"My mother, who taught kindergarten and first grade before her marriage, said that I was the stubbornest child she had ever known. I would say that my stubbornness has been to a great extent responsible for whatever success I have had in mathematics. But then it is a common trait among mathematicians." Julia Robinson
Constance Reid. "The Autobiography of Julia Robinson," Coll.Math.J. 17 (January 1986) 4.
Ch6 quote 5 :
"Some intervention of intuition issuing from the unconscious is necessary at least to initiate the logical work."
Jacques Hadamard
The Psychology of Invention in the Mathematical Field, Princeton: Princeton University Press, 1945, p112.
Ch6 quote 6 :
"The first rule of discovery is to have brains and good luck. The second rule is to sit tight and wait till you get a bright idea." George Polya
How to solve it, Princeton: Princeton University Press, 1945, p.158
Ch6 quote 8 :
"Why should a mathematician care for plausible reasoning? His science is the only one that can rely on demonstrative reasoning alone. The physicist needs inductive evidence, the lawyer has to rely on circumstantial evidence, the historian on documentary evidence, the economist on statistical evidence. These kinds of evidence may carry strong conviction, attain a high level of plausibility, and justly so, but can never attain the force of a strict demonstration. ... Perhaps it is silly to discuss plausible grounds in mathematical matters. Yet I do not think so. Mathematics has two faces. Presented in finished form, mathematics appears as a purely demonstrative science, but mathematics in the making is sort of an experimental science. A correctly written mathematical paper is supposed to contain strict demonstrations only, but the creative work of the mathematician resembles the creative work of the naturalist: observation, analogy, and conjectural generalizations, or mere guesses, if you prefer to say so, play an essential role in both. A mathematical theorem must be guessed before it is proved. The idea of a demonstration must be guessed before the details are carried through." George Polya
"On Plausible Reasoning," in Proceedings of the International Congress of Mathematicians-1950, Vol 1, Providence, RI: American Mathematical Society, 1952, p.739.
Ch6 quote 11 :
"Intuition implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytical apparatus of one's craft. It is the intuitive mode that yields hypotheses quickly, that produces interesting combinations of ideas before their worth is known. It precedes proof: indeed, it is what the techniques of analysis and proof are designed to test and check. It is founded on a kind of combinatorial playfulness that is only possible when the consequences of error are not overpowering or sinful. Above all, it is a form of activity that depends upon confidence in the worthwhileness of the process of mathematical activity rather than upon the importance of right answers at all times." Jerome Bruner
"On Learning Mathematics," Math. Teacher 53 (December 1960) 613.
Ch6 quote 14 :
"Mathematics -- this may surprise you or shock you some -- is never deductive in its creation. The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and he becomes convinced of their truth long before he can write down a logical proof. The conviction is not likely to come early -- it usually comes after many attempts, many failures, many discouragements, many false starts. It often happens that months of work result in the proof that the method of attack they were based on cannot possibly work, and the process of guessing, visualizing and conclusion- jumping begins again. . . . The deductive stage, writing the result down, and writing down its Rogers proof are relatively trivial once the real insight arrives; it is more like the draftsman's work, not the architect's." Paul Halmos
"Mathematics as a Creative Art," Amer.Scientist 56 (Winter 1968) 380.
Ch6 quote 24 :
"There are many things you can do with problems besides solving them. First you must define them, pose them. But then of course you can also refine them, depose them, or expose them or even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn't lead to it, so you found a new goal they did lead to. It's called play. Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings." D. Hawkins
"The Spirit of Play," in Necia Grant Cooper (ed.), From Cardinals to Chaos, Cambridge: Cambridge University Press, 1988, p.44.
Ch10 quote 11 :
"A mathematician, like a painter or a poet, is a maker of patterns." G.H.Hardy
A Mathematician's Apology, Cambridge: Cambridge University Press, 1940, p24.
Ch10 quote 13 :
"The ideas chosen by my unconscious are those which reach my consciousness, and I see that they are those which agree with my aesthetic sense." Jacques Hadamard
The Psychology of Invention in the Mathematical Field, Princeton: Princeton University Press, 1945, p39.
Chapter 6/ quote18
"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity . . . . The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasonings."
ALAN TURING
Quote from, "Alan Turing the Enigma", New York:Simon & Schuster, 1983, p.144.
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