Creativity Carlo Tomasi Bibliography • J. Hadamard, The Psychology of Invention in the Mathematical Field, Dover, New York, NY, 1954 (original 1945). • N. Wiener, Invention -- The Care and Feeding of Ideas, MIT Press, Cambridge, MA, 1993. • J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas, Perseus Books Group, New York, NY, 2001. • J. E. Oliver, The Incomplete Guide to the Art of Discovery, Columbia University Press, New York, NY, 1991. • H. Simon, The Sciences of the Artificial, MIT Press, Cambridge, MA, 1969. Outline and a Warning • Can we do without creativity in academia? • Where does creativity come from in mathematics, art science? • Poincaré’s introspections analyzed by Hadamard • But can we learn from this? Is CS a science? • Simon’s Sciences of the Artificial • Yes. Yes. • There will be small fonts, as this talk is based on readings Not Quite Necessary for Academic Success • For every idea however wrong there are data for which the idea will work • For every paper however useless there is a venue that will publish it • For every talent however weak there is a school that will hire it • The band-wagon method: Apply today’s method to anything at hand • The naked-emperor method: State it in a complicated way, and few will argue • The clique method: Find enough peers like you, and you have a forum • Resume padding and the LPU (Least Publishable-Unit) • This is no fun! Creativity and Discovery • Novelty + usefulness • 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] Jules Henri Poincaré, 1854—1912 • Brother of Raymond Poincaré, prime minister then president of France in WWI • Originator of algebraic topology, theory of analytic functions of several complex variables • Major contributor to algebraic geometry, number theory, physics, applied math • Looked at his own thought processes in a 1908 lecture on Mathematical invention to the Institute Général Psychologique in Paris Poincaré’s Anecdotes • Describes his discovery of a theory of so-called Fuchsian groups • Not very important to know what these are • Started by thinking that A tessellation of the unit disc Fuchsian functions cannot with hyperbolically isometric regions. exist Tiles are related to each other by Fuchsian transformations, which form a group. Drawing by Maurits Cornelis (Another creative fellow) → Escher (1898-1972) . 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. [Poincaré] 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 he 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. [Poincaré] Jacques Hadamard,1865 – 1963 • French mathematician • The number of primes <n grows as fast as n /ln n • Conjectured by Lebesgue (1798) and Gauss (1849) • Hadamard’s proof is based on his theory of integral functions applied to the Riemann zeta function The shortest path between two truths in J. H.real domain passes through the complex domain. the 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. Hadamard’s 1945 Book • Introspection, his own, Poincaré’s, and that of others • Preparation, incubation, intimation, illumination, formulation • Follows a theory by English social psychologist Graham Wallas (1925, cited by J. H.) Graham Wallas, 1858--1932 preparation, incubation, intimation, illumination, formulation • Conscious, long hard work. • 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] preparation, incubation, intimation, illumination, formulation Hadamard’s (or Wallas’) proposal: • Invention or discovery takes place by combining ideas • There are very many combinations, most useless, few fruitful • The mind constructs many possible combinations, unconsciously and at random • The conscious mind only chooses some that could be fruitful Un/subconscious creation of ideas as… • Divergent, as opposed to convergent thinking (Joy Paul Guilford, 1897–1988, US psychologist) • Generate, as opposed to test J. P. Guilford (Allen Newell, 1927—1992, H. Simon A. Newell Herbert Simon, 1916—2001, fathers of AI, CMU) • Unconscious or subconscious? Unconscious or Subconscious? • Fringe-subconscious • 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.‖ Hippolyte Taine preparation, incubation, intimation, illumination, formulation • ―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.‖ Paul Valéry preparation, incubation, intimation, illumination, formulation • ―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.‖ C. F. Gauss Illumination:To Invent Is to Choose • ―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] • Divergent, then convergent thinking (Guilford) • Generate, then test (Newell and Simon) Illumination: Choice Criteria 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 esthetic feeling that all true mathematicians know, and surely it belongs to emotional sensibility.‖ [Poincaré] preparation, incubation, intimation, illumination, formulation • Or formalization: express by writing • Conscious, disciplined, focused, painstaking • To verify. ―The feeling of absolute certitude which accompanies the inspiration generally corresponds to reality; but it may happen that it has deceived us.‖ [Hadamard] • To ―precise.‖ ―It never happens that the unconscious work gives us the results of a somewhat long calculation already solved in its entirety.” [Poincaré] • To form ―relay results‖ so that ―the outcome is not the end of the research but one stage of it, so that we think of utilizing it.‖ [Hadamard] Summary of Hadamard • Preparation: No free lunch. Expect frustration. • Incubation: Random search. Give it time. • Intimation, Illumination: Eureka! • Formulation: Down to business. • Louis Pasteur, 1822—1895: "Chance favors only the prepared L. Pasteur mind.― • Thomas A. Edison, 1847—1931: ―Genius is one per cent inspiration and ninety-nine per cent perspiration.‖ T. A. Edison Reality Check • Few scientists complete their careers without making a discovery of some sort, yet few make discoveries of major significance. [Oliver] • All scientists struggle through prolonged intervals of absence of discovery and the accompanying anxiety about a future that threatens to be discovery free. [Oliver] Math, Science, and CS • Do we care about all of this? • Is CS a science? • Is CS mathematics? • Frank Harary, 1921—2005, UMich (math) and NMSU (CS), graph theorist: ―Any field that has the word science in its name is guaranteed thereby not to be a science.‖ CS as a Science • H. Simon, The Sciences of the Artificial, MIT Press, 1969 • Artificial: contingent to the goals or purposes of their designer • Sciences study an object (artificial or natural) in relation to its environment: – what function or purpose it serves (clock tells time), – its interaction with the outer environment (ship buffeting, moisture) – how it achieves that purpose (inner environment: gears or quartz) • Human rationality in the artificial is akin to natural selection in the natural: they both make objects adapt to purpose • Study functions and interactions by simulation • Simulation shows hard-to-infer consequences of well understood laws and premises (e.g., weather prediction) • Poorly understood systems can be simulated by abstracting only the relevant Abstract Study with the Computer 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] Abstract Study of the Computer • ―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] Empirical Study of the Computer “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] Summary of Simon • Science studies – Internal environment (how it works) – External environment (how it interacts) – What function it serves • Science is both mathematical and empirical, and proceeds by abstraction • Computers can be used to simulate by abstraction • Computers can be studied mathematically and empirically, often normatively (design) rather than descriptively (discovery) • Computer Science is a science – As the science of symbol systems – As the mathematical and empirical study of computing systems Creativity in CS • Creativity is fundamental in computer science when – Posing a question – Formulating an abstract model – Conjecturing an answer – ―Solving‖ a model – Designing a system – Designing an experiment (about a natural or artificial system) – Writing a paper • There is no free lunch: creativity is hard work, courage, and intellectual generosity Bibliography • J. Hadamard, The Psychology of Invention in the Mathematical Field, Dover, New York, NY, 1954 (original 1945). • N. Wiener, Invention -- The Care and Feeding of Ideas, MIT Press, Cambridge, MA, 1993. • J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas, Perseus Books Group, New York, NY, 2001. • J. E. Oliver, The Incomplete Guide to the Art of Discovery, Columbia University Press, New York, NY, 1991. • H. Simon, The Sciences of the Artificial, MIT Press, Cambridge, MA, 1969. Be Obsessed!
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