"Nothing in our experience suggests the introduction of [complex numbers]. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius." (Eugene P Wigner,"The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)
"Number knows no limitations, either from the side of the infinitely great or from the side of the infinitely small, and the facility it offers for generalization is too great for us not to be tempted by it." (Félix E Borel, "Space and Time", 1960)
"Numbers are the landmarks which enable us to speak in a language common to all men, of successive moments of duration." (Félix E Borel, "Space and Time", 1960)
"[…] to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of quantum mechanics." (Eugene Wigner,"The Unreasonable Effectiveness of Mathematics in the Natural Sciences", 1960)
"The purpose of computing is insight, not numbers […] sometimes […] the purpose of computing numbers is not yet in sight." (Richard Hamming, [Motto for the book] "Numerical Methods for Scientists and Engineers", 1962)
"Analysis is primarily concerned with limit processes and continuity, so it is not surprising that mathematicians thinking along these lines soon found themselves studying (and generalizing) two elementary concepts: that of a convergent sequence of real or complex numbers, and that of a continuous function of a real or complex variable." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)
"It seems to me that a worthwhile distinction can be drawn between two types of pure mathematics. The first - which unfortunately is somewhat out of style at present - centers attention on particular functions and theorems which are rich in meaning and history, like the gamma function and the prime number theorem, or on juicy individual facts […] The second is concerned primarily with form and structure." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)
"[…] numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things." (Richard Dedekind, "Essays on the Theory of Numbers", 1963)
"The modern era has uncovered for combinatorics a wide range of fascinating new problems. These have arisen in abstract algebra, topology, the foundations of mathematics, graph theory, game theory, linear programming, and in many other areas. Combinatorics has always been diversified. During our day this diversification has increased manifold. Nor are its many and varied problems successfully attacked in terms of a unified theory. Much of what we have said up to now applies with equal force to the theory of numbers. In fact, combinatorics and number theory are sister disciplines. They share a certain intersection of common knowledge, and each genuinely enriches the other." (Herbert J Ryser, "Combinatorial Mathematics", 1963)
"[…] it took men about five thousand years, counting from the beginning of number symbols, to think of a symbol for nothing." (Isaac Asimov, "Of Time and Space and Other Things", 1965)
"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)
"Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number π." (William L Schaaf, "Nature and History of π", 1967)
"It would not be an exaggeration to say that all of mathematics derives from the concept of infinity. In mathematics, as a rule, we are not interested in individual objects" (numbers, geometric figures), but in whole classes of such objects: all natural numbers, all triangles, and so on. But such a collection consists of an infinite number of individual objects." (Naum Ya. Vilenkin, "Stories about Sets", 1968)
"[…] random numbers should not be generated with a method chosen at random. Some theory should be used." (Donald E. Knuth, "The Art of Computer Programming" Vol. II, 1968)
"Computer science is at once abstract and pragmatic. The focus on actual computers introduces the pragmatic component: our central questions are economic ones like the relations among speed, accuracy, and cost of a proposed computation, and the hardware and software organization required. The (often) better understood questions of existence and theoretical computability - however fundamental - remain in the background. On the other hand, the medium of computer science - information - is an abstract one. The meaning of symbols and numbers may change from application to application, either in mathematics or in computer science. Like mathematics, one goal of computer science is to create a basic structure in terms of inherently defined concepts that is independent of any particular application." (George E Forsythe, "What to do till the computer scientist comes", 1968)
"The generation of random numbers is too important to be left to chance." (Robert R. Coveyou, 1969)
"The modern definition of a function in the context of real numbers is that it is a relationship, usually a formula, by which a correspondence is established between two sets A" (the domain) and B" (the range) of real numbers in such a manner that to every number in set A there corresponds only one number in set B." (Alan Jeffrey, "Mathematics for Engineers and Scientists", 1969)
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