I am emotional, but with great pleasure, with the duty of expressing my deep gratitude for the high honour that the University of Seville has conferred upon me, by giving me the distinction of the award of its Honoris Causa Doctorate.
I was born and educated in Spain, but transplanted to South America, on the banks of the estuary of the Rio de la Plata. Having reached the age when there are already few hopes and many memories, my feelings are drawn by the impressions of the years of youth in the homeland of my origin and those which followed them, and many more than the first, in the South American country of my wife, my daughters, and my grandchildren. They are opposing forces that pull me both from roots anchored in the past and from the stresses of the future, stronger and stronger forces, which generate deeper and deeper feelings. Among them, the gratitude for signs of affection and friendship, always bound together, becomes deep and very meaningful, while in cases like today it is mixed with the pleasure of finding friends of many years and new ones that will last forever.
Those who have had a life divided between this side and the other side of the seas, always living comparing actions, customs, landscapes and scenarios, we are marked by the sign of an eternal longing, which pushes us into the past and intensely sensitises us to any situation that arouses memories or evokes the past. That is why, and also at this time, as the world prepares to commemorate the fifth centenary of the discovery of America, as well as the place, this bright Andalusian land, where the deed was planned and started, I am particularly moved today. I feel part of, and an interpreter of, the generations that during these centuries emigrated to those coasts, always hospitable, that like Aissa to her enchanted prince, they knew how to give us "children, a little earth, woman and a lot of love", but that never erased all in our intimate background, the remnants of the distant land.
As is customary in receiving a high distinction, such as the present one, I will allow myself to deliver some thoughts on the science of my specialty, mathematics and the successive dissections of the same until locating the place that occupies the special chapter in which I have worked, the so-called integral or stochastic geometry. It has been a work of more than 50 years, enough to have witnessed the origin and development of these disciplines, until the present, in which its extinction is already envisioned, not by disappearance without trace, but by its dissolution within more broad and unifying theories, lending to them the same effort and help to continue marching, together, towards new forms and new perspectives.
This particular fact is an example of the frenzy with which all branches of science advance today. In a few decades we see the birth, development and oblivion of theories that are quickly replaced by others, which are sometimes the same with different clothes, always leaving a positive result and the universe of knowledge expands at a speed difficult to follow, if not through successive posts and incessant relays of the creative teams.
The amount of new knowledge now taking place in all branches of learning continues to grow exponentially without precedent, and mathematics does not escape this general tendency, rather it is one of its most significant examples. A rough estimate leads to the fact that if all the work of mathematics published in the world during the year 1989 (reviewed in the journal 'Mathematical Reviews'), was put together and collected into volumes of 1,000 pages each, there would be a collection of 350 volumes. This causes serious problems of information and storage, as both the spaces and the budgets for the journal libraries are insufficient. Moreover, this huge sea of publications is not proportional to its usefulness, for surely many of them that could be useful in other areas of mathematics or in other branches of knowledge, go unnoticed. It is impossible to be aware of everything that could interest each one, the only hope is modern computers, which allow us to store in very little space a lot of information and also allow us to identify and find the data sought.
It is possible to ask for the causes of this enormous scientific production. One of them, of course, is the general law governing population growth, according to which production engenders more production, as a consequence of the fact that the slope of growth is proportional to the volume of what is created. But another important cause is the widespread emergence of scientific research as a profession in all countries in the second half of the 20th century. In earlier times, researchers were engaged in a certain profession, providing them with support and with which they fulfilled society, and "in addition", if they felt motivated by their vocation, engaged in research as a complementary, non-binding task which was unpaid. For this reason, scientific researchers were considered to be people who sacrificed themselves for the common good and for that reason deserved the greatest gratitude and admiration.
Nowadays, practically since the Second World War (1939-45), things have changed, since most of the researchers perform their tasks in the exercise of a salaried profession, that is as a freely chosen and adequately remunerated obligation. This means that scientific research has become, as Ortega i Gasset said, a dimension of human life before which there is no sense of fuss or of awe in the public or of presumption in the performer and, on the other hand, explains the exorbitant growth of scientific production, in particular of mathematics, which makes it difficult to characterize the specific subject in which each one works, by the luxuriance of the branches of science and its incessant and irregular growth, in an overlapping and intertwined way.
Let us look at a little history to see how the general idea of what we call mathematics has been differentiated and grown by successive divisions and by the birth in them of new branches with complicated interconnections.
When man began to feel the need or the curiosity to know his surroundings and to understand the world in which he lived, two fundamental activities arose: to count and to measure. With counting were born numbers and the operations with them: it was the calculation. With measurement the shapes and figures were outlined: it was geometry. Both activities constituted the branch of knowledge that was called mathematics. From the beginning it was seen that mathematics could be useful to act, as a tool to be handled, and also to provide a way to structure and order thinking and help creativity. The first gave rise to the so-called applied mathematics and the second to pure mathematics, two aspects whose boundaries were never, and have become less and less, very precise, but which have been remained through the centuries.
Plato (428 BC, 347 BC) in his Republic, clearly distinguishes both aspects. For him there is the calculation, which he recommends be prescribed to those who are to perform the most important functions of the city:
... so that it draws the mind upwards by pure intelligence and forces it to argue about numbers in themselves and will not be put off by attempts to confine the argument to collections of visible or tangible objects and the calculation of the merchants and shopkeepers, who cultivate it for purchases and sales (525, d).An analogous difference is made to geometry
... which should lead the mind to contemplate the essence and for the sake of knowledge, instead of handling material objects and reasoning with a view to the practice of squaring, applying and adding. (527, b)There remained therefore a well defined pure mathematics for philosophers and connoisseurs, whose medium is that of ideas, and the other a practical or applied mathematics, for the needs inherent to the real world. However, this difference between pure mathematics and applied mathematics has never been as clear as Plato supposed. Less than a century after the Republic, Eratosthenes appears (284 BC, 192 BC) who uses mathematics to measure the radius of the Earth and also Archimedes (287 BC, 212 BC) who at the same time make fine speculations about pure mathematics (the method of exhaustion) applied the same to concrete problems of statistics or hydrostatics. In both his and Claudius Ptolemy's (90, 168) creations in trigonometry and his world system, it is difficult to decide which parts correspond to pure mathematics and which to the applied one.
The first systematic work of pure geometry was the Euclid's Elements (3rd century BC) whose purity refers both to its constituent elements (points, lines, planes) that are simple and perfect, obtained by idealization of visual forms discernible by the senses, to the axiomatic construction, which served as a model for all subsequent mathematics, and also to the common notions with which the congruence of figures is introduced through the movements of the plane. Modifying these different components from which the construction of Euclid begins, many other possible geometries appeared over time.
Modifying the postulates led to the birth first of non-Euclidean geometries (early nineteenth century) and then a number of other geometries that appeared only to replace the postulates by others, a work that culminated in 1899 with the Foundations of Geometry by David Hilbert (1852-1925), a work that can be considered as the endpoint of Euclid's style of geometry. With respect to the "common notions", replacing the group of the movements with other groups, many other geometries were born that were ordered by Félix Klein (1849-1925) in his famous 'Erlangen Program' (1872).
It was also possible to extend geometry by replacing the basic elements of Euclid (contained in their definitions: point, straight, plane), thought of as models of spatial intuition, by more complex forms. Thus Descartes (1596-1650) and Fermat (1601-1665) identified the "points" with pairs of real numbers and the "lines" with linear equations between them, creating coordinate geometry.
Without losing the geometric model of Nature, one can think of a change and extension of geometry assuming a different intuition from the classical one. If our eyes were telescopes or electron microscopes, our intuition of the outside world would be very different and so equally different would have been the geometry built upon it. The simplifications introduced by an intuition appropriate to the senses of man, resulted in the fact that for centuries only very regular curves, formed by arcs with continuous tangents were studied. Only in the last century were curves considered without tangents at any point (Weierstrass, 1815-1897) or curves that fill an area (Peano, 1858-1932), which were long considered pathological examples, not encompassed as important elements of a geometry. Since the fourth decade of the present century, however, these types of geometric entities have begun to be studied systematically and applications in different parts of science have been sought. They were born in what Mandelbrot has called fractals, being sets whose dimension, conveniently defined by Hausdorff and Besicovitch, is a fractional number. These new entities have been able to be represented graphically and studied in some detail thanks to the possibilities offered by modern electronic computers, having appeared in many applications to different problems of solids physics and biology.
All these generalizations and extensions of geometry have come from successive modifications of the basics upon which Greek geometry, symbolized by Euclid, was built. But mathematics has another mode of growth, which arises by drawing bridges between separate parts of the same, creating hybrids, which are sometimes of interest in themselves and others for each of the parts involved. A typical example is that of geometric probabilities, a result of the cross between geometry and probability, two well differentiated fields until the middle of the eighteenth century.
The idea of probability, although it is implicit in any game of chance and therefore goes back to at least the tenth century BC of the Trojan War, in which, according to Sophocles, the besiegers were entertaining themselves by playing dice, it was not considered by mathematicians until 1654, in a correspondence between Fermat and Pascal (1623-1662) about an apparent paradox also arising from a particular set of dice. That is to say, the theory of probabilities was born through the problem of playing a game, an activity always censured from the moral point of view, and it is possible that this "unholy" origin, despite the aristocratic patronage of Fermat and Pascal, was the cause that for more than two centuries, the calculus of probabilities would be excluded from the academic cloisters.
However, this calculus was progressing, especially thanks to the posthumous work of Jacob Bernoulli (1654-1705), entitled Ars Conjectandi; it appeared in 1713, and when it had already reached a high degree of development, in 1777 appeared the curious book Essai D'Arithmetiqué Morale by George Louis Leclerc (1707-1788), Count Buffon, author of a famous Natural History in 36 volumes, in one of which coming with a special supplement, appeared the following test.
Buffon was a naturalist, with Linnaeus the most important of the eighteenth century, and among living beings he considered that he should study man, not only in his anatomy and physiology, but also in his spirit, his hopes, his fears, and his passions. One of the most generalized passions, according to Buffon, was gambling and therefore he proposed to analyze it with the idea of seeing its influence on man's behaviour on himself and on society. This is how he was led to give "moral" value to numbers, especially to the amount of money, because who has twice as much fortune as another is not necessarily twice as happy. He then analyzes some games of chance and observes that
... analysis has been the only instrument to date used in the science of probabilities, as if geometry were not indicated for those purposes, when in fact a little attention is enough to observe that the advantage of analysis over geometry is only accidental and that randomness is as characteristic of analysis as of geometry.Then he adds,
... to put geometry in its rightful position in the science of chance, it will be enough to invent games that are based on extension and their relations.As an example, he presents the case of a needle that is thrown randomly onto a plane in which equidistant parallel lines have been drawn, the distance between them being greater than the length of the needle, with the agreement that the player wins if the needle does not cut any parallel and lose in the opposite case. In order to calculate the prize that the player will receive in the case of winning, it is necessary to calculate the probability that the needle does not cut any parallel. Assuming that the bet is equal to unity, for the game to be equitable, the prize must be equal to the inverse of that probability. This is what Buffon calculates, with what is considered to have given rise to the theory of geometric probabilities. As we see, this is a problem for which it is not possible to "count" favourable and possible cases, as in discrete games of chance, such as those based on dice or coins, but must "measure" those cases. The difference between counting and measuring is precisely what distinguishes arithmetic from geometry.
Thus, geometrical probabilities, as well as probabilities in general, originated in a game of chance, confirming a phrase of Leibniz in a letter to Montmort in 1715, in which he says
... men are never more ingenious than in the invention of games: there the spirit feels quite at home.Note also that the joint consideration of geometric ideas and probabilities gave rise to interesting results. For example, using the same result of Buffon's needle, practically performing the experiment and using the fact that frequency tends to probability, we have a method to find experimentally the length of the needle and also the number π, the ratio of the circumference of a circle to its diameter. Such is the origin of the so-called Monte Carlo method to obtain results by chance.
A generalization of the problem of Buffon's needle did not have long to wait. Laplace (1749-1827) in his Analytical Theory of Probabilities (1812), considers the plane divided into congruent rectangles by two series of parallel lines and calculates the probability that a needle thrown at random on the plane does not cut any of those straight lines (problem of the Laplace needle). If, instead of a needle, any curve of finite length is thrown onto the plane, the mean value of the number of points of intersection of the curve with the rectangle grid can be calculated.
When you have a theory based on the symbiosis of two others, the evolution can be whole or separately according to each component. In the case under consideration, evolution ending in stochastic geometry, there were two branches with the same roots and many analogies, but also with different purposes and forms of presentation.
The problems of Buffon or Laplace needle motivate the problem of measuring sets of lines in the plane. Although there have always been measured sets of points (length of curves or areas of domain) however the measure of sets of straight lines had not been considered. Necessity arose from geometric probabilities, and at the outset certain paradoxes appeared, such as the classic by J Bertrand (Calculus of Probabilities, 1889) quoted in many classic texts of probabilities. The Englishman M W Crofton (1826-1915) was the first to realize that this was due to
... certain inaccuracies of the instrument used and that basic concepts, like any new subject, had to be checked patiently and repeatedly, testing and correcting them in the light of experiences and comparisons to purge them of any mistakes.He then defined a density to measure sets of lines and found the curious fact that the measure of the lines that cut a convex set is equal to the length of its contour. That is to say, just as the measure of the points of a convex domain is equal to its area, the measure of the lines that cut it (always in the plane) is equal to its perimeter, a curious duality that has since been extended to great generality by R V Ambartzumian (Combinatorial Integral Geometry, 1982).
Crofton obtained his results in an intuitive way, but came to interesting formulas related to convex sets, of pure geometrical interest, which appeared as exemplification of probabilistic problems. The rigorous justification of the way Crofton proceeded was given later by E Cartan (1869-1951) in 1896 and H Lebesgue (1875-1941) in 1912.
In the 1930s, W Blashke (1885-1962) and his Hamburg school resumed study of the results of Crofton-Cartan Lebesgue to generalize them to more dimensions and systematize them into a new chapter of geometry, which was called integral geometry. The results were of pure geometric interest, and the primitive idea of probability that had given rise to them had virtually disappeared. On the other hand, the ideas derived from group theory, such as the Erlangen Program, became exploited in the new geometry, giving rise to affine integral geometry, projective integral geometry, non-Euclidean integral geometry, etc. Lie group theory became fundamental for the definition of invariant densities for these geometries. The theory was also linked with the classic harmonic analysis and the measurement in groups.
The probabilistic branch of the primitive theory of geometric probabilities took longer to develop, which took place in the decade of the 60s, mainly by Roger E Miles who enriched it with the incorporation of the stochastic processes, giving rise to what was later called Stochastic Geometry, whose basic work is currently the 'Stochastic Geometry and its Applications' of D Stoyan, W J Kendall and J Mecke (1987). An interesting chapter of this geometry is that of random mosaics, an area in which science and art are mixed, from the mosaics of the Alhambra in Granada to Escher's paintings, through the crystallographic groups.
At the beginning we mentioned the two aspects, pure and applied, of mathematics. In the cases of Integral Geometry and Stochastic Geometry, the two aspects separately present their interest. They are theories generally elaborated by pure mathematicians, with views only to problems of the world of ideas, with no other guide than the beauty of them and the interest to understand and order their achievements. Then there are applications, which have originated in the so-called Stereology, which is a useful technique in different branches of knowledge, such as metallurgy, mineralogy, botany, anatomy,
... it is a set of methods for the exploration of three-dimensional space or projections on planes.Its theoretical bases are the integral and stochastic geometry and its techniques related to microscopy. One of the main creators is the Spaniard Luis M Cruz Orive, of the University of Bern.
Stereology is a first, very simplified, approximation of the general problem of computerized tomography, whose roots also belong to integral geometry in the harder sense of Helgason and Gelfand, which was initiated by the mathematician Radon in 1917, led to practical use by the physicist A M Cormack (1963) and industrialized by the engineer G N Hounsfield (1972), receiving for these two last ones the Nobel prize for Medicine in 1979. It is a clear example of the unity of science because for the joint effort of a mathematician, a physicist and an engineer, still separated in time and space, each working in his field, achieved a transcendental result in medicine.
This duality, pure and applied, or philosophy and tool, of mathematics, which turns it into an interweaving of paths and bridges between the world of ideas and that of our real environment, is precisely what gives it universality and permanence, for its wide spectrum of flavours and variety of forms. Perhaps that is why mathematics is at the same time the most conservative and most creative and changing science, which keeps fresh and alive its most ancient roots, in renewed harmony with the most recent and revolutionary shoots. We have tried to make it clear with a simple and very limited example, but surely an analogous evolution can be found in many other chapters of that science.