"How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature? Perhaps we see equations as simple because they are easily expressed in terms of mathematical notation already invented at an earlier stage of development of the science, and thus what appears to us as elegance of description really reflects the interconnectedness of Nature’s laws at different levels." (Murray Gell-Mann, 1969)
"In spite of its universality and good precision the linear model is very elementary in its means which are mainly those of linear algebra, so even people with very modest mathematical training can understand and master it. The last is very important for a creative and non-routine use of the analytical means which are given by the model." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives," 1975)
"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)
"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)
"If there are paradoxes in mathematics, think of how many paradoxes there must be in ordinary speech. The existence of paradoxes involves logic. This led naturally to the question of local logics. It seems to me that it is no more natural to expect that a universal logic will hold than that a universal geometry holds." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)
"While the equations represent the discernment of eternal and universal truths, however, the manner in which they are written is strictly, provincially human. That is what makes them so much like poems, wonderfully artful attempts to make infinite realities comprehensible to finite beings." (Michael Guillen," Five Equations That Changed the World", 1995)
"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness. (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Mathematicians, like the rest of us, cherish clever ideas; in particular they delight in an ingenious picture. But this appreciation does not overwhelm a prevailing skepticism. After all, a diagram is - at best - just a special case and so can't establish a general theorem. Even worse, it can be downright misleading. Though not universal, the prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important - but they prove nothing. I want to oppose this view and to make a case for pictures having a legitimate role to play as evidence and justification - a role well beyond the heuristic. In short, pictures can prove theorems." (James R Brown, "Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures", 1999)
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