Beyond the History of Mathematics II

"I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history." (James W L Glaisher, [opening address] 1890)

 “In most sciences, one generation tears down what another has built and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure” (Hermann Hankel, “Die Entwicklung der Mathematik in den letzten Jahrhunderten, 1884)

"The true method of foreseeing the future of mathematics is to study its history and its actual state." (Henri Poincaré, "Science and Method", 1908)

"One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics." (Henri Poincaré)

“Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought. I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him.” (Alfred N Whitehead, “Mathematics as an Element in the History of Thought” in “Science and the Modern World”, 1925)

“Mathematicians study their problems on account of their intrinsic interest, and develop their theories on account of their beauty. History shows that some of these mathematical theories which were developed without any chance of immediate use later on found very important applications.” (Karl Menger, “What is calculus of variations and what are its applications?”, The Scientific Monthly 45, 1937)

“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)

“Although the study of the history of mathematics has an intrinsic appeal of its own, its chief raison d'être is surely the illumination of mathematics itself.” (Charles H Edwards Jr, “The Historical Development of the Calculus”, 1979)

“Mathematical research should be as broad and as original as possible, with very long range-goals. We expect history to repeat itself: we expect that the most profound and useful future applications of mathematics cannot be predicted today, since they will arise from mathematics yet to be discovered." (Arthur Jaffe, “Ordering the universe: the role of mathematics”, SIAM Review Vol 26. No 4, 1984)

"One of the lessons that the history of mathematics clearly teaches us is that the search for solutions to unsolved problems, whether solvable or unsolvable, invariably leads to important discoveries along the way. (Carl B Boyer & Uta C Merzbach, “A History of Mathematics”, 1991)

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On Equations VI (Figurative Equations I)

“If equations are trains threading the landscape of numbers, then no train stops at pi.” (Richard Preston)

“Science is a differential equation. Religion is a boundary condition.” (Alan Turing)

“Equations are more important to me, because politics is for the present, but an equation is something for eternity.” (Albert Einstein)

“The idea that the world exists is like adding an extra term to an equation that doesn’t belong there." (Marvin Minsky)

“Life is and will ever remain an equation incapable of solution, but it contains certain known factors.” (Nikola Tesla, "A Machine to End War”, 1937)

"What truly is logic? Who decides reason? […] It's only in the mysterious equations of love that any logical reasons can be found." (John F Nash Jr)

“Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.” (Neil deGrasse Tyson)

“An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn’t care.” (Anon)

“Words are a pretty fuzzy substitute for mathematical equations.” (Isaac Asimov)

On Problem Solving IV: Solvability

"A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution." (George Pólya, "How to Solve It", 1945)

"We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but if such a problem could exist, it would be insoluble. In fact, when solving a problem, we should always profit from previously solved problems, using their result or their method, or the experience acquired in solving them." (George Polya, "How to Solve It", 1945)

"The answer to the question ‘Can there be a general method for solving all mathematical problems?’ is no! Perhaps, in a world of unsolved and apparently unsolvable problems, we would have thought that the desirable answer to this question from any point of view, would be yes. But from the point of view of mathematicians a yes would have been far less satisfying than a no is. […] Not only are the problems of mathematics infinite and hence inexhaustible, but mathematics itself is inexhaustible." (Constance Reid, "Introduction to Higher Mathematics for the General Reader", 1959)

"Some problems are just too complicated for rational logical solutions. They admit of insights, not answers." (Jerome B Wiesner, The New Yorker, 1963)

"A problem will be difficult if there are no procedures for generating possible solutions that are guaranteed (or at least likely) to generate the actual solution rather early in the game. But for such a procedure to exist, there must be some kind of structural relation, at least approximate, between the possible solutions as named by the solution-generating process and these same solutions as named in the language of the problem statement." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"Deep in the human nature there is an almost irresistible tendency to concentrate physical and mental energy on attempts at solving problems that seem to be unsolvable." (Ragnar Frisch, "From Utopian Theory to Practical Applications", [Nobel lecture] 1970)

"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis." (Lotfi A Zadeh, "Fuzzy languages and their relation to human intelligence", 1972)

"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)

"A common mistake in problem solving is to encompass too much territory, which dilutes any solutions chance of success. [...] However, the opposite error occurs more frequently." (Terry Richey, "The Marketer's Visual Tool Kit", 1994)

"[…] the meaning of the word 'solve' has undergone a series of major changes. First that word meant 'find a formula'. Then its meaning changed to 'find approximate numbers'. Finally, it has in effect become 'tell me what the solutions look like'. In place of quantitative answers, we seek qualitative ones." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems." (Carl Pomerance, MAA, 2000)

"Heuristics are rules of thumb that help constrain the problem in certain ways (in other words they help you to avoid falling back on blind trial and error), but they don't guarantee that you will find a solution. Heuristics are often contrasted with algorithms that will guarantee that you find a solution - it may take forever, but if the problem is algorithmic you will get there. However, heuristics are also algorithms." (S Ian Robertson, "Problem Solving", 2001)

"Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley." (Yōko Ogawa, "The Housekeeper and the Professor", 2003)

"Knowing a solution is at hand is a huge advantage; it’s like not having a 'none of the above' option. Anyone with reasonable competence and adequate resources can solve a puzzle when it is presented as something to be solved. We can skip the subtle evaluations and move directly to plugging in possible solutions until we hit upon a promising one. Uncertainty is far more challenging." (Garry Kasparov, "How Life Imitates Chess", 2007)

"Mathematical good taste, then, consists of using intelligently the concepts and results available in the ambient mathematical culture for the solution of new problems. And the culture evolves because its key concepts and results change, slowly or brutally, to be replaced by new mathematical beacons." (David Ruelle, "The Mathematician's Brain", 2007)

"The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists!" (David Ruelle, "The Mathematician's Brain", 2007)

"Every problem has a solution; it may sometimes just need another perspective." (Rebecca Mallery et al, "NLP for Rookies", 2009)

"Don't mistake a solution method for a problem definition - especially if it’s your own solution method." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"The really important thing in dealing with problems is to know that the question is never answered, but that it doesn't matter, as long as you keep asking. It's only when you fool yourself into thinking you have the final problem definition - the final, true answer - that you can be fooled into thinking you have the final solution. And if you think that, you're always wrong, because there is no such thing as a 'final solution'." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"I have not seen any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated." (Paul Anderson)

"It is efficient to look for beautiful solutions first and settle for ugly ones only as a last resort. [...] It is a good rule of thumb that the more beautiful the guess, the more likely it is to survive." (Timothy Gowers)

"One is always a long way from solving a problem until one actually has the answer." (Stephen Hawking)

"The best way to escape from a problem is to solve it." (Brendan Francis)

"The worst thing you can do to a problem is solve it completely." (Daniel Kleitman)

"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by reason, for in mathematics there is no ignorabimus." (David Hilbert)

"We can not solve our problems with the same level of thinking that created them." (Albert Einstein) 

"When the answer to a mathematical problem cannot be found, then the reason is frequently that we have not recognized the general idea from which the given problem only appears as a link in a chain of related problems." (David Hilbert) 

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