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The Constructed Objectivity of Mathematics and the Cognitive Subject
, 2001
"... Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out the mathematical con ..."
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Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out the mathematical conceptualization from any other form of knowledge, and give unity to it. Yet, this very conceptualization is deeply rooted in our "acts of experience", as Weyl says, beginning with our presence in the world, first in space and time as living beings, up to the most complex attempts we make by language to give an account of it. I will try to sketch the origin of some key steps in organizing perception and knowledge by "mathematical tools", as mathematics is one of the many practical and conceptual instruments by which we categorize, organise and "give a structure" to the world. It is conceived on the "interface" between us and the world, or, to put it in husserlian terminology, it is "de
Carnap's remarks on Impredicative Definitions and the Genericity Theorem
 IN LOGIC, METHODOLOGY AND PHILOSOPHY OF SCIENCE: LOGIC IN FLORENCE
, 1997
"... In a short, but relevant paper [Car31], Rudolf Carnap summarizes the logicist foundation of mathematics, largely following Frege and Russell 's view. Carnap moves away though from Russell's approach on a crucial aspect: a detailed justification of impredicative definitions (a formal version of Russe ..."
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In a short, but relevant paper [Car31], Rudolf Carnap summarizes the logicist foundation of mathematics, largely following Frege and Russell 's view. Carnap moves away though from Russell's approach on a crucial aspect: a detailed justification of impredicative definitions (a formal version of Russell's "vicious circle"), that he accepts. In this note we revisit Carnap's justification of impredicativity, within the frame of impredicative Type Theory. More precisely, we recall the treatment of impredicativity given in Girard's System F and justify it by reference to a recent result, the Genericity Theorem in [LMS93], which may help to set on mathematical grounds Carnap's informal remark. We then discuss the logical complexity of (the proof of) that theorem. Finally, the role of the Genericity Theorem in understanding the surprising "uniformities" of the consistency proof of Arithmetic, via System F, is hinted. The problem A definition is said to be impredicative, if it defines a concep...
Prototype Proofs in Type Theory
, 2000
"... The proofs of universally quantified statements, in mathematics, are given as "schemata" or as "prototypes" which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. ..."
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The proofs of universally quantified statements, in mathematics, are given as "schemata" or as "prototypes" which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of \Gammacalculus and act as "proofschemata", as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i.e. Girard's system F, where typequantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context.
From Hypocomputation to Hypercomputation
, 2008
"... Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of ..."
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Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of metamathematics, and so many of the previous results have been folded into other theories, that we are in danger of loosing an appreciation of the broader structure of formal theories. As an aid to those looking to develop hypercomputational theories, we will briefly survey the known landmarks both inside and outside the borders of computational theory. We will not focus in this paper on why the structure of formal theory looks the way it does. Instead we will focus on what this structure looks like, moving from hypocomputational, through traditional computational theories, and then beyond to hypercomputational theories.
The Origin of Mathematics
, 1994
"... In every science one finds a hard seed of reality and beautiful flowers of logical imagination. The former, the origin is rooted in our intuition and common experience. As the origin, it precedes any later development and lies beyond it. Differentiation of a culture is the process in which various a ..."
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In every science one finds a hard seed of reality and beautiful flowers of logical imagination. The former, the origin is rooted in our intuition and common experience. As the origin, it precedes any later development and lies beyond it. Differentiation of a culture is the process in which various areas of common experience are
DOI 10.1007/s112290119883y What are numbers?
"... Abstract This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs existnwise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the socalled Caesar objection will be answered by ex ..."
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Abstract This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs existnwise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the socalled Caesar objection will be answered by explaining exactly what kind of manner or mode numbers are. And then what we shall call the Functionality of Cardinality objection will be answered by establishing the fact that for any numbers m and n, if there are exactly mFs and also there are exactly nFs, then m = n.