"[...] it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning." (David Hilbert, "Mathematical Problems", 1900)
"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)
"Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? […] If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism." (Henri Poincaré, "Science and Hypothesis", 1901)
"No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only give us immediately evident truths borrowed from direct intuition; it would only be an intermediary parasite." (Henri Poincaré, "Science and Hypothesis", 1901)
"Syllogistic reasoning remains incapable of adding anything to the data that are given it; the data are reduced to axioms, and that is all we should find in the conclusions." (Henri Poincaré, "Science and Hypothesis", 1901)
"The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinite number of syllogisms. […] We see, then, that in reasoning by recurrence we confine ourselves to the enunciation of the minor of the first syllogism, and the general formula which contains as particular cases all the majors. This unending series of syllogisms is thus reduced to a phrase of a few lines." (Henri Poincaré, "Science and Hypothesis", 1901)
"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)
"Treatises on mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is convention, and what is hypothesis." (Henri Poincaré, "Science and Hypothesis", 1901)
"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)
"By no process of sound reasoning can a conclusion drawn from limited data have more than a limited application." (Jospeph W Mellor, "Higher Mathematics for Students of Chemistry and Physics", 1902)
"We imagine cases, place mental diagrams before our mind's eye, and multiply these cases, until a habit is formed of expecting that always to turn out the case, which has been seen to be the result in all the diagrams. To appeal to such a habit is a very different thing from appealing to any immediate instinct of rationality. That the process of forming a habit of reasoning by the use of diagrams is often performed there is no room for doubt. It is perfectly open to consciousness." (Charles S Peirce, "Fallibility of Reasoning and the Feeling of Rationality", cca. 1902)
"An axiom is a self-evident truth, the statement of which is superfluous to the conclusiveness of the reasoning, and which only serves to show a principle involved in the reasoning. It is generally a truth of observation; such as the assertion that something is true." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)
"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)
"Diagrammatic reasoning is the only really fertile reasoning. If logicians would only embrace this method, we should no longer see attempts to base their science on the fragile foundations of metaphysics or a psychology not based on logical theory; and there would soon be such an advance in logic that every science would feel the benefit of it." (Charles S Peirce, "Prolegomena to an Apology for Pragmaticism", Monist 16(4), 1906)
"The requisites for the axioms are various. They should be simple, in the sense that each axiom should enumerate one and only one statement. The total number of axioms should be few. A set of axioms must be consistent, that is to say, it must not be possible to deduce the contradictory of any axiom from the other axioms. According to the logical 'Law of Contradiction,' a set of entities cannot satisfy inconsistent axioms. Thus the existence theorem for a set of axioms proves their consistency. Seemingly this is the only possible method of proof of consistency. "Alfred N Whitehead, "The axioms of projective geometry, 1906)
"Theory is not solely an economical representation of experimental laws; it is also a classification of these laws. […] theory, by developing the numerous ramifications of the deductive reasoning which connects principles to experimental laws, establishes an order and a classification among these laws. It brings some laws together, closely arranged in the same group; it separates some of the others by placing them in two groups very far apart. Theory gives, so to speak, the table of contents and the chapter headings under which the science to be studied will be methodically divided, and it indicates the laws which are to be arranged under each of these chapters." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)
"The type of reasoning found in mathematics seems thus not only available but essentially interwoven with every inference in non-mathematical reasoning, being always used in one of its two steps ; facility in making the other step, the more difficult one, must be attained through other than purely mathematical training." (Jacob W A Young, "The Teaching of Mathematics", 1907)
“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)
“We should always aim toward the economy of thought. It is not enough to give models for imitation. It must be possible to pass beyond these models and, in place of repeating their reasoning at length each time, to sum this in a few words." (Jules H Poincaré, 1909)
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