外文翻譯---數(shù)學直覺和認知的根源-全文預覽
【正文】 ell as confusion of the nineteenth—a century having produced many results, among which several were false or unproven or badly stated—a program, however, which excluded from foundational analysis any scienti?c examination of the constitutive process of mathematical concepts and structures. Such is the aim of the new project regarding the cognitive foundations of Mathematics, which is also an epistemological project. It is certainly not a question of discarding proof, with its logical and formal ponents, but simply of steering away from the formal(and putational) monomania, formerly justi?ed and which dominated the previous century. It produced these marvelous logica formal machines surrounding us, acting without meaning. The analysis of the constitution of sense and meaning in Mathematics, through cognition and history, is the purpose of the investigations in Mathematical Cognition, also in order to pensate the provable inpleteness of formalisms.In order to grasp this issue, it is necessary to be precise regarding the term ‘‘formal’’, proper to the formal systems extensively taken to be the only locus for foundation. There exists a very widespread ambiguity, particularly in the ?eld of physics: for a physicist, to ‘‘formalize’’ a physical process means to mathematize it. Since Hilbert’s program, formal instead has meant: a system given by ?nite sequences of meaningless signs, governed by rules, themselves being ?nite sequences of signs, which only operate by purely mechanical ‘‘sequencematching’’ and ‘‘sequencereplacement’’—as do lambdacalculus and Turing machines, for instance, two paradigms for any effective formalism, following the equivalence results. To be fair, some recentrevitalization of Hilbert’s program try to extend this notion of ‘‘formal’’, sometimes in reference to writings by Hilbert as well. One may surely propose new notions, yet the de?nition of what a formal system is, is formally given by inpleteness theorem and Turing’s work. AsGo 168。 Mathematical Intuition and the Cognitive Roots of MathematicalConceptsGiuseppe Longo ? Arnaud ViarougePublished online: 20 January 2010Springer Science+Business Media . 2010AbstractThe foundation of Mathematics is both a logica formal issue and an epistemological one. By the ?rst, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This ‘‘genealogy of concepts’’, is necessary both in order to enrich the foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable inpleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition.Keywords: Numerical cognition Mathematical intuition Foundations of mathematics1 From
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