An Introduction to Mathematics

"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..."

"[T]he question arose as to what are the forces which preserve the planets in this motion. According to the old false view, held by Kepler, the actual velocity itself required preservation by force. Thus he looked for tangential forces, as in the accompanying figure (4). But according to the Newtonian law, apart from some force the planet would move for ever with its existing velocity in a straight line, and thus depart entirely from the sun."

"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."

"The science of mechanics rose among the Greeks from a consideration of the theory of the mechanical advantage obtained by the lever, and also from a consideration of various problems connected with the weights of bodies. It was finally put on its true basis at the end of the sixteenth and during the seventeenth centuries... partly with the view of explaining the theory of falling bodies, but chiefly in order to give a scientific theory of planetary motions."

"[[w:Dynamics (mechanics)|[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."

"[A]according to the laws of motion a force is fully represented by the vector acceleration it produces in a body of given mass. Accordingly, forces will be said to be added when their joint effect is to be reckoned according to the parallelogram law. Hence for the fundamental vectors of science, namely transportations, velocities, and forces, the addition of any two of the same kind is the production of a "resultant" vector according to the rule of the parallelogram law."

"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases... mental power... Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that... [a] whole population... could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility."

"[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."

"Mathematics is often considered a difficult and mysterious science, because of the numerous symbols which it employs. ...[T]echnical terms of any profession or trade are incomprehensible to those who have never been trained to use them. But this is not because they are difficult in themselves. On the contrary, they have invariably been introduced to make things easy. So in mathematics, granted that we are giving... attention to... ideas, the symbolism is invariably an immense simplification. ...[I]t represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other."

"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."

"Science and the Modern World (...SMW...) was one of the first products of Whitehead's professional philosophical career... The book was a popular muse upon philosophical and cultural aspects of science, including mathematics... He used historical examples and situations quite regularly, but the historical background was itself popular, based upon his reading of others rather than personal investigation. ...the second chapter, entitled 'Mathematics as an element in the history of thought' ...Whitehead was influenced by the mathematical logic and philosophy of mathematics that he had developed with Bertrand Russell in Principia Mathematica... at the same time of SMW a second editing [of Principia] was being prepared, by Russell alone... When the first edition was finished, Whitehead had written an article on mathematics for the Encyclopedia Britannica... and a popular book Introduction to Mathematics... and they too left some mark on SMW, especially the chapters of the book on variables, periodicity, and trigonometry. In the intervening years he had worked quite notably on mathematics education..."