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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 today 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 this 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 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 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 one-sided
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! 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 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.2
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 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 finding 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
Euler-Mascheroni constant C, or the existence of an infinite
number of prime numbers of the form 2n + 1. However
unapproachable these problems may seem to us and however 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;3
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. |