Algebraic geometry, real and complex analysis, mathematical physics and , as well as ) were not solved. At the moment, 16 problems out of 23 have been solved. Another 2 are not correct mathematical problems (one is formulated too vaguely to understand whether it has been solved or not, the other, far from being solved, is physical, not mathematical). Of the remaining 5 problems, three are not solved, and two are solved only for some cases.
List of problems
| 1 | resolved | Cantor's problem on the power of the continuum () |
| 2 | resolved | Consistency of axioms of arithmetic |
| 3 | resolved | Equivalence of equal sizes |
| 4 | too vague | List the lines in which the lines are geodesics |
| 5 | resolved | Are all continuous? |
| 6 | not mathematical | Mathematical presentation of the axioms of physics |
| 7 | resolved | If a≠ 0, 1 - , and b- algebraic, but irrational, is it true that a b - |
| 8 | open | Problem prime numbers( And ) |
| 9 | partially resolved | The proof is most common law reciprocity in any number field |
| 10 | resolved | Solvability problem |
| 11 | resolved | Study of quadratic forms with arbitrary algebraic numerical coefficients |
| 12 | open | Extension of Kronecker's theorem on Abelian fields to an arbitrary algebraic domain of rationality |
| 13 | resolved | Impossibility of solution general equation seventh power using functions that depend on only two variables |
| 14 | resolved | Proof of the finite generation of the algebra of invariants of an algebraic group |
| 15 | resolved | Rigorous justification of Schubert's calculative geometry |
| 16 | partially resolved | The number and location of ovals of a real algebraic curve of a given degree on the plane; number and location of polynomial limit cycles vector field given degree on the plane |
| 17 | resolved | Representation of certain shapes as a sum of squares |
| 18 | partially resolved | Irregular fillings of space with congruent polyhedra. The most dense packing of balls |
| 19 | resolved | Are regular variational solutions always analytical? |
| 20 | resolved | General task about boundary conditions (?) |
| 21 | resolved | Proof of the existence of linear differential equations with a given monodromy group |
| 22 | resolved | Uniformization of analytical dependencies using automorphic functions |
| 23 | resolved | Development of methods of calculus of variations |
Footnotes
- Cohen's result shows that neither the continuum hypothesis nor its negation contradicts (the standard system of set theory axioms). Thus, the continuum hypothesis in this system of axioms can neither be proven nor disproved.
- According to Rowe and Gray (see below), most of the problems have been resolved. Some of them were not formulated precisely enough, but the results achieved allow us to consider them as “solved”. Moat and Gray refer to the fourth problem as one that is too vague to judge whether it has been solved or not.
- Rove and Gray also call problem #18 "open" in their 2000 book because the ball packing problem (also known as Kepler's problem) had not been solved at that time, but is now reported to have been solved (see below). Advances in solving problem No. 16 have been made in recent times, as well as in the 1990s.
- Problem #8 contains two known issues, both of which remain unresolved. The first of these is one of seven Millennium Prize Problems that have been designated as "Hilbert Problems" for the 21st century.
- Problem #9 has been solved for the Abelian case; the non-Abelian case remains unsolved.
- The statement about the finite generation of the algebra of invariants is proven for reductive groups. Nagata in 1958 constructed a counterexample for general case. It is also proved that if the algebra of invariants of any (finite-dimensional) representation of an algebraic group is finitely generated, then the group is reductive.
- The first (algebraic) part of problem No. 16 is more precisely formulated as follows. Harnack proved that maximum number ovals is equal to M=(n-1)(n-2)/2+1, and that such curves exist - they are called M-curves. How can the ovals of the M-curve be arranged? This problem has been done up to degree n=6 inclusive, and for degree n=8 quite a lot is known (although it has not been completed yet). In addition, there are general statements that limit how the ovals of M-curves can be arranged - see the works of Gudkov, Arnold, Roon, Hilbert himself (however, it is worth considering that there is an error in Hilbert's proof for n=6: one of the cases , which he considered impossible, turned out to be possible and was built by Gudkov). The second (differential) part remains open even for quadratic vector fields - it is not even known how many there may be, and even that an upper bound exists. Even the individual finiteness theorem (that every polynomial vector field has a finite number of limit cycles) was only recently proven. It was considered proven by Dulac, but an error was discovered in his proof, and this theorem was finally proven by Ilyashenko and Ecal - for which each of them had to write a book.
(standard system of axioms of set theory). Thus, the continuum hypothesis in this axiom system can neither be proven nor disproved (provided that this axiom system is consistent).
A. A. Bolibrukh. Hilbert's problems (100 years later)
Hilbert's first problem: the continuum hypothesis
The continuum conjecture, Hilbert's first problem, relates to problems in the foundations of mathematics and set theory. It is closely related to such simple and natural questions as “How much?”, “More or less?”, and almost any high school student can understand what this problem is. However, we will need some additional information to formulate it.
Set equivalence
Consider the following example. There is a dance party at school. How to determine who is more present at this evening: girls or boys?
You can, of course, count both of them and compare the two obtained numbers. But it’s much easier to answer when the orchestra starts playing a waltz and all the dancers break into pairs. Then, if everyone present is dancing, it means that everyone has found a pair, i.e. there are the same number of boys and girls. If only boys remain, then there are more boys, and vice versa.
This method, sometimes more natural than direct recalculation, is called principle of pairing, or one-to-one correspondence principle.
Let us now consider a collection of objects of arbitrary nature --- a bunch of. Objects included in a set are called its elements. If element x included in the set X, this is denoted as follows: x X. If the set X 1 contained in many X 2, i.e. all elements of the set X 1 are also elements X 2, then they say that X 1--- subset X 2, and briefly write it like this: X 1 X 2.
A bunch of Certainly, if it has a finite number of elements. Sets can be either finite (for example, a set of students in a class) or infinite (for example, --- a bunch of all natural numbers 1,2,3,... ). Sets whose elements are numbers are called numerical.
Let X And Y--- two sets. They say that between these sets it is established one-to-one correspondence, if all elements of these two sets are divided into pairs of the form (x,y), Where x X, yY, and each element from X and each element from Y participates in exactly one pair.
An example is when all the girls and boys at a dance party are paired up, and there is an example of a one-to-one match between many girls and many boys.
Sets between which one-to-one correspondence can be established are called equivalent or equally powerful. Two finite sets are equivalent if and only if they have the same number of elements. Therefore, it is natural to assume that if one infinite set is equivalent to another, then it has “the same number” of elements. However, based on this definition of equivalence, one can obtain very unexpected properties of infinite sets.
Infinite sets
Let us consider any finite set and any of its own (non-empty and not coinciding with itself) subset. Then the elements in the subset less, than in the set itself, i.e. part is less than the whole.
Do infinite sets have this property? And can it make sense to say that one infinite set has “fewer” elements than another, also infinite? After all, about two infinite sets we can only say for now whether they are equivalent or not. Do non-equivalent infinite sets exist at all?
Below we will answer all these questions one by one. Let's start with a funny one fantastic story from the book "Stories about Sets" by N. Ya. Vilenkin. The action takes place in the distant future, when inhabitants of different galaxies can meet each other. Therefore, for all those traveling through space, a huge hotel was built, stretching across several galaxies.
In this hotel infinitely many numbers(rooms), but, as expected, all rooms are numbered, and for any natural number n there is a room with this number.
Once a congress of cosmozoologists was held in this hotel, in which representatives of all galaxies participated. Since there are also an infinite number of galaxies, all the places in the hotel were occupied. But at this time his friend came to the hotel director and asked to put him in this hotel.
“After some thought, the director turned to the administrator and said:
Place him in #1.
Where will I put the tenant of this room? --- the administrator asked in surprise.
And move him to #2. Send the tenant from #2 to #3, from #3 to #4, etc.”
In general, let the guest living in the room k, will move into the room k+1, as shown in the following figure:
Then everyone will have their own number again, and #1 will be free.
Thus, we managed to accommodate the new guest --- precisely because there are infinitely many rooms in the hotel.
Initially, the congress participants occupied all the hotel rooms, therefore, between many cosmozoologists and many a one-to-one correspondence was established: each cosmozoologist was given a number, on the door of which the corresponding natural number was written. It is natural to assume that there were “as many” delegates as there are natural numbers. But another person arrived, he was also accommodated, and the number of residents increased by 1. But again there were “the same number” of them as there were natural numbers: after all, everyone fit into the hotel! And if we denote the number of cosmozoologists by 0 , then we get the "identity" 0 = 0 +1 . For no end 0 it, of course, was not fulfilled.
We came to a surprising conclusion: if you add one more element to a set that is equivalent, you get a set that is again equivalent. But it’s absolutely clear what the cosmozoological delegates represent Part of the many people who settled in the hotel after the arrival of the new guest. This means that in this case the part is not “less” than the whole, but “equal” to the whole!
So, from the definition of equivalence (which does not lead to any “oddities” in the case of finite sets) it follows that part of an infinite set can be equivalent to the entire set.
It's possible that famous mathematician Bolzano, who tried to apply the principle of one-to-one correspondence in his reasoning, was afraid of such unusual effects and therefore did not further develop this theory. It seemed completely absurd to him. But Georg Cantor in the second half of the 19th century again became interested in this issue, began to study it and created set theory, an important section of the foundations of mathematics.
Let's continue our story about the endless hotel.
The new guest “was not surprised when the next morning he was offered to move to # 1,000,000 . It’s just that belated cosmozoologists from the VSK-3472 galaxy arrived at the hotel, and it was necessary to accommodate more 999,999 tenants."
But then some kind of mishap occurred, and philatelists came to the same hotel for the congress. There were also an infinite number of them --- one representative from each galaxy. How to place them all?
This task turned out to be very difficult. But even in this case, there was a way out.
“First of all, the administrator ordered the tenant to be moved from #1 to #2.
And move the tenant from #2 to #4, from #3 to #6, in general, from the room n--- to the room 2n.
Now his plan became clear: in this way he freed up an infinite number of odd numbers and could accommodate philatelists in them. As a result, even numbers turned out to be occupied by cosmozoologists, and odd numbers by philatelists... A philatelist standing in line n-m, occupied the room 2n-1". And again everyone managed to be accommodated in a hotel. So, an even more amazing effect: when combining two sets, each of which is equivalent , we again obtain a set equivalent . I.e. even when we “double” the set, we get a set equivalent to the original one!
Countable and uncountable sets
Consider the following chain: . ( --- is a set of integers, and --- set of rational numbers, i.e. set of numbers of the form p/q, Where p And q--- whole, q0.) All these sets are infinite. Let us consider the question of their equivalence.
Let us establish a one-to-one correspondence between And : we form pairs of the form (n,2n) And (-n,2n+1), n, as well as a couple (0,1) (in first place in each pair the number from , and on the second --- from ).
There is another way to establish this correspondence, for example, write out all the integers in a table, as shown in the figure, and, going around it along the arrows, assign a certain number to each integer. Thus, we " let's recalculate" all integers: each z some natural number (number) is compared and for each number there is an integer to which this number is assigned. In this case, it is not necessary to write out an explicit formula.
Thus, equivalent .
Any set equivalent to the set of natural numbers is called countable. Such a set can be “recounted”: all its elements can be numbered natural numbers.
At first glance, there are “much more” rational numbers on the line than integers. They are located dense everywhere: in any arbitrarily small interval there are infinitely many of them. But it turns out that many also countable. Let us first prove the countability + (the set of all positive rational numbers).
Let's write down all the elements + into the following table: in the first line - all numbers with a denominator of 1 (i.e., integers), in the second - with a denominator of 2, etc. (see figure). Every positive rational number will definitely appear in this table, and more than once ( for example, number 1====... occurs in every row of this table ) .
Now we will recalculate these numbers: following the arrows, we assign a number to each number (or skip this number if we have already encountered it before in another entry). Since we are moving along the diagonals, we will go around the entire table (that is, sooner or later we will get to any of the numbers).
So, we have indicated a way to number all the numbers from + , i.e. they proved that + countable.
Note that this method of numbering does not preserve order: of two rational numbers, the larger one may appear earlier, or perhaps later.
What about negative rational numbers and zero? Just like with cosmozoologists and philatelists in the endless hotel. Let's number + not all natural numbers, but only even ones (giving them numbers not 1, 2, 3, ..., but 2, 4, 6, ...), we assign the number 1 to zero, and assign the number 1 to all negative rational numbers (by the same scheme as positive) odd numbers, starting with 3.
That's it rational numbers are numbered with naturals, therefore, countable.
A natural question arises: Maybe all infinite sets are countable?
It turned out that --- the set of all points on the number line is uncountable. This result, obtained by Cantor in the last century, made a very strong impression on mathematicians.
Let us prove this fact in the same way as Cantor did: with the help diagonal process.
As we know, every real number x can be written in the form decimal:
x=A, 1 2 ... n ...,
Where A--- an integer, not necessarily positive, but 1, 2, ..., n, ... are numbers (from 0 to 9). This idea is ambiguous: for example,
½=0.50000...=0.49999...
(in one version of the notation, starting from the second digit after the decimal point, there are only zeros, and in the other - only nines). To make the record unambiguous, in such cases we will always choose the first option. Then each number corresponds to exactly one of its decimal notations.
Let us now assume that we have succeeded in recalculating all real numbers. Then they can be arranged in order:
x 1 =A, 1 2 3 4 ...
x 2 =B, 1 2 3 4 ...
x 3 =C, 1 2 3 4 ...
x 4 =D, 1 2 3 4 ...
To come to a contradiction, let's construct the following number y, which not counted, i.e. not contained in this table.
For any number a let's determine the number in the following way:
=
Let's put 
(this number k The -th digit after the decimal point is 1 or 2, depending on which digit appears on k-th place after the decimal point in decimal notation numbers x k).
For example, if
x 1 = 2.1345...
x 2 = -3.4215...
x 3 = 10.5146...
x 4 = -13.6781...
.....................
That =0,2112...
So, using the diagonal process we got a real number y, which does not coincide with any of the numbers in the table, because y different from everyone x k at least k the th digit of the decimal expansion, and different records, as we know, correspond to different numbers.
To prove the continuum hypothesis means to derive it from these axioms. To refute it means to show that if it is added to this system of axioms, it will turn out contradictory a set of statements.
Solution
It turned out that Hilbert's first problem had a completely unexpected solution.
In 1963, the American mathematician Paul Cohen proved that the continuum hypothesis can neither be proven nor disproved.
This means that if we take the standard system of Zermelo---Frenkel axioms ( ZF) and add to it the continuum hypothesis as another axiom, then it turns out consistent approval system. But if to ZF add negation continuum hypothesis (i.e. the opposite statement), then again we get consistent approval system.
Thus, neither the continuum hypothesis nor its negation it is forbidden withdraw from standard system axiom.
This conclusion was very strong effect and was even reflected in literature (see epigraph).
How to deal with this hypothesis? Usually it is simply attached to the Zermelo-Frenkel axiom system. But every time they prove something based on a continuum hypothesis, they must indicate that it was used in the proof.
The second of the famous mathematical problems that David Hilbert put forward in 1900 in Paris on II International Congress mathematicians. There is still no consensus among the mathematical community as to whether it has been solved or not. The problem sounds like this: Are the axioms of arithmetic contradictory or not? Kurt Gödel proved that the consistency of the axioms of arithmetic cannot be proven from the axioms of arithmetic themselves (unless arithmetic is actually inconsistent). Besides Gödel, many others outstanding mathematicians dealt with this problem.
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PREFACE
The collection offered to the reader’s attention contains the text of Hilbert’s famous report “Mathematical Problems” translated into Russian for the first time, delivered at the II International Congress of Mathematicians, held in Paris from August 6 to 12, 1900.226 people took part in the Congress: 90 people from France, 25 from Germany, 17 from the United States, 15 from Italy, 13 from Belgium, 9 from Russia, 8 each from Austria and Switzerland, 7 each from England and Sweden, 4 from Denmark, 3 each from Holland, Spain and Romania, 2 each from Serbia and Portugal, 4 from South America, Turkey, Greece, Norway, Canada, Japan and Mexico sent one delegate each.
The main languages of the Congress were English, French, German and Italian.
Henri Poincaré was elected Chairman of the Congress, Charles Hermite (1822 - 1901), who was absent, was elected honorary chairman, E. Chuber (Vienna), K. Geyser (Zurich), P. Gordan (Erlangen), A. Greenhill (London) were elected vice-chairmen. , L. Lindelof (Helsingfors), F. Lindemann (Munich), G. Mittag-Leffler (Stockholm), absent E. Moore (Chicago), M. A. Tikhomandritsky (Kharkov), V. Volterra (Turin), G. Zeiten (Copenhagen), secretaries of the Congress - I. Bendikson (Stockholm), A. Capelli (Naples), G. Minkowski (Zurich), I. L. Ptashitsky (St. Petersburg), and the absent A. Whitehead (Cambridge).
E. Duporcq (Paris) was elected Secretary General of the Congress.
There were six sections: 1) arithmetic and algebra (chairman D. Hilbert, secretary E. Cartan),
The 5th and 6th sections sat together.On the opening day of the Congress general meeting two hour-long reports took place: M. Cantor “On the historiography of mathematics,” in which he reviewed works on the history of mathematics, starting with J. Montucl and G. Libri, and V. Volterra about scientific activity E. Betti, F. Brioschi, and F. Casorati.
Then the breakout sessions began, at which 46 reports were made, including by L. Dixon, G. Mittag-Leffler, D. Gilbert, J. Hadamard, A. Capelli, I. Fredholm, I. Bendixson, V. Volterra and others .
Russian mathematics was represented at the Congress by a single message from M.A. Tikhomandritsky "On the disappearance of the function N several variables."
At the final general meeting, G. Mittag-Leffler spoke, who spoke about recent years the life of Weierstrass according to his letters to S.V. Kovalevskaya, and A. Poincaré, who made a report “On the role of intuition and logic in mathematics.”
This is how the Congress took place, where on August 8, at a joint meeting of the 5th and 6th sections, D. Hilbert read his report “Mathematical Problems”.
As D. Sintsov* writes, "Hilbert's message caused a number of comments from those present, who indicated that some of the problems listed by Hilbert were completely or partially resolved by them"**. By that time, Hilbert, a 38-year-old professor at Göttingen, was already widely known for his work on the theory of invariants and the theory algebraic numbers. In 1899, his famous “Foundations of Geometry” were published, which constituted an era in the foundations of mathematics. The amazing versatility and generalizing power of Gilbert's talent allowed him to easily navigate various areas mathematics, in almost all of which he obtained outstanding results and posed a number of important problems.
* D. M. Sintsov, Second International Mathematical Congress, Phys.-Math. Sciences (2) 1, No. 5 (1901), 129-137.
** Probably the number of problems in the original text of the report exceeded twenty-three.
The most interesting problems, according to Hilbert, are “the study of which can significantly stimulate the further development of science”, This is what he suggested to mathematicians in his report. Two-thirds of a century has passed since then. Hilbert’s problems remained relevant throughout this period; efforts were made to solve them the most talented mathematicians. The development of ideas related to the content of these problems made up a significant part of mathematics in the 20th century.
The translation of the main part of the report (excluding the text of the 15th and 23rd problems and the conclusion) was carried out by M. G. Shestopal from the text published in Gottinger Nachrichten (1900, 253-297), and reviewed by I. N. Bronstein and I. M Yaglom, who made a number of editorial amendments and changes to it. The text of the 15th and 23rd problems, as well as the final part of the report, was translated by A. V. Dorofeeva. The translation includes additions made by Hilbert for the publication of a report placed in the third volume of his Collected Works (Gesammelte Abhandlungen, Berlin, Springer, 1932-1935) - in the text they are enclosed in square brackets. The translation was checked with English translation(Bull. Amer. Math. Soc. 8, No. 10 (1902), 403-479), also with a translation carried out in the office of the history of mathematics and mechanics of Moscow State University by A. V. Dorofeeva and M. V. Chirikov *.
* This translation served as the beginning of work on the historical and mathematical analysis of Hilbert’s problems, carried out in the office of the history of mathematics and mechanics of Moscow State University under the guidance of prof. K. A. Rybnikova.
A known difficulty was the translation of some old mathematical terms. In some cases the German term is placed in parentheses next to the translation, and in one case the term (Polarenprocess) is left without translation. The translators worked hard to convey to the Russian reader the peculiar, sometimes even pathetic language of Hilbert’s report. The authors of the issue comments kindly agreed to review the translations of the relevant issues and made a number of significant corrections.
Assess the outstanding significance that Hilbert's report played for mathematics in the 20th century. will allow, we hope, comments on the problems that make up the second part of the collection. The creation of such commentaries, containing an overview of the main results achieved in the direction of solving Hilbert's problems, has already been undertaken by individual authors *. However, work of this kind with the involvement of well-known specialists in the relevant fields of mathematics is being carried out, as far as we know, for the first time.
* L. Bieberbach, Dber die Einfluss von Hilbert Pariser Vortrag liber "Mathematische Probleme", auf die Entwicklung der Matbematik in den letzen dreissig Jabren, Naturwissenschaften 18 (1930), 1101-1111; S.S. Demidov, On the history of Hilbert's problems. IMI, vol. 17, "Science", 1967, 91-121.
The publication of this book was greatly facilitated by the attention and assistance of many people, among whom it is necessary to note the participants in the seminar on the history of mathematics and mechanics of Moscow State University, especially its leaders, Professors I.G. Bashmakov, K.A. Rybnikova, A.P. Yushkevich, the late S.A. Yanovskaya, as well as an employee of the Mathematical Institute named after V.A. Steklov Academy of Sciences of the USSR A.N. Parshin, whose advice and help greatly helped improve the publication.
S. S. Demidov
A FEW WORDS ABOUT HILBERT'S PROBLEMS
At the International Mathematical Congress in Paris in 1900, the outstanding German mathematician David Hilbert gave a presentation entitled “Mathematical Problems.” This report was then published several times in the original and in translations *; The latest edition of the original is found in the third volume of Gilbert's collected works **.* First published in Arcbiv f. Math. u Phys., Ill series, 1 (1901), 44-63, 213-237.
** D. Hilbert, Gesammelte Abhandlungen, vol. Ill, 1935, 290-329.
The Russian translation of Gilbert's report is printed on the following pages.
Neither before Hilbert's 1900 report nor after this report did mathematicians, as far as I know, come forward with scientific reports, covering problems of mathematics in general *. Thus, Hilbert's report turns out to be a completely unique phenomenon in the history of mathematics and in mathematical literature. And now, almost 70 years after Hilbert gave his report, it retains its interest and significance.
* The report of the American mathematician J. von Neumann at the International Mathematical Congress in Amsterdam in 1954 is not a refutation of this statement: it is true that von Neumann’s report was called “Unsolved Problems in Mathematics,” but the speaker began his report with a statement that he would consider imitating madness Hilbert speaks about the problems of mathematics in general, but intends to limit himself only to problems in some areas of mathematics (mainly in areas close to functional analysis). Von Neumann's report was not published - the only thing that was published about it in the Proceedings of the Amsterdam Congress was that the manuscript of the report was not available to the publishers; apparently it doesn't exist. Therefore, this report can currently be judged only by the recollections of those who listened to it.
Hilbert had an exceptional influence on the entire development of modern mathematics, covering almost all areas of mathematical thought; this is explained by the fact that Hilbert was a mathematician in whom the power of mathematical thought was combined with rare breadth and versatility. This versatility was, so to speak, quite conscious: Hilbert constantly emphasizes that mathematics is unified, that its various parts are in constant interaction with each other and with the natural sciences, and that in this interaction is not only the key to understanding the essence itself mathematics, but also the best remedy against the splitting of mathematics into separate, unconnected parts - a danger that in our time of enormous quantitative growth and frightening specialization of mathematical research
constantly makes you think about yourself. WITH great strength and Hilbert speaks with conviction, especially at the end of his remarkable report, about the holistic nature of mathematics as the basis of all accurate natural science knowledge. His conviction in this serves to a large extent as the guiding thread of this report as a whole and, undoubtedly, in many cases guided the author in the selection of the mathematical problems he put forward.The report begins with an interesting, I would say inspired, written general introductory part, which speaks not only about the significance for mathematics of a “well-posed” special problem, but also makes judgments about mathematical rigor, about the connection of mathematics with natural science, and about other things related to to every mathematician who actively thinks about his science. At the conclusion of this introductory part, Hilbert, with striking difference and conviction, expresses his main thesis, the “axiom” of decidability in in a broad sense the words of any mathematical problem are a thesis, the content of which is deep confidence in the unlimited power of human knowledge and an irreconcilable struggle against all agnosticism - against the absurd "Ignorabimus" *, as Gilbert says elsewhere.
* "Ignorabimus"(lat.) - "we won't know"- one of famous speeches physiologist E. Dubois-Reymond ended (as applied to some unclear scientific questions) with the exclamation: “Ignoramus et ignorabimus” - we don’t know and won’t know!
Next come the problems themselves. They begin with set theory (the continuum problem) and the foundation of mathematics, move on to the foundations of geometry, the theory of continuous groups (the famous fifth problem about the liberation of the concept of a continuous group from the requirement of differentiability), to number theory, algebra and algebraic geometry and end with analysis (differential equations, especially with partial derivatives, calculus of variations). Special place occupies the sixth problem - about the axiomatics of probability theory and mechanics.
By their nature, Hilbert's problems are very heterogeneous. Sometimes this is a specifically posed question to which an unambiguous answer is sought - yes or no - such as, for example, the geometric third problem or the arithmetic seventh problem about transcendental numbers. Sometimes the problem is posed less clearly, as, for example, in the twelfth problem (Hilbert gave special attention to it important), in which it is required to find both the generalization of Kronecker’s theorem itself and the corresponding class of functions that should replace the exponential and modular ones.
The fifteenth problem is, in essence, the problem of substantiating the entire theory of algebraic varieties.
Sometimes the problem under this number actually contains several different, although closely related, problems. Finally, the twenty-third problem is, in essence, the problem of the further development of the calculus of variations.
Now, many years after Hilbert posed his problems, we can say that they were posed well. They turned out to be a suitable object for focusing the creative efforts of mathematicians of various scientific directions and schools. What these efforts were and what results they led to, which of Hilbert’s problems have been solved and which have not yet - the reader can learn about this, although not in full detail, from the comments to these problems.
The nature of these comments is somewhat heterogeneous (which is largely dictated by the nature of the problems themselves) - some of them can be understood by a reader familiar with mathematics in the first two courses of mechanics-mathematics or physics-mathematics faculties of universities or pedagogical institutes, while others require quite high mathematical culture. I think, in any case, that the reader will be grateful to the authors of the comments,
which significantly facilitated familiarization with that truly outstanding work of general mathematical literature that is Hilbert’s report; In addition, from the comments one can, it seems to me, understand the impact this report had on the further development of mathematics.P. S. Alexandrov
Who among us would not want to lift the veil behind which our future is hidden, in order to penetrate at least with one glance into the upcoming successes of our knowledge and the secrets of its development in the coming centuries? What will be the special goals that the leading mathematical minds of the next generation will set for themselves? What new methods and new facts will be discovered in the new century on the wide and rich field of mathematical thought?
History teaches that the development of science is continuous. We know that every age has its own problems, which the subsequent era either solves or pushes aside as fruitless in order to replace them with new ones. To imagine the possible nature of development mathematical knowledge in the near future, we must turn over in our imagination the questions that still remain open, survey the problems that pose modern science, and the solutions of which we expect from the future. Such a review of problems seems to me today, at the turn of the new century, to be especially timely. After all, big dates not only make us look back at the past, but also direct our thoughts into the unknown future.
It is impossible to deny the profound significance that certain problems have for the advancement of mathematical science in general and important role, which they play in the work of an individual researcher. Any scientific field is viable as long as it has an abundance of new problems. Lack of new problems means withering away or cessation independent development. Just as in general every human endeavor is connected with one goal or another, so mathematical creativity is connected with the formulation of problems. The strength of the researcher is learned in solving problems: he finds new methods, new points of view, he opens wider and freer horizons.
It is difficult, and often impossible, to correctly assess the significance of a particular task in advance; because ultimately its value will be determined by the benefits it brings to science. This begs the question: Are there common features that characterize a good math problem?
One old French mathematician said: " Mathematical theory can be considered perfect only when you have made it so clear that you undertake to explain its contents to the first person you meet." This requirement of clarity and easy accessibility, which is stated so sharply here in relation to a mathematical theory, I would put even more sharply in relation to a mathematical problem, if it claims to be perfect; after all, clarity and easy accessibility attract us, while complexity and intricacy repel us.
A mathematical problem, further, must be so difficult as to attract us, and at the same time not completely inaccessible, so as not to render our efforts hopeless; it should be a guiding sign on the tangled paths leading to hidden truths; and she should then reward us with the joy of finding a solution.
Mathematicians of the last century devoted themselves with passionate zeal to solving individual difficult problems; they knew the value of a difficult task. I will only recall the one posed by Johann Bernoulli problem about the line of fastest fall.“As experience shows,” says Bernoulli, announcing his task, “nothing so powerfully motivates high minds to work on enriching knowledge as the formulation of a difficult and at the same time useful task"And so he hopes to earn gratitude mathematical world, if he, following the example of such men as Mersenne, Pascal, Fermat, Viviani and others who (before him) did the same, proposes the problem to the outstanding analysts of his time, so that they can test it as a touchstone the merits of your methods and measure your strengths. The calculus of variations owes its origin to this Bernoulli problem and other similar problems.
Fermat’s well-known statement is that the Diophantine equation
x n + y n = z n
undecidable in integers x, y, z, barring some obvious exceptions. The problem of proving this undecidability provides a striking example of the stimulating influence a special and, at first glance, insignificant problem can have on science. For, prompted by Fermat's problem, Kummer came to the introduction of ideal numbers and the discovery of a theorem on the unique decomposition of numbers in cyclotomic fields into ideal prime factors- a theorem which, thanks to generalizations to any algebraic number domain obtained by Dedekind and Kronecker, is now central to modern theory numbers and whose significance goes far beyond number theory into the field of algebra and function theory.
Let me remind you of another interesting problem - three body problem. The fact that Poincaré undertook a new consideration and significantly advanced this difficult task, led to the fruitful methods and far-reaching principles introduced by these scientists into celestial mechanics, methods and principles which are now recognized and applied also in practical astronomy.
Both problems mentioned - Fermat's problem and the three-body problem - are, in our stock of problems, as if opposite poles: the first represents free achievement pure reason, belonging to the field of abstract number theory, the second is put forward by astronomy and is necessary for the knowledge of the simplest basic phenomena of nature.
It often happens, however, that the same special problem appears in very different areas of mathematics. So, shortest line problem plays an important historical and fundamental role simultaneously in the foundations of geometry, in the theory of curves and surfaces, in mechanics and in the calculus of variations. And as F. Klein convincingly demonstrates in his book on the icosahedron *, problem about regular polyhedra is important at the same time for elementary geometry, group theory, algebraic theory and linear differential equation theory!
* F. Klein, Vorlesungen uber das Ikosaeder und die Auflosung der Gleichungen von funften Grade, Leipzig, 1884.- Note ed.
To highlight the importance individual problems, I will also allow myself to refer to Weierstrass, who considered it a great success for himself that the combination of circumstances allowed him to tackle such a significant problem at the beginning of his scientific career, like the Jacobi problem on the inversion of an elliptic integral.
After we have considered general meaning problems in mathematics, let us turn to the question of from what source does mathematics draw its problems. There is no doubt that the first and oldest problems of each mathematical field of knowledge arose from experience and were presented to us by the world of external phenomena. Even the rules for counting with integers were discovered at an early stage along this path. cultural development of humanity, just as now a child learns the application of these rules empirical method. The same applies to the first problems of geometry - the problems of doubling the cube, squaring the circle, which came to us from ancient times, as well as oldest problems the theory of numerical equations, the theory of curves, differential and integral calculus, calculus of variations, the theory of Fourier series and potential theory, not to mention the whole wealth of problems in mechanics proper, astronomy and physics.
With the further development of any mathematical discipline, the human mind, encouraged by success, already shows independence; he himself poses new and fruitful problems, often without noticeable influence outside world, with the help of only logical comparison, generalization, specialization, successful division and grouping of concepts, and then he himself comes to the fore as a statement of problems. This is how they arose prime number problem and other problems of arithmetic, Galois theory, the theory of algebraic invariants, the theory of Abelian and automorphic functions, and so almost arose in general all subtle questions of modern number theory and function theory.
Meanwhile, during the action creative power pure thinking, the outside world again insists on its rights: it imposes new questions on us with its real facts and opens up to us new areas of mathematical knowledge. And in the process of bringing these new fields of knowledge into the realm of pure thought, we often find answers to old unsolved problems and in this way best advance old theories. On this constantly repeating and changing play between thinking and experience, it seems to me, are based those numerous and striking analogies and that seeming pre-established harmony that the mathematician so often discovers in problems, methods and concepts of various fields of knowledge.
Let us dwell briefly on the question of what may be the general requirements that we have the right to present to the solution of a mathematical problem. I mean first of all the requirements that make it possible to verify the correctness of the answer using finite number conclusions and, moreover, on the basis of a finite number of premises that form the basis of each task and which must be precisely formulated in each case. This requirement of logical deduction with the help of a finite number of conclusions is nothing more than a requirement for the rigor of evidence. Indeed, the requirement of rigor, which has already become proverbial in mathematics, corresponds to the general philosophical need of our mind; on the other hand, only the fulfillment of this requirement leads to the identification of the full significance of the essence of the task and its fruitfulness. A new task, especially if it is brought to life by phenomena of the external world, is like a young shoot that can grow and bear fruit only if it is carefully and according to the strict rules of the art of gardening nurtured on an old trunk - the solid foundation of our mathematical knowledge.
Will big mistake think at the same time that rigor in proof is the enemy of simplicity. Numerous examples convince us of the opposite: strict methods are at the same time the simplest and most accessible. The desire for rigor leads precisely to the search for the simplest proofs. This same desire often paves the way to methods that prove more fruitful than the older, less rigorous methods. Thus, the theory of algebraic curves, thanks to more strict methods the theory of functions of a complex variable and the expedient use of transcendental means has been significantly simplified and acquired greater integrity. Further, a proof of the legality of applying four elementary arithmetic operations to power series, as well as term-by-term differentiation and integration of these series and the recognition of a power series based on this [as a tool for mathematical analysis - P.A. ], undoubtedly greatly simplified the whole analysis, in particular the theory of exclusion and the theory of differential equations (together with its existence theorems).
But a particularly striking example that illustrates my point is the calculus of variations. The study of the first and second variations of a definite integral led to extremely complex calculations, and the corresponding studies of old mathematicians lacked the necessary rigor. Weierstrass showed us the way to a new and completely reliable foundation for the calculus of variations. Using the example of a simple and double integral I will briefly outline at the end of my report how following this path leads at the same time to an amazing simplification of the calculus of variations due to the fact that in order to establish the necessary and sufficient criteria for maximum and minimum, the calculation of the second variation becomes unnecessary and even partially eliminates the need for tedious inferences related to to the first variation. I'm not even talking about the advantages that arise from the fact that there is no need to consider only those variations for which the values of the derivatives of the functions change insignificantly.
Presenting to full solution problem of the requirement of rigor in proof, I would like, on the other hand, to refute the opinion that completely rigorous reasoning is applicable only to the concepts of analysis or even arithmetic alone. I consider this opinion, sometimes supported by outstanding minds, to be completely false. Such a one-sided interpretation of the requirement of rigor quickly leads to ignoring all concepts arising from geometry, mechanics, physics, and stops the flow of [to mathematics - P.A. ] new material from the outside world and, in the end, even leads to the rejection of the concept of continuum and irrational number. Is there a more important vital nerve than the one that would be cut off from mathematics if geometry and mathematical physics were removed from it? I, on the contrary, believe that whenever mathematical concepts originate from the theory of knowledge or in geometry, or in natural science theories, mathematics is faced with the task of exploring the principles underlying these concepts, and so substantiating these concepts with the help of a complete and simple systems of axioms so that the rigor of the new concepts and their applicability to deduction is in no way inferior to the old arithmetic concepts.
New concepts also include new designations. We choose them in such a way that they resemble the phenomena that served as the reason for the formation of these concepts. Thus, geometric figures are images for recalling spatial concepts and as such are used by all mathematicians. Who does not connect with two inequalities a>b>c between three quantities a, b, c, image of a trio of rectilinearly located and next friend behind each other of points as a geometric interpretation of the concept “between”? Who does not use the image of segments and rectangles nested within each other if one needs to carry out a complete and rigorous proof of a difficult theorem on the continuity of functions or the existence of a limit point? Who can do without the figure of a triangle, a circle with a given center, or without a trio of mutually perpendicular axes? Or who would want to abandon the image of a vector field or a family of curves, or surfaces with their envelopes - concepts that play such an essential role in differential geometry, in the theory of differential equations, in the foundations of the calculus of variations and in other purely mathematical fields of knowledge?
Arithmetic signs are written geometric figures, and geometric figures are drawn formulas, and no mathematician could do without these drawn formulas, just as he could not refuse to put in brackets or open them or use other analytical signs when calculating .
The use of geometric figures as a rigorous means of proof presupposes exact knowledge and complete mastery of those axioms that underlie the theory of these figures, and therefore, in order for these geometric figures to be included in the general treasury of mathematical signs, a strict axiomatic study of their visual content is necessary.
Just as when adding two numbers you cannot sign the digits of the terms in the wrong order, but you must strictly follow the rules, i.e. those axioms of arithmetic that govern arithmetic operations, so operations on geometric images are determined by those axioms that underlie geometric concepts and connections between them.
The similarity between geometric and arithmetic thinking is also manifested in the fact that in arithmetic studies we, just as little as in geometric considerations, trace the chain of logical reasoning to the end, right down to the axioms. On the contrary, especially in the first approach to a problem, in arithmetic, just like in geometry, we first use some fleeting, unconscious, not entirely clear combination, based on trust in some arithmetical instinct, in the effectiveness of arithmetic signs, - without that we could not advance in arithmetic just as we cannot advance in geometry without relying on the powers of the geometric imagination. An example of an arithmetic theory that operates in a strict manner with geometric concepts and signs * can serve as Minkowski’s work “Geometry of Numbers” **.
** Leipzig, 1896.
Let us make a few more remarks about the difficulties that mathematical problems can present and about overcoming these difficulties.
If we fail to find a solution to a mathematical problem, the reason for this is often that we have not yet acquired a sufficiently general point of view from which the problem under consideration appears to be only a separate link in a chain of related problems. Having found this point of view, we often not only make the given problem more accessible to research, but also master a method applicable to related problems. Examples include integration along a curvilinear path introduced by Cauchy into the theory of a definite integral and Kummer's establishment of the concept of an ideal in number theory. This way of finding general methods is the most convenient and reliable, because if one is looking for general methods without having any specific task in mind, then these searches are, for the most part, in vain.
In the study of mathematical problems, specialization plays, I believe, an even more important role than generalization. It is possible that in most cases, when we search in vain for an answer to a question, the reason for our failure is that simpler and easier problems than this one have not yet been solved or have not been completely solved. Then the whole point is to find those easier problems and implement their solution by the most advanced means, with the help of concepts that can be generalized. This rule is one of the most powerful levers for overcoming mathematical difficulties, and it seems to me that in most cases this lever is put into action, sometimes unconsciously.
At the same time, it also happens that we achieve an answer with insufficient prerequisites, or going in the wrong direction, and as a result we do not achieve the goal. Then the task arises of proving the unsolvability of this problem under the accepted premises and the chosen direction. Such proofs of impossibility were carried out by old mathematicians, for example, when they discovered that the ratio of the hypotenuse of an isosceles right triangle to its side is irrational number. In modern mathematics, proofs of the impossibility of solutions to certain problems play outstanding role; there we state that such old and difficult problems, as a proof of the axiom of parallels, as the squaring of a circle or the solution of an equation of the fifth degree in radicals, we still received a rigorous solution that completely satisfies us, although in a different direction than the one that was first assumed.
This amazing fact, along with other philosophical foundations, creates in us a confidence that is undoubtedly shared by every mathematician, but which no one has yet confirmed with proof - the confidence that every specific mathematical problem must certainly be accessible strict decision* either in the sense that it is possible to obtain an answer to the question posed, or in the sense that the impossibility of solving it will be established and at the same time the inevitability of failure of all attempts to solve it will be proven.
* We consider it necessary to present this statement, which is so decisive for Hilbert’s entire scientific worldview, in the original "...die uberzeugung, dass ein jedes bestimmte mathematische Problem einer strengen Erieitung notwendig fahig sein muss." - Note P.A.
Let's imagine some unsolved problem, say, the question of the irrationality of the constant WITH Euler - Mascheroni or the question of the existence of an infinite number of prime numbers of the form 2n + 1 . No matter how inaccessible these problems seem to us and no matter how helpless we now stand before them, we still have firm belief that their solution with the help of a finite number of logical conclusions must still succeed.
Is this axiom of solvability of each given problem a characteristic feature only mathematical thinking or, perhaps, there is a general law relating to the inner essence of our mind, according to which all the questions that it poses can be resolved by it? After all, in other areas of knowledge there are old problems that have been resolved in the most satisfactory manner and to the greatest benefit of science by proving the impossibility of their solution. I remember the problem about perpetuum mobile(perpetual motion machine) *. After futile attempts to design perpetual motion machine began, on the contrary, to explore the relationships that must exist between the forces of nature, under the assumption that perpetuum mobile impossible. And this formulation of the inverse problem led to the discovery of the law of conservation of energy, from which the impossibility follows perpetuum mobile in the original understanding of its meaning.
This belief in the solvability of every mathematical problem is a great help for us in our work; We hear a constant call within ourselves: when there is a problem, look for a solution. You can find it through pure thinking; for in mathematics there is no Ignorabimus! **
* Wed. H. HeImholtz, Uber die Wechselwirkung der Naturkrafte und die darauf bezuglichen neuesten ErmittIungen der Physik, report in Konigsberg, 1854 (Russian translation: “On the interaction of the forces of nature,” in the collection G. Helmholtz, Popular Speeches, ed. 2, part 1, St. Petersburg. , 1898. - Note ed. ).
**See footnote. - Note ed.
There are countless problems in mathematics, and once one problem is solved, countless new problems pop up to take its place. Allow me in the future, as if as a test, to name several specific problems from various mathematical disciplines, problems the study of which can significantly stimulate the further development of science.
Let's turn to the basics of analysis and geometry. The most significant and important events of the last century in this field are, it seems to me, the arithmetic mastery of the concept of continuum in the works of Cauchy, Bolzano, Cantor and the discovery of non-Euclidean geometry by Gauss, Bolyai and Lobachevsky. I therefore draw your attention to some problems belonging to these areas.<...>
1. Cantor's problem about the power of the continuum<...>The problems mentioned are just examples of the problems; but they are enough to show how rich, diverse and broad mathematical science is already; We are faced with the question of whether mathematics will ever experience what has been happening to other sciences for a long time, whether it will not disintegrate into separate private sciences, the representatives of which will barely understand each other and the connection between which will therefore become less and less.2. Consistency of arithmetic axioms
3. Equality of two tetrahedra with equal bases and equal heights.
4. The problem about direct how shortest connection two points.
5. The concept of a continuous group of Lie transformations, without the assumption of differentiability of the functions defining the group.
6. Mathematical presentation of the axioms of physics.
7. Irrationality and transcendence of some numbers.
8. The problem of prime numbers.
9. Proof of the most general law of reciprocity in any number field.
10. The problem of the solvability of the Diophantine equation.
11. Quadratic shapes with arbitrary algebraic numerical coefficients.
12. Extension of Kronecker’s theorem on Abelian fields to an arbitrary algebraic domain of rationality.
13. The impossibility of solving a general seventh-degree equation using a function that depends only on two variables.
14. Proof of the finiteness of some complete system functions.
15. Rigorous justification of Schubert’s calculative geometry.
16. The problem of topology of algebraic curves and surfaces.
17. Presentation certain forms as a sum of squares.
18. Construction of space from congruent polyhedra.
19. Are solutions to a regular variational problem necessarily analytical?
20. General problem on boundary conditions.
21. Proof of the existence of linear differential equations with a given monodromy group.
22. Uniformization of analytical dependencies using automorphic functions.
23. Development of methods of calculus of variations
I don't believe in it and I don't want it. Mathematical Science in my opinion, it represents an indivisible whole, an organism, the viability of which is determined by the coherence of its parts. Indeed, despite all the differences in mathematical material in particular, we still very clearly see the identity of logical auxiliary means, the similarity of the formation of ideas in mathematics as a whole and numerous analogies in its various fields. We also notice that the further mathematical theory develops, the more harmonious and unified its structure takes shape, and unexpected connections open up between hitherto separated areas. It so happens that with the expansion of mathematics, its unified character is not lost, but becomes more and more distinct.
But - we ask - with the expansion of mathematical knowledge, does it not eventually become impossible for the individual researcher to cover all its parts? By way of answer I wish to refer to the fact that the nature of mathematical science is such that every real success in it goes hand in hand with the discovery of stronger auxiliaries and simpler methods, which at once facilitate the understanding of earlier theories and eliminate the difficulties of old reasonings; therefore, the individual researcher, thanks to the fact that he will internalize these stronger aids and simpler methods, it will be easier to navigate various areas of mathematics than is the case for any other science.
The unified nature of mathematics is due to inner being this science; After all, mathematics is the basis of all exact science. And in order to perfectly fulfill this high purpose, may it find in the coming century brilliant masters and numerous adherents burning with noble zeal *.
* In the original these words sound like this: "Der einheitliche Charakter der Mathematik liegt im inneren Wesen dieser Wissenschaft begrundet; denn die Mathematik ist die Grundlage alles exakten naturwissenschaftlichen Erkennens. Damit sie diese hohe Bestimmung vollkommen erfulle, mogen ihr im neuen Jahrhundert geniale Meister erstehen und za hlreiche in edlem Eifer ergluhende Jungerl" - Note ed.