"If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. […] The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. We must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed." (David Hilbert, 1900)
"[…] it is an error to believe that rigor in the proof is the enemy of simplicity." (David Hilbert, "Mathematical Problems", [lecture at Paris International Congress] 1900)
"[...] 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", [lecture at Paris International Congress] 1900)
"Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock our efforts. It should be to us a guidepost on the mazy path to hidden truths, and ultimately a reminder of our pleasure in the successful solution." (David Hilbert, "Mathematical Problems", [lecture at Paris International Congress] 1900)
"Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent pan, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts." (David Hilbert, "Mathematical Problems", [lecture at Paris International Congress] 1900)
“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)"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." (David Hilbert, 1902)
“With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and to cast aside some more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.” (David Hilbert, “Mathematical Problems”, Bulletin of the American Mathematical Society Vol. 8, 1902)"Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.” (David Hilbert, “Natur und Mathematisches Erkennen”, 1919–20)
“To reach our goal [of proving consistency], we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves.” (David Hilbert, 1922)
"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; “Die logischen Grundlagen der Mathematik.“ Mathematische Annalen 88 (1), 1923)
“For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers.” (David Hilbert, “Anschauliche Geometrie”, 1932)
“In mathematics […] we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials [….] being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.” (David Hilbert, “Geometry and the imagination”, 1952)
“Mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers.” (David Hilbert, “Geometry and the Imagination”, 1952)
“In topology we are concerned with geometrical facts that do not even involve the concepts of a straight line or plane but only the continuous connectiveness between points of a figure.” (David Hilbert, “Geometry and Imagination”, 1952)
“The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.” (David Hilbert)
“When the answer to a mathematical problem cannot be found, then the reason is frequently that we have not recognized the general idea from which the given problem only appears as a link in a chain of related problems.” (David Hilbert)
“This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by reason, for in mathematics there is no ignorabimus." (David Hilbert)
“The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena." (David Hilbert)
"For, compared with the immense expanse of modern mathematics, what would the wretched remnants mean, the few isolated results incomplete and unrelated, that the intuitionists have obtained." (David Hilbert)
"As long as a branch of science offers an abundance of problems, so long is it alive." (David Hilbert)
"There is nothing more practical than a good theory.” ([attributed to] David Hilbert)
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