist attempts to study an electron; he finds that he cannot observe it without, by the very act of observing, altering its position or its velocity or both. I find myself compelled to think of a realm which includes both mathematics and mathematicians as an organic whole of which the two parts cannot be separated, and cannot be thought of separately without damage to our reasoning.
In short, it reduces to a matter of the nature of addition or composition. Can you fruitfully or safely think of Pythagoras apart from his theorem, or of the theorem without Pythagoras? (I speak a little mythically, in order to be able to make concessions to those who see in Pythagoras a group such as Bourbaki in our own time.)
Let us try to look into this a little more closely. I have already reminded you that according to Professor Koerner the statement "1 + 1 = 2" belongs to pure mathematics, while the statement "one apple and one apple make two apples" belongs to applied mathematics. I should like to carry this thought a little further and to suggest that consideration of such forms of addition as adding one dog and one cat belongs to philosophy of mathematics.
We once had a kindly neighbour whose vocabulary was somewhat limited. This sometimes gave her a power of graphic description denied to those of us who are more loquacious. She had tabby-and-white cat that was rather nervous. One day she said "Miss So-and-so came in yesterday afternoon. She had her little dog with her. It was all right till the cat went up the chimney." The situation in that previously peaceful drawing room was transformed by adding one dog to one cat. The sudden disappearance of the cat by the wrong exit; the vociferous barking of the little dog, alarmed by this unprecedented manoeuvre; the old lady pulling on the tail of the sooty and protesting cat, while her visitor tried to control her dog - all this made of the local environment of the two animals something which could not be obtained by any arithmetical, or even applied mathematical, addition of their previous separate environments.
Something as dynamic, if not as explosive, pertains, I believe, to the worlds of mathematicians and of mathematics. The mathematical world is surely more than just a set of mathematically-minded human beings; more also than just a set of mathematical propositions; and the combination of these two sets is more than their arithmetical sum, just as the mixing of two separately innocuous chemicals may constitute an explosive mixture.
Some of you may remember that two years ago at our meeting here an Oxford scholar attempted to answer the question "What is mathematics?" by analysing mathematics. He found that it consisted of such and such a percentage of logic, such and such a percentage of something else - four elements in all - and ended, leaving the implication that it was nothing more.
I attempted to show the inadequacy of this reasoning by pointing out that the same four elements, though in different proportions, could be discerned in my wife. The implication - though I did not state it explicitly - was that that was not the whole story.
I do not want to go so far as to elevate mathematics to the category of personality. It has not the faculty of responsiveness in the way that human personality has. But, as Henri Poincaré has shown (and as Professor Temple quoted in his presidential address in this college five years ago), it does partake of the activities of the subconscious mind. It offers neither an adamant obstacle nor a treacle resistance to our attempted advances. It calls for and responds to our questionings just as our body will demand and respond to alternations of exercise, work and rest. It will not be astonishing therefore to find that principle of simultaneous service and supremacy, which we discern in the development of mathematics, operating in the mathematical world - the world of mathematicians and their mathematics - as a whole.
And the thought goes even deeper still, because this mathematical world is itself an intrinsic part of nature, compounded of the same stuff.
This has been brought out well, and much better than words of mine can do, in another passage in Norbert Wiener's autobiography where he describes how from the Massachusetts Institute of Technology he used to look out on the River Charles and watch "the mass of ever shifting ripples and waves" and to reflect that "the highest destiny of mathematics (is) the discovery of order among disorder," and to wonder how this could be done for the waves. What descriptive language could he use?
The problem of the waves was clearly one for averaging and statistics, and in this way was closely related to the Lebesgue integral which I was studying at the time. Thus I came to see that the mathematical tool for which I was seeking was one suitable to the description of nature, and I grew ever more aware that it was within nature itself that I must seek the language and the problems of my mathematical investigations.That is it. A particular piece of mathematics - of Hardy's "real mathematics" - may be completely abstract, impersonal and objective. But mathematical developments cannot be; they cannot be divorced either from the men with whom they originate or from the world of nature of which men and mathematics are parts.
Similarly mathematical teaching, rightly undertaken, will be a function both of the teacher and of the taught. I know that there are distinguished mathematicians who can give a lecture on a particular topic which is a model of deduction and exposition - and may we ever have some of these people among us. But their lectures are almost completely independent of their audiences, and that is not teaching as I understand it. I remember that in my early days at this college there was one year in which I had to lecture on intermediate trigonometry to three different classes on the same day of each week. The first class consisted of Arts and Science students, the second of Engineering students, and the third was a class of evening students - mainly in Arts. The syllabus was the same for all, but never were two of the lectures the same. And so also for mathematical discovery. I cannot believe that the most accomplished mathematicians carry out their research with mechanical precision, "faultily faultless, icily regular, splendidly null, dead perfection, no more," and that there is in their work nothing of the waste and false starts and prodigality of fresh attempts that characterise nature.
And so while I would not claim that we can study the world and thence conclude, either inductively or deductively, there is one and only one way of life open to us as or as an Association, I do maintain that we are part and parcel of this mathematical world, that it has its laws and that these are laws of the nature that is intrinsic to all; that consequently we shall do well to work with them and not ignore them, and that if we cooperate in this way it will be for the benefit of our subject and its future in this country.
I hope I have been able to give you a glimpse of this characteristic feature of mathematical development - the constant interplay of sovereignty and service - and so to persuade you that a readiness for such activity should characterise our Association in its policy.
We need not fear to take this principle as our guide. We must refuse to give up freedom, either for others or for ourselves; we must refuse to allow the mathematical world to become a mere tool for the world of machines, money and materialism. That would be slavery. But when we have our freedom we should not be afraid or disdain to serve. Only those who are free can serve, and there is a service that is perfect freedom. And that service - the service of one's neighbour as one's self - brings the intrinsic right to existence, to individuality, to personality, to continued freedom, and to the only kind of sovereignty worth having.
This, I believe, is as true for an Association as for an individual.
It is in this belief, and with its associated hope, that your Council and I, though often unconsciously, have accepted the challenge of the year that is just ending, and I commend it to you as a trustworthy guide for the years to come.
J. T. C.
King's College, London.
Last Updated June 2021