"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors.” (Richard von Mises, “Positivism: A Study in Human Understanding”, 1951)
"The constructions of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow-mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity." (Hermann Weyl, "A Half-Century of Mathematics", The American Mathematical Monthly, 1951)
“The history of mathematics shows that the introduction of better and better symbolism and operations has made a commonplace of processes that would have been impossible with the unimproved techniques.” (Morris Kline, “Mathematics in Western culture”, 1953)
"The study of the history of mathematics shows clearly enough that after each period of research and extension there follows a period of review and synthesis during which more general methods are evolved and the foundation of mathematics consolidated." (Gustave Choquet, "What is Modern Mathematics", 1963)
“It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands.” (Philip J Davis, “The Mathematics of Matrices”, 1965)
"Mathematics is a vast adventure of ideas; its history reflects some of the noblest thoughts of countless generations." (Dirk J Struik, "A Concise History of Mathematics", 1967)
"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making." (Morris Kline, "Mathematics for the Nonmathematician", 1967)
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