"Mathematics is a way of finding out, step by step, facts which are inherent in the statement of the problem but which are not immediately obvious. Usually, in applying mathematics one must first hit on the facts and then verify them by proof. Here we come upon a knotty problem, for the proofs which satisfied mathematicians of an earlier day do not satisfy modem mathematicians."
"Mathematicians start out with certain assumptions and definitions, and then by means of mathematical arguments or proofs they are able to show that certain statements or theorems are true."
"No proof is final. New counterexamples undermine old proofs. The proofs are then revised and mistakenly considered proven for all time. But history tells us that this merely means that the time has not yet come for a critical examination of the proof." (Morris Kline, "Mathematics: The Loss of Certainty", 1980)
"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true. The author tends to belong to this second school of thought." (Richard W Hamming, "Coding and Information Theory", 1980)
"The axiom of choice has many important consequences in set theory. It is used in the proof that every infinite set has a denumerable subset, and in the proof that every set has at least one well-ordering. From the latter, it follows that the power of every set is an aleph. Since any two alephs are comparable, so are any two transfinite powers of sets. The axiom of choice is also essential in the arithmetic of transfinite numbers." (R Bunn, "Developments in the Foundations of Mathematics, 1870-1910", 1980)
"We are now compelled to accept the fact that there is no such thing as an absolute proof or a universally acceptable proof. We know that, if we question the statements we accept on an intuitive basis, we shall be able to prove them only if we accept others on an intuitive basis." (Morris Kline, "Mathematics: The loss of certainty", 1980)
"We become quite convinced that a theorem is correct if we prove it on the basis of reasonably sound statements about numbers or geometrical figures which are intuitively more acceptable than the one we prove." (Morris Kline, "Mathematics: The loss of certainty", 1980)
"When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the result doesn’t make sense intuitively." (Morris Kline, "Mathematics: The Loss of Certainty", 1980)
"When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the result doesn’t make sense intuitively." (Morris Kline, "Mathematics: The Loss of Certainty", 1980)
"A proof transmits conviction from its premises down to its conclusion, so it must start with premises […] for which there already is conviction; otherwise, there will be nothing to transmit." (Robert Nozick, Philosophical Explanations, 1981)
"We become quite convinced that a theorem is correct if we prove it on the basis of reasonably sound statements about numbers or geometrical figures which are intuitively more acceptable than the one we prove." (Morris Kline, "Mathematics: The loss of certainty", 1980)
"When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the result doesn’t make sense intuitively." (Morris Kline, "Mathematics: The Loss of Certainty", 1980)
"When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the result doesn’t make sense intuitively." (Morris Kline, "Mathematics: The Loss of Certainty", 1980)
"A proof transmits conviction from its premises down to its conclusion, so it must start with premises […] for which there already is conviction; otherwise, there will be nothing to transmit." (Robert Nozick, Philosophical Explanations, 1981)
"If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book […] it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way." (George Pólya, "Mathematical Discovery: on Understanding, Learning, and Teaching Problem Solving", 1981)
"We often hear that mathematics consists mainly in ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences’? A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks." (Gian-Carlo Rota, “Complicating Mathematics” in "Discrete Thoughts", 1981)
"We often hear that mathematics consists mainly in ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences’? A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks." (Gian-Carlo Rota, “Complicating Mathematics” in "Discrete Thoughts", 1981)
"In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)
"The psychological core of understanding consists in your having a 'working model' of the phenomenon in your mind. If you understand inflation, a mathematical proof, the way a computer works, DNA or a divorce, then you have a mental representation that serves as a model of an entity in much the same way as, say, a clock functions as a model of the earth's rotation." (Philip Johnson-Laird, "Mental Models: Towards a Cognitive Science of Language, Inference and Consciousness", 1983)
"Most mathematics books are filled with finished theorems and polished proofs, and to a surprising extent they ignore the methods used to create mathematics. It is as if you merely walked through a picture gallery and never told how to mix paints, how to compose pictures, or all the other ‘tricks of the trade’." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"The assumption of rationality has a favored position in economics. It is accorded all the methodological privileges of a self-evident truth, a reasonable idealization, a tautology, and a null hypothesis. Each of these interpretations either puts the hypothesis of rational action beyond question or places the burden of proof squarely on any alternative analysis of belief and choice. The advantage of the rational model is compounded because no other theory of judgment and decision can ever match it in scope, power, and simplicity." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)
"The use of even the most sophisticated forms of mathematics can never be considered as a guarantee of quality. Mathematics is, and can only be, a means of expression and reasoning. The real substance on which the economist works remains economic and social. Indeed, one must avoid the development of a complex mathematical apparatus whenever it is not strictly indispensable. Genuine progress never consists in a purely formal exposition, but always in the discovery of the guiding ideas which underlie any proof. It is these basic ideas which must be explicitly stated and discussed." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Noble lecture] 1988)
"People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it. Yet the conception of a new argument is hardly likely actually to proceed in this way. There is a globality and seemingly vague conceptual content that is necessary in the construction of a mathematical argument; and this can bear little relation to the time that it would seem to take in order fully to appreciate a serially presented proof" (Roger Penrose, "The Emperor’s New Mind", 1989)
"People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it. Yet the conception of a new argument is hardly likely actually to proceed in this way. There is a globality and seemingly vague conceptual content that is necessary in the construction of a mathematical argument; and this can bear little relation to the time that it would seem to take in order fully to appreciate a serially presented proof" (Roger Penrose, "The Emperor’s New Mind", 1989)
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