By Gilles Dowek
Computation is revolutionizing our international, even the internal global of the "pure" mathematician. Mathematical tools - in particular the idea of facts - that experience their roots in classical antiquity have visible a thorough transformation because the Seventies, as successive advances have challenged the concern of cause over computation. Like many revolutions, this one comes from inside. Computation, calculation, algorithms - all have performed a tremendous position in mathematical growth from the start - yet backstage, their contribution was once obscured within the enduring mathematical literature. to appreciate the way forward for arithmetic, this attention-grabbing ebook returns to its earlier, tracing the hidden background that follows the thread of computation. alongside how it invitations us to reassess the conversation among arithmetic and the typical sciences, in addition to the connection among arithmetic and computing device technology. It additionally sheds new mild on philosophical innovations, akin to the notions of analytic and artificial judgment. eventually, it brings us to the edge of the recent age, within which computer intelligence bargains new methods of fixing mathematical difficulties formerly inaccessible. This ebook is the 2007 Winner of the Grand Prix de Philosophie de l'Académie Française.
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Additional info for Computation, Proof, Machine: Mathematics Enters a New Age
The same year, Alfred Tarski put forward a similar algorithm applicable to all propositions in the theory of real numbers with both addition and multiplication. ” Presburger’s algorithm allows us to determine the provability of propositions not just of this form, but of the forms “there exist and such that ,” “for all , there exists such that ,” “there exist and such that ,” and so forth. A consequence of Tarski’s theorem, then, is that all geometry problems can be solved by computation. Whereas the Greeks had introduced reasoning to solve problems – especially in geometry – that they could not solve by computing, Tarski showed that, in the case of geometry at least, moving from computation to reasoning is after all unnecessary, because an algorithm that the Greeks had not foreseen can be substituted for reasoning.
The proposition turns into the proposition “if then ” and , the proposition can in turn be similarly transformed, and so on, and so on. As a result, although this transformation process succeeds when the proposition is provable, it doesn’t fail when the proposition is not provable but keeps searching endlessly for a solution. The method defined by the rule “ turns into ,” however, is not, because the expression would be transformed into , then and so on, without ever reaching a result. Hilbert’s decision problem was not whether reasoning could be replaced by a computation method, because the answer to that is obviously yes: all you need to do is replace each inference rule with computation rule.
For Frege, the notion of “set” was intertwined with that of “concept”: the set containing all roses is nothing more than the concept “to be a rose,” and a concept is defined by a proposition. Thus did Frege reopen an issue that had been more or less dormant since antiquity, namely, clarification of the grammar of mathematical propositions and rules of inference. Unlike the Stoics and like the logicians of the Middle Ages, Frege decomposed atomic propositions; however, he decomposed them not into two elements (a predicate and its subject), but into a “relational predicate” and the several complements it links (among which is the subject).