Mathematics, from the points of view of the Mathematician and of the Physicist

"If we were to question a man of average education, or even one... [in] the cultured class, as to what he conceives to be the nature of Mathematical Science, and as to what he thinks are the aims... we should probably receive a somewhat vague... impression that Mathematics is concerned with calculations... involving a copious use of symbols and diagrams entirely unintelligible to the uninitiated. ...of no interest to anyone except a few individuals who have an unaccountable taste for such things, and happen to be endowed with a peculiar transcendental faculty... unnecessary for other people, and which he... is quite happy without... [O]ur friend would probably admit that... subjects, such as Arithmetic and Mensuration... are extremely useful... and that anything beyond them is of little or no concern to the world in general."

"In the scientific world... decidedly vague and narrow conceptions of the functions of Mathematical thinking are current. In such circles, the notion is extremely common that the sole function of Mathematics is to provide the means of carrying out... calculations... and thus that Mathematics plays in them a comparatively humble part analogous to that of a mechanical tool."

"Mathematical thinking...has played a most important part in the formation of the concepts with which the Physical Sciences work... [I]t has reduced the originally vague conceptions which arise in connection with physical observation to precise forms in which they can be exhibited as measurable quantities."

"Mathematical thinking, in a more or less explicit form, pervades every department of human activity. The grocer... The Engineer... The Philosopher, in his reflections on spatial and temporal relations, on number and quantity, on matter and motion, is in a region of thought in which the boundary between his own domain and that of the Mathematician is almost non-existent. The Epistemologist has always to take Mathematical knowledge as a kind of touchstone on which to test his theories of the nature of knowledge. The dominant views in various departments of philosophical thinking have been modified in important points by the results of recent Mathematical research, and will... in the future, be further modified..."

"Mathematical thought is... the most all-pervading and the most highly specialized department of mental activity."

"The more closely men scrutinized natural phenomena, at first for practical reasons, and later from intellectual curiosity, the more things and processes they found to have aspects which are measurable, and the more they were able to employ their developing Mathematical processes and concepts for the precise characterization of various aspects of the world of phenomena."

"But... the natural development of Mathematical thought, starting as it did in connection with the more obvious aspects of sensuous experience, under the pressure of physical needs, brings it to a region reaching far beyond that in which the primitive intuitions of time, space, and matter formed the exclusive subject matter of the Science."

"[T]he Engineer, like the Physicist, has constantly to make use of Mathematical methods; but as his ultimate aim is to harness the forces of nature and use them to obtain practical results, rather than to bring their relations under general laws and concepts as the theoretical Physicist does, he is perhaps less directly concerned than the Physicist with the part which Mathematical Science has played in the formation of the concepts... He uses applied Mathematics, and applied Physics, and is apt to take both of them more or less as ready-made products, although he cannot do so beyond a certain point without grave danger to his efficiency as a scientific engineer."

"In former times the Mathematician and the Physicist were usually one and the same man. ...it was in the nineteenth century that the increasing complexity of both Sciences produced that separation ...which has become continually more marked, and has reached its extreme... in our own time. ...The chief drawback is that each specialist, from lack of interest in, and knowledge of, the progress of the other great department, is apt to miss that large source of inspiration in his own study which is supplied by the other one."

"I remember... at a Board meeting at Cambridge, the subject of Bessel's functions came into the discussion... to include them in an examination syllabus. Their utility in connection with Applied Mathematics having been referred to, a very great Pure Mathematician who was present ejaculated—"Yes, Bessel's functions are very beautiful functions, in spite of their having practical applications." It would have been interesting to have heard what this great man would have said if he had known that Professor Perry would one day propose the desecration of these beautiful functions by recommending them as suitable playthings for young boys."

"Speaking... from personal experience, one of the effects of prolonged study of some of the more abstract branches of Mathematics, as for example the Theory of Functions, is that one begins to take the greatest interest in, and to be most attracted by... aspects of the subject which are most remote from the interests of the Physicist. One gets into an attitude... in which the kind of well-behaved functions, without abnormal singularities... appear to have a somewhat bourgeois aspect, in their comparatively uninteresting respectability. In the mind of one who makes a minute and prolonged study of the peculiarities which Fourier's series may present, a[n]... effect of that study is that the ordinary Fourier's series, which converge everywhere... normally, begin to acquire a certain tameness... which deprives them of interest. The failure of convergence of Taylor's series becomes to some Mathematical students a matter of greater interest than that presented by the series in the ordinary cases in which... they are fitted for purposes of application."

"Mathematical thinking has played a very important part in the formation of the fundamental concepts of the Physicist; very often this part has been a dominant one. Many of these concepts could only have received a precise meaning and... taken definite forms as the result of the work of Mathematicians... the result of a long train of previous Mathematical thinking. For example, the conception of Energy, and the exact meaning of the... law of the Conservation of Energy, emerged as results of the development of the abstract side of molar mechanics, which determined the mode in which the of moving bodies and as work are defined as measurable quantities. Only by the transference and extension of these notions to the molecular domain did the conception involved in the modern doctrine become possible. The doctrine... had been established before Joule and Mayer commenced their work, and was a necessary presupposition of their further development. Joule was able to determine the only owing to the fact that mechanical work was already regarded as a measurable quantity, measured in a manner which had been fixed in the course of the development of the older Mathematical Mechanics. The notion of Potential, fundamental in Electrical Science, and which every Physicist, and every Electrical Engineer, constantly employs, was first developed as a Mathematical conception during the eighteenth century in connection with the theory of the attractions of gravitating bodies. It was transferred to the electrical domain by George Green and others, together with a good deal of detailed mathematics connected with it which had previously been applied to the function."

"The ultimate aim of the Physicist, even [the]... experimental[ist], is much higher than that of attaining to a merely empirical knowledge of facts. His real object is to classify facts in such a way as to refer them to general laws which are of a more or less abstract character, and which involve concepts of schematic representations that require... the aid of the Mathematician."

"The man of true Physical instincts, endowed with the great faculty of scientific imagination, possessed for example by Lord Kelvin in a very remarkable degree, is for ever imagining models which shall enable him... to represent and depict the course of actual physical processes. The possibility and consistency of such models require Mathematical Analysis for their investigation. The Mathematician may also, by tracing the necessary consequences of the postulation of a model of a particular type, formulate crucial tests in accordance with which further experiments will decide whether a... model can be retained at least provisionally, or whether it must be rejected as inadequate... and must give place to some other model..."

"Perhaps the most striking example of the services which have been rendered to Science by the contemplation of various models, many or all of which have ultimately been found to be inadequate for complete representation, is to be found in the history of Optics. The various forms of the corpuscular theory, and of the wave theory, of Light were all attempts to represent the phenomena by models, the value of which had to be estimated by developing their Mathematical consequences, and comparing these consequences with the results of experiments. The adynamical theory of Fresnel, the elastic solid theory of the ether developed by Navier, Cauchy, Poisson, and Green, the labile ether theory developed by Cauchy and Kelvin, and the rotational ether theory of MacCullagh were all efforts of the kind... indicated; they were all successful in some greater or less degree in the representation of the phenomena, and they all stimulated Physicists to further efforts to obtain more minute knowledge of those phenomena. Even such an inadequate theory as that of Fresnel led to the very interesting observation by Humphry Lloyd of the phenomenon of conical refraction in crystals, as the result of the prediction by Rowan Hamilton that the phenomenon was a necessary consequence of the Mathematical fact that Fresnel's wave surface in a biaxal crystal possesses four conical points."

"Although the theoretical Physicist has for his real aim the formulation of abstract schemes for the description and correlation of the physical phenomena which he observes, with the Mathematician processes of abstraction must go very much further than with the Physicist."

"A strong tendency of Mathematics in its later developments is to split up notions, originally undivided, into components, and to proceed to deal with these components in isolation, and often in separate branches of study."

"[D]uring the last half-century, number and measurable quantity have been separated... the idea of number alone has been recognized as the foundation upon which Mathematical Analysis rests, and the theory of extensive magnitude is now regarded as a separate department in which the methods of Analysis are applicable, but as no longer forming part of the foundation upon which Analysis itself rests. For the purposes of analysing the implications of the methods employed, and of pushing those methods to the highest possible degree of development, this kind of separation is indispensable, and has led to the very abstract form..."

"For the Physicist on the other hand, it is essential that abstraction should not go nearly so far... Too much abstraction... would entail the penalty that he would lose his way in a field which is barren for his purposes, and would lead to a loss of contact with the phenomenal world. To do what the Mathematician does, and must do... would be fatal to the Physicist, to whom above all things a large degree of concreteness in his conceptions is indispensable."

"Not only did Mathematical Science take its origin in the necessities and interests... in the physical world, but at every stage of its development the problems of Physics have been the source of the ideas which have directed the Mathematician, and from which new paths of investigation have been suggested..."

"But every great problem... from the physical side... has given rise to a train of ideas... and has started a host of questions... [which] have led him in most cases far beyond the original domain..."

"The most abstract branches of modern Mathematics, the theory of functions, real and complex, the theories of groups and of s—all arose originally from physical beginnings, but have reached out into vast developments... remote from the physical region. At any moment one... of these developments may become urgently necessary for the purposes of Physics, and may thus be in a position to pay back some of the debt they owe to the parent from whose side they have wandered so far."

"The question is often asked... why Mathematicians cannot restrict themselves more to those aspects of their Science which bring them in contact with Physics, and which are concerned with what often receives the question-begging and ambiguous name of reality; a word that has an indefinite number of shades of meaning, varying with every difference in philosophical view, but which in this connection is generally associated with... the physical world. Why... do modern Mathematicians... wander away from the source... from which its ever-renewed inspiration has been received, in order to lose themselves in a transcendentalism which, in its aloofness from physical investigation, condemns them to an endless and barren immersion in abstractions of their own creation? ...cut ...off from the roots of the Science?"

"To stop short at a point dictated by considerations of applicability to Physics is impossible to those to whom clear and thoroughly defined conceptions are a desideratum, the lack of which in any department of their study leaves them no rest. To attempt to confine the activities of Mathematicians by imposing... a restriction of the nature... above... would be to attempt to strangle the Science as a progressive development."

"Mathematics can in the long run be developed to the highest degree of perfection, not only from the point of view of specialists within its own domain, but also as constituting an essential component of the intellectual life and stock of ideas of the world, only on the condition that it is allowed full freedom of self-expression."

"The utilitarian notion... has the fatal limitation that it attempts to assign limits to what is, or may in the future become, useful, in accordance with a more or less arbitrarily restricted standard of what constitutes utility."

"When the exigencies of Physics suggest to the Mathematician some... special problem for solution, he is impelled to search for some generalization, some law, under which a whole class of analogous processes or problems can be subsumed. ...The Physicist also ...is really occupied in attempting to exhibit ...some general law under which a whole class of phenomena can be subsumed. A narrow utilitarianism would be as fatal to the growth of Physics as to that of Mathematics."

"The Mathematical Physicist plays a part of supreme importance as an intermediary and interpreter between the Pure Mathematician and the experimental Physicist. ...he must follow... progress both in Mathematics... and in experimental Physics. ...[I]n spite of some brilliant exceptions, the Mathematical Physicist does not... take as prominent a part as was formerly the case, especially during the nineteenth century, the age of Maxwell, Kelvin, Stokes, Helmholtz. ...In earlier times, when ...molar mechanics, and especially , occupied the centre of the interests... the passage from the observation of concrete phenomena to their abstract Mathematical representation was comparatively easy. The observational work was simpler and less technical... highly equipped physical laboratories had not yet come into existence... The Mathematical Physicists and Astronomers of the eighteenth century were largely engaged in working out the detailed implications of the law of gravitation, and had commenced, largely under the influence of the idea of , to work out problems such as... the vibrations of strings and other bodies."

"In the... [early] nineteenth century the centre of physical interest passed on to such subjects as Hydrodynamics, the Conduction of Heat, and Elasticity, in which an abstract representation of a body as a continuous plenum... made the problems readily accessible to continuous Mathematical Analysis. ...[M]uch attention was given to... Electricity and Magnetism, and much of the Mathematical Analysis which had been devised for...dealing with problems of gravitational attractions, vibrations, etc., was found, with further development, to be applicable to the new problems... Much of the work, such as that of Ampère... was still carried out under the influence of the idea of , first brought into prominence in connection with the Newtonian law of gravitation, but the idea of the continuous medium gradually became the dominating notion."

"The period in which Physical Mathematics was applied with such great success to continuous media probably reached its culmination in Maxwell's equations of Electrodynamics which are now usually regarded as representing the average effects exhibited when actual discreteness is smoothed out."

"In our own time the centre of physical interest has transferred itself, in connection with Electromagnetism, to the molecular and sub-molecular domain, in which discrete objects become the subject of scrutiny."

"The boundaries between Physics and Chemistry have been broken down. In this region of investigation... Physicists... have been rewarded by the discovery, during the last two decades, of a crowd of remarkable facts, probably destined to have the most far-reaching influence upon our conceptions of the material world."

"It has been said that the Theory of Numbers is a subject which has never been soiled by any practical application. Who can be absolutely sure that even so apparently transcendental a branch of thought as this will always remain undefiled by the contaminating touch of physical application?"

"In the history of Science it is possible to find many cases in which the tendency of Mathematics to express itself in the most abstract forms has proved to be of ultimate service in the physical order of ideas. Perhaps the most striking example is to be found in the development of abstract Dynamics. The greatest treatise which the world has seen, on this subject, is Lagrange's Mécanique Analytique, published in 1788. ...conceived in the purely abstract Mathematical spirit ...Lagrange's idea of reducing the investigation of the motion of a dynamical system to a form dependent upon a single function of the of the system was further developed by Hamilton and Jacobi into forms in which the equations of motion of a system represent the conditions for a stationary value of an integral of a single function. The extension by Routh and Helmholtz to the case in which "ignored co-ordinates" are taken into account, was a long step in the direction of the desirable unification which would be obtained if the notion of potential energy were removed by means of its interpretation as dependent upon the kinetic energy of concealed motions included in the dynamical system. The whole scheme of abstract Dynamics thus developed upon the basis of Lagrange's work has been of immense value in theoretical Physics, and particularly in statistical Mechanics... But the most striking use of Lagrange's conception of generalized co-ordinates was made by Clerk Maxwell, who in this order of ideas, and inspired on the physical side by... Faraday, conceived and developed his dynamical theory of the Electromagnetic field, and obtained his celebrated equations. The form of Maxwell's equations enabled him to perceive that oscillations could be propagated in the electromagnetic field with the velocity of light, and suggested to him the Electromagnetic theory of light. Heinrich Herz, under the direct inspiration of Maxwell's ideas, demonstrated the possibility of setting up electromagnetic waves differing from those of light only in respect of their enormously greater length. We thus see that Lagrange's work... was an essential link in a chain of investigation of which one result... gladdens the heart of the practical man, viz. wireless telegraphy."