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Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
"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.
A Theory of Explicit Mathematics Equivalent to ID_1
"... We show that the addition of name induction to the theory EETJ + (LEM I N ) of explicit elementary types with join yields a theory prooftheoretically equivalent to ID_1. ..."
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We show that the addition of name induction to the theory EETJ + (LEM I N ) of explicit elementary types with join yields a theory prooftheoretically equivalent to ID_1.
Forcing for Hat Inductive Definitions in Arithmetic — One of the Simplest Applications of Forcing —
"... By forcing, we give a direct interpretation of ÎDω into Avigad’s FP. To the best of the author’s knowledge, this is one of the simplest applications of forcing to “real problems”. 1 ..."
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By forcing, we give a direct interpretation of ÎDω into Avigad’s FP. To the best of the author’s knowledge, this is one of the simplest applications of forcing to “real problems”. 1
BETWEEN THE FINITARY AND THE IDEAL
"... Within contemporary philosophy of mathematics there is a trend focussing on how mathematics is done and how it evolves, rather than how it should be done or how it should evolve. This fact is somewhat contrary to the philosophy of mathematics in the 20th century, which to a large extent was dominate ..."
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Within contemporary philosophy of mathematics there is a trend focussing on how mathematics is done and how it evolves, rather than how it should be done or how it should evolve. This fact is somewhat contrary to the philosophy of mathematics in the 20th century, which to a large extent was dominated by views developed during the socalled foundational crisis in the beginning of that very century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justication and consistency of mathematics. Mathematicians and philosophers like Gottlob Frege (1848{1925), Bertrand Russell (1872{1970), David Hilbert (1862{1943), Kurt Godel (1906{1978) and others were very successful in their development of logic from around 1890 to, say, 1940 and they had a huge impact on the philosophy of mathematics of those days. Most probably it was Hilbert's program, rise and fall, which was the most single in
uential factor of the foundational studies until 1960's. Among the results was the widespread conception that the proper { if not the only { approach to philosophy of mathematics was through