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Foundational and mathematical uses of higher types
 REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN
, 1999
"... In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles ..."
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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles which generalize (and for n = 0 coincide with) the socalled `weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context)
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we sho ..."
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Cited by 9 (3 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
The computational content of classical arithmetic
, 2009
"... Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in c ..."
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Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical firstorder arithmetic, and reflects on some of the relationships between them. Variants of the GödelGentzen doublenegation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory.
REFLECTIONS ON FORMALISM AND REDUCTIONISM IN LOGIC AND COMPUTER SCIENCE
"... This report contains a preprint (paper 1) and a reprint (paper 2). The first develops some epistemological views which were hinted in the second, in particular by stressing the need of a greater role of geometric insight and images in foundational studies and in approaches to cognition. The second p ..."
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This report contains a preprint (paper 1) and a reprint (paper 2). The first develops some epistemological views which were hinted in the second, in particular by stressing the need of a greater role of geometric insight and images in foundational studies and in approaches to cognition. The second paper is the "philosophical " part of a lecture in Type Theory, whose technical sections, omitted here, have been largely subsumed by
Beyond Matter and Spirit to an
"... Science has rendered the matter and spirit conceptual framework obsolete. This paper advocates its replacement with structure (what we can understand through science) and essence (what we experience in stream of consciousness). The Totality Axiom is introduced to integrate structure and essence ..."
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Science has rendered the matter and spirit conceptual framework obsolete. This paper advocates its replacement with structure (what we can understand through science) and essence (what we experience in stream of consciousness). The Totality Axiom is introduced to integrate structure and essence and thus provide a starting point for integrating scientific understanding with spiritual intuition. Using this axiom, the paper shows how the mathematical result known as Godel's Incompleteness Theorem can be applied to the evolution of consciousness giving substance to spiritual intuitions. This is an important example of how the Totality Axiom lays the groundwork for an objective spirituality with the potential to develop at a rate adequate to cope with the enormous power that science and technology is creating through the objective guidance of experiment.