On Intuition (1900-1925)

"New ideas emerge dimly into intuition, come into consciousness from nobody knows where, and become the material on which the mind operates, forging them gradually into consistent doctrine, which can be welded on to existing domains of knowledge. But this process is never complete: a crude connection can always be pointed to by a logician as an indication of the imperfection of human constructions." (Henri Poincaré, "Science and Hypothesis", 1902)

"The chief end of mathematical instruction is to develop certain powers of the mind, and among  these the intuition is not the least precious. By it the mathematical world comes in contact with the real world, and even if pure mathematics could do without it, it would always be necessary to turn to it to bridge the gulf between symbol and reality. The practician will always need it, and for one mathematician there are a hundred practicians. However, for the mathematician himself the power is necessary, for while we demonstrate by logic, we create by intuition; and we have more to do than to criticize others’ theorems, we must invent new ones, this art, intuition teaches us.” (Jules H Poincaré, "The Value of Science", 1905)

“We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under this same word?” (Henri Poincaré , ”Intuition and Logic in Mathematics”, 1905)

"First [...] a new theory is attacked as absurd; then it is admitted to be true, but obvious and insignificant; finally it is seen to be so important that its adversaries claim they themselves discovered it." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1907)

 "Theories thus become instruments, not answers to enigmas, in which we can rest. We don't lie back upon them, we move forward, and, on occasion, make nature over again by their aid." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1907)

“It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.” (Henri Poincaré, "Science and Method", 1908)

"I do in no wise share this view [that the axioms are arbitrary propositions which we assume wholly at will, and that in like manner the fundamental conceptions are in the end only arbitrary symbols with which we operate] but consider it the death of all science: in my judgment the axioms of geometry are not arbitrary, but reasonable propositions which generally have the origin in space intuition and whose separate content and sequence is controlled by reasons of expediency." (Felix Klein, "Elementarmathematik vom hoheren Standpunkte aus", 1909)

“Mathematics is merely a shorthand method of recording physical intuition and physical reasoning, but it should not be a formalism leading from nowhere to nowhere, as it is likely to be made by one who does not realize its purpose as a tool.” (Charles P Steinmetz, “Transactions of the American Institute of Electrical Engineers”, 1909)

"Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition." (Graham Fitch, "The Fourth Dimension simply Explained", 1910)

“Knowledge is the distilled essence of our intuitions, corroborated by experience.” (Elbert Hubbard, “A Thousand & One Epigrams, 1911)

“In fact, the opposition of instinct and reason is mainly illusory. Instinct, intuition, or insight is what first leads to the beliefs which subsequent reason confirms or confutes; but the confirmation, where it is possible, consists, in the last analysis, of agreement with other beliefs no less instinctive. Reason is a harmonising, controlling force rather than a creative one. Even in the most purely logical realms, it is insight that first arrives at what is new.“ (Bertrand Russell, “Our Knowledge of the External World”, 1914)

"The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them."(Albert Einstein, “Principles of Research”, 1918)

“We must have a logical intuition of the probable relations between propositions. Once the existence of this relation between evidence and conclusion, the latter becomes the subject of the degree of belief.” (John M Keynes, “Treatise on Probability”, 1921)

"By intuition is frequently understood perception, or the knowledge of actual reality, the apprehension of something as real. […] Intuition is the undifferentiated unity of the perception of the real and of the simple image of the possible. " (Benedetto Croce, “The Essence of Æsthetic”, 1921)