Discrete Thoughts: Essays on Mathematics, Science and Philosophy
Springer Science & Business Media, Jul 1, 2009 - Mathematics - 266 pages
as anywhere today, it is becoming more d- ficult to tell the truth. To be sure, our store of accurate facts is more plentiful now than it has ever been, and the minutest details of history are being thoroughly recorded. Scientists, - men and scholars vie with each other in publishing excruciatingly definitive accounts of all that happens on the natural, political and historical scenes. Unfortunately, telling the truth is not quite the same thing as reciting a rosary of facts. Jos6 Ortega y Gasset, in an adm- able lesson summarized by Antonio Machado's three-line poem, prophetically warned us that the reason people so often lie is that they lack imagination: they don't realize that the truth, too, is a matter of invention. Sometime, in a future that is knocking at our door, we shall have to retrain ourselves or our children to properly tell the truth. The exercise will be particularly painful in mathematics. The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics. Shocking as it may be to a conservative logician, the day will come when currently MATHEMATICS, IN vague concepts such as motivation and purpose will be made formal and accepted as constituents of a revamped logic, where they will at last be allotted the equal status they deserve, si- by-side with axioms and theorems.
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Discrete Thoughts: ESSAYS ON MATHEMATICS, Science, and Philosophy
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ability algebraic algorithm analysis applied argument artiﬁcial intelligence audio axioms become Brownian motion CAI language CAI systems calculation century coalitions color combinatorial theory complex computer science computer’s concept continuum hypothesis courseware coursewriting deﬁned deﬁnition differential geometry difﬁcult diophantine equations economic elementary equations equilibrium example fact ﬁeld ﬁgure ﬁnd ﬁnite number ﬁrms ﬁrst ﬁxed formal function genetic geometry given graph graphic Hilbert Husserl ideas inﬁnite input instructions integers intellectual interaction machine material mathe mathematical logic mathematicians matics mechanics memory ment molecules move Neumann objects participants pattern payoff payoff matrix perhaps philosophers physics players possible present probability problem production programming languages proof puter real numbers reason result scientiﬁc scientists second law sequence set theory signiﬁcant space speciﬁc square Stan Ulam statistical structure student suboptimal symmetric game techniques theorem tion today’s Turing machine University