*Some aspects of mathematics*, a version of which we give below:

Very Honorable Sir,

Magnificent and Most Excellent Sir,

Your Excellencies and Most Illustrious Sirs,

Faculty Members,

Ladies and Gentlemen:

I will refer to some aspects of mathematics, far from conventional teaching, but very important and extremely attractive from an intellectual and aesthetic point of view.

I have always felt that mathematics is not only science but also art and that it contains the kind of beauty in which poetry also participates. For this reason I will cite some words that the poet from Granada, Federico Garcia Lorca, wrote to Gerardo Diego. He said:

Regarding mathematics, some have felt something analogous to what Garcia Lorca expresses in relation to poetry, so they have tried to find out what they were handling and, hence, over many years, have tried to clarify accurately the fundamentals of mathematics. On the other hand, and in line with what Garcia Lorca says about the need for technique, I must add that work in mathematics sometimes requires considerable effort. Even specific mental habits have to be acquired beforehand, and even more, my experience tells me that knowledge becomes really clear only when incorporated into one's own life. The same sense of rigour does not seem to be a natural thing to man. Poincaré said that rigour in mathematics requires learning. One may wonder whether the effort to acquire certain knowledge or achieve certain outcomes can be disproportionate. Bertrand Russell, in hisBut what shall I say of poetry, what shall I say of those clouds, of that sky? Look, look at them, look at them, and nothing more. You will understand that a poet can not say anything about poetry.

Here it is, look, I have the fire in my hands. I understand and work with it perfectly, but I can not speak of it without literature. I can not speak of my poetry, and not because I am unaware of what I am doing. If it is true that I am a poet by the grace of God - or of the devil - it is also that I am by the grace of technique and effort, able to realize at all what a poem is.

*Autobiography*, calls the period in which he was more actively engaged in mathematical logic, his intellectual honeymoon; but, on the other hand, he declares that he abandoned mathematics because he did not feel up to doing such a hard job.

When referring to the fundamentals of mathematics, we must mention the so-called *Elements* of Euclid that occupy one of the first places among books that have been written throughout all time. It is a great monument to the Greek intellect, more important than the contributions which the Greeks contributed to literature or philosophy. The basis of all our mathematics can be said to be found in the *Elements* of Euclid.

Euclid was director of the famous Museum of Alexandria when Egypt was ruled by Ptolemy I and published his work, later called the *Elements* of Euclid, at the age of thirty, in the year 300 BC.

In the *Elements* the axiomatic method is initiated, which has had so much importance and repercussion and still has in the so-called modern mathematics. It can be said that Euclid was the systematiser of almost all the mathematical results known in his time, ordering them in a masterly way in a deductive system, demonstrating from a few simple geometric properties, self-evident and not requiring proof, according to the spirit of the time, all that follows as logical consequences of the former.

The *Elements* of Euclid are made up of thirteen books, three of which contain arithmetic and the rest geometry. There are collected in them many investigations of the Pythagoreans of the fourth and fifth centuries, like those of Hippocrates of Chios, Eudoxus and Thaetetus. For this reason I would like to say a few words about Pythagoras and the Pythagoreans. Indeed, Pythagoras was a very important person from the intellectual point of view, of great influence in both ancient and modern times. With him begin the idea of proof in mathematics and deductive-demonstrative arguments. Pythagoras was born on the Ionian island of Samos, in front of Miletus, and had its flourishing in period up to the year 530 BC. From 535 BC. to 515 BC. Samos was ruled by the tyrant Polycrates, who had absolutely no moral scruples: he even had a fleet engaged in piracy; nevertheless, he sponsored the arts and among its beneficiaries was the famous poet Anacreon. Pythagoras, who disagreed with the conduct of Polycrates, left Samos, visited Egypt and finally settled in Crotone, in what is now southern Italy, Magna Grecia. In the year 515 BC. Polycrates, deceived by the Persians was captured and, following the custom of the time, was crucified.

In Crotone, Pythagoras founded a society where he and his disciples dedicated themselves to philosophy and mathematics; property was common and scientific discoveries as well. Pythagoras used to mix mysticism with science; he believed in transmigration and the immortality of the soul, and affirmed that the greatest purification is obtained by dedicating oneself to disinterested science, and the man who does so can, more effectively, free himself from the wheel of birth.

An important discovery of Pythagoras, or one of his disciples, was the mathematical proposition that states that in a right-angled triangle, the area of the square built on the hypotenuse is equal to the sum of the areas of squares built on each of the other two sides. Already the Egyptians knew that the triangle whose sides have lengths 3, 4 and 5 has a right angle; but the Greeks were the first to observe the equality 3^{2} + 4^{2} = 5^{2} and, probably, inspired by it, reached the general proposition that we have quoted and which is known as the Pythagorean theorem.

As I have said before, the Pythagoreans introduced into mathematics deductive arguments, which are essential in the exposition and development of this science: every mathematician is aware that where one truly learns mathematics is in the demonstrations of theorems.

One of the greatest mathematicians of antiquity was the Pythagorean Archytas of Tarentum (440-360 BC), a worthy predecessor of Archimedes, who applied geometry to mechanics by constructing mechanisms analogous to those which later served Archimedes for the defence of Syracuse. All later Greeks who had a voice in mathematics were, directly or indirectly, his disciples. To Archytas is due a very ingenious solution to the famous problem of the duplication of the cube, which according to tradition was posed as follows: some priests of a temple erected to Apollo on the island of Delos were informed by the oracle that the god wanted the altar, which had a cubic shape, to be changed to another whose volume was double. At first, they thought to double the lengths of all edges, but they saw that the volume was eight times greater, so they addressed the mathematicians. The problem then arose in two different ways. The first consists of the following: given the edge of the cube, the problem is to construct, using as drawing instruments only ruler and compass, a segment that is the edge of a cube whose volume is twice the first. In the second, which Archytas resolved, it was not necessary to restrict himself to rule and compass.

Other classic, historically important, problems are the quadrature of the circle and the trisection of the angle. The quadrature of the circle consists of given a circle in the plane, construct from it, using only ruler and compass, a square whose area coincides with the area of the circle. In the trisection of the angle, we try to divide a given angle into three equal angles, using only ruler and compass.

Duplication of the cube, squaring of the circle and trisection of the angle can not be done using only ruler and compass, but this negative conclusion was reached only in the nineteenth century, after more than two thousand years of work and thanks to the progress in algebra and mathematical analysis.

A disciple of Archytas, Eudoxus of Cnidus (400-347 BC), developed a geometrical theory for the study of irrational numbers, contained in Book V of the 'Elements' of Euclid. This theory was perfected in the eleventh century by the Persian poet and mathematician Omar Kayyam, author of the collection of poems the Rubaiyat. Eudoxus's exposition is of extraordinary logical perfection, precursor to the rigorous mathematics of the nineteenth century and, in particular, the theory of cuts for the construction of real numbers introduced by Dedekind.

Mathematical entities have an exact character, which is not the case with sensible objects, so, for example, a line drawn, no matter how perfect the rule that is used, will appear with irregularities, hence the Pythagoreans came to the conclusion that exact reasoning is done with ideal objects whose reality is eternal, and that of all human activities, the intellectual is the noblest. One more step led to the result that mathematical entities are God's thoughts, hence that Plato's later claim that God is a geometer is perfectly justified. On the other hand, Plato goes much further by inventing his well-known theory of ideas, whose general character is broader than mathematics; Plato also constructs his reminiscence theory of knowledge which explains how certain knowledge is nothing more than memories of things known in previous existences. This is very well set forth in the Meno or dialogue on virtue, where Socrates addresses Meno, asserting that there is no teaching but reminiscence, and to prove it, he tells him to call one of his slaves; a dialogue is then established between Socrates and the slave in which, although the slave knows nothing of mathematics, he obtains the proof of the Pythagorean theorem in a particular case: when the base angles of the right-angled triangle are equal. In the course of this demonstration there is a moment when Socrates addresses Meno and says: *You see, Meno, that I do not teach you anything: I just ask you about it.* In this Socratic dialogue, although it is not possible to conclude what is virtue, its reading is delightful. This happens with all of Plato's writings, except perhaps with the longest dialogue: *The Republic*.

I will now say something more about the *Elements* of Euclid. They begin with twenty-three definitions, among which I cite the following:

2) The line has length but has no width.

15) A circle is a plane figure bounded by a line whose points

are at the same distance from a fixed point: the centre.

23) Parallel lines are those placed in a plane that however far

they are produced in either direction they do not intersect.

^{st}) Through any two points a straight line passes.

2

^{nd}) A straight line can be produced indefinitely.

3

^{rd}) You can draw a circle with given centre and radius.

4

^{th}) All right angles are equal.

^{th}) In a plane, from a point not on a given line, you can draw

a single parallel to the given line.

^{st}) Two things which are equal to a third are equal to each other.

2

^{nd}) Adding equal things to equal things, the sums are the same.

3

^{rd}) Subtracting equal things from equal things, the remainders are equal.

4

^{th}) Things that coincide with one another equal one another.

5

^{th}) The whole is greater than the part.

It is obvious that the postulates have a mathematical character whereas the axioms have some more general nuances, being able to be used in the foundations of other sciences. In modern mathematics, this Greek difference does not exist and the word 'axiom' is used with the same meaning as 'postulate'.

In Book I of the *Elements* the fifth postulate is not used until proposition twenty-nine. The previous propositions belong to what is now called absolute geometry. This geometry is that which consists of the geometrical properties that are obtained without using the postulate of the parallels. It is understandable that from antiquity it was tried to prove proposition twenty-nine without using the fifth postulate, or what is equivalent, to improve the geometry of Euclid by not admitting the property of the parallels without demonstration, that it is as a postulate, but to try to prove it from the remaining axioms, implicitly or explicitly accepted in the *Elements*, or what is equivalent, to see that said postulate is a proposition within absolute geometry.

For twenty centuries there have been numerous attempts to prove the fifth postulate, but the mathematician who for the first time went further on this subject was the Italian Jesuit Saccheri (1667-1733), a professor of mathematics at the University of Pavia who, in 1733, published an important investigation on this postulate. It does not seem that Saccheri's publication attracted much attention in his time, and was quickly forgotten; his work was not known either by the founders of non-Euclidean geometry: Gauss, Lobachevsky and János Bolyai. However, it is necessary to place Saccheri among the great mathematicians who have contributed in a clear way to the development of this geometry.

Saccheri begins to demonstrate the fifth postulate of Euclid from the remaining postulates by building a certain quadrilateral, now called a Saccheri quadrilateral, and reasoning from it to come to the conclusion. In the background, Saccheri is using a method that in mathematics is called 'reduction to the absurd', starting from the hypothesis that is equivalent to that in the plane, through a point not on a given straight line more than one straight line parallel to the given straight line can be drawn and, reasoning from here, to try to come to a contradiction. He constructs a long chain of propositions, some of which are very curious, for example, that the sum of the angles of a triangle is less than two right angles, but in spite of all the work he did, he failed to reach his goal. It is now known that Saccheri could never have reached a contradiction. On the other hand, when he produced his chain of propositions, he was unwittingly developing a new science: non-Euclidean geometry. This geometry is constructed from the postulates that appear explicitly or implicitly in the 'Elements' of Euclid, suppressing the postulate of the parallels and putting in its place the following: "In a plane, through any point not on a given straight line, can be drawn more than one parallel to the given line." Many of the propositions of this geometry contradict other results of Euclidean geometry. It might be believed, in view of this, that if Euclidean geometry is true then non-Euclidean geometry is false, but if one understands the concept of true as free from contradiction, then it is proved by suitable models that if Euclidean geometry is true, Non-Euclidean geometry is also true.

We can conclude from all this that the efforts of two thousand years to try to prove the postulate of the parallels could not achieve their goal, because this was impossible, but served to achieve more important goals such as the deep study of the nature of mathematics and the discovery of non-Euclidean geometry.

Gauss, a German of Göttingen, perhaps the most important mathematician of all time, was the first to have a clear idea of a geometry other than that of Euclid.

When he was about twenty years old he began to study the theory of parallels and for almost thirty years he continued those studies. After many fundamental reflections he developed, in part, the new geometry, which he called non-Euclidean.

In 1824, he wrote a letter to his friend Franz Adolph Taurinus, in which he says:

And he adds in his letter:The hypothesis that the sum of the angles of a triangle is less than two right angles leads to a very curious geometry, very different from ours, but totally consistent, and I have developed it to my satisfaction. The theorems of this geometry seem paradoxical and, for the uninitiated, absurd, although a little serene reflection shows that they have nothing impossible.

The reason why Gauss did not dare to publish his discovery is due to the great authority of Emmanuel Kant, who died in 1804.This is a private communication that you can not publish or do anything that contributes to making it known.

Kant wrote a science book about earthquakes following the famous earthquake in Lisbon in 1755; but Kant was not a scientist, he was a philosopher. Nevertheless, his ideas exerted a great influence in all the fields. Kant was the founder of German idealism. In his book *The Critique of Pure Reason*, the most important that he wrote, published in 1781, he tries to prove that our knowledge comes not only from experience but also a part of it is 'a priori', which is not obtained from the exterior world. The logic and all propositions of pure mathematics are for him 'a priori'. For Kant, space and time are subjective and are part of our apparatus of perception. Plato said that God is a geometer; but Kant goes much further by claiming that space is Euclidean. Given the immense authority of Kant, Gauss did not dare to publish his discoveries that would occupy him for a time that he was not prepared to lose. Before 1831, he did not write a brief summary of the new geometry; this summary was found among his papers upon his death.

A friend of Gauss, Wolgang Bolyai, a Hungarian mathematician, who studied in Göttingen, published in 1832 a treatise on geometry in two volumes. His son János contributed to this work by adding a twenty-six-page appendix which collected his research on geometry, which he had begun ten years earlier when he was twenty-one years old. This appendix contains the fundamental results of non-Euclidean geometry. A copy of it was sent by Wolgang Bolyai to his friend Gauss in 1832, and resulted in Gauss abandoning his project of writing his own investigations in a detailed manner.

János Bolyai studied the consequences that derive from the *Elements* of Euclid by suppressing the fifth postulate and replace it with the following:

He thus obtained many propositions among which are found others obtained previously by the Giovanni Saccheri, with the difference that Saccheri tried to arrive at a contradiction whereas Bolyai knew that was developing a new geometry.There is in the plane a line and a point not on it through which it is possible to draw more than one parallel to that line.

Gauss wrote a letter to Wolgang Bolyai in which, among other things, he told him:

On the other hand, a Russian mathematician, a professor at the University of Kazan, Nicolai Ivanovich Lobachevsky, who had been rector of the University for twenty years, had also discovered non-Euclidean geometry and published his results in 1829, before János Bolyai.The whole content of the work, the way your son proceeded, the results he came to, coincide almost entirely with my own meditations, in my mind for thirty or thirty-five years, so I was completely astonished. As for my personal work, which I have so far entrusted to the paper, it was my intention not to let anything be published during my life. Most men do not have clear ideas about the issues to which we are referring, and I have found very few people who have a special interest in what I have told you about such a matter. It was my idea to write, over time, so that at least it did not perish with me. And it is a pleasant surprise for me to see that this fatigue can be avoided now, and I am extremely pleased that it is precisely the son of my old friend who is there to take it forward in such a remarkable way.

I must now say that geometries apply to the world, but that the spatial properties of the world are not exactly the properties studied by man in their geometries, and so it turns out that Euclidean space approaches locally to ordinary space and, therefore, Euclid's geometry can be used in many engineering constructions, but when it comes to applying it to the Universe, things are not so clear. On the other hand, I have to say that mathematics has been applied a great deal in other sciences, for example in physics, with mathematical models that are sometimes very successful, but also sometimes with failures. I would like to refer here to one of the most famous scientists of our century: Albert Einstein. It seemed that Einstein conceived of physical research as a kind of game between God and the investigator, so that God had created laws of the Universe, which he kept more or less hidden, and the investigator tried to expose them. For this reason, once they presented him with a mathematical model of certain physical phenomena, an extremely difficult model, of an almost superhuman complication, Einstein, with a certain sense of humour, said:

Returning to geometry, it was natural for Gauss to raise the question as to whether real space would be Euclidean or not. To find the answer, he measured near Göttingen a triangle whose vertices were mountain summits and whose sides were about fifty kilometres long: if he could prove that the sum of the angles of this triangle was less than two right angles, then ordinary space would not be Euclidean. He made the measurements and the calculations and concluded that the difference between two right angles and the sum of the angles of said triangle could be due to the errors of the measuring instruments. So his question then remained unanswered.God is subtle but he is not malicious.

None of the founders of the new geometry solved the problem of their logical compatibility. It seems that Bolyai feared that by extending his studies to three-dimensional space he would find incompatibilities; Lobachevsky thought something analogous. It was Beltrami, an Italian mathematician, who, in 1868, published an article interpreting the non-Euclidean plane, locally, as a surface, the pseudoesphere: the straight lines are the geodesics of that surface. In this way he obtained a Euclidean model of non-Euclidean geometry. Therefore, if Euclid's geometry was true, so was non-Euclidean geometry, hence the new geometry was as compatible as the old geometry.

In both Euclidean and non-Euclidean geometries, each straight line is considered infinite. However if one does not look at the line as infinite, one can obtain a geometry, that of Riemann, in which there are no parallels and the sum of the angles of a triangle is greater than two right angles. It seems that Riemann's geometry is closer to real space than Euclid's; Albert Einstein uses it in his theory of general relativity.

Felix Klein gave to the geometry that we have considered the names that they have at the moment, that is to say, the geometry of Saccheri, Gauss, Bolyai and Lobachevsky receives the name of hyperbolic geometry, the one of Riemann, elliptical geometry, and that of Euclid, parabolic geometry.

The detailed study of geometry led some mathematicians of the nineteenth century to a deep and exhaustive analysis of the fundamentals of mathematics. The main artifice was by David Hilbert, a German mathematician, who in 1899 published his famous work 'Foundations of Geometry'. Of course, important works on the fundamentals of geometry, such as those of Meray, Pasch, Peano, Veronese and Enriques, were published before Hilbert's work, but it was Hilbert who succeeded in establishing a definitive axiomatic system, independence and consistency of axioms, by reducing geometry to arithmetic. Hilbert pointed out that if in arithmetic there are no contradiction, then no contradictions are reached in geometry.

In mathematics, an axiomatic system is said to be incomplete when there is some statement that makes sense in the theory deduced from that system and which is undecidable, that is, its truth or falsity can not be demonstrated from the axioms. For example, the axiom system of absolute geometry is incomplete, since it can not be proved from these axioms whether the statement of the parallels is true or false, that is, it can not be proved from such axioms if the following is true or false:

It is natural to ask, then, whether the system of Hilbert's axioms, which give rise to Euclidean geometry, is incomplete or not. As Hilbert reduces geometry to arithmetic, we can ask the question about arithmetic. The answer to this was given by the Austrian Kurt Gödel, the most important logician of this century and one of the greatest thinkers of all time. He was born in Brünn (Moravia) in 1906. He studied mathematics at the University of Vienna and also obtained a doctorate from that university. In his doctoral thesis he demonstrated the sufficiency of the axioms of first order logic, the first-order predicate calculus. It is probably the shortest doctoral thesis ever written: it only has eleven pages. Gödel participated in the activities of the famous Vienna Circle. In 1939, fleeing from the Nazis he went to the United States where he remained until his death in 1978. He was a member of the Institute of Advanced Studies at Princeton from 1940. He not only investigated in logic and philosophy of mathematics but also in the general theory of relativity, influenced by his colleague Einstein, who was also a member of the Institute. Well, Gödel, in 1930, published his famous theorem of the incompleteness of arithmetic, where he shows that any system of axioms containing elementary arithmetic is incomplete.In a plane, through a point not on a straight line, you can draw a single straight line parallel to it.

One aspect of mathematics that is much discussed over time is that of applications. It is difficult to know for the results of pure mathematics if they will ever be applied, for history tells us how difficult it is to answer this, as properties of conics can serve as an example. A Pythagorean mathematician of the fourth century BC, Menaechmus, a disciple of Eudoxus, found the method of coordinates rediscovered in the seventeenth century by Descartes, and while studying the problem of the duplication of the cube, began the theory of conics. Later, Apollonius, a Greek mathematician, wrote in the third century BC his masterly work on conics. Apollonius thought that his investigations were pure learned writing that would never be applied, but in the seventeenth century Kepler, using the numerous data from the observations made by the astronomer Tycho Brahe, discovered his famous laws, the first of which states that each planet describes an ellipse such that the sun is located in one of its foci, the second says that the rectilinear segment connecting the sun to one planet sweeps out equal areas in equal times, and the third expresses, for two different planets, the proportionality between the squares of their periods around the sun and the cubes of average distances to the sun. Later Newton introduces the law of universal gravitation and adopting the simplified hypothesis of two bodies, the sun and a single planet, was able from that to obtain mathematically the three laws of Kepler. It is natural to think that if Kepler had not known about conics and, in particular, he had not known what an ellipse was, he would hardly have come to enunciate those laws.

Another example is that of a Hungarian mathematician, nationalised American, John von Neumann, who dealt with the foundations of mathematics by introducing in 1925 a famous axiomisation of set theory, where he used the primitive concept of class. He also worked on hydrodynamics problems, having to handle certain partial differential equations whose solutions were difficult to study. He felt that there was a need to obtain numerical data for these solutions, whose observation illuminated a path leading to the creation of a theory. This forced him to examine the problem of calculation with electronic machines. During the years 1944 and 1945, von Neumann provided important discoveries about computing.

The Polish mathematician Stanislaw Ulam, a nationalised American, friend of von Neumann and who worked with him in Los Alamos on the atomic research project, says that von Neumann's contributions to computer theory are inspired by the articles he wrote about the fundamentals of mathematics. This shows that even the most abstract part of mathematics can be applied.

In any case, mathematics has to take more into account than purely utilitarian aspects. Thus, the theory of infinite sets, created by the German mathematician Georg Cantor, at the end of the last century and beginning of this, has been the basis of the great development in Mathematical Logic, one of the greatest intellectual achievements of our century. On the other hand, the beauty of set theory is extraordinary. When the German Gottlob Frege published his work in which he constructed mathematics on the basis of certain principles of logic, Bertrand Russell communicated to Frege a contradiction which was deduced from the principles used by him. This contradiction is Russell's famous paradox. Before Russell, paradoxes in set theory had been detected by Cantor himself and by Burali-Forti, but Russell's paradox, which the German mathematician Zermelo had found independently, is so direct and clear that it is no wonder that it caused a great commotion in mathematics. It seemed that the building built by Cantor was shaking. Fraenkel says that someone communicated to David Hilbert his concern about the existence of paradoxes; Hilbert's reply was as follows:

I would like to cite here the Polish mathematician Waclaw Sierpinski, who dedicated his life to the investigation of infinite sets. His grave, in Warsaw, according to his desire expressed many years before his death, has the following inscription:Cantor built with his set theory a paradise for mathematicians, and there will be no one capable of driving us out of it.

Now I will say a few words about my personal attitude. I have not been content just to learn mathematics, but I have also been concerned with acquiring strong mental habits in this field, and even more, I have incorporated mathematics into my life. I have loved my profession deeply, with an almost religious warmth. I could address the mathematics I know like Antonio Machado (a Spanish poet, 1875-1939) to the poplars on the banks of the Duero, and I would tell them:Waclaw Sierpinski,1882-1969, explorer of infinity.

I end by thanking you for the great honour you have granted me and also for the attention you have given me.... you go with me, I carry you in my heart.