ROGER PENROSE
Although I did not expect to become a mathematician when I was growing up—my first desire had been to be a train driver, and later it was (secretly) to be a brain surgeon—mathematics had intrigued and excited me from a young age. My father was not a professional mathematician, but he used mathematics in original ways in his statistical work in human genetics. He clearly enjoyed mathematics for its own sake, and he would often engage me and my two brothers with mathematical puzzles and with aspects of the physical and biological world that had a mathematical character. To me, it had become clear that mathematics was something to be enjoyed. It was evidently also something that played an important part in the workings of the world, and one could see this basic role not only in physics and astronomy, but also in many aspects of biology.
I learned much of the beauties of geometry from him, and together we constructed from cardboard not only the five Platonic solids but also many of their Archimedean and rhombic cousins. This activity arose from one occasion when, at some quite early age, I had been studying a floor or table surface, tiled with a repeating pattern of ceramic regular hexagons. I had wondered, somewhat doubtfully, whether they might, if continued far enough into the distance, be able to cover an entire spherical surface. My father assured me that they could not but told me that regular pentagons, on the other hand, would do so. Perhaps there was some seed of a thought of a possible converse to this, planted early in my mind, about a possibility of using regular pentagons in a tiling of the plane, that found itself realized about one third of a century later!
My earliest encounter with algebra came about also at an early age, when, having long been intrigued by the identity 2 + 2 = 2 × 2, I had hit upon 1
+ 3 = 1 × 3. Wondering whether there might be other examples, and using some geometrical consideration concerning squares and rectangles, or something—I had never done any algebra—I hit upon some rather too-elaborate formula for what I had guessed might be a general expression for the solution to this problem. Upon my showing this to my older brother Oliver, he immediately showed me how my formula could be reduced to 
, and he explained to me how this formula indeed provided the general solution to a + b = a × b. I was amazed by this power of simple algebra to transform and simplify expressions, and this basic demonstration opened my eyes to the wonders of the world of algebra.
Much later, when I was about 15, I told my father that my mathematics teacher had informed us that we would be starting calculus on the following day. Upon hearing this, a desperate expression came over his face, and he immediately took me aside to explain calculus to me, which he did very well. I could see that he had almost deprived himself of the opportunity to be the first to introduce to me the joy and the magic of calculus. I think that almost as great as my immediate fascination with this wonderful subject was my father’s passionate need to relate to me in this mathematically important way. This method (and through other intellectual pursuits such as biology, art, music, puzzles, and games) seems to have been his only emotional route to his sons. To try to communicate with him on personal matters was a virtual impossibility. It was with this background that I had grown up to be comfortable with mathematics and to regard it as a friend and as a recreation, and not something to be frightened or deterred by.
Yet there was an irony in store for me. Both my parents had been medically trained and had decided that of their three sons, I was the one to take over the family concerns with medicine (and, after all, I had my secret ambition to be a brain surgeon). This possibility went by the wayside, however, because of a decision that I had had to make at school that entailed my giving up biology in favor of mathematics, much to the displeasure of my parents. (It was my much younger sister Shirley who eventually took up the banner of medicine, eventually becoming a professor of cancer genetics.) My father was even less keen when I later expressed the desire to study mathematics, pure and simple, at university. He seemed to be of the opinion that to do just mathematics, without necessarily applying it to some other scientific area of study, one had to be a strange, introverted sort of person, with no other interests but mathematics itself. In his desires and ambitions for his sons, my father was, indeed, an emotionally complicated individual!
In fact, I think that initially mathematics alone was my true main interest, with no necessity for it to relate to any other science or to any aspect of the external physical world. Nor had I any great desire to communicate my mathematical understandings to others. Yet, as things developed, I began to feel a greater and greater need to relate my mathematical interests to the workings of the outside world and also, eventually, to communicate what understandings I had acquired to the general public. I have, indeed, come to recognize the importance of trying to convey to others an appreciation of not only the unique value of mathematics but also its remarkable aesthetic qualities. Few other people have had the kind of advantages that I have myself had, with regard to mathematics, arising from my own curiously distinctive mathematical background.
This volume serves such a purpose, providing accounts of many of the achievements of mathematics. It is the fourth of a series of compendia of previously published articles, aimed at introducing to the general public various areas of mathematics and its multifarious applications. It is, in addition, aimed also at other mathematicians, who may themselves work in areas other than the ones being described here. With regard to this latter purpose, I can vouch for its success. For in my own case, I write as a mathematician whose professional interests lie in areas almost entirely outside those described here, and upon reading these articles I have certainly had my perspectives broadened in several ways that I had not expected.
The breadth of the ideas that we find here is considerable, ranging over many areas, such as the philosophy of mathematics, the issue of why mathematics is so important in education and society, and whether its public perception has changed in recent years; perhaps it should now be taught fundamentally differently, and there is the issue of the extent to which the modern technological world might even have thoroughly changed the very nature of our subject. We also find fascinating historical accounts, from achievements made a thousand years or so before the ancient Greeks, to the deep insights and occasional surprising errors made in more modern historical times, and of the wonderful mathematical instruments that played important roles in their societies. We find unexpected connections with geometry, both simple and highly sophisticated, in artistic creations of imposing magnitude and to the fashionable adornment of individual human beings. We learn of the roles of symmetry in animals and in art, and of the use of art in illustrating the value of mathematical rigor. There is much here on the role of randomness and how it is treated by statistics, which is a subject of ubiquitous importance throughout science and of importance also in everyday life.
Yet I was somewhat surprised that, throughout this great breadth of mathematical application, I find no mention of that particular area of the roles of mathematics that I myself find so extraordinarily remarkable, and to which I have devoted so much of my own mathematical energies. This area is the application of sophisticated mathematics to the inner workings of the physical world wherein we find ourselves. It is true that with many situations in physics, very large numbers of elementary constituents (e.g., particles) are involved. This truth applies to the thermodynamic concepts of heat and entropy and to the detailed behavior of gases and other fluids. Many of the relevant concepts are then governed by the laws of large numbers—that is, by the principles of statistics—and this issue is indeed addressed here from various different perspectives in several of these articles.
However, it is often the case that such statistical considerations leave us very far from what is needed, and a proper understanding of the underlying laws themselves is fundamentally needed. Indeed, in appropriate circumstances (i.e., when the physical behavior is in sufficiently “clean” systems), a precision is found that is extraordinary between the observed physical behavior and the calculated behavior that is expected from the known physical laws. This precision already exists, for example, in modern treatments that use powerful computer techniques for the ancient 17th century laws of Isaac Newton. But in appropriate circumstances, the agreement can be far more impressive when the appropriate mathematically sophisticated laws of general relativity or quantum field theory are brought into play.
These matters are often hard to explain in the general terms that could meet the criteria of this collection, and one can understand the omission here of such extraordinary achievements of mathematics. It must be admitted, also, that there is much current activity of considerable mathematical sophistication that, though it is ostensibly concerned with the workings of the actual physical world, has little, if any, direct observational connection with it. Accordingly, despite the considerable mathematical work that is currently being done in these areas—much of it of admittedly great interest with regard to the mathematics itself—this work may be considered from the physical point of view to be somewhat dubious, or tenuous at best because it has no observational support as things stand now. Nevertheless, it is within physics, and its related areas, such as chemistry, metallurgy, and astronomy, that we are beginning to witness the deep and overreaching command of mathematics, when it is aided by the computational power of modern electronic devices.