Percy MacMahon addresses the British Association in 1901, Part 2

Percy A MacMahon was President of Section A of the British Association for the Advancement of Science in 1901. The Association met in Glasgow in September and MacMahon addressed Section A - Mathematical and Physical Sciences. Below is the second part of his lecture.

To read the first part of MacMahon's lecture, follow the link: British Association 1901, Part 1

Percy A MacMahon, the President, continued his Address:

At the beginning of the nineteenth century it was possible for most workers to be well acquainted with nearly all important theories in any division of science; the number of workers was not great, and the results of their labours were for the most part concentrated in treatises and in a few publications especially devoted to science; it was comparatively easy to follow what was being done. At the present time the state of affairs is different. The number of workers is very large; the treatises and periodical scientific journals are very numerous; the ramifications of investigation are so complicated that it is scarcely possible to acquire a competent knowledge of the progress that is being made in more than a few of the subdivisions of any branch of science. Hence the so-called specialist has come into being.

Evident though it be that this is necessarily an age of specialists, it is curious to note that the word 'specialist' is often used as a term of opprobrium, or as a symbol of narrow-mindedness. It has been stated that most specialists run after scientific truth in intellectual blinkers; that they wilfully restrain themselves from observing the work of others who may be even in the immediate neighbourhood; that even when the line of pursuit intersects obviously other lines, such intersection is passed by without remark; that no attention is paid to the existence or the construction of connecting lines; that the necessity for collaboration is overlooked; that the general advance of the body of scientific truth is treated as of no concern; that absolute independence of aim is the thing most to be desired. I propose to inquire into the possibility of such an individual existing as a scientific man.

I take as a provisional definition of a specialist in science one who devotes a very large proportion of his energies to original research in a particular subdivision of his subject. It will be sufficient to consider the subjects that come under the purview of Section A, though it will be obvious that a similar train of reasoning would have equal validity in connection with the subjects included in any of the other sections. I take the word 'specialist' to denote a man who makes original discoveries in some branch or science, and I deny that any other man has the right, in the modern meaning of the word, to be called by others, or to call himself, a specialist. I would not wish to be understood to imply a belief that a truly scientific man is necessarily a specialist; I do believe that a scientific man of high type is almost invariably an original discoverer in one or more special branches of science; but I can conceive that a man may study the mutual relations of different sciences and of different branches of the same science and may throw such an amount of light upon the underlying principles as to be in the highest degree scientific. I will now advance the proposition that, with this exception, all scientific workers are specialists; it is merely a question of degree. An extreme specialist is that man who makes discoveries in only one branch, perhaps a very narrow branch, of his subject. I shall consider that in defending him I am à fortiori defending the man who is a specialist, but not of this extreme character.

A subject of study may acquire the reputation of being narrow either because it has for some reason or other not attracted workers, and is in reality virgin soil only awaiting the arrival of a husbandman with the necessary skill; or because it is an extremely difficult subject which has resisted previous attempts to elucidate it. In the latter case, it is not likely that a scientific man will obstinately persist in trying to force an entrance through a bare blank wall. Either from weariness in striving, or from the exercise of his judgment, he will turn to some other subdivision which appears to give greater promise of success. When the subject is narrow merely because it has been overlooked, the specialist has a grand opportunity for widening and freeing it from the reproach of being narrow; when it is narrow from its inherent difficulty he has the opportunity of exerting his full strength to pierce the barriers which close the way to discoveries. In either case the specialist, before he can determine the particular subject which is to engage his thoughts, must have a fairly wide knowledge of the whole of his subject. If be does not possess this be will most likely make a bad choice of particular subjects, or, having made a wise selection, will lack an essential part of the mental equipment necessary for a successful investigation. Again, though the subject may be a narrow one, it by no means follows that the appropriate or possible methods of research are prescribed within narrow limits. I will instance the Theory of Numbers which, in comparatively recent times, was a subject of small extent and of restricted application to other branches of science. The problems that presented themselves naturally, or were brought into prominence by the imaginations of great intellects, were fraught with difficulty. There seemed to be an absence, partial or complete, of the law and order that investigators had been accustomed to find in the wide realm of continuous quantity. The country as explored was found to be full of pitfalls for the unwary. Many a lesson concerning the danger of hasty generalisation had to be learnt and taken to heart. Many a false step had to be retraced. Many a road which a first reconnaissance had shown to be straight for a short distance, was found on further exploration, to suddenly change its direction and to break up into a number of paths which wandered in a fitful manner in country of increasing natural difficulty. There were few vanishing points in the perspective. Few, also, and insignificant were the peaks from which a general view could be gathered of any considerable portion of the country. The surveying instruments were inadequate to cope with the physical characters of the land. The province of the Theory of Numbers was forbidding. Many a man returned empty-handed and baffled from the pursuit, or else was drawn into the vortex of a kind of Maelström and had his heart crushed out of him. But early in the last century the dawn of a brighter day was breaking. A combination of great intellects Legendre, Gauss, Eisenstein, Stephen Smith, etc. - succeeded in adapting some of the existing instruments of research in continuous quantity to effective use in discontinuous quantity. These adaptations are of so difficult and ingenious a nature that they are to-day, at the commencement of a new century, the wonder and, I may add, the delight of beholders. True it is that the beholders are few. To attain to the point of vantage is an arduous task demanding alike devotion and courage. I am reminded, to take a geographical analogy, of the Hamilton Falls, near Hamilton Inlet, in Labrador. I have been informed that to obtain a view of this wonderful natural feature demands so much time and intrepidity, and necessitates so many collateral arrangements, that a few years ago only nine white men had feasted their eyes on falls which are finer than those of Niagara. The labours of the mathematicians named have resulted in the formation of a large body of doctrine in the Theory of Numbers. Much that, to the superficial observer, appears to lie on the threshold of the subject is found to be deeply set in it and to be only capable of attack after problems at first sight much more complicated have been solved. The mirage that distorted the scenery and obscured the perspective has been to some extent dissipated; certain vanishing points have been ascertained; certain elevated spots giving extensive views have been either found or constructed. The point I wish to urge is, that these specialists in the Theory of Numbers were successful for the reason that they were not specialists at all in any narrow meaning of the word. Success was only possible because of the wide learning of the investigator; because of his accurate knowledge of the instruments that had been made effective in other branches; and because he had grasped the underlying principles which caused those instruments to be effective in particular cases. I am confident that many a worker who, from the supposed extremely special character of his researches has been the mark of sneer and of sarcasm, would be found to have devoted the larger portion of his time to the study of methods which had been available in other branches, perhaps remote from the one which was particularly attracting his attention. He would be found to have realised that analogy is often the finger-post that points the way to useful advance; that his mind had been trained, and his work assisted, by studying exhaustively the successes and failures of his fellow-workers. But it is not only existing methods that may be available in a special research.

Furthermore, a special study frequently creates new methods which may be subsequently found applicable to other branches. Of this the Theory of Numbers furnishes several beautiful illustrations. Generally, the method is more important than the immediate result. Though the result is the offspring of the method, the method is the offspring of the search after the result. The Law of Quadratic Reciprocity, a corner-stone of the edifice, stands out not only for the influence it has exerted in many branches, but also for the number of new methods to which it has given birth, which are now a portion of the stock-in-trade of a mathematician. Euler, Legendre, Gauss, Eisenstein, Jacobi, Kronecker, Poincaré, and Klein are great names that will be for ever associated with it. Who can forget the work of H J S Smith on homogeneous forms and on the five-square theorem, work which gave rise to processes that have proved invaluable over a wide field, and which supplied many connecting links between departments which were previously in more or less complete isolation?

In this connection I will further mention two branches with which I have a more special acquaintance - the theory of invariants, and the combinatorial analysis. The theory of invariants was evolved by the combined efforts of Boole, Cayley, Sylvester, and Salmon, and has progressed during the last sixty years with the co-operation, amongst others, of Aronhold, Clebsch, Gordan, Brioschi, Lie, Klein, Poincaré, Forsyth, Hilbert, Elliott, and Young. It involves a principle which is of wide significance in all the subject-matters of inorganic science or organic science, and of mental, moral and political philosophy. In any subject of inquiry there are certain entities, the mutual relations of which under various conditions it is desirable to ascertain. A certain combination of these entities may be found to have an unalterable value when the entities are submitted to certain processes or are made the subjects of certain operations. The theory of invariants in its widest scientific meaning determines these combinations, elucidates their properties, and expresses results when possible in terms of them. Many of the general principles of political science and economics can be expressed by means of invariantive relations connecting the factors which enter as entities into the special problems. The great principle of chemical science which asserts that when elementary or compound bodies combine with one another the total weight of the materials is unchanged, is another case in point. Again, in physics, a given mass of gas under the operation of varying pressure and temperature has the well-known invariant, pressure multiplied by volume and divided by absolute temperature. Examples might be multiplied. In mathematics the entities under examination may be arithmetic, algebraic, or geometric; the processes to which they are subjected may be any of those which are met with in mathematical work. It is the principle which is so valuable. It is the idea of invariance that pervades to-day all branches of mathematics. It is found that in investigations the invariantive fractions are those which persist in presenting themselves, even when the processes involved are not such as to ensure the invariance of those functions. Guided by analogy may we not anticipate similar phenomena in other fields of work?

The combinatorial analysis may be described as occupying an extensive region between the algebras of discontinuous and continuous quantity. It is to a certain extent a science of enumeration, of measurement by means of integers, as opposed to measurement of quantities which vary by infinitesimal increments. It is also concerned with arrangements in which differences of quality and relative position in one, two, or three dimensions, are factors. Its chief problem is the formation of connecting roads between the sciences of discontinuous and continuous quantity. To enable, on the one hand, the treatment of quantities which vary per saltum, either in magnitude or position, by the methods of the science of continuously varying quantity and position, and on the other hand to reduce problems of continuity to the resources available for the management of discontinuity. These two roads of research should be regarded as penetrating deeply into the domains which they connect.

In the early days of the revival of mathematical learning in Europe the subject of 'combinations' cannot be said to have rested upon a scientific basis. It was brought forward in the shape of a number of isolated questions of arrangement, which were solved by mere counting. Their solutions did not further the general progress, but were merely valuable in connection with the special problems. Life and form, however, were infused when it was recognised by De Moivre, Bernoulli, and others that it was possible to create a science of probability on the basis of enumeration and arrangement. Jacob Bernoulli, in his Ars Conjectandi, 1713, established the fundamental principles of the Calculus of Probabilities. A systematic advance in certain questions which depend upon the partitions of numbers was only possible when Euler showed that the identity xa.xb =xa+b reduced arithmetical addition to algebraical multiplication and vice versa. Starting with this notion, Euler developed a theory of generating functions on the expansion of which depended the formal solutions of many problems. The subsequent work of Cayley and Sylvester rested on the same idea, and gave rise to many improvements. The combinations under enumeration had all to do with what may be termed arrangements on a line subject to certain laws. The results were important algebraically as throwing light on the theory of Algebraic series, but another large class of problems remained untouched, and was considered as being both outside the scope and beyond the power of the method. I propose to give some account of these problems, and to add a short history of the way in which a method of solution has been reached. It will be gathered from remarks made above that I regard any department of scientific work, which seems to be narrow or isolated, as a proper subject for research. I do not believe in any branch of science, or subject of scientific work, being destitute of connection with other branches. If it appears to be so, it is especially marked out for investigation by the very unity of science. There is no necessarily pathless desert separating different regions. Now a department of pure mathematics which appeared to be somewhat in this forlorn condition a few years ago, was that which included problems of the nature of the magic square of the ancients. Conceive a rectangular lattice or generalised chess board (cf. 'Gitter,' Klein), whose compartments are situations for given numbers or quantities, so that there is a rectangular array of certain entities. The general problem is the enumeration of the arrays when both the rows and the columns of the lattice satisfy certain conditions. With the simplest of such problems certain progress had undoubtedly been made. The article on Magic Squares in the Encyclopaedia Britannica, and others on the same subject in various scientific publications, are examples of such progress, but the position of isolation was not sensibly ameliorated. Again the well-known 'problème des rencontres' is an instance in point. Here the problem is to place a number of different entities in an assigned order in a line and beneath them the same entities in a different order subject to the condition that the entities in the same vertical line are to be different. This easy question has been solved by generating functions, finite differences, and in many other ways. In fact when the number of rows is restricted to two, the difficulties inherent in the problem when more than two rows are in question do not present themselves. The problem of the Latin Square is concerned with a square of order n and n different quantities which have to be placed one in each of the n2 compartments in such wise that each row and each column contains each of the quantities. The enumeration of such arrangements was studied by mathematicians from Euler to Cayley without any real progress being made. In reply to the remark 'Cui bono?' I should say that such arrangements have presented themselves for investigation in other branches of mathematics. Symbolical algebras, and in particular the theory of discontinuous groups of operations, have their laws defined by what Cayley has termed a multiplication table. Such multiplication tables are necessarily Latin Squares, though it is not conversely true that every Latin Square corresponds to a multiplication table. One of the most important questions awaiting solution in connection with the theory of finite discontinuous groups is the enumeration of the types of groups of given order, or of Latin Squares which satisfy additional conditions. It thus comes about that the subject of Latin Squares is important in mathematics, and some new method of dealing with them seems imperative.

A fundamental idea was that it might be possible to find some mathematical operation of which a particular Latin Square might be the diagrammatic representative. If, then, a one-to-one correspondence could be established between such mathematical operations and the Latin Squares, the enumeration might conceivably follow. Bearing this notion in mind, consider the differentiation of xn with regard to x. Noticing that the result is n.xn-1 (n an integer), let us inquire whether we can break up the operation of differentiation into n elementary portions, each of which will contribute a unit to the resulting coefficient n. If we write down xn as the product of n letters, viz., x.x.x.x ..., it is obvious that if we substitute unity in place of a single x in all possible ways, and add together the results, we shall obtain n.xn-1. We have, therefore, n different elementary operations, each of which consists in substituting unity for x. We may denote these diagrammatically by

and from this point of view d/dx is a combinatorial symbol, and denotes by the coefficient n the number of ways of selecting one out of n different things.

Similarly, the higher differentiations give rise to diagrams of two or more rows, the numbers of which are given by the coefficients which result from such differentiations. Following up this clue much progress has been made. For a particular problem success depends upon the design, on the one hand, of a function, on the other hand, of an operation such that diagrams make their appearance which have a one-to-one correspondence with the entities whose enumeration is sought. For a general investigation, however, it is more scientific to start by designing functions and operations, and then to ascertain the problems of which the solution is furnished. The difficulties connected with the Latin Square and with other more general questions have in this way been completely overcome.

The second new method in analysis that I desire to bring before the Section had its origin in the theory of partition. Diophantus was accustomed to consider algebraical questions in which the symbols of quantity were subject to certain conditions, such, for instance, that they must denote positive numbers or integer numbers. A usual condition with him was that the quantities must denote positive integers. All such problems and particularly those last specified are qualified by the adjective Diophantine. The partition of numbers is then on all fours with the Diophantine equation

a + b + g + ... + u = n,
a further condition being that one solution only is given by a group of numbers satisfying the equation; that in fact permutations amongst the quantities a, b, g ... are not to be taken into account. This further condition is brought in analytically by adding the Diophantine inequalities
abg ≥ ... ≥ u ≥ 0
u in number. The importation of this idea leads to valuable results in the theory of the subject which suggested it. A generating function can be formed which involves in its construction the Diophantine equation and inequalities, and leads after treatment to a representative, as well as enumerative, solution of the problem. It enables further the establishment of a group of fundamental parts of the partitions from which all possible partitions of numbers can be formed by addition with repetition. In the case of simple unrestricted partition it gives directly the composition by rows of units which is in fact carried out by the Ferrers-Sylvester graphical representation, and led in the hands of the latter to important results connection with algebraical series which present themselves in elliptic functions and in other departments of mathematics. Other branches of analysis and geometry supply instances of the value of extreme specialisation.

What we require is not the disparagement of the specialist, but the stamping out of narrow-mindedness and of ignorance of the nature of the scientific spirit and of the life-work of those who devote their lives to scientific research. The specialist who wishes to accomplish work of the highest excellence must be learned in the resources of science and have constantly in mind its unity and its grandeur.

JOC/EFR April 2007

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