"Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar,—we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one."

“Nature consists of the elements given by the senses. Primitive man first takes out of them certain complexes of these elements that present themselves with a certain stability and are most important to him. The first and oldest words are names for "things". … The sensations are no "symbols of things". On the contrary the "thing" is a mental symbol for a sensation-complex of relative stability. Not the things, the bodies, but colours, sounds, pressures, times are the true elements of the world.”

“Personally, people know themselves very poorly.”

“I see the expression of... economy clearly in the gradual reduction of the statical laws of machines to a single one, viz. , the principle of virtual work: in the replacement of Kepler's laws by Newton's single law... and in the reduction, simplification and clarification of the laws of dynamics. I see clearly the biological-economical adaptation of ideas, which takes place by the principles of continuity and of adequate definition and splits the concept 'heat' into the two concepts of 'temperature' and 'quantity of heat'; and I see how the concept 'quantity of heat' leads on to 'latent heat', and to the concepts of 'energy' and 'entropy'.”

“There is no problem in all mathematics that cannot be solved by direct counting. But with the present implements of mathematics many operations can be performed in a few minutes which without mathematical methods would take a lifetime.”

“The student of mathematics often finds it hard to throw off the uncomfortable feeling that his science, in the person of his pencil, surpasses him in intelligence,—an impression which the great Euler confessed he often could not get rid of. This feeling finds a sort of justification when we reflect that the majority of the ideas we deal with were conceived by others, often centuries ago. In a great measure it is really the intelligence of other people that confronts us in science.—Mach, Ernst.”

“The mental operation by which one achieves new concepts and which one denotes generally by the inadequate name of induction is not a simple but rather a very complicated process. Above all, it is not a logical process although such processes can be inserted as intermediary and auxiliary links. The principle effort that leads to the discovery of new knowledge is due to abstraction and imagination.”