"Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us." (Henri Poincaré, "Science and Hypothesis", 1901)
"Let us notice first of all, that every generalization implies in some measure the belief in the unity and simplicity of nature." (Jules H Poincaré, "Science and Hypothesis", 1901)
"Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof." (David Hilbert,"Mathematical Problems", Bulletin of the American Mathematical Society Vol. 8, 1902)
"To undertake the calculation of any probability, and even for that calculation to have any meaning at all, we must admit, as a point of departure, an hypothesis or convention which has always something arbitrary about it. In the choice of this convention we can be guided only by the principle of sufficient reason. Unfortunately, this principle is very vague and very elastic, and in the cursory examination we have just made we have seen it assume different forms. The form under which we meet it most often is the belief in continuity, a belief which it would be difficult to justify by apodeictic reasoning, but without which all science would be impossible. Finally, the problems to which the calculus of probabilities may be applied with profit are those in which the result is independent of the hypothesis made at the outset, provided only that this hypothesis satisfies the condition of continuity." (Henri Poincaré, "Science and Hypothesis", 1901)
"A belief in a proposition is a controlled and contented habit of acting in ways that will be productive of desired results only if the proposition is true; An affirmation is an act of an utterer of a proposition to an interpreter, and consists, in the first place, in the deliberate exercise, in uttering the proposition, of a force tending to determine a belief in it in the mind of the interpreter." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)
"The most violent revolutions in an individual's beliefs leave most of his old order standing. Time and space, cause and effect, nature and history, and one's own biography remain untouched. New truth is always a go-between, a smoother-over of transitions. It marries old opinion to new fact so as ever to show a minimum of jolt, a maximum of continuity." (William James, "What Pragmatism Means", 1907)
"[…] the chief reason in favor of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reasons rather from them, than for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." (Alfred N Whitehead, "Principia Mathematica", 1910)
"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality." (Cassius J Keyser, "The Humanization of the Teaching of Mathematics", 1912)
"It is experience which has given us our first real knowledge of Nature and her laws. It is experience, in the shape of observation and experiment, which has given us the raw material out of which hypothesis and inference have slowly elaborated that richer conception of the material world which constitutes perhaps the chief, and certainly the most characteristic, glory of the modern mind." (Arthur J Balfour, "The Foundations of Belief", 1912)
"The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics." (Cassius J Keyser, Science, New Series, Vol. 35 (904), 1912)
"It is only when one looks not towards the outside, at their utility, but within mathematics itself at the relationship among the unused parts that one sees the other, real face of this science. It is not goal-oriented but uneconomical and passionate… (The mathematician) believes that what he is doing will probably eventually lead to some practical cash value, but this is not what spurs him on; he serves the truth, which is to say his destiny, not its purpose. The result may be economical a thousand times over; what is immanent is a total surrender and a passionate devotion." (Robert Musil, "The Mathematical Man", 1913)
"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 mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another." (Henri Poincaré, 1913)
"It seems rather futile, if such be the normal history of hypothetical models, to inflict on us the labor of learning abstruse hypotheses which continually revamp old metaphysical terms and merely dress them up in new transcendental symbols. It is a valuable exercise to strip hypotheses of their technical phraseology; to change those words which deceive our minds into believing that a clear idea has been conveyed, when, in fact, they have merely been wrenched from any real significance." (Louis T More, "The Limitations of Science", 1915)
"It [science] involves an intelligent and persistent endeavor to revise current beliefs so as to weed out what is erroneous, to add to their accuracy, and, above all, to give them such shape that the dependencies of the various facts upon one another may be as obvious as possible." (John Dewey, "Democracy and Education", 1916)
"Science herself consults her heart when she lays it down that the infinite ascertainment of fact and correction of false belief are the supreme goods for man." (William James, "Selected Papers on Philosophy", 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)
"A hypothesis or theory is clear, decisive, and positive, but it is believed by no one but the man who created it. Experimental findings, on the other hand, are messy, inexact things, which are believed by everyone except the man who did the work." (Harlow Shapley, "Review of Scientific Instruments" Vol. 6, 1922)
"The question whether any branch of science can ever become purely deductive is easily answered. It cannot. If science deals with the external world, as we believe it does, and not merely with the relations of propositions then no branch of science can ever be purely deductive. Deductive reasoning by itself can never tell us about facts. The use of deduction in science is to serve as a calculus to make our observations go further, not to take the place of observation." (Arthur D Ritchie, "Scientific Method: An Inquiry into the Character and Validity of Natural Laws", 1923)
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