Mathematics Quotes
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Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent to a pair of ears. Mathematics can only be "seen" with the "eyes of the mind". It is as if we had no sense of hearing, so that only someone able to sight read music would be able to appreciate its patterns and harmonies.
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The indispensability argument says (roughly) that if you have ample reason to accept an empirical scientific theory that makes indispensable use of mathematics, and that theory entails that numbers exist, then you have ample reason to accept that numbers exist. The argument affirms the antecedent of this conditional, and concludes that you have ample reason to believe that numbers exist. What is striking about this argument is that it seems to show that the empirical reasons that suffice for accepting a scientific theory also suffice for accepting a metaphysical claim.
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Though determinants and matrices received a great deal of attention in the nineteenth century and thousands of papers were written on these subjects, they do not constitute great innovations in mathematics.... Neither determinants nor matrices have influenced deeply the course of mathematics despite their utility as compact expressions and despite the suggestiveness of matrices as concrete groups for the discernment of general theorems of group theory.
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Mathematics is a body of knowledge, but it contains no truths.
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Mathematics is the art of accurate reasoning on inaccurately-drawn figures... let that be our motto.
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My first contact with game theory was a popular article in 'Fortune Magazine' which I read in my last high school year. I was immediately attracted to the subject matter, and when I studied mathematics, I found the fundamental book by von Neumann and Morgenstern in the library and studied it.
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There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
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Mathematics is not just a language. Mathematics is a language plus reasoning.
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I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that people learn about and gather together. Then there is a sort of inevitable development in those fields. You get to the point where a certain theorem is bound to be proved, independent of any particular individual, because it is just in the path of development.
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We humans have a wide range of abilities that help us perceive and analyze mathematical content. We perceive abstract notions not just through seeing but also by hearing, by feeling, by our sense of body motion and position. Our geometric and spatial skills are highly trainable, just as in other high-performance activities. In mathematics we can use the modules of our minds in flexible ways - even metaphorically. A whole-mind approach to mathematical thinking is vastly more effective than the common approach that manipulates only symbols.
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My first degree was in mathematics. That was great, but it didn't help with many of the things that puzzled me. I became a philosopher because I wanted to understand everything, especially those things that didn't make sense. And that has continued to be my philosophical motivation. That's one reason I have such a roving philosophical eye - once I have figured out a philosophical topic to my satisfaction, I find myself moving on to new problems.
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The infinite in mathematics is always unruly unless it is properly treated.
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You treat world history as a mathematician does mathematics, in which nothing but laws and formulas exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.
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... though mathematics may teach a man how to build a bridge, it is what the Scotch Universities call the humanities, that teach him to be civil and sweet-tempered.
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Of Belief Human mathematics, so to speak, like the length of life, are subject to the doctrine of chances.
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The unreasonable efficiency of mathematics in science is a gift we neither understand nor deserve.
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Logic has virtually nothing to do with the way we think.
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It has been a fortunate fact in the modern history of physical science that the scientist constructing a new theoretical system has nearly always found that the mathematics. . . required. . . had already been worked out by pure mathematicians for their own amusement. . . . The moral for statesmen would seem to be that, for proper scientific "planning", pure mathematics should be endowed fifty years ahead of scientists.
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The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation of it.
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I came into history from a primary concern with mathematics and science. This has been a tremendous help to me as a person and as a historian, although it must be admitted it has served to make my historical interpretations less conventional than may be acceptable of many of my colleagues in the field.
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Perhaps the most surprising thing about mathematics is that it is so surprising.
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The manuscript looks chaotic, even by mathematics standards.
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Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.
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Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language.
