---------------------------------------------------------------- Why can't somebody give us a list of things that everybody thinks and nobody says, and another list of things that everybody says and nobody thinks? Oliver Wendell Holmes ---------------------------------------------------------------- There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain. A. N. Whitehead ---------------------------------------------------------------- A marveilous newtrality have these things mathematicall, and also a strange participation between things supernaturall and things naturall. John Dee ---------------------------------------------------------------- http://www.mathpages.com/home/quotes.htm ---------------------------------------------------------------- % It is the first duty of a hypothesis to be intelligible. % -- Huxley, Thomas Henry (1825-1895). ---------------------------------------------------------------- % I have often noticed that when people come to understand a mathematical % proposition in some other way than that of the ordinary demonstration, they % promptly say, "Oh, I see. That's how it must be." This is a sign that they % explain it to themselves from within their own system. % -- Georg Christoph Lichtenberg(1742-1799). ---------------------------------------------------------------- There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This is of course quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer. E. J. Mcshane ---------------------------------------------------------------- Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions. Felix Klein ---------------------------------------------------------------- There are two ways to do great mathematics. The first way is to be smarter than everybody else. The second way is to be stupider than everybody else -- but persistent. Raoul Bott ---------------------------------------------------------------- Minus times minus equals plus; The reason for this we won't discuss. --- Anonymous "God does not care about our mathematical difficulties. He integrates empirically." ------ Albert Einstein "It is of course easier to supply the proof when we have previously acquired some knowledge of the question by the method, than it is to find it without any previous knowledge." --Archimedes (in his *Method* letter to Eratosthenes) Most of the difficulties students have learning calculus can be traced to weak algebra skills. -- Leonhard Euler "His book is a marvel of laboured detail. No expositor could take more pains with his reader, space being held of no moment if clearness had to be secured. As might be expected, therefore, all that is really worth preserving of his work is but a small fraction of the 264 pages which he occupies in exposition." --Muir on Desnanot's work on determinants Failure is instructive. The person who really thinks learns quite as much from his failures as from his successes. ---John Dewey, quoted in Contemporary Abstract Algebra, by J. Gallian. ---------------------------------------------------------------- "If you find a path with no obstacles, it probably doesn't lead anywhere." -- Frank A. Clark ...for no human inquiry can be called science unless it pursues its path through mathematical exposition and demonstration. --- Leonardo da Vinci Reasoning draws a conclusion, but does not make the conclusion certain, unless the mind discovers it by the path of experience. --- Roger Bacon It is self-evident that any and all paths must be open to a researcher during the actual course of his [or her] investigations. --- Karl Weierstrass It is very difficult to write mathematics books nowadays. If one doesn't take pains with the fine points of theorems, explanations, proofs and corollaries, then it won't be a mathematics book; but if one does these things, then the reading of it will be extremely boring. --- J. Kepler It is downright sinful to teach the abstract before the concrete. Z. A. Melzak The heart of mathematics consists of concrete examples and concrete problems. P. R. Halmos ---------------------------------------------------------------- Audience, level, and treatment--- a description of such matters is what prefaces are supposed to be about. P. R. Halmos ---------------------------------------------------------------- [On the concept of group:] ... what a wealth, what a grandeur of thought may spring from what slight beginnings. H.F. Baker ================================================================ tensors: We are so accustomed to disguise ourselves to others that in the end we become disguised to ourselves. ~Francois Duc de La Rochefoucauld ---------------------------------------------------------------- Love is a spirit all compact of fire. William Shakespeare quotes (English Dramatist, Playwright and Poet, 1564-1616) ---------------------------------------------------------------- Society is always trying in some way to grind us down to a single flat surface. Oliver Wendell Holmes quotes (American Physician, Poet, Writer, Humorist and Professor at Harvard, 1809-1894) ---------------------------------------------------------------- The more important the subject and the closer it cuts to the bone of our hopes and needs, the more we are likely to err in establishing a framework for analysis. Stephen Jay Gould quotes Some people say, "How can you live without knowing?" I do not know what they mean. I always live without knowing. That is easy. How you get to know is what I want to know. --- Richard Feynman