The study of mathematics is apt to commence in disappointment. ...[L]ike the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it—'"Tis here, 'tis there, 'tis gone"...—and too noble for our gross methods.
"A show of violence,"...may surely be "offered" to the trivial results which occupy the pages of some elementary mathematical treatises.
[I]ts fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. ...[T]he unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception.
[I]t is... an error to confine attention to technical processes, excluding consideration of general ideas. Here lies the road to pedantry.
The object of the following chapters is not to teach mathematics, but to enable students from the very beginning... to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena.
Arithmetic... will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science.
[A]rithmetic... applies to everything... of all things it is true that two and two make four. Thus... mathematics... deals with properties and ideas which are applicable to things just because they are things, and apart from... feelings, or emotions, or sensations... This is what is meant by calling mathematics an abstract science.
The progress of science consists in observing... interconnections and in showing with a patient ingenuity that the events of this evershifting world are but examples of a few general connections or relations called laws.
To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought.
In the eye of science, the fall of an apple, the motion of a planet round a sun, and the clinging of the atmosphere to the earth are all seen as examples of the law of gravity. This possibility of disentangling the most complex evanescent circumstances into various examples of permanent laws is the controlling idea of modern thought.
[W]e ascribe the origin of... sensations to relations between the things which form the external world.
[W]e... endeavour to imagine the world as one connected set of things which underlies all the perceptions of all people.
[W]e hear and we touch the same world as we see. ...[W]e want to describe the connections between these external things in some way which does not depend on any particular sensations, nor even on all the sensations of any particular person.
The laws... are to be described, if possible, in a neutral universal fashion, the same for blind men as for deaf men, and the same for beings with faculties beyond our ken as for normal human beings.
But when we have put aside our immediate sensations, the most serviceable part—from its clearness, definiteness, and universality—of what is left is composed of our general ideas of the abstract formal properties of things... the abstract mathematical ideas...
[S]tep by step, and not realizing the full meaning of the process, mankind has been led to search for a mathematical description of the properties of the universe, because in this way only can a general idea of the course of events be formed, freed from reference to particular persons or to particular types of sensation.
[S]cience seeks to describe an apple in terms of the positions and motions of molecules, a description which ignores me and you and him, and also ignores sight and touch and taste and smell.
Mathematics as a science commenced when first someone... proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science.
The ideas of any and of some are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic.
[I]n the place of saying that 3 > 2, we generalize and say that if . ...[T]his latter assumption... is of vital importance, both to philosophy and to mathematics; for by it the notion of infinity is introduced.
After the rise of algebra the differential calculus was invented by Newton and Leibniz... a pause in the progress of the philosophy of mathematical thought occurred so far as these notions are concerned; and it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics...
One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations.
[I]t is more instructive to dwell upon the vast amount of preparatory thought, the product of many minds and many centuries, which was necessary before this exact law could be formulated. In the first place, the mathematical habit of mind and the mathematical procedure... had to be generated; otherwise Newton could never have thought of a formula representing the force between any two masses at any distance.
The really profound changes in human life all have their ultimate origin in knowledge pursued for its own sake.
The use of the was not introduced into Europe till the end of the twelfth century A.D., more than 3000 years after its first use in China. The importance which the science of electromagnetism has since assumed in every department of human life is not due to the superior practical bias of Europeans, but to the fact that [these] phenomena were studied by men who were dominated by abstract theoretic interests.
Mathematical ideas now appear. During the decade 1780 to 1789, Coulomb... proved that magnetic poles attract or repel each other, in proportion to the inverse square of their distances, and also that the same law holds for electric charges—laws curiously analogous to that of gravitation.
In 1820, Oersted... discovered that electric currents exert a force on magnets, and almost immediately afterwards the mathematical law of the force was correctly formulated by Ampere... who also proved that two electric currents exerted forces on each other.
As a result of his mathematical investigations, Maxwell recognized that, under certain conditions, electrical vibrations ought to be propagated. He at once suggested that the vibrations which form light are electrical. This suggestion has since been verified, so that now the whole theory of light is nothing but a branch of the great science of electricity.
Herz... in 1888, following on Maxwell's ideas, succeeded in producing electric vibrations by direct electrical methods. His experiments are the basis of our .
[B]y the gradual introduction of the relevant theoretic ideas, suggested by experiment and themselves suggesting fresh experiments, a whole mass of isolated and even trivial phenomena are welded together into one coherent science, in which the results of abstract mathematical deductions, starting from a few simple assumed laws, supply the explanation to the complex tangle of the course of events.
[L]and-surveys had produced geometry, and the observations of the heavens disclosed the exact regularity of the solar system. Some of the later Greeks, such as Archimedes, had just views on the elementary phenomena of and optics. Indeed, Archimedes, who combined a genius for mathematics with physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics.
The world had to wait for eighteen hundred years till the Greek mathematical physicists found successors. In the sixteenth and seventeenth centuries... Leonardo da Vinci... and Galileo... rediscovered the secret, known to Archimedes, of relating abstract mathematical ideas with the experimental investigation of natural phenomena.
[T]he slow advance of mathematics and the accumulation of accurate astronomical knowledge had placed natural philosophers in a much more advantageous position for research. ...[T]he very egoistic self-assertion of that age, its greediness for personal experience, led its thinkers to want to see for themselves... and the secret of the relation of mathematical theory and experiment in inductive reasoning was practically discovered.
There are always men of thought and men of action; mathematical physics is the product of an age which combined in the same men impulses to thought with impulses to action.
The true Newtonian law takes diametrically the opposite point of view. The state of a body unacted on by force is that of uniform motion in a straight line, and no external force or influence is to be looked for as the cause, or, if you like to put it so, as the invariable accompaniment of this uniform rectilinear motion. Rest is merely a particular case of such motion, merely when the velocity is and remains zero.
[W]hen a body is moving, we do not seek for any external influence except to explain changes in the rate of the velocity or changes in its direction. So long as the body is moving at the same rate and in the same direction there is no need to invoke the aid of any forces.
The difference between the two points of view is well seen by reference to the theory of the motion of the planets. Copernicus... showed how much simpler it was to conceive the planets, including the earth, as revolving round the sun in orbits which are nearly circular; and later, Kepler... in... 1609 proved that... the orbits are practically ellipses...
Newton, therefore, had to search for a force which would bend the motion round into its elliptical orbit. This he showed must be a force directed towards the sun, as in the next figure (5). ...[T]he force is the gravitational attraction of the sun acting according to the law of the inverse square of the distance... above.
[D]ynamics... now claims to be the ultimate science of which the others are but branches. ...that the various qualities of things perceptible to the senses are merely our peculiar mode of appreciating changes in position on the part of things existing in space.
[A]ccording to modern science, heat is nothing but the agitation of the molecules of a body. ...[S]ound is nothing but the result of motions of the air striking on the drum of the ear.
[T]he endeavour to give a dynamical explanation of phenomena is the attempt to explain them by statements of the general form, that such and such a substance or body was in this place and is now in that place.
[W]e arrive at the great basal idea of modern science, that all our sensations are the result of comparisons of the changed configurations of things in space at various times. It follows, therefore, that the laws of motion, that is, the laws of the changes of configurations of things, are the ultimate laws of physical science.
In the application of mathematics to the investigation of natural philosophy, science does systematically what ordinary thought does casually.
When we talk of a chair, we usually mean something which we have been seeing or feeling... [I]n mathematical physics the opposite course is taken. The chair is conceived without any reference to... modes of perception. ...[T]he chair becomes in thought a set of molecules in space, or a group of electrons, a portion of the ether in motion, or however the current scientific ideas describe it.
[S]cience reduces the chair to things moving in space and influencing each other's motions.
[T]he various elements or factors which enter into a set of circumstances... are merely... lengths of lines, sizes of angles, areas, and volumes, by which the positions of bodies in space can be settled.
[T]he fact of motion and change necessitates the introduction of the rates of changes of such elements, that is to say, velocities, angular velocities, accelerations, and suchlike...
Accordingly, mathematical physics deals with correlations between variable numbers which are supposed to represent the correlations which exist in nature between the measures of these geometrical elements and of their rates of change.
[A]ways the mathematical laws deal with variables, and it is only in the occasional testing of the laws by reference to experiments, or in the use of the laws for special predictions, that definite numbers are substituted.
[T]he events of such an abstract world are sufficient to "explain" our sensations.
When we hear a sound, the molecules of the air have been agitated in a certain way: given the agitation, or air-waves as they are called, all normal people hear sound; and if there are no air-waves, there is no sound.
Our very thoughts appear to correspond to conformations and motions of the brain; injure the brain and you injure the thoughts.
[T]he events of this physical universe succeed each other according to the mathematical laws which ignore all special sensations and thoughts and emotions.
[T]his is the general aspect of the relation of the world of mathematical physics to our emotions, sensations, and thoughts; and a great deal of controversy has been occasioned by it and much ink spilled.
The whole situation has arisen... from the endeavour to describe an external world "explanatory" of our various individual sensations and emotions, but... not essentially dependent upon any particular sensations or upon any particular individual.
[I]f in truth there be such a world, it ought to submit itself to an exact description, which determines accurately its various parts and their mutual relations.
[E]xtension of the notation to decimal fractions was not accomplished till the seventeenth century. Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation.
If any one doubts the utility of symbols, let him write out in full, without any symbol whatever, the whole meaning of some of the fundamental laws of algebra...the commutative and associative laws for addition... and multiplication, and... the distributive law...
[B]y the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain.