Reflections on the biography of Mathematics

Below we give a version of the talk by Pedro Abellanas Cebollero delivered on the opening of the 1979-80 session of the Universidad Complutense de Madrid:


Right Honourable rector, ladies and gentlemen:

Today begins a new session in this University and it was left to a professor of Geometry to take charge of the inaugural lecture. This means that this lecture will deal with Geometry. It is evident that the act that brings us here has come to have a symbolic character. It would be very difficult to obtain the participation of all the professors and students of the University, but these difficulties are not the greatest inconvenience to the realization of such a meeting. What really prevents us all from meeting, even if only once a year, is the degree of intellectual distance reached between them, as a consequence of the superspecialization that has happened in the University. The study of the problems that constitute the living part of science leads to theories, and these need, for their proper formulation and development, techniques. The fact that has occurred in this century, unprecedented in the history of thought, has been, on the one hand, the constant proliferation of theories and techniques, and, on the other hand, the increase of the speed of dating of the one and the other, especially of the latter. If the University is dedicated to teaching theories and techniques, it may find itself in a senseless situation, since, in addition to requiring a budget incompatible with the possibilities of the Nation, it would lead to the production of graduates who, in the best of cases , would be specialists in expired techniques. Therefore, this act, although symbolic, has the value of a testimony of the will of the University to find a solution to this serious problem of our time.

It would seem natural, in a first lecture, to begin by giving them a definition of Geometry. This was possible when Geometry was less developed, but now we are not able to say what Geometry is. For a very simple reason: because Geometry is a living subject and therefore changing. The aspect that it presents today is different from yesterday and tomorrow. So I think it is worth it to contemplate Geometry at various times in its life, so that perhaps you can get some information about what is what for twenty-three centuries called Geometry; and this with the secret hope on my part that this contemplation of Mathematics may serve as an example for other specialists to compare, in order to begin the path that may lead us to find out, and, therefore, to interest ourselves, in our reciprocal problems. With this spirit I invite you to reflect with me on the aspects of Mathematics that I have considered more interesting.


  1. GEOMETRY.
    Geometry, like living beings, has a temporal dimension, depends on the variable time, but not on the mode of historical facts, but exactly with the characteristics of an authentic living being with its own genetic code. Its birth is the Elements of Euclid. The quantity and quality of mathematical knowledge at the beginning of the third century (BC) required an ordering for use by a limited memoir. This ordering is the Elements. The language of this work is geometric, but it also exposes ideas and problems which are arithmetic, algebraic, and what today is called mathematical analysis. The Elements are all the Mathematics of its time; or, in other words: Greek Mathematics was Geometry. In the Elements all the fundamental problems of Mathematics appear. In this first picture of the new being all the features of the adult being are accurately recognized. It is not a waste of time that we take a few minutes to distinguish them. Already on the first page of the first book we find these definitions:
    Def 2. A line is a length without width.

    Def 5. A surface is what has only length and width.

    Def 6. The ends of a surface are lines.

    Definition 2, with today's loose language, corresponds to saying that a line is the result of deforming, without breaking, a piece of a straight line, which in mathematical terminology is expressed by saying that a line is a subset of points of space homeomorphic to a line segment. Definition 6 is not properly a definition, but a proposition. In order to understand it, it is necessary, first of all, to keep in mind that for the Greeks infinite figures were meaningless; throughout the whole work, a straight line is always a segment capable of being prolonged, so it is constantly said: "extending the line AB". Therefore, surfaces, for the Greeks, always had a border, when they were not was closed, like the spherical surface. With current language, definition 6 would be stated: the intersection of two surfaces is a line. The equivalent condition of definitions 2 and 6 constitutes one of the fundamental theorems of present Mathematics: the theorem of the implicit function.

    Book V, attributed to Eudoxus, is probably the most beautiful and finished book of the Elements. In it the theory of magnitudes is constructed with great perfection, in a completely modern way, and certainly much superior to that which appeared in the textbooks of fifty years ago. A magnitude is a set whose elements are called quantities and that have the form, for example, of 5 m, π m, etc; where a number and an amount appear. With the quantities two operations are defined: addition: 3 m + 5 m = 8 m, and multiplication by numbers: 2 (3 m) = (2 × 3) m = 6 m, which have well-known properties. These sets are now referred to as vector semispaces. The greatest difficulty of the construction of magnitudes is in defining the set of numbers, which must be that of the real numbers. What is surprising is that the concept of real numbers has been one of the most difficult to establish precisely, and to Dedekind, at the end of the last century, we owe the first correct definition of these numbers. Dedekind, in the preface to his memoir: Was sind and was sollen die Zahlen, confesses that the idea developed in that memoir is not his own, that he has limited himself to expressing in modern language the construction given in Book V of Euclid. It's amazing! Euclid's book V has been unintelligible for more than twenty-two centuries.

    But this fact is not unusual, it is repeated many times, but not over such a long period of time. The numbers that the Greeks define in Book V are not the positive real numbers, as Dedekind thought; but contains them as a particular case, as Krull saw. The set of negative numbers took a long time to reach, and during the eighteenth century there were still great controversies about negative numbers. What is now called Linear Algebra, is therefore perfectly constructed, for vector semispaces of dimension 1, in the aforementioned book V. But there is more: multilinear algebra, built on the idea of the tensor product, which appears in the mid-twentieth century as one of the most modern theories of Mathematics, is perfectly developed, always on vector spaces of dimension 1, in books II, VII, VIII and IX, and this is not because the Greeks, as we have just pointed out, did not work with numbers, but with magnitudes, so they were forced, in order to define multiplication, to use the tensor product. In the just mentioned books VII, VIII and IX there is also obtained fundamental results of the Theory of Numbers, studying divisibility and arriving at giving a beautiful demonstration of the infinity of prime numbers. In Book IX a geometric progression is added.

    Books I, II, III and IV contain the traditional geometrical part relating to the plane, and books XI, XII and XIII the geometry of space; parallelism and perpendicularity in space, polyhedral angles, volumes of prisms, pyramids, round bodies and the construction and calculation of elements of regular polyhedra.

    Let us take a moment of attention to Book X. Let us begin by remembering that the Greeks said that the segment AB was commensurable with the CD segment when there was an integral (and positive) multiple of AH equal to another multiple of CO. If this was not the case, it was said that AB was immeasurable with CD. The latter is also expressed by saying that the measure of AB with the CD unit is an irrational number. The Pythagoreans already knew that the diagonal of the square was immeasurable with its side, but what Euclid proposes in this book is to see how many irrational numbers exist. Note that irrational numbers cannot be represented by a finite number of figures, and for the Greeks, the modern infinity was unthinkable, so these numbers were surprising; but nevertheless, as they could be constructed by a finite number of geometrical constructions, they could be admitted without difficulty, but only those that admitted constructions of this type. Hence the interest to fully analyse this set. This is the purpose of Book X, and the fundamental result of it is to demonstrate that there are infinitely many irrational numbers, that is, to show that, given several irrational quantities, they are able to construct a different one from the previous collection. With current terminology what is achieved in this book is to construct quadratic or biquadratic equations whose roots are real and positive. The beauty of this book and the ingenuity employed in it are extraordinary. However, over the centuries many have wondered what the book is for, the results of which are not applied at any time, since the whole effort to study the problem of the real quadratic irrational cannot be considered justified by the application to the calculation of the elements of the convex regular polyhedra of Book XIII. Twenty-two centuries later, from Gauss's Disquisitiones Arithmeticae, one can understand the meaning of it. The problem of this book is of the strictest actuality, since it is one of the problems that are at present under discussion: the study of real algebraic properties. It is very difficult to continue our meditations without opening a small parenthesis to ask the question: "what would have happened if Euclid had to fill out forms to justify the importance of his research?" The parentheses are closed.

    Let us continue our observation of the newborn Geometry. The problem of measuring the volume of prisms is discussed at the end of Book XI. The idea is simply this: two prisms have the same volume when they can be cut into pieces, so that when recomposing all of the pieces in a convenient way the result in both cases the same cube. This problem is fully resolved in that book. In Book XII it is attempted to do the same with the pyramid. But this is no longer possible. That is, you cannot chop a pyramid into a finite number of parts such that when you reassemble them you get a cube. To solve this problem it is necessary to use the most important idea of Mathematics, which is the concept of a limit. This concept is practically contained in the method of EXHAUSTION, which is also attributed to Eudoxus.

    If this was so, the genius of Hellenic antiquity would be Eudoxus, since he solved the two most difficult problems: creation of real numbers and concept of limit. The concept of limit is the foundation of Newton's and Leibniz's Infinitesimal Calculus, so that the method of exhaustion can be considered as the germ of this important chapter of Mathematics. With the method of exhaustion the volumes of pyramids and round bodies are calculated in Book XII. But there is more: what is really achieved in this chapter is to reduce the calculation of a volume to the calculation of areas, or with current language, the calculation is reduced from that of a triple integral to that of a double integral, which is Stokes' important mathematical theorem of our day.

    Finally, Book XIII is dedicated to the construction of the five convex regular polyhedra. And again you can ask the question: "what are these polyhedra for?" Excluding the cube, it can be said that at that time for nothing. However, from the introduction of the idea of a group by Galois in the last century, regular polyhedra have provided models of groups with applications to algebra, the theory of automorphic functions, crystallography, and so on.

    We see, then, that in this picture of Geometry in its infancy we observe all the fundamental characteristics that will appear developed in its adult age. The most important thing is to observe that the basic problems of Mathematics are perfectly posed in the Elements, that is, linear and multilinear problems, number theory, problems of local approximation and problems of measurement.

  2. ABSTRACT STRUCTURES.
    Let us pass over eighteen centuries and observe the aspect of Mathematics in the Renaissance, concretely in the sixteenth century. Subsequently to Euclid new results were obtained, especially with the contributions of Archimedes, Ptolemy and Apollonius, but it does not vary the fundamental conception of reasoning on concrete magnitudes. The important fact that occurs in the Renaissance could be said to be an oblivion. We have seen that quantities consisted of two parts: a number, its coefficient, and a magnitude, which is taken as unity. Well, in the Renaissance they forget this second part, unity, and it is replaced by a sign, for example, the letter x, about which nothing is known, it is only said how one can operate with it and establishes that with that symbol x is operated as if it were a natural number. This simple fact produces an extraordinary simplification, because when forgetting the quantities, which were vectors, and that could not be multiplied as numbers, what required one to resort to introducing a multiplication of the tensor product, considering the symbol having empty content and operating with it as if it were a number, we immediately obtain a set, whose elements we now call polynomials, with which we operate as with integers. But this is not all that happens, but this cheerful and trusting spirit leads us to work with negative numbers without knowing exactly what these numbers are, without excessive rigorous concerns; simply because things go better by using those numbers. Then the controversy and the difficulties will come. With current language it can be said that the great contribution of the Renaissance was to replace the tensor algebra of the Greeks by the algebra of the polynomials. Obviously this is a trap and we will see that until very recently this arbitrariness has not come to light, but inexorably all falsehood is discovered sooner or later, and so it has happened with this one that could be called the protest attitude of the first youth of Mathematics. However, this phenomenon of forgetting, this falsehood, was tremendously positive for the development of Mathematics, because it allowed it to move forward by substituting the tensor algebra, which is there, in the phenomena of the Universe, by a simpler one: the algebra of polynomials. It is curious that the commentators that I know do not realize the fact that it actually occurs and think that it is simply to have discovered an adequate notation. The important thing is to admit that with the symbol x can be operated as with a number, which in the case of magnitudes is not true. The process of forgetting some properties and keeping a part of them is called abstraction. Therefore, in the Renaissance a second process of abstraction takes place. This process, as a result of what is said, does not properly affect Geometry, but to the non-geometric part of the Elements, The Algebra, which in the Elements is treated geometrically, becomes independent of Geometry.

    The problem of quadratic irrationalities studied in Euclid's Book X now admits, within the algebra of polynomials, a precise and simpler formulation, with which it advances remarkably and obtains the formulas of resolution of the general equations of third and fourth degrees in which Scipio del Ferro intervened using methods of the XVI century, and Cardano and Viète with their schools along the same lines. With this it was clear that not every equation has roots that are quadratic irrational, that the third and fourth degree equations have roots that are cubic irrational and that, therefore, cannot be constructed graphically with ruler and compass. This is the reason why the Greeks could not go beyond what they had done.

    Every process of independence is followed by another process of annexation and submission. It is natural that Algebra soon began to invade Geometry. This process was carried out in the following century by Fermat and Descartes. The introduction of coordinates by these sages, and the systematic development made by Descartes, allowed one to obtain geometric properties from algebraic properties, initiating a new process of unity in Mathematics contrary to that of the Greek period. But it still took a long time to get back to the unit. To summarize: the first youth of Mathematics is characterized by the appearance of the first abstract structure: the algebra of the polynomials.

  3. THE PROBLEM OF APPROXIMATION.
    The trigonometric ratios, which offered no difficulty in handling the theory of magnitudes in Book V, presented a serious problem of calculation in removing them from that dizziness and considering them as numerical functions, since then, as the irrational numbers require for their representation infinitely many numbers, and there is no possibility of writing so many figures, it becomes necessary to go to the concept of approximation. Since, on the other hand, the only operations known to be performed are addition and multiplication, which are the only ones involved in polynomials, to calculate non-polynomial function tables, a general process was needed to substitute for a non-polynomial function another that it was polynomial, and that it approached the first as closely as desired. This is the so-called local approximation problem. If one reflects a little, it will be seen that the solution of this problem was key to progress not only of mathematics, but in all experimental sciences. The honour of constructing the theory of local approximation corresponded, as is well known, to Newton and Leibniz, and the theory was called infinitesimal calculus. An infinitesimal is simply a function defined in the vicinity of a point, which is generally the origin of coordinates, which has the property of having a limit when the variable approaches that point. The fundamental concept, therefore, that intervenes here is that of limit. It is thanks to this concept that Mathematics could continue to overcome the limitations of Greek thought, which came as close as possible with the method of exhaustion, as we have seen before, but failed to overcome this stage. The concept of limit allowed one to compare infinitesimals. Two infinitesimals are called equivalent when the limit of their quotient was a number other than zero. With this, and taking as a canonical family of infinitesimal powers of the variable, we can assign to each infinitesimal a number, which was called its order, and in this way we obtained a measure of the local approximation of two functions, which is the essential point of the theory of this approach. As a particular case, two problems that have been systematically solved since the time of Apollonius were solved: the problem of tracing the tangent to a curve at one of its points and the problem of calculating the area of an enclosure bounded by a curve; they were the reverse of each other.

    There is no doubt that Newton and Leibniz clearly had the idea of a limit, but the development of Mathematics at that time did not allow an adequate formulation of it. It took almost two centuries for the correct definition of a limit to appear, due to Cauchy. However, from Newton and Leibniz, mathematics and physics acquired an bursting development as has never been previously known.

    Limiting to our theme of Geometry, let us see how it is affected by this great discovery of Newton and Leibniz, and what it looks like at this time, which would correspond more or less to the XVII, XVIII and first third of the XIX century, and which could be considered as the second youth of the Mathematics and to represent it by the name of Newton. At the beginning of the seventeenth century an extraordinary geometrician appears, Desargues, who constructs a new space, that later was called projective, suitable to study global properties of figures of the first and second order. The mathematical environment of the time is not prepared to accept this work, the reason why this happens is lost in the libraries. The infinitesimal calculus allowed the local study of curves at all the points of these where it was applicable, that is, in those in which there was a single tangent. At the points of the curves with more than one tangent, the method is not applicable, and it was the same Newton who gave the method to study these points; a method that is currently in use, but one that has not been found bettered, which is further proof of Newton's genius. The most important contribution to the study of Geometry through the methods of infinitesimal calculus is due to the great mathematician of the last century, Gauss, who can be considered as the founder of what we know today as differential geometry. But geometry in the sense of the Greeks was not resigned to dying. In addition to the work of Desargues, which we have just mentioned and which did not directly influence matters, there continued to be cultivated the topic that was called pure geometry, highlighted by the work of Monge, the creator of descriptive geometry. But the most important contribution to pure geometry, that is, geometry without numbers, was due to Poncelet (1818), who rediscovered the projective space of Desargues, but now in a more complete way and with full awareness of the purpose of that space, which is to study global properties of figures defined by algebraic equations. For the study of these global properties Poncelet enunciated the so-called "principle of continuity," also known as the "principle of the conservation of numbers," which he could not formulate precisely because, as on previous occasions, he had no mathematical structure adequate to do so. The controversies about this principle were numerous, not always carried out with a scientific spirit, and they continued. The creation of Poncelet was a respite for pure geometry, which lasted a short time, because Plücker soon introduced coordinates to study projective space. In short, Geometry advances, like all science, in a substantial way, but progress is achieved mainly by the use of coordinates and differential calculus. Attempts to keep Geometry independent of the number idea are quickly reabsorbed by the analytical methods of coordinates.

  4. THE CRISIS.
    The transition from youth to the maturity of Mathematics occurs through a deep crisis that originates in Geometry. The process was precipitated, like that of all crises. In 1813, Gauss communicates by letter to a colleague who has obtained a consistent model of geometry without the postulate of Euclid's parallels. In 1826, Lobachevsky, and simultaneously Bolyai, constructed other models of non-Euclidean geometries, and so did Riemann in 1852 when studying Geometry on a surface. On the other hand, von Staudt published in 1848 his treatise on projective Geometry, making a rigorous construction of this geometry based on Euclidean geometry, with what seemed to consolidate pure geometry, independent of the idea of number. However, pure geometry was criticized for not being able to graphically represent points, lines and imaginary planes. The construction of Staudt was perfect, within the possibilities of rigour of the time; but what Staudt demonstrated by his construction was the exact opposite of what he intended, which was to free Geometry from the idea of number, for it proved the identity of pure geometry with analytical geometry over the field of complex numbers. This transcendental result submerged Geometry within Mathematics. Then there was the problem of saving the personality of Geometry within the Mathematics that had absorbed it. Neither Staudt nor many of the later geometers realized the significance of the result achieved. Fortunately, Cayley, in the middle of the last century, devoted himself to a profound study of projective geometry with the use of coordinates and advanced in its development, being able to see that many properties of Euclidean Geometry could be obtained as particular cases of projective properties, which led him to exclaim: "Projective Geometry is all Geometry."

    Later, Klein elaborated a little more the idea of Cayley, arriving at the formulation of his definition of Geometry in its famous 'Erlanger Programm'; according to Klein, geometry was that part of Mathematics that studied the properties of the invariants figures with respect to a group of transformations. With this it seemed that, although Geometry had ceased to represent the main role, it had not lost its personality. Klein's definition leaves out a large number of geometric properties outside Geometry, but had the virtue of developing an important theory of Mathematics: the theory of invariants; since this definition reduced Geometry to the study of the invariants of the group of transformations.

    These ideas led many mathematicians to look for invariants of the group of transformations that were well known, which was that of projectivities. This chase for invariants led to increasingly complicated symbolic calculations indicating that it was poorly posed. Because a young mathematician at the turn of the century, David Hilbert, posed and solved the fundamental problem of proving that all projective invariants could be obtained from a finite number of them, this made sense of these studies and closed the problem.

    Another way to limit the neighbourhood of Geometry was to give a new and more rigorous axiomatization of it. The first to follow this path was Pasch in 1882, who published his modern geometry. This is the first time that the term modern in mathematics appears and is really the indicator of what was to be called later modern mathematics. In fact, the problem that had arisen with all these events was the need for a new ordering of Mathematics, and this had to do with the fundamentals. The book of Pash does not contain any new properties of Geometry, it is a matter of providing Geometry with a more solid foundation. Pasch's axiomatics still have many deficiencies, and Hilbert later formulated a rigorous axiomatization of Geometry. But this was not enough, because the problem was not only Geometry, it was all Mathematics. It was a crisis of growth, which required revising all mathematics. Peano, with very fine intuition, realized that not only Geometry, but also Arithmetic needed a solid foundation and elaborated an axiomatization of the natural number system which is still in use. But with all this, it was necessary to deal with the complete problem of obtaining a rigorous foundation for Mathematics. Fortunately, Cantor had begun to develop a theory of sets, which had suitable characteristics to support the entire mathematical edifice. And so it was done. It started from the concept of a set as a primitive and undefined concept, but paradoxes began to emerge. Some of these paradoxes were attributed to the use of the principle of "tertium non datur", so that the so-called intuitionist school of Brouwer and his disciple Heyting emerged, which eliminated that principle. For Brouwer only constructive demonstrations were valid. It is evident that constructive demonstrations are most desirable both from the utilitarian and the aesthetic points of view; but unfortunately, within intuitionism, mathematics is reduced to almost nothing, since a problem as elementary as that of finding out whether or not two planes are the same plane does not always have an answer. However, most of the paradoxes that were presented had their origin in working with sets that were not well defined, as is the case of the paradox of the bridge that they presented to our Mr D Sancho, the best governor of islands that the world has seen, which he solved by applying the wise and prudent teachings of his master and teacher, which are in accordance with Hilbert's solution in his article "On the Infinite" of the Mathematische Annalen, in which he said: "No one will force us from the paradise Cantor has created for us. If there arise paradoxes we will apply ourselves with all the enthusiasm to analyse them, to detect their origin and eliminate the causes." This position of Hilbert led him to attack frontally the problem of the foundation of Mathematics. Hilbert noted that the difficulties presented in Mathematics came from either managing the actual infinity, as well as using infinite processes. Hence Hilbert departed from what he called finite formal systems. Consider a system F formed by a finite number of symbols, logical laws and axioms, which contained the system Z of natural numbers defined by Peano. The problem was to show that F was not contradictory. Despite his efforts, for more than thirty years Hilbert did not achieve his goal. It was his disciple Gödel, who came to prove that inside F could not prove the consistency of F. This result, although negative, closes the period of crisis and opens a new permanent field of work in Mathematics: the Fundamentals. We find that at the end of the whole crisis of Mathematics the solution of the same coincides with that of our governor of the island Barataria, so based on the fact that every problem has a solution, we must apply all our effort to find it.

  5. RETURN TO CLASSICAL MATHEMATICS.
    Fermat had argued that, for n > 2, the diophantine equation xn + yn = zn had no solution in whole numbers. This theorem, known by the name of Fermat's last theorem, has not yet been proved, but there have been numerous mathematicians of all ages who have dealt with it. One of them was Kummer, in the seventies of the last century. Studying this problem introduced Kummer into numbers, which he called ideal numbers, which allowed him to demonstrate the uniqueness of the factorial decomposition in cyclic fields. These numbers were generalized by Dedekind for the study of the algebraic functions of one variable, creating so-called ideals, which allowed him to elaborate an arithmetic theory of algebraic functions of one variable, which systematized the study that was carried out at the end of the last century with transcendent methods, that is, using the integrals that Abel had introduced on an algebraic curve. In the same volume of the year 1882 of the Crelle's Journal appeared the Dedekind theory of algebraic functions, supported by the ideals, and another algebraization of the same due to Kronecker. These two memoirs can be considered as the origin of today's mathematics, which we could designate as the maturity of our living being. The ideals introduced by Dedekind allowed the unification of the study of the algebraic theory of numbers and the algebraic functions of one variable, and the most interesting case is that the ideals have the same structure as the magnitudes of the Greeks, which had been forgotten since the Renaissance, although not its name, since it continued being denominated in the books of Analysis like magnitudes to the numbers. The physicists, however, who are in permanent contact with the Universe, could not at any time apply the functor to the magnitudes; for this reason, vector spaces and the tensor product of vector spaces as we conceive them today appeared in Physics much earlier than in Mathematics. However, Dedekind's new ideas were slow to develop beyond the realm in which they were conceived. It was through his disciple Emmy Nother and her school (E Artin, W Krull, van der Waerden) that they came to constitute what is now called Commutative Algebra. Parallel to non-commutative algebra was developed the permutation groups, introduced by Galois to study the general problem of the reduction of algebraic equations and the groups of continuous transformations introduced by S Lie. This period of forty years, in which the ideas of Dedekind, Kronecker, Galois, and S Lie remained latent, the development of Algebra was matched to the complement of Topology. In 1914 Hausdorff published his Theory of Sets. which promoted the systematic and rigorous development of the construction of spaces, as we conceive them today, called topological, and, with it, another fundamental structure of today's mathematics, the Topology. Around the year 1930 the first two books were published according to the new order of Mathematics: 'Modern Algebra', by van der Waerden, within the school of E Nöther, and the 'Introduction to Topology', of Seifert-Threlfall. These two books facilitated the formation of mathematics students in those years according to the new mathematical ordering and allowed this new generation of mathematicians to assume the work of the moment, what was the reconstruction of Mathematics from algebraic and topological structures, which allowed rigorous demonstrations. All the problems had to be considered from the beginning, to find new formulations of them within these structures and to give correct demonstrations of all the known theorems. A group of young French mathematicians, in the previously mentioned 1930s, who organised themselves under the pseudonym of Nicolas Bourbaki, contributed significantly to this work. The work plan that was proposed by this group was very ambitious: they began by writing a chapter on Set Theory, continued with the chapter of Linear Algebra, another on Topology, others on Function Theory, Algebra Commutative, Algebra and Lie groups, Differential and Analytical Varieties, etc., as new chapters still appear. They gave to the work, justly, the name of 'Elements of Mathematics', since they intended, and if possible have succeeded, to be the substitute of the 'Elements' of Euclid for the future. The contribution of this monumental work, some of whose chapters consist of ten volumes, has been extraordinary for mathematics. The group's work has consisted of periodic meetings of the group, about four per year, which they call Seminars, which present and study the results, both their own and others, which they considered of interest for the progress of Mathematics, and that in their day may constitute a new chapter or a new issue of an already existing chapter.

    The group has been renewed over the years and tries to incorporate similar young mathematicians with great creative ability. In the last Seminar of June, the "exposé" number 542 was reached. Fortunately for Mathematics, and although N Bourbaki's idea was to keep the book up to date by updating the successive editions, this is practically unfeasible as a consequence of the constant progress of mathematical thinking, but, in any case, it constitutes a solid point of reference for the work of the mathematician.

    But let's not forget that we are in a Geometry class, so let's get back to our subject. The first thing we have to ask is whether Geometry has survived all these profound structural changes or whether it has been disrupted in the storm and scattered its foundations. We have already seen that, following the involuntary contribution of Staudt, Geometry was incorporated into Mathematics, Cayley and Klein thought to save its personality as projective Geometry, and that after the crisis we have not found it again. This has been intentional, because we needed previously the current framework of Mathematics to be able to locate in it the sudden changes of Geometry. What died with the incorporation of Geometry into Mathematics was called pure geometry, but in the last half of the last century Geometry continued to develop. On the one hand, we studied figures and transformations defined by polynomials; on the other hand, the study of properties of curves and surfaces was continued with the aid of Mathematical Analysis. The first studies came to constitute Algebraic Geometry and the others Differential Geometry.

    The new contributions of Dedekind and Kronecker implied an algebraization of Algebraic Geometry, which allowed, in the hands of van der Waerden and Zariski, its solid and rigorous construction.

    The first thing that was achieved was to solve the problem posed on the first page of the Elements of Euclid relative to the definition VI of a curve in the case of varieties defined by polynomials, reaching a correct and usable definition of algebraic varieties and, consequently, of algebraic correspondences. On the other hand, in Differential Geometry the same problem was solved in the definition of differentiable varieties. The concept of a differentiable variety, by extension, gave rise to that of a topological variety and, by specialization, that of an analytical variety.

    In the study of analytic varieties, the concept of a structure sheaf (Cartan-Leray) emerges naturally, which Serre moves to the case of algebraic varieties, thus solving in a complete way the problem of finding the relation between the two definitions of curves of the 'Elements'. With all this, the first definitions of the Elements of Euclid had only been obtained, but Geometry remained to be done. It was now able to begin to demonstrate the great amount of results obtained between 1850 and 1950 in both the field of Algebraic Geometry, by the Italian and German geometric schools and. using transcendent methods. by the French school, and in the field of Differential Geometry, by French and Italians. In the field of Algebraic Geometry arises in the middle of this century an extraordinary mathematician, A Grothendíeck. which proposed, on its own, the immense task of organizing all existing knowledge of Algebraic Geometry and begins to publish a monumental work, whose title carries the word "elements" again. These are the Elements of Algebraic Geometry. In the introduction to this book the author says that his goal is to come to understand him the Geometry of the Italian school. Serre's works already provide him with the concept of algebraic variety, which he generalizes by replacing it with "schema". But to study Geometry we must add something, corresponding to the magnitudes that the Greeks had used in Euclidean space, and this adequate concept to be able to formulate and demonstrate the results of the Italian school turns out to be the one of modules. Now, a bundle of modules is but a generalization of the idea of magnitude of the Greeks, with what closes the cycle and we are with the surprising fact that to reconstruct the Geometry after the great crisis of principles of century returns to take the same aspect as the theme for the Greeks.

    This return of Geometry to its original structure makes it possible to predict that it will again play an important role in the study of Physics and, indeed, has already begun to realize this conjecture. since a little over a year ago. Atiyah got, with surprise from both mathematicians and physicists, interesting applications of current Algebraic Geometry to the study of an important physical phenomenon.

    Indeed, Geometry is not dead. What happens is that in order to be able to say what Geometry is, a hundred words are not enough. It is necessary to use many books of many pages. just as to define a person is necessary to see him act during his entire life. The present Mathematics is more like Greek Mathematics than that of the three preceding centuries. so his most appropriate name is Classical Mathematics.

  6. EPILOGUE.
    The new foundation of Mathematics is based on a very simple principle, which can be stated as follows: "In order to be able to demonstrate the propositions with rigour, it is necessary to empty the concepts of primitive content, limited to establishing the allowed logical mechanism and the fundamental relations between such concepts."

    The situation is the same as that with card games: a game is characterized by its rules and can be played with either French or Spanish cards interchangeably, being independent of the meaning attributed to the cards. In these conditions it is always possible to check if a play is correct or not, with the only condition that the rules are not contradictory. As we have seen above, these sets, formed by symbols or names that represent concepts, by the laws of logic used and by the relationships between those signs, are called Hilbert formal systems. Therefore, mathematics works with formal systems. Although, in principle, any non-contradictory formal system can be studied in Mathematics, only those that have been extracted from a problem of interest within already constructed mathematics or the experimental sciences are considered as systems of scientific interest.

    Formal systems are important because they have allowed the new ordering of mathematical knowledge, as well as achieving verifiable demonstrations, but it should not be forgotten that they are not the essential part of mathematics. We have seen that mathematics has a permanent feature: that of studying and solving problems. These problems are grouped, throughout the life of Mathematics, in the following three major problems:

    1. Linear problems. - Essentially these problems try to find out if a set of vectors, or elements of a module, are linearly dependent. This problem begins with the theory of proportionality in Euclid's book V and with the multilinear problems we have pointed out earlier. They continue with the linear problems of Descartes' Analytical Geometry, as well as the linear problems of Staudt's Projective Geometry, linear systems of Italian Algebraic Geometry, coherent modules of Grothendieck's Algebraic Geometry, cohomology groups, abelian groups, algebraic theory of numbers, topological vector spaces, etc.

    2. Problem of measurement. - It begins with the theory of magnitudes in Book V of Euclid and the tensor product of books VI, VII, VIII and IX; the calculation of areas of polygons and volumes of polyhedra, already mentioned in the Elements, with the method of exhaustion. It continues with the integral calculus of Newton and Leibniz, and Stokes' theorem. Its complete formulation gives rise to the theory of measurement. On the other hand, within this problem of measurement is the Pythagorean theorem, metric spaces, and so on.

    3. The problem of local approximation. - This problem appears as soon as decimal numbers are used in arithmetic operations. Its first formulation and general resolution gave rise to the differential calculus of Newton and Leibniz, with Taylor's theorem and later those of Weierstrass's implicit and preparatory function, the local study of differential equations, the problems of the dimensions in the theory transcendental numbers. The more general formulation of the problem gave rise to topological spaces.

    It is evident that many important theories of today's mathematics could be mentioned that would surely be difficult to include in one of these three great problems, for example, the theory of finite groups, of Lie groups, etc. But nonetheless, I believe that these three problems contain that part of Mathematics with more applications both in Mathematics and in other sciences, so that on them should be focus in training mathematicians and the users of Mathematics. Another issue is that of the specialist in a certain type of problems.

    The new structure of Mathematics raised the problem of adapting teaching to it at all levels. Since 1960, the number of international meetings for the study of this problem has grown dramatically, and important curriculum reforms have been carried out in many countries, from kindergarten to university. Sufficient time has elapsed before a criticism can be made of the results obtained with this reform. Evidently there have been many differences in reform from one country to another; but, however, negative criticism has the same characteristics in all of them, varying naturally in their the peculiarities and their gravity in each country. The following facts can be stated schematically:

    1.) In primary and secondary education, we went from dogmatic and rote learning methods supported by materials based on the Elements of Euclid to others of the same characteristics supported in the theory of sets.
    2.) In the above mentioned grades, the teaching was divided into two parts without any connection between them; one that was called modern mathematics and that of the traditional teaching, which only managed to lengthen the programmes and cause more dispersion of knowledge.
    3.) Higher education was broken up into the independent study of new theories, unrelated to each other and away from the problems that had originated them.
    4.) In the primary and secondary schools, the new plans were implemented without time so that teachers could assimilate and adapt to the new techniques.
    The consequence of all this: the problem of teaching Mathematics needs a permanent study. There are no magic formulas to solve human problems, but there is a possibility of correcting defects using the experience of failure. The best lessons that can be learnt from the reforms that have been carried out is that we must avoid precipitation and jumps into the void. Let's try to train better teachers and the rest will follow as the result. At this point, one should begin by specifying what goals one should attain in each grade level and patiently study how to achieve them. This study should be jointly undertaken by teachers of all levels, without hurry and without pauses. Here, students of the University, an important work awaits you, that you cannot finish, nor the students of your students, but that therefore should never be abandoned.

    The new order of Mathematics has greatly extended the field of questions that are expected to be studied and provides more accurate and satisfactory tools to attack them, so initial research, that is the one corresponding to doctoral theses, has progressed and improved considerably. In Spain, without being able to think of triumphalists, this aspect hasn't been neglected in the development of doctoral theses of a good international level thanks to a few individuals, scattered throughout our land, who made the effort to catch up with the times, despite being, as will have been observed, the most active times in the life of Mathematics. But next to this research there is that of the important problems. Now, what are the important problems? This question was asked by HiIbert in his communication to the International Congress of Mathematicians held in 1900 in Paris.

    Hilbert begins by recognizing that: "It is difficult, and often impossible, to judge a priori the value of a problem, for the ultimate prize depends on the gain that science obtains from its solution." However, later Hilbert says: "Let me try to formulate below some concrete problems, in several branches of Mathematics, from whose discussion we can expect an advancement of Science." Hilbert then proposed twenty-three problems. The importance of these problems has been proven a posteriori. The best talents have been taking care of them, but many of the problems remain to be solved. Most of them have contributed effectively to the progress of Mathematics. At the last Bourbaki Seminar, last June, there was an exposition on the problem number sixteen. The solution to this problem is not yet envisaged, which is worth mentioning as an example. Hilbert says: "I find it of great interest to investigate the relative position of the real components of a plane algebraic curve of the real projective plane when the number of them is maximum." All this proves that the few creators of scientific thought that appear in the world have a deep vision of science that allows them to see what problems are important. Corroborating this view, confirmed in the case of Hilbert, the American Mathematical Society published two years ago a couple of volumes in which the current state of the study of twenty-three problems of Hilbert was given and there was also proposed twenty-seven new problems, concerning all fields of Mathematics, presented by outstanding mathematicians today, whose solution is considered important at this time for the development of Mathematics. This type of research in the great problems of our time is the one that, in my opinion, we do not have yet in our homeland. Studying these problems is a challenge for today's youth. It is necessary ingenious ideas are launched with impetus and enthusiasm to this company; young people with will and talent, and, if we are to heed Don Santiago, the will is enough.

    Nothing else for today. See you tomorrow.


JOC/EFR November 2017

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