An Introduction to Mathematics

„The first acquaintance which most people have with mathematics is through arithmetic. […] The nature of the things is perfectly indifferent, of all things it is true that two and two make four. Thus we write down as the leading characteristic of mathematics that it deals with properties and ideas which are applicable to things just because they are things, and apart from any particular feelings, or emotions, or sensations, in any way connected with them. This is what is meant by calling mathematics an abstract science.”

„It is natural to think that an abstract science cannot be of much importance in the affairs of human life, because it has omitted from its consideration everything of real interest.”

„The object of the solution of the equation is the determination of the unknown.”

„The conclusion of no argument can be more certain than the assumptions from which it starts.”

„The really profound changes in human life all have their ultimate origin in knowledge pursued for its own sake. […] 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 in the West electrical and magnetic phenomena were studied by men who were dominated by abstract theoretic interests.”

„All our sensations are the result of comparisons of the changed configurations of things in space at various times.”

„Our very thoughts appear to correspond to conformations and motions of the brain; injure the brain and you injure the thoughts. Meanwhile the events of this physical universe succeed each other according to the mathematical laws which ignore all special sensations and thoughts and emotions.”

„By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power.”

„Nothing is more incomprehensible than a symbolism which we do not understand.”

„This example shows that, by 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.”

„It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”

„One very important property for symbolism to possess is that it should be concise, so as to be visible at one glance of the eye and to be rapidly written.”

„What the mathematician is seeking is Generality.”

„Any moving body has a certain magnitude of velocity in a certain direction, that is to say, its velocity is a directed magnitude, a vector. Again a force has a certain magnitude and has a definite direction.”

„The whole life of Nature is dominated by the existence of periodic events, that is, by the existence of successive events so analogous to each other that, without any straining of language, they may be termed recurrences of the same event.”

„Men go on groping for centuries, guided merely by a dim instinct and a puzzled curiosity, till at last «some great truth is loosened».”

„Physical science reposes on the main ideas of number, quantity, space, and time.”

„No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature.”

„The genius who has the good fortune to produce the final idea which transforms a whole region of thought, does not necessarily excel all his predecessors who have worked at the preliminary formation of ideas. In considering the history of science, it is both silly and ungrateful to confine our admiration with a gaping wonder to those men who have made the final advances towards a new epoch.”