1902.] HILBERT : MATHEMATICAL PROBLEMS

1902.] HILBERT : MATHEMATICAL PROBLEMS. 437 node curve is fundamental in this theory. A number of Cremona's theorems on ruled surfaces with straight line directrices are generalized to apply to all ruled surfaces. Br. Wilczynski gives both analytic and synthetic proofs of these theorems. The third covariant furnishes another congruence associated with a given surface, and in particular a third ruled surface associated with the original one and the one already mentioned. A few brief remarks are made, showing how these covariant surfaces may serve to simplify the integration of the original system of differential equations. This paper will be combined with the previous paper on covariants for publication in the Transactions. E. J. WILCZYNSKI. MATHEMATICAL PKOBLEMS.* LECTURE DELIVERED BEFORE THE INTERNATIONAL CONGRESS OF MATHEMATICIANS AT PARIS IN 1900. BY PROFESSOR DAVID HILBERT. W H O of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries ? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive ? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose ? History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of to-day sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future. * Translated for the BULLETIN, with the author's permission, by Dr. MARY WINSTON NEWSON. The original appeared in the Göttinger Nachrichten, 1900, pp. 253-297, and in the Archiv der Mathernatik una Physik, 3dser., vol. 1 (1901), pp. 44-63 and 213-237. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4 3 8 HILBERT : MATHEMATICAL PROBLEMS. [Juty? The deep significance of certain problems for the advance of mathematical science in general and the important rôle which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive ; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel ; he finds new methods and new outlooks, and gains a wider and freer horizon. It is difficult and often impossible to judge the value of a problem correctly in advance ; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: " A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect ; for what is clear and easily comprehended attracts, the complicated repels us. Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution. The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult jpartieular problems with passionate zeal. They knew the value of difficult problems. I remind you only of the " problem of the line of quickest descent/ ' proposed by John Bernoulli. Experience teaches, explains Bernoulli in the public announcement of this problem, that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems, and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne, Pascal, Fermât, Viviani and others and laying before the distinguished analysts of his time a problem by which, as a touchstone, they may test the value of their methods and measure their strength. The calculus of variations owes its origin to this problem of Bernoulli and to similar problems. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT : MATHEMATICAL PROBLEMS. 439 Fermât had asserted, as is well known, that the diophantine equation x n + y n=z n (x, y and z integers) is unsolvable—except in certain selfevident cases. The attempt to prove this impossibility offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. For Kummer, incited by Fermât's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circular field into ideal prime factors—a law which to-day, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions. To speak of a very different region of research, I remind you of the problem of three bodies. The fruitful methods and the far-reaching principles which Poincaré has brought into celestial mechanics and which are to-day recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution. The two last mentioned problems—that of Fermât and the problem of the three bodies—seem to us almost like opposite poles—the formera free invention of pure reason, belonging to the region of abstract number theory, the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature. But it often happens also that the same special problem finds application in the most unlike branches of mathematical knowledge. So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the'theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly has F. Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations. In order to throw light on the importance of certain problems, I may also refer to Weierstrass, who spoke of it as his happy fortune that he found at the outset of his scientific career a problem so important as Jacobins problem of inversion on which to work. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 440 HILBERT : MATHEMATICAL PROBLEMS. [July, Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization, just as the child of to-day learns the application of these laws by empirical methods. The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle ; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential—to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics. But, in the further development of a branch of mathematics, the human mind, encouraged by the success of its solutions, becomes conscious of its independence. It evolves from itself alone, often without appreciable influence from without, by means of logical combination, generalization, specialization, by separating and collecting ideas in fortunate ways, new and fruitful problems, and appears then itself as the real questioner. Thus arose the problem of prime numbers and the other problems of number theory, Galois's theory of equations, the theory of algebraic invariants, the theory of abelian and automorphic functions ; indeed almost all the nicer questions of modern arithmetic and function theory arise in this way. In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new branches of mathematics, and while we seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus at the same time advance most successfully the old theories. And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions, methods and ideas of the various branches of his science, have their origin in thisf ever-recurring interplay between thought and experience. It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT : MATHEMATICAL PROBLEMS. 44 1 problem. I should say first of all, this : that 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. Indeed the requirement of: rigor, which has become proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding ; and, on the other hand, only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full effect. A new problem, especially when it comes from the world of outer experience, is like a young twig, which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem, the established achievements of our mathematical science. Besides it is an error to believe that rigor in the proof 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. It also frequently leads the way to methods which are more capable of development than the old methods of less rigor. Thus the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of the more rigorous function-theoretical methods and the consistent introduction of transcendental devices. Further, the proof that the power series permits the application of the four elementary arithmetical operations as well as the term by term differentiation and integration, and the recognition of the utility of the power series depending upon this proof contributed materially to the simplification of all analysis, particularly of the theory of elimination and the theory of differential equations, and also of the existence proofs demanded in those theories. But the most striking example for my statement is the calculus of variations. The treatment of the first and second variations of definite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations. By the examples of the simple and double integral I will show briefly, at the close of my lecture, how this way leads at once to a surprising simplification of License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 442 HILBERT : MATHEMATICAL PROBLEMS. [July , the calculus of variations. For in the demonstration of the necessary and sufficient criteria for the occurrence of a maximum and minimum, the calculation of the second variation and in part, indeed, the wearisome reasoning connected with the first variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the differential coefficients of the function vary but slightly. While insisting on rigor in the proof as a requirement for a perfect solution of a problem, I should like, on the other hand, to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are susceptible of a fully rigorous treatment. This opinion, occasionally advocated by eminent men, I consider entirely erroneous. Such a onesided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry, mechanics and physics, to a stoppage of the flow of new material from the outside world, and finally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number. But what an important nerve, vital to mathematical science, would be cut by the extirpation of geometry and mathematical physics I On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts. To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts. So the geometrical figures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians. Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea " between''? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation ? Who could dispense with the figure of the triangle, the circle with its center, or with the cross of three perpendicular License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT : MATHEMATICAL PROBLEMS. 443 axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences ? The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas ; and no mathematician could spare these graphic formulas, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs. The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie those figures ; and in order that these geometrical figures may be incorporated in the general treasure of mathematical signs, there is necessary a rigorous axiomatic investigation of their conceptual content. Just as in adding two numbers, one must place the digits under each other in the right order, so that only the rules of calculation, i. e., the axioms of arithmetic, determine the correct use of the digits, so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations. The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski's work, Die Geometrie der Zahlen. * Some remarks upon the difficulties which mathematical problems may offer, and the means of surmounting them, may be in place here. If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which * Leipzig, 1896. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 444 HILBEET: MATHEMATICAL PROBLEMS. [July, is applicable also to related problems. The introduction of complex paths of integration by Cauchy and of the notion of the IDEALS in number theory by Kummer may serve as examples. This way for fin ding general methods is certainly the most practicable and the most certain ; for he who seeks for methods without having a definite problem in mind seeks for the most part in vain. In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends, then, on finding out these easier problems, and on solving them by means of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties and it seems to me that it is used almost always, though perhaps unconsciously. 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 right triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, 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 the 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. Take any definite unsolved problem, such as the question as to the irrationality of the EulerMascheroni constant C, or the existence of an infinite number of prime numbers of the form 2W+ 1. However unapproachable these problems may seem to us and howLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT : MATHEMATICAL PROBLEMS. 44 5 ever helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes. Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable ? For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility. I instance the problem of perpetual motion. After seeking in vain for the construction of a perpetual motion machine, the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible ; * and this inverted question led to the discovery of the law of the conservation of energy, which, again, explained the impossibility of perpetual motion in the sense originally intended. 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 pure reason, for in mathematics there is no ignorabimus. The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected. Let us look at the principles of analysis and geometry. The most suggestive and notable achievements of the last century in this field are, as it seems to me, the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-euclidean geometry by Gauss, Bolyai, and Lobachevsky. I therefore first direct your attention to some problems belonging to these fields. 1. CANTOR'S PROBLEM OF THE CARDINAL NUMBER OF THE CONTINUUM. Two systems, i. e , two assemblages of ordinary real numbers or points, are said to be (according to Cantor) equivalent or of equal cardinal number, if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one defi- *See Helmholtz, " Ueber die Weohselwirkung der Naturkrâefte iiüd die darauf bezüglichen neuesten Ermittelungen der Physik"; Vortrag, gehalten in Königsberg, 1854. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4 4 6 HILBERT : MATHEMATICAL PROBLEMS. [July , nite number of the other. The investigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving. This is the theorem : Every system of infinitely many real numbers, i. e., every assemblage of numbers (or points), is either equivalent to the assemblage of natural integers, 1, 2, 3,.•• or to the assemblage of all real numbers and therefore to the continuum, that is, to the points of a line ; as regards equivalence there are, therefore, only two assemblages of numbers, the countable assemblage and the continuum. From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage ; the proof of this theorem would, therefore, form a new bridge between the countable assemblage and the continuum. Let me mention another very remarkable statement of Cantor's which stands in the closest connection with the theorem mentioned and which, perhaps, offers the key to its proof. Any system of real numbers is said to be ordered, if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that, if a is before 6 and b is before c, then a always comes before c. The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger. But there are, as is easily seen, infinitely many other ways in which the numbers of a system may be arranged. If we think of a definite arrangement of numbers and select from them a particular system of these numbers, a so-called partial system or assemblage, this partial system will also prove to be ordered. Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way, that not only in the assemblage itself but also in every partial assemblage there exists a first number. The system of integers 1, 2, 3, ••• in their natural order is evidently a well ordered assemblage. On the other hand the system of all real numbers, L e., the continuum in its natural order, is evidently not well ordered. For, if we think of the points of a segment of a straight line, with its initial point excluded, as our partial assemblage, it will have no first element. The question now arises whether the totality of all numbers may not be arranged in another manner so that every License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT : MATHEMATICAL PROBLEMS. 447 partial assemblage may have a first element, L e., whether the continuum cannot be considered as a well ordered assemblage—a question which Cantor thinks must be answered in the affirmative. It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor's, perhaps by actually giving an arrangement of numbers such that in every partial system a first number can be pointed out. 2. TH E COMPATIBILITY OF THE ARITHMETICAL AXIOMS. "When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas ; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. Upon closer consideration the question arises : Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another. But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms : To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results. In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. Any contradiction in the deductions from thegeometrical axioms mustthereupon be recognizable in the arithmetic of this field of numbers. In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms. On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms. The axioms of arithmetic are essentially nothing else than the known rules of calculation, with the addition of the axiom of continuity. I recently collected them * and in so doing * Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 8 (1900), p. 180. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 448 HILBERT : MATHEMATICAL PROBLEMS. [July, replaced the axiom of continuity by two simpler axioms, namely, the well-known axiom of Archimedes, and a new axiom essentially as follows : that numbers form a system of things which is capable of no further extension, as long as all the other axioms hold (axiom of completeness). I am convinced that it must be possible to find a direct proof for the compatibility of the arithmetical axioms, by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers. To show the significance of the problem from another point of view, I add the following observation : If contradictory attributes be assigned to a concept, I say, that mathematically the concept oes not exist So, for example, a real number whose square is — 1 does not exist mathematically. But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of a finite number of logical processes, I say that the mathematical existence of the concept (for example, of a number or a function which satisfies certain conditions) is thereby proved. In the case before us, where we are concerned with the axioms of real numbers in arithmetic, the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum. Indeed, when the proof for the compatibility of the axioms shall be fully accomplished, the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless. The totality of real numbers, i. e., the continuum according to the point of view just indicated, is not the totality of all possible series in decimal fractions, or of all possible laws according to which the elements of a fundamental sequence may proceed. It is rather a system of things whose mutual relations are governed by the axioms set up and for which all propositions, and only those, are true which can be derived from the axioms by a finite number of logical processes. In my opinion, the concept of the continuum is strictly logically tenable in this sense only. It seems to me, indeed, that this corresponds best also to what experience and intuition tell us. The concept of the continuum or even that of the system of all functions exists, then, in exactly the same sense as the system of integral, rational numbers, for example, or as Cantor's higher classes of numbers and cardinal numbers. For I am convinced that the existence of the latter, just as that of the continuum, can be proved in License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT I MATHEMATICAL PROBLEMS. 449 the sense I have described ; unlike the system of all cardinal numbers or of all Cantor's alephs, for which, as may be shown, a system of axioms, compatible in my sense, cannot be set up. Either of these systems is, therefore, according to my terminology, mathematically non-existent. From the field of the foundations of geometry I should like to mention the following problem : 3. TH E EQUALITY OF THE VOLUMES OF Two TETRAHEDRA OF EQUAL BASES AND EQUAL ALTITUDES. In two letters to Gerling, Gauss * expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i. e., in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved.f Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into con* gruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra. % 4. PROBLEM OF THE STRAIGHT LINE AS THE SHORTEST DISTANCE BETWEEN TWO POINTS. Another problem relating to the foundations of geometry is this : If from among the axioms necessary to establish ordinary euclidean geometry, we exclude the axiom of parallels, or assume it as not satisfied, but retain all other axioms, we obtain, as is well known, the geometry of Lobachevsky (hyperbolic geometry). We may therefore say that this is a geometry standing next to * Werke, vol. 8, pp. 241 and 244. f Cf., beside earlier literature, Hubert, Grundlagen der Geometrie, Leipzig, 1899, eh. 4. [Translation by Townsend, Chicago, 1902.] % Since this was written Herr Dehn has succeeded in proving this impossibility. See his note: "Ueber raumgleiche Polyeder," in Nachrichten d. K. Gesellsch. d. Wiss. zu Q'ôttingen, 1900, and a paper soon to appear in the Math. Annalen [vol. 55, pp. 465-478], License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 450 HILBEBT: MATHEMATICAL PROBLEMS. [July, euclidean geometry. If we require further that that axiom be not satisfied whereby, of three points of a straight line, one and only one lies between the other two, we obtain Biemann's (elliptic) geometry, so that this geometry appears to be the next after Lobachevsky's. If we wish to carry out a similar investigation with respect to the axiom of Archimedes, we must look upon this as not satisfied, and we arrive thereby at the non-archimedean geometries which have been investigated by Veronese and myself. The more general question now arises : Whether from other suggestive standpoints geometries may not be devised which, with equal right, stand next to euclidean geometry. Here I should like to direct your attention to a theorem which has, indeed, been employed by many authors as a definition of a straight line, viz., that the straight line is the shortest distance between two points. The essential content of this statement reduces to the theorem of Euclid that in a triangle the sum of two sides is always greater than the third side—a theorem which, as is easily seen, deals solely with elementary concepts, i. e., with such as are derived directly from the axioms, and is therefore more accessible to logical investigation. Euclid proved this theorem, with the help of the theorem of the exterior angle, on the basis of the congruence theorems. Now it is readily shown that this theorem of Euclid cannot be proved solely on the basis of those congruence theorems which relate to the application of segments and angles, but that one of the theorems on the congruence of triangles is necessary. We are asking, then, for a geometry in which all the axioms of ordinary euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom. One finds that such a geometry really exists and is no other than that which Minkowski constructed in his book, Geometrie der Zahlen,* and made the basis of his arithmetical investigations. Minkowski's is therefore also a geometry standing next to the ordinary euclidean geometry; it is essentially characterized by the following stipulations : 1. The points which are at equal distances from a fixed point 0 lie on a convex closed surface of the ordinary euclidean space with 0 as a center. * Leipzig, 1896. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT I MATHEMATICAL PROBLEMS. 45 1 2. Two segments are said to be equal when one can be carried into the other by a translation of the ordinary euclidean space. In Minkowski's geometry the axiom of parallels also holds. By studying the theorem of the straight line as the shortest distance between two points, I arrived* at a geometry in which the parallel axiom does not hold, while all other axioms of Minkowski7 s geometry are satisfied. The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e. g.9 upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seem to me desirable. 5. LIE'S CONCEPT OF A CONTINUOUS GROUP OF TRANSFORMATIONS WITHOUT THE ASSUMPTION OF THE DIFFERENTIABILITY OF THE FUNCTIONS DEFINING THE GROUP. It is well known that Lie, with the aid of the concept of continuous groups of transformations, has set up a system of geometrical axioms and, from the standpoint of his theory of groups, has proved that this system of axioms suffices for geometry. But since Lie assumes, in the very foundation of his theory, that the functions defining his group can be differentiated, it remains undecided in Lie's development, whether the assumption of the differentiability in connection with the question as to the axioms of geometry is actually unavoidable, or whether it may not appear rather as a consequence of the group concept and the other geometrical axioms. This consideration, as well as certain other problems in connection with the arithmetical axioms, brings before us the more general question : How far Lie1 s concept of continuous groups of transformations is approachable in our investigations without the assumption of the differentiability of the functions. * Math. Annalen, vol. 46, p. 91. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 452 HILBERT : MATHEMATICAL PROBLEMS. [Juty, Lie defines a finite continuous group of transformations as a system of transformations x ! =fi(x v -> x n ; a v -> ar) (* = h - , *0 having the property that any two arbitrarily chosen transformations of the system, as X i :=fi{XV '"•) X n 5 aV "*> a r)? applied sucessively result in a transformation which also belongs to the system, and which is therefore expressible in the form */W«{/i(*><0> ->ƒ„(>> a ) ; 6i> -> M = /i(% -^ w 5 ci> "*>c r)> where cv •••, cr are certain functions of av •••, ar and 6X, •••, 6r . The group property thus finds its full expression in a system of functional equations and of itself imposes no additional restrictions upon the functions fv —,/w; ov •••, cr . Yet Lie's further treatment of these functional equations, viz., the derivation of the well-known fundamental differential equations, assumes necessarily the continuity and differentiability of the functions defining the group. As regards continuity : this postulate will certainly be retained for the present—if only with a view to the geometrical and arithmetical applications, in which the continuity of the functions in question appears as a consequence of the axiom of continuity. On the other hand the differentiability of the functions defining the group contains a postulate which, in the geometrical axioms, can be expressed only in a rather forced and complicated manner. Hence there arises the question whether, through the introduction of suitable new variables and parameters, the group can always be transformed into one whose defining functions are differentiable; or whether, at least with the help of certain simple assumptions, a transformation is possible into groups admitting Lie's methods. A reduction to analytic groups is, according to a theorem announced by Lie * but first proved by Schur,f always possible when the group is transitive and the existence of the first and certain second derivatives of the functions defining the group is assumed. * Lie-Engel, Theorie der Transformationsgruppen, vol. 3, Leipzig, 1893, §§82, 144. t " Ueber den analytischen Charakter der eine endliche Kontinuierliehe Transformationsgruppen darstellenden Funktionen," Math. Annalen, vol. 41. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT : MATHEMATICAL PROBLEMS. 453 For infinite groups the investigation of the corresponding question is, I believe, also of interest. Moreover we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel* with so much ingenuity, the difference equations, and other equations occurring in the literature of mathematics, do not directly involve anything which necessitates the requirement of the differentiability of the accompanying functions. In the search for certain existence proofs in the calculus of variations I came directly upon the problem : To prove the differentiability of the function under consideration from the existence of a difference equation. In all these cases, then, the problem arises : In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption f It may be further remarked that H. Minkowski in his above-mentioned Geometrie der Zahlen starts with the functional equation M + VV "'J Xn + Vn) =/Ol > '"J Xn) + KVv ~> Vn) and from this actually succeeds in proving the existence of certain differential quotients for the function in question. On the other hand I wish to emphasize the fact that there certainly exist analytical functional equations whose sole solutions are non-differentiable functions. For example a uniform continuous non-differentiable function

i-4)+*(4-4)-<>- where X, F, and Z are rational integral homogeneous functions of the nth degree in x, y, z, and the latter are to be determined as functions of the parameter t. 17. EXPRESSION OF DEFINITE FORMS BY SQUARES. A rational integral function or form in any number of variables with real coefficients such that it becomes negative for no real values of these variables, is said to be definite. The system of all definite forms is invariant with respect to the operations of addition and multiplication, but the *Cf. Eohn, ''Flachen vierter Ordnung," Preisschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig, 1886. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4 6 6 HILBERT : MATHEMATICAL PROBLEMS. [July, quotient of two definite forms—in case it should be an integral function of the variables—is also a definite form. The square of any form is evidently always a definite form. But since, as I have shown,* not every definite form can be compounded by addition from squares of forms, the question arises—which I have answered affirmatively for ternary forms f—whether every definite form may not be expressed as a quotient of sums of squares of forms. At the same time it is desirable, for certain questions as to the possibility of certain geometrical constructions, to know whether the coefficients of the forms to be used in the expression may always be taken from the realm of rationality given by the coefficients of the form represented, x I mention one more geometrical problem : 18. BUILDING UP OF SPACE FROM CONGRUENT POLYHEDRA. If we enquire for those groups of motions in the plane for which a fundamental region exists, we obtain various answers, according as the plane considered is Riemann's (elliptic), Euclid's, orLobachevsky's (hyperbolic). In the case of the elliptic plane there is a finite number of essentially different kinds of fundamental regions, and a finite number of congruent regions suffices for a complete covering of the whole plane ; the group consists indeed of a finite number of motions only. In the case of the hyperbolic plane there is an infinite number of essentially different kinds of fundamental regions, namely, the well-known Poincaré polygons. For the complete covering of the plane an infinite number of congruent regions is necessary. The case of Euclid's plane stands between these ; for in this case there is only a finite number of essentially different kinds of groups of motions with fundamental regions, but for a complete covering of the whole plane an infinite number of congruent regions is necessary. Exactly the corresponding facts are found in space of three dimensions. The fact of the finiteness of the groups of motions in elliptic space is an immediate consequence of a fundamental theorem of C. Jordan, § whereby the number of * Math. Annalen, vol. 32. f Acta Mathematica, vol. 17. % Cf. Hilbert : Grundlagen der Geometrie, Leipzig, 1899, Chap. 7 and in particular \ 38. \ Crelle's Journal, vol. 84 (1878), and Atti d. Reale Acad, di Napoli 1880. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT: MATHEMATICAL PROBLEMS. 467 essentially different kinds of finite groups of linear substitutions in n variables does not surpass a certain finite limit dependent upon n. The groups of motions with fundamental regions in hyperbolic space have been investigated by Fricke and Klein in the lectures on the theory of automorphic functions,* and finally Fedorov,f Schoenflies % and lately Rohn § have given the proof that there are, in euclidean space, only a finite number of essentially different kinds of groups of motions with a fundamental region. Now, while the results and methods of proof applicable to elliptic and hyperbolic space hold directly for ^-dimensional space also, the generalization of the theorem for euclidean space seems to offer decided difficulties. The investigation of the following question is therefore desirable : Is there in n-dimensional euclidean space also only a finite number of essentially different kinds of groups of motions with a fundamental region f A fundamental region of each group of motions, together with the congruent regions arising from the group, evidently fills up space completely. The question arises : Whether polyhedra also exist which do not appear as fundamental regions oj groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling up of all space is possible. I point out the following question, related to the preceding one, and important to number theory and perhaps sometimes useful to physics and chemistry: How can one arrange most densely in space an infinite number of equal solids of given form, e. o F itself being an analytic function. W-e shall call this sort of problem a regular variation problem. It is chiefly the regular variation problems that play a rôle in geometry, in mechanics, and in mathematical physics ; and the question naturally arises, whether all solutions of regular variation problems must necessarily be analytic functions. In other words, does every lagrangian partial differential equation of a regular variation problem have the property of admitting analytic integrals exclusively t And is this the case even when the function is constrained to assume, as, e. F *+(&-*)«;} d x = Ç\Fyôy + ôyxFp + (y. -jp ) àF,\ dx = 3J+ Ç\y,-p)àF,dx. We derive from this, namely, the following facts : If we construct any simple family of integral curves of the ordinary differential equation (1) of the second order and then form an ordinary differential equation of the first order ( 2 ) yx = p(%,y) which also admits these integral curves as solutions, then the function p(x, y) is always an integral of the partial differential equation (1*) of the first order ; and conversely, if p(x, y) denotes any solution of the partial differential equation (1*) of the first order, all the non-singular integrals of the ordinary differential equation (2) of the first order are at the same time integrals of the differential equation (1) of the second order, or in short if yx= s p(xJ y) is an integral equation of the first order of the differential equation (1) of the second order, p(x, y) represents an License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT : MATHEMATICAL PROBLEMS, 47 5 integral of the partial differential equation (1*) and conversely ; the integral curves of the ordinary differential equation of the second order are therefore, at the same time, the characteristics of the partial differential equation (1*) of the first order. In the present case we may find the same result by means of a simple calculation ; for this gives us the differential equations (1) and (1*) in question in the form (1} vF 4- v F 4- F —7^= 0 v y ifxx yxyx » iJx yxy • y^x y v ? (1*) ( A + PPy) F„ + pFpy + Fpx - Fy = 0, where the lower indices indicate the partial derivatives with respect to x, y, p, yx. The correctness of the affirmed relation is clear from this. The close relation derived before and just proved between the ordinary differential equation (1) of the second order and the partial differential equation (1*) of the first order, is, as it seems to me, of fundamental significance for the calculus of variations. For, from the fact that the integral J* is independent of the path of integration it follows that (3) JTV(P) + (yx-p)Fp(p)\dx= £FQx)dx, if we think of the left hand integral as taken along any path y and the right hand integral along an integral curve y of the differential equation ya = 2>0&, y) . With the help of equation (3) we arrive at Weierstrass's formula (*) ÇhF(yx)dx - ÇFÇyx)dx = f £( y> *> F(yx, P) = F{yx) - F{p) - (y. - p) Fp(p). Since, therefore, the solution depends only on finding an integral p (x, y) which is single valued and continuous in a certain neighborhood of the integral curve y, which we are considering, the developments just indicated lead immediately —without the introduction of the second variation, but only License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4 7 6 HILBERT : MATHEMATICAL PROBLEMS. [July , by the application of the polar process to the differential equation (1)—to the expression of Jacobi's condition and to the answer to the question : How far this condition of Jacobi's in conjunction with Weierstrass's condition JEJ> 0 is necessary and sufficient for the occurrence of a minimum. The developments indicated may be transferred without necessitating further calculation to the case of two or more required functions, and also to the case of a double or a multiple integral. So, for example, in the case of a double integral J=JF(zx,zy,z;x,y)d«,, [«.-£, *,= | ] to be extended over a given region w, the vanishing of the first variation (to be understood in the usual sense) <5JT=0 gives the well-known differential equation of the second order dFz dFg r F - dF F - d F F-dFl for the required function « of a; and y. On the other hand we consider the integral J* = ƒ \F + (zx-p)Fp + (s, - q)Ft \ da,, [F=F(P,q,z;X,y),FP = dF^**>*>y\ „ _dF(p,q,z;x,y)1 dq J' and inquire, how p and q are to be taken as functions of x, y and z in order that the value of this integral may be independent of the choice of the surface passing through the given closed twisted curve, i. e., of the choice of the function z of the variables x and y. The integral J * has the form License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] HILBERT : MATHEMATICAL PROBLEMS. 477 and the vanishing of the first variation <5J* = 0, in the sense which the new formulation of the question demands, gives the equation dx dy dz ' i. e., we find for the functions p and q of the three variables x, y and z the differential equation of the first order dFp 3Fq d(pF, + gFt-F) = Q^ dx dy dx If we add to this differential equation the partial differential equation (I*) P* + 2P* — ?. + «fo resulting from the equations the partial differential equation (I) for the function z of the two variables x and y and the simultaneous system of the two partial differential equations of the first order (I*) for the two functions p and q of the three variables x, y, and z stand toward one another in a relation exactly analogous to that in which the differential equations (1) and (1 *) stood in the case of the simple integral. It follows from the fact that the integral J * is independent of the choice of the surface of integration z that f\F(p, q) + (*u-p)F,(p, q) + (zy-q)Fq(p, q)}du> if we think of the right hand integral as taken over an integral surface i of the partial differential equations and with the help of this formula we arrive at once at the formula License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4 7 8 HILBERT : MATHEMATICAL PROBLEMS. [July , (IV) fF(zx, ss)d - ƒ E( [£(>„, %„ p, q) = F(zx, zy) - F(p, q) - (zx -p)Fp(p, q) -(*y-q)Fq(p,q)], which plays the same rôle for the variation of double integrals as the previously given formula (4) for simple integrals. With the help of this formula we can now answer the question how far Jacobrs condition in conjunction with Weierstrass's condition U > 0 is necessary and sufficient for the occurrence of a minimum. Connected with these developments is the modified form in which A. Kneser,* beginning from other points of view, has presented Weierstrass'stheory. While Weierstrass employed to derive sufficient conditions for the extreme tehos integral curves of equation (1) which pass through a fixed point, Kneser on the other hand makes use of any simple family of such curves and constructs for every such family a solution, characteristic for that family, of that partial differential equation which is to be considered as a generalization of the Jacobi-Hamilton equation. The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of to-day is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly. *Cf. his above-mentioned textbook, J J 14, 15, 19 and 20. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1902.] REPLY TO MU. COOLIDGE. 479 But, we ask, 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 cast aside older 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. 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. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples. REPLY TO MR. J. L. COOLIDGE'S REVIEW OF HILL'S EUCLID. I DESIRE to thank the editors of the BULLETIN for their courtesy in acceding to my request that they should insert a reply to the review of my edition of the fifth and sixth books of Euclid's Elements by Mr. Coolidge, published in the February number of the BULLETIN, as it contains statements which give an erroneous impression of the contents of the book. The book differs from previous editions in two important particulars. These are : 1. The explanations of the fundamental definitions of the fifth book of Euclid. 2. The removal of the indirectness from Euclid's line of argument. The second of these matters, though emphasized by italics on page viii of the preface, has been passed over without notice by the reviewer. The discovery of this indirectness and the possibility of removing it, were published by me in the Cambridge Philosophical Transactions, volume 16, part 4 ; and the importance of the work was recognized in the review of that paper in the Jahrbuch uber die Fortschritte der Mathematik, volume 28 (1897), page 152. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use