No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical.
Hans FreudenthalRelated topics
affected beauty complex hot idea invention inversion matter problem procedure solution teaching turn upside wayRelated quotes
Remembering that I'll be dead soon is the most important tool I've ever encountered to help me make the big choices in life. Because almost everything - all external expectations, all pride, all fear of embarrassment or failure - these things just fall away in the face of death, leaving only what is truly important. Remembering that you are going to die is the best way I know to avoid the trap of thinking you have something to lose. You are already naked. There is no reason not to follow your heart.
Steve Jobs
There was then no professorship in mathematics and no opening for a mathematician to a career at Cambridge; and so Wallis reluctantly left the university. In 1649 he was appointed to the Savilian chair of geometry at Oxford, where he lived until his death on Oct. 28, 1703. It was there that all his mathematical works were published. Besides those he wrote on theology, logic, and philosophy; and was the first to devise a system for teaching deaf mutes.
John Wallis
Early in my professional life, I found that many areas of economics attracted me. I started working and publishing in price theory by 1938. In 1946, I published an early work on linear programming (The Cost of Subsistence) which solved the problem only approximately; George Dantzig soon presented the exact solution. In the 1940s, I began empirical work on price theory, starting with a test of the kinked oligopoly demand curve theory of rigid prices.
George Stigler
The term A[rt] resembles a chemist's graduated glass. Each age contributes its own quantity: for example, 5 grams of the perfume 'Coty' to tickle the nostrils of the fine gentry. Or another example, 10 cc of sulfuric acid to be thrown into the face of the ruling classes. Or, 15 cc: of some kind of metallic solution that later changes into a new source of light.. ..[as Lissitzky goes on to say] This A[rt] is an invention of the mind, i.e. a complex, where rationality is fused with imagination.
El Lissitsky
There is something that even now strikes me as strange. Originally I was not possessed by gloomy thoughts. My concern, what confronted me with my real problem, was beauty alone. But I do not think that the war affected me by filling my mind with gloomy thoughts. When people concentrate on the idea of beauty, they are, without realizing it, confronted with the darkest thoughts that exist in this world. That, I suppose, is how human beings are made.
Yukio Mishima
