On Proofs (1960-1969)

"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." (Jerome S Bruner,"On Learning Mathematics", Mathematics Teacher Vol. 53, 1960)

"A mathematical proof, as usually written down, is a sequence of expressions in the state space. But we may also think of the proof as consisting of the sequence of justifications of consecutive proof steps - i.e., the references to axioms, previously-proved theorems, and rules of inference that legitimize the writing down of the proof steps. From this point of view, the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"The most natural way to give an independence proof is to establish a model with the required properties. This is not the only way to proceed since one can attempt to deal directly and analyze the structure of proofs. However, such an approach to set theoretic questions is unnatural since all our intuition come from our belief in the natural, almost physical model of the mathematical universe." (Paul J Cohen, "Set Theory and the Continuum Hypothesis", 1966)

"For foundations it is important to know what we are talking about; we make the subject as specific as possible. In this way we have a chance to make strong assertions. For practice, to make a proof intelligible, we want to eliminate all properties which are not relevant to the result proved, in other words, we make the subject matter less specific." (Georg Kreisel & Jean-Louis Krivine, "Elements of Mathematical Logic: Model Theory", 1967)

"Foundations provide an analysis of practice. To deserve this name, foundations must be expected to introduce notions which do not occur in practice. Thus in foundations of set theory, types of sets are treated explicitly while in practice they are generally absent; and in foundations of constructive mathematics, the analysis of the logical operations involves (intuitive) proofs while in practice there is no explicit mention of the latter." (Georg Kreisel & Jean-Louis Krivine, "Elements of Mathematical Logic: Model Theory", 1967)

"It is characteristic of science that the full explanations are often seized in their essence by the percipient scientist long in advance of any possible proof." (John Desmond Bernal, "The Origin of Life", 1967)

"Now a mathematician has a matchless advantage over general scientists, historians, politicians, and exponents of other professions: He can be wrong. A fortiori, he can also be right. [...] A mistake made by a mathematician, even a great one, is not a 'difference of a point of view' or 'another interpretation of the data' or a 'dictate of a conflicting ideology', it is a mistake. The greatest of all mathematicians, those who have discovered the greatest quantities of mathematical truths, are also those who have published the greatest numbers of lacunary proofs, insufficiently qualified assertions, and flat mistakes." (Clifford Truesdell, "Late Baroque Mechanics to Success, Conjecture, Error, and Failure in Newton's Principia" [in "Essays in the History of Mechanics"], 1968)