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- Herausgeber: Science & Technology Publishing
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Mathematical Problems is a seminal work by David Hilbert, one of the most influential mathematicians of the late 19th and early 20th centuries. Originally presented as a lecture at the International Congress of Mathematicians in Paris in 1900, this book outlines 23 unsolved problems that Hilbert believed would shape the future of mathematical research. The problems span a wide range of mathematical disciplines, including number theory, algebra, geometry, analysis, and mathematical logic, reflecting Hilbert’s deep understanding of the interconnectedness of mathematical ideas. In this work, Hilbert not only presents the problems themselves but also discusses their significance, the progress made up to his time, and the potential impact their solutions could have on the broader field of mathematics. His clear and visionary exposition inspired generations of mathematicians, setting the agenda for much of 20th-century mathematical research. Many of the problems have since been solved, while others remain open, continuing to challenge and inspire mathematicians today. Mathematical Problems is more than a list of questions; it is a profound reflection on the nature of mathematical inquiry and the pursuit of knowledge. Hilbert’s optimism about the solvability of mathematical problems and his belief in the unity and universality of mathematics are evident throughout the text. This book is essential reading for anyone interested in the history of mathematics, the development of mathematical thought, and the enduring quest to unravel the mysteries of the mathematical universe.
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Seitenzahl: 78
Veröffentlichungsjahr: 2025
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BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
CONTINUATION OF THE BULLETIN OF THE NEW YORK MATHEMATICAL SOCIETY.
A HISTORICAL AND CRITICAL REVIEW OF MATHEMATICAL SCIENCE
EDITED BY F.N. COLE, ALEXANDER ZIWET, F. MORLEY, E.O. LOVETT, C.L. BOUTON, D.E. SMITH.
VOL. VIII.
OCTOBER 1901 TO JULY 1902.
PUBLISHED FOR THE SOCIETYBY THE MACMILLAN COMPANY,LANCASTER, PA., AND NEW YORK,1902.
MATHEMATICAL PROBLEMS
LECTURE DELIVERED BEFORE THE INTERNATIONAL CONGRESS OF MATHEMATICIANS AT PARIS IN 1900.
CONTENTS
PROBLEM
PAGE
1.
CANTOR'S PROBLEM OF THE CARDINAL NUMBER OF THE CONTINUUM.
10
2.
THE COMPATIBILITY OF THE ARITHMETICAL AXIOMS.
11
3.
THE EQUALITY OF THE VOLUMES OF TWO TETRAHEDRA OF EQUAL BASES AND EQUAL ALTITUDES.
13
4.
PROBLEM OF THE STRAIGHT LINE AS THE SHORTEST DISTANCE BETWEEN TWO POINTS.
13
5.
LIE'S CONCEPT OF A CONTINUOUS GROUP OF TRANSFORMATIONS WITHOUT THE ASSUMPTION OF THE DIFFERENTIABILITY OF THE FUNCTIONS DEFINING THE GROUP.
15
6.
MATHEMATICAL TREATMENT OF THE AXIOMS OF PHYSICS.
18
7.
IRRATIONALITY AND TRANSCENDENCE OF CERTAIN NUMBERS.
19
8.
PROBLEMS OF PRIME NUMBERS.
20
9.
PROOF OF THE MOST GENERAL LAW OF RECIPROCITY IN ANY NUMBER FIELD.
21
10.
DETERMINATION OF THE SOLVABILITY OF A DIOPHANTINE EQUATION.
22
11.
QUADRATIC FORMS WITH ANY ALGEBRAIC NUMERICAL COEFFICIENTS.
22
12.
EXTENSION OF KRONECKER'S THEOREM ON ABELIAN FIELDS TO ANY ALGEBRAIC REALM OF RATIONALITY.
22
13.
IMPOSSIBILITY OF THE SOLUTION OF THE GENERAL EQUATION OF THE 7TH DEGREE BY MEANS OF FUNCTIONS OF ONLY TWO ARGUMENTS.
25
14.
PROOF OF THE FINITENESS OF CERTAIN COMPLETE SYSTEMS OF FUNCTIONS.
26
15.
RIGOROUS FOUNDATION OF SCHUBERT'S ENUMERATIVE CALCULUS.
28
16.
PROBLEM OF THE TOPOLOGY OF ALGEBRAIC CURVES AND SURFACES.
28
17.
EXPRESSION OF DEFINITE FORMS BY SQUARES.
29
18.
BUILDING UP OF SPACE FROM CONGRUENT POLYHEDRA.
30
19.
ARE THE SOLUTIONS OF REGULAR PROBLEMS IN THE CALCULUS OF VARIATIONS ALWAYS NECESSARILY ANALYTIC?
32
20.
THE GENERAL PROBLEM OF BOUNDARY VALUES.
34
21.
PROOF OF THE EXISTENCE OF LINEAR DIFFERENTIAL EQUATIONS HAVING A PRESCRIBED MONODROMIC GROUP.
34
22.
MATHEMATICAL PROBLEMS[1]
Who 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.
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 particular 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, Fermat, 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.
Fermat had asserted, as is well known, that the diophantine equation () is unsolvable—except in certain self-evident 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 Fermat'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 Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a 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 Jacobi's problem of inversion on which to work.
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.
