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The reasonable effectiveness of Mathematics and its Cognitive roots
, 2001
"... this paper, Mathematics is viewed as a "three dimensional manifold" grounded on logic, formalisms and invariants of space; we will appreciate by 1 ..."
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this paper, Mathematics is viewed as a "three dimensional manifold" grounded on logic, formalisms and invariants of space; we will appreciate by 1
The reasonable effectiveness of Mathematics and its Cognitive roots 1
"... “At the beginning, Nature set up matters its own way and, later, it constructed human intelligence in such a way that [this intelligence] could understand it” [Galileo Galilei, 1632 (Opere, p. 298)]. “The applicability of our science [mathematics] seems then as a symptom of its rooting, not as a mea ..."
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“At the beginning, Nature set up matters its own way and, later, it constructed human intelligence in such a way that [this intelligence] could understand it” [Galileo Galilei, 1632 (Opere, p. 298)]. “The applicability of our science [mathematics] seems then as a symptom of its rooting, not as a measure of its value. Mathematics, as a tree which freely develops his top, draws its strength by the thousands roots in a ground of intuitions of real representations; it would be disastrous to cut them off, in view of a shortsided utilitarism, or to uproot them from the ground from which they rose ” [H. Weyl, 1910]. Summary. Mathematics stems out from our ways of making the world intelligible by its peculiar conceptual stability and unity; we invented it and used it to single out key regularities of space and language. This is exemplified and summarised below in references to the main foundational approaches to Mathematics, as proposed in the last 150 years. Its unity is also stressed: in this paper, Mathematics is viewed as a "three dimensional
Reflections on Concrete Incompleteness?
"... Abstract. How do we prove true, but unprovable propositions? Gödel produced a statement whose undecidability derives from its “ad hoc” construction. Concrete or mathematical incompleteness results, instead, are interesting unprovable statements of Formal Arithmetic. We point out where exactly lays ..."
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Abstract. How do we prove true, but unprovable propositions? Gödel produced a statement whose undecidability derives from its “ad hoc” construction. Concrete or mathematical incompleteness results, instead, are interesting unprovable statements of Formal Arithmetic. We point out where exactly lays the unprovability along the ordinary mathematical proofs of two (very) interesting formally unprovable propositions, KruskalFriedman theorem on trees and Girard’s Normalization Theorem in Type Theory. Their validity is based on robust cognitive performances, which ground mathematics on our relation to space and time, such as symmetries and order, or on the generality of Herbrands notion of prototype proof. Introduction: some history, some philosophy Suppose that you were asked to give the result of the sum of the first n integers. There exist many proofs of this simple fact (see [Nelsen93] for this and more examples), an immediate one (allegedly (re)invented by Gauss at the age of 7