*Aesthetic and intellectual aspects of mathematics* by Manuel Valdivia

Manuel Valdivia was given an Honorary Doctorate by Alicante University on 2 May 2000. In reply he made the speech

*Aesthetic and intellectual aspects of mathematics*, a version of which we give below:I am deeply grateful for the great honour bestowed on me by this academic act. I thank you with all my heart. I have been linked to the University of Alicante since its birth, many years ago, and I have been fortunate to find in it extraordinary disciples with whom I have worked with great enthusiasm, thinking at all times of the University, always needful of the efforts of everybody. And I have been fortunate also that many of my disciples became close friends.

In my university work, and throughout my life, I have been encouraged by two feelings: the first, an immense gratitude towards my teachers, and the second, a great hope in relation to my disciples. As the years have passed, this hope was more than fulfilled, so I am proud of my disciples. Now my great hope is renewed in relation to the disciples of my disciples.

I am a mathematician who came from engineering, for that reason, and as an agronomist, my work was initially oriented towards the applications of mathematics to agriculture. Later, at the University, I focused on basic research.

No one doubts the numerous applications of mathematics to other sciences and, especially to physics, to which, on the other hand, we owe a lot of mathematics. But there are other aesthetic and intellectual aspects to which I will dedicate some words.

It is very difficult not to be drawn to the beauty of mathematics. Mathematics, apart from being a science, is undoubtedly an art. There have been many people who have compared mathematics with poetry or music. The famous English mathematician G H Hardy said that a mathematician, as a poet, is a creator of aesthetic expressions. If the mathematical expressions are more permanent than the poetic ones, it is because mathematics is made with ideas, whereas in poetry words have more importance, the way of saying things, and the words with time wear out more than the ideas.

Novalis [Friedrich Freiherr von Hardenberg (2 May 1772 - 25 March 1801)], the great German poet, said that algebra is poetry. The poet Paul Valery felt a great passion for mathematics. In fact, in the 1930s, despite being one of France's most prestigious poets, he had long abandoned poetry to devote himself to mathematics.

At first, mathematics had an empirical character, and it was in the sixth century BC when it was organized as a science, and this was thanks to Pythagoras, a person very important from the intellectual point of view, of great influence both in ancient times as in modern ones. Pythagoras used to mix mysticism with science; he believed in the transmigration of souls, conceived as a punishment as the soul is forced to live several lives, passing from one person to another, and even dwelling in animals or plants. That is why he affirmed that the greatest purification is obtained by dedicating oneself to disinterested science, and the man who does so can free himself from the wheel of birth.

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 mathematical reasoning is done with ideal objects whose reality is eternal, and that of all human activities, the intellectual is the noblest. Much later, Plato, in Athens, would go further by inventing his well-known theory of ideas, whose general character transcends mathematics.

Mixing mysticism with mathematics was very common among the Pythagoreans. One of the mystical symbols of this school was the regular pentagon. In fact, this geometric figure was later used, in the Middle Ages, to exorcise the devil. Well, a Pythagorean of the fifth century BC, Hippasus of Metapontum, felt obliged to study the geometrical properties of this symbol, and thus discovered that a side and a diagonal in the regular pentagon are two immeasurable segments, that is to say, that it is impossible to find a third segment that is taken as a unit, so that both the side and the diagonal have whole measures. This was an extremely important discovery, and is nothing more than the discovery of irrational numbers.

Throughout the centuries, many have studied mathematics to know about astronomy and finally to devote themselves to astrology, animated by the conviction that stars, planets and comets influence human affairs. If celestial phenomena occurring at the birth of a person affect the development of his life, it is obvious that astrologers who took their profession seriously had to perform, before making their predictions, numerous calculations on positions of planets and other celestial bodies. Also, then, superstition influenced the development of mathematics.

I will now refer to the existence of mathematical ideas and concepts in relation to the real world. And here Plato, in spite of the many centuries that separate us from him, exerts a clear influence. Platonism is still alive in much of modern mathematicians. According to Plato, mathematical objects, although they are neither physical nor material, have a real existence, independent of man, exist outside space and time. The work of the mathematician does not consist in inventing them, since they already exist, but only in discovering their properties. It turns out that the greatest logician of the twentieth century, Kurt Gödel, was a totally convinced Platonist. Gödel tells us that when he entered the University of Vienna in 1924, he intended to graduate in physics, but when he received lessons from the mathematician Phillipp Furtwängler, a specialist in number theory, he was so impressed that he switched from physics to mathematics. A master of Gödel, Hans Hahn, whom we know very well if we work in functional analysis, as his name goes together with that of Stefan Banach, in the so-called Hahn-Banach theorem, put Gödel in contact with the famous Vienna Circle, among whose members were relevant scientists, one of them Hahn himself. In 1926, when Gödel joined this group, they met in a seminar of the Faculty of Mathematics of the University of Vienna. There they dedicated themselves to constructing a philosophy of science that is now known as "logical positivism" which holds that for a statement to make sense it has to be verifiable by physical experience. This was, of course, a collision with Gödel's Platonism. On the other hand, the members of the Vienna circle eliminated the concept of God. Then Gödel, although he greatly respected them scientifically, as he was deeply religious, abandoned them.

Gödel says that his Platonism helped him a lot in his investigations. Especially in his work on the incompleteness of arithmetic. Gödel's discovery of the incompleteness of any axiomatic system containing elementary arithmetic is one of the most profound and important of mathematical logic.

At present, there are many mathematicians who are not Platonists but who act as if they were Platonists. While they are working, the relationships and concepts which they handle they consider to be as tangible as the world around them. And they are constantly using the infinite, a concept that, from the time of the Greeks, is usual in mathematics. And although infinity has been the source of numerous paradoxes, beginning with Zeno of Elea's classic about Achilles and the tortoise, that of Galileo, seeing that there are as many natural numbers as perfect squares, that of infinitesimals, made clear by Bishop Berkeley, of Dirac, with his famous Delta, which was originally introduced as a function, which did not exist but was useful, and so on. However, when infinity caused major problems was at the end of the 19th century, with the work of the German mathematician Georg Cantor creating his transfinite arithmetic, which began in 1874 by demonstrating that the set of real numbers can not be put into one to one correspondence with the set of positive integers. This result is very important because it expresses, for the first time, that there are sets that are not countable.

The German mathematician Leopold Kronecker, who harshly criticized Cantor's works, said the following, referring to mathematics:

*God made the natural numbers, the rest is the work of man.*

I am not going to say anything about this, because it has a religious character, but I will say that if they accept the natural numbers as a set and, therefore, with a cardinal number, the aleph zero, there is no reason not to admit infinitely larger sets and, therefore, not to accept all the transfinite cardinals of Cantor.

The theory of sets is very abstract and is used to support mathematics. But apart from its utility and great beauty I must add that thanks to it, mathematical logic has developed, one of the greatest intellectual achievements of the twentieth century.

A Hungarian mathematician, John von Neumann, a nationalized American, took up the foundations of mathematics, introducing in 1925 a famous axiomatic set theory, where he uses the primitive concept of class. He also worked on hydrodynamics problems, dealing with partial differential equations whose solutions were difficult to study. This forced him to examine the problem of calculation with electronic machines. During the years 1944 and 1945, von Neumann made important discoveries about computing. The American mathematician Stanislaw Ulam, a friend of von Neumann's, who worked with him in Los Alamos, on the atomic research project, says von Neumann's contributions to computer theory were inspired by the articles he wrote on the foundations of mathematics. This shows that even the most abstract parts of mathematics can be, directly or indirectly, applied.

Increasingly, mathematics is applied to the most diverse fields of science, and this is very flattering to mathematicians. But, on the other hand, we must be aware of our limitations. I think that God has made an extremely complex world that can encompass our science. In this regard, I like to quote William Shakespeare, in his famous tragedy Hamlet, when Hamlet addressing Horace, says the following;

*There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.*

Nothing else. Thank you very much.