Discovery is thrill, excitement, and euphoria. Discovery is the difference between victory and defeat, between satisfaction and disappointment, between success and failure. [Oliver]
Discovery may be fickle and unpredictable to the point of exasperation and frustration. It may be elusive to the point of dismay or the destruction of a career. It may be addictive to the point of dereliction of duty. [Oliver]
Describes his discovery of a theory of so-called Fuchsian groups
Not very important to know what these are
Started by thinking that Fuchsian functions cannot exist
A tessellation of the unit disc with hyperbolically isometric regions. Tiles are related to each other by Fuchsian transformations, which form a group. Drawing by Maurits Cornelis Escher (1898-1972) . (Another creative fellow) ->
I wanted to represent [Fuchsian] functions by the quotient of two series; this idea was very conscious and deliberate; the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and succeeded without difficulty in forming the series I have called thetafuchsian.
Just at that time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those on non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the results at my leisure.
Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristic of brevity, suddenness, and immediate certainty, that the arithmetic transformations of indefinite ternary quadratic forms were identical with those of non-Euclidean geometry. […]
Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable.
Hadamard’s proof is based on his theory of integral functions applied to the Riemann zeta function
Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle. J. H. The shortest path between two truths in the real domain passes through the complex domain. J. H.
Attack all questions “ carrying all the outworks, one after the other. There was one, however, that still held out, whose fall would involve that of the whole place. But all my efforts only served at first the better to show me the difficulty, which indeed was something. ” [Poincaré]
Errors, dead ends, frustration.
Result: “ After this shaking-up imposed upon them by our will, these atoms do not return to their primitive rest. They freely continue their dance. ” [Poincaré]
Taine, French historian and psychologist, 1828—1893: “ You may compare the mind of a man to the stage of a theatre, very narrow at the footlights but constantly broadening as it goes back. At the footlights, there is hardly room for more than one actor […] As one goes further and further away from the footlights, there are other figures less and less distinct as they are more distant from the lights. And beyond these groups, in the wings and altogether in the background, are innumerable obscure shapes that a sudden call may bring forward and even within direct range of the footlights. Undefined evolutions constantly take place throughout this seething mass of actors of all kinds, to furnish the chorus leaders who in turn, as in a magic lantern picture, pass before your eyes .”
“ For some thinkers, while engaged in a creative work, illumination may be preceded by a kind of warning by which they are made aware that something of that nature is imminent without knowing exactly what it will be .” [Hadamard]
Paul Valéry, French Poet, 1871—1945: “ Sometimes I have observed this moment when a sensation arrives at the mind; it is as a gleam of light, not so much illuminating as dazzling. This arrival calls attention, points, rather than illuminates, and in fine , is itself an enigma which carries with it the assurance that it can be postponed. You say `I see, and then tomorrow I shall see more.’ There is an activity, a special sensitization; soon you will go into the dark-room, and the picture will be seen to emerge .”
“ Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work .” [Poincaré]
Carl Friedrich Gauss, 1777—1855, on a theorem he had tried to prove for years: “ Finally, two days ago, I succeeded, not on account of my painful efforts, but by the grace of God. Like a sudden flash of lightning, the riddle happened to be solved. I myself cannot say what was the conducting thread which connected what I previously knew with what made my success possible .”
“ It takes two to invent anything. The one makes up combinations, the other one chooses, recognizes what he wishes and what is important to him in the mass of the things that the former has imparted to him. What we call genius is much less the work of the first one than the readiness of the second one to grasp the value of what has been laid before him and to choose it .” [Valéry]
The rules of choice “ are extremely fine and delicate. It is almost impossible to state them precisely; they are felt rather than formulated. Under these conditions, how can we imagine a sieve capable of applying them mechanically? The privileged unconscious phenomena, those susceptible to becoming conscious, are those which, directly or indirectly, affect most profoundly our emotional sensibility . It may be surprising to see emotional sensibility invoked à propos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true aesthetic feeling that all true mathematicians know, and surely it belongs to emotional sensibility .” [Poincaré]
The more we are willing to abstract from the detail of a set of phenomena, the easier it becomes to simulate the phenomena. Moreover we do not have to know, or guess at, all the internal structure of the system but only that part of it that is crucial to the abstraction. It is fortunate that this is so, for if it were not, the topdown strategy that built the natural sciences over the past three centuries would have been infeasible. We knew a great deal about the gross physical and chemical behavior of matter before we had a knowledge of molecules, a great deal about molecular chemistry before we had an atomic theory, and a great deal about atoms before we had any theory of elementary particles— if indeed we have such a theory today. This skyhook-skyscraper construction of science from the roof down to the yet unconstructed foundations was possible because the behavior of the system at each level depended on only a very approximate, simplified, abstracted characterization of the system at the level next beneath. This is lucky, else the safety of bridges and airplanes might depend on the correctness of the “Eightfold Way” of looking at elementary particles. [Simon]
“ No artifact devised by man is so convenient for this kind of functional description as a digital computer. It is truly protean , for almost the only ones of its properties that are detectable in its behavior (when it is operating properly!) are the organizational properties .” [Simon]
“ The computer is a member of an important family of artifacts called symbol systems , or more explicitly, physical symbol systems. Another important member of the family (some of us think, anthropomorphically, it is the most important) is the human mind and brain. [Simon]
Symbol systems are almost the quintessential artifacts, for adaptivity to an environment is their whole raison d’être . They are goal-seeking, information-processing systems, usually enlisted in the service of the larger systems in which they are incorporated .” [Simon]
“ The research that was done to design computer time-sharing systems is a good example of the study of computer behavior as an empirical phenomenon. Only fragments of theory were available to guide the design of a time-sharing system or to predict how a system of a specified design would actually behave in an environment of users who placed their several demands upon it. Most actual designs turned out initially to exhibit serious deficiencies, and most predictions of performance were startlingly inaccurate. Under these circumstances the main route open to the development and improvement of time-sharing systems was to build them and see how they behaved. And this is what was done. They were built, modified, and improved in successive stages. Perhaps theory could have anticipated these experiments and made them unnecessary. In fact it didn’t, and I don’t know anyone intimately acquainted with these exceedingly complex systems who has very specific ideas as to how it might have done so. To understand them, the systems had to be constructed, and their behavior observed.” [Simon]