How do mathematicians think and solve problems
Who of us would not like to lift the veil under which the future lies, to glimpse the impending advances in our science and the mysteries of its development over the centuries to come! What special goals will the leading mathematical minds of the coming generations pursue? What new methods and new facts will the new centuries discover - in the wide and rich field of mathematical thought?
History teaches the continuity of the development of science. We know that every age has its own problems which the coming age solves or pushes aside as sterile and replaces them with new problems. If we want to get an idea of the presumed development of mathematical knowledge in the near future, we have to let the open questions pass before our minds and survey the problems that current science poses and whose solution we expect from the future. Today, at the turn of the century, seems to me well suited for such a survey of the problems; because the great periods of time not only ask us to look back into the past, but they also direct our thoughts to the unknown to come.
The great importance of certain problems in the advancement of mathematical science in general, and the important role they play in the work of the individual researcher, is undeniable. As long as there is an abundance of problems in a branch of knowledge, it is vital; Lack of problems means death or cessation of independent development. Just as every human company pursues goals, mathematical research needs problems. Solving problems increases the strength of the researcher; he finds new methods and perspectives, he gains a broader and freer horizon.
It is difficult, and often impossible, to properly assess the value of a problem beforehand; after all, the profit that science owes to the problem is decisive. Still, we can ask whether there are general features that characterize a good math problem.
An old French mathematician said: A mathematical theory cannot be considered perfect until you have made it clear enough to explain it to the first man you meet on the street. This clarity and easy comprehension, as it is so drastically demanded here for a mathematical theory, I would like to demand much more of a mathematical problem if the same is to be perfect; for what is clear and easy to grasp attracts us, what is intricate frightens us off.
A mathematical problem is also difficult, so that it excites us, and yet not completely inaccessible, so that it does not mock our exertion; it is a landmark for us on the winding paths to hidden truths - afterwards it is rewarding for us with the joy of the successful solution.
The mathematicians of earlier centuries used to devote themselves to solving difficult problems; they knew the value of difficult problems. I only remember what JOHANN BERNOULLI asked Fastest fall line problem. Experience has shown, as BERNOULLI explains in the public announcement of this problem, that noble spirits are no longer driven to work on the increase of knowledge than when they are presented with difficult and at the same time useful tasks, and so he hopes to thank them for themselves of the mathematical world if, following the example of men like MERSENNE, PASCAL, FERMAT, VIVIANI and others who did the same thing before him, he presented the excellent analysts of his time with a task so that they could test the quality of theirs Assess methods and measure their strength. The calculus of variations owes its origin to the problem mentioned by BERNOULLI and similar problems.
As is well known, FERMAT had asserted that the Diophantine equation - except in certain obvious cases -
is unsolvable in whole numbers; the problem of proving this impossibility, offers a striking example of how beneficial a very special and seemingly insignificant problem can have on science. Because, stimulated by Fermat's problem, KUMMER succeeded in introducing ideal numbers and in discovering the theorem of the unambiguous decomposition of the numbers of a circular field into ideal prime factors - a theorem that today is generalized to arbitrary algebraic number ranges by DEDEKIND and KRONECKER is at the center of modern number theory and its importance extends far beyond the limits of number theory into the field of algebra and function theory.
To speak of a completely different area of research, let me remind you of that Three-body problem. To the fact that POINCARÉ undertook to treat this difficult problem again and to bring it closer to the solution, we owe the fruitful methods and the far-reaching principles which this scholar of heavenly mechanics developed and which the practical astronomer also recognizes and uses today.
The two problems mentioned above, Fermat's problem and the three-body problem, appear to us almost like opposite poles in the supply of problems: the first is a free invention of the pure understanding, belonging to the region of abstract number theory; the other forced upon us by astronomy and necessary for the knowledge of the simplest fundamental natural phenomena.
But it also often happens that the same special problem intervenes in the most diverse disciplines of mathematical knowledge. That's how it plays Shortest line problem at the same time an important historical and fundamental role in the fundamentals of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly did F. KLEIN describe the meaning of the icosahedron in his book Problem of the regular polyhedra in elementary geometry, in group and equation theory and in the theory of linear differential equations.
In order to highlight the importance of certain problems, I may also refer to WEIERSTRASS, who described it as a fortunate coincidence that he encountered such an important problem as this at the beginning of his scientific career Jacobian reversal problem was to work on.
Now that we have understood the general meaning of problems in mathematics, let us turn to the question of what sources mathematics draws its problems from. Certainly the first and oldest problems in any mathematical branch of knowledge stem from experience and were stimulated by the world of external appearances. Even the rules of the Arithmetic with whole numbers were discovered at a lower level of human culture in this way, just as children still learn to apply these laws using the empirical method. The same is true of the first problems of geometry: the problems of cube doubling, the quadrature of the circle and the oldest problems from the theory of the resolution of numerical equations, from the theory of curves and differential and integral calculus, from the calculus of variations, the theory of Fourier series and the theory of potential - even not to mention the further abundance of the actual problems from mechanics, astronomy and physics.
In the further development of a mathematical discipline, however, the human mind, encouraged by the success of the solutions, becomes aware of its independence; he creates new and fruitful problems of his own accord, often without any recognizable external stimulus, solely through logical combination, generalization, specialization, separating and collecting the terms, and then himself comes to the fore as the real questioner. That's how it came about Prime number problem and the rest of the problems of arithmetic, Galois equation theory, the theory of algebraic invariants, the theory of Abelian and automopic functions, and so came about in general almost all of the finer questions of modern number and function theory.
In the meantime, while the creative power of pure thinking is at work, the outside world comes into its own again, forces new questions on us through the real phenomena, opens up new areas of mathematical knowledge and, by trying to acquire these new areas of knowledge for the realm of pure thought , we often find the answers to old unsolved problems and at the same time best promote the old theories. It seems to me that the numerous and surprising analogies and that seemingly pre-established harmony that mathematicians so often perceive in the questions, methods and concepts of various fields of knowledge are based on this constantly repeating and changing game between thinking and experience.
We shall briefly discuss what justified general demands are to be made on the solution of a mathematical problem: I mean above all that the correctness of the answer can be demonstrated by a finite number of inferences, on the basis of a finite number of assumptions which lie in the problem and which must be precisely formulated each time. This requirement of logical deduction by means of a finite number of inferences is nothing other than the requirement of rigor in the argumentation. Indeed, the requirement of rigor, which is known to have become of proverbial importance in mathematics, corresponds to a general philosophical need of our understanding, and on the other hand, it is only through its fulfillment that the intellectual content and the fruitfulness of the problem come into full effect. A new problem, especially if it comes from the external world, is like a young rice, which only thrives and bears fruit if it is grafted carefully and according to the strict rules of the gardener onto the old trunk, the safe possession of our mathematical knowledge becomes.
It is also a mistake to believe that rigor in reasoning is the enemy of simplicity. On the contrary, in numerous examples we find confirmation that the strict method is at the same time the simpler and easier to grasp. Striving for severity forces us to find simpler modes of inference; also it often paves the way for us to methods that are more viable than the old methods of less rigor. Thus, the theory of algebraic curves experienced a considerable simplification and greater uniformity through the stricter function-theoretical method and the consequent introduction of transcendent aids. The proof, furthermore, that the power series allows the use of the four elementary types of calculation as well as the differentiation and integration in terms of terms and the knowledge of the meaning of the power series based on it, contributed considerably to the simplification of the entire analysis, in particular the theory of elimination and the theory of differential equations as well as the in the same to lead evidence of existence. The most striking example for my assertion is the calculus of variations. The treatment of the first and second variations of certain integrals sometimes involved extremely complicated calculations, and the relevant developments of the ancient mathematicians lacked the necessary rigor. WEIERSTRASS showed us the way to a new and reliable justification of the calculus of variations. Using the example of the simple integral and the double integral, at the end of my lecture I will briefly indicate how the pursuit of this path also leads to a surprising simplification of the calculus of variations, in that the calculation to prove the necessary and sufficient criteria for the occurrence of a maximum and minimum the second variation and in some cases even the laborious conclusions linked to the first variation become completely dispensable - not to mention the progress that lies in the abolition of the restriction to those variations for which the differential quotients of the function vary only slightly.
If I present rigor in the proofs as a requirement for a perfect solution of a problem, then I would at the same time refute the opinion that only the concepts of analysis or even only those of arithmetic are capable of completely strict treatment. I consider such an opinion, sometimes held by prominent quarters, to be absolutely erroneous; Such a one-sided interpretation of the strictness requirement soon leads to an ignoring of all concepts originating from geometry, mechanics and physics, to a prevention of the inflow of new material from the outside world and ultimately even to a rejection of the concepts of the continuum and the Irrational number. But what important vital nerve would mathematics be cut off by an exstripation of geometry and mathematical physics? On the contrary, wherever mathematical concepts emerge from the epistemological side or in geometry or from the theories of natural science, the task of mathematics is to investigate the principles underlying these concepts and to fix them in such a way by a simple and complete system of axioms that the sharpness of the new concepts and their usability for deduction is in no way inferior to the old arithmetic concepts.
The new concepts necessarily also include new signs: we choose these in such a way that they remind us of the phenomena which gave rise to the formation of the new concepts. The geometric figures are symbols for the memory images of spatial perception and are used as such by all mathematicians. Who does not always use the image of three points lying one behind the other on a straight line with the double equation for three quantities as the geometric symbol of the term "between"? Who does not use the drawing of interlaced lines and rectangles when it is a matter of proving a difficult sentence about the continuity of functions or the existence of compression points in full rigor? Who could do without the figure of the triangle, the circle with its center, who could do without the cross of three mutually perpendicular axes, or who wanted to do without the idea of the vector field or the image of a family of curves and surfaces with their envelopes, which in differential geometry, plays such an important role in the theory of differential equations, in the justification of the calculus of variations and other purely mathematical branches of knowledge?
The arithmetic signs are written figures and the geometrical figures are drawn formulas, and no mathematician could do without these drawn formulas, any more than in arithmetic, for example, the formation and resolution of brackets or the use of other analytical symbols.
The use of geometrical signs as strict evidence presupposes the exact knowledge and complete mastery of the axioms on which those figures are based, and so that these geometrical figures may be incorporated into the general treasure of mathematical symbols, a strict axiomatic investigation of their intuitive content is necessary. Just as when adding two numbers one must not incorrectly compare one another, but rather only the rules of calculation, i.e. the axioms of arithmetic, determine the correct operation with the digits, so the operation with the geometrical symbols is determined by the axioms of the geometrical concepts and their connection.
The correspondence between geometrical and arithmetic thinking is also shown in the fact that in arithmetic research just as little as in geometrical considerations we follow the chain of thought operations down to the axioms at every moment; Rather, especially when first tackling a problem, in arithmetic, just as in geometry, we first apply a rapid, unconscious, not definitely sure combination, trusting a certain arithmetic feeling for the mode of operation of the arithmetic symbols, without which we are in arithmetic would just as little progress as in geometry without the geometric imagination. I call MINKOWSKI's work "Geometry of Numbers" (Leipzig 1896) as a model for an arithmetic theory that operates strictly with geometric concepts and symbols.
I would like to make a few remarks about the difficulties which mathematical problems can present and how to overcome such difficulties.
When we fail to answer a mathematical problem, the reason is often that we have not yet recognized the more general point of view from which the problem presented appears only as a single link in a chain of related problems. After finding this point of view, not only does the problem presented become more accessible to our research, but we also come into possession of a method that can be applied to the related problems.The introduction of complex integration paths in the theory of definite integrals by CAUCHY and the establishment of the ideal concept in number theory by KUMMER serve as an example. This way of finding general methods is certainly the most convenient and safest; because if you are looking for methods without having a specific problem in mind, your search is usually in vain.
I believe that specialization plays an even more important role than generalization when dealing with mathematical problems. Perhaps in most cases where we look in vain for the answer to a question, the cause of failure is that we have not yet dealt with simpler and easier problems than the one presented, or with them incompletely. It then all depends on finding these easier problems and solving them with the most perfect tools possible and with generalizable terms. This rule is one of the most important levers for overcoming mathematical difficulties, and it seems to me that this lever is mostly used, albeit unconsciously.
Sometimes it happens that we strive for the answer under insufficient conditions or in an incorrect sense and as a result do not achieve the goal. The task then arises to prove the impossibility of solving the problem under the given conditions and in the required sense. Such impossibilities were already carried out by the ancients by showing, for example, that the hypotenuse of an isosceles right triangle stands in an irrational relationship to the cathetus. In modern mathematics the question of the impossibility of certain solutions plays a prominent part, and we are so aware that old difficult problems such as proving the axiom of parallels, squaring the circle or solving the equations of the fifth degree by taking the roots, albeit in other than the original meaning, have nevertheless found a completely satisfactory and strict solution.
It is this curious fact, along with other philosophical reasons, which gives rise to a conviction in us which every mathematician certainly shares, but which up to now at least no one has supported by evidence - I mean the conviction that every particular mathematical problem has to be dealt with strictly must necessarily be capable, whether it is that the answer to the question posed is successfully given, or that the impossibility of its solution and thus the necessity of all attempts to fail is demonstrated. Some specific unsolved problem is presented, for example the question of the irrationality of the Euler-Mascheronian constants or the question of whether there are infinitely many prime numbers of the form. As inaccessible as these problems seem to us and as perplexed as we are at the moment, we are nevertheless convinced that their solution must be achieved through a finite number of purely logical conclusions.
Is this axiom of the solvability of every problem a peculiarity only characteristic of mathematical thinking, or is it perhaps a general law clinging to the inner essence of our understanding that all questions which it poses can also be answered through it? We encounter old problems in other sciences too, which have been settled in the most satisfactory manner and for the highest benefit of science by proving the impossibility. I recall the problem of the perpetual motion machine.
After unsuccessful attempts to construct a perpetual motion machine, research was carried out into the relationships that must exist between the forces of nature if a perpetual motion machine is to be impossible (cf. Lecture given in Königsberg 1854.) and this reverse question led to the discovery of the law of the conservation of energy, which in turn explains the impossibility of the perpetual motion machine in the originally required sense.
This conviction of the solubility of every mathematical problem is a powerful incentive for us during our work; we hear the constant call within us: there is the problem, look for the solution. You can find it through pure thinking; because there is no ignorance in mathematics!
The number of problems in mathematics is immeasurable, and as soon as one problem is solved, countless new problems arise in its place. In the following, allow me, as a test, as it were, to name specific problems from various mathematical disciplines, the treatment of which can be expected to promote science.
Let us review the principles of analysis and geometry. The most stimulating and significant events in this field of the last century, it seems to me, are the arithmetic acquisition of the concept of the continuum in the works of CAUCHY, BOLZANO, and CANTOR and the discovery of non-Euclidean geometry by GAUSS, BOLYAI, LOBACEVSKIJ. I therefore first draw your attention to some of the problems related to these areas.
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