자료유형  

 학위논문

Control Number  

0017164334

International Standard Book Number  

9798346387954

Dewey Decimal Classification Number  

510

Main Entry-Personal Name  

Mackereth, Stephen Gary.

Publication, Distribution, etc. (Imprint  

[S.l.] : University of Pittsburgh., 2024

Publication, Distribution, etc. (Imprint  

Ann Arbor : ProQuest Dissertations & Theses, 2024

Physical Description  

216 p.

General Note  

Source: Dissertations Abstracts International, Volume: 86-05, Section: B.

General Note  

Advisor: Walsh, Sean;Avigad, Jeremy;Ricketts, Thomas;Shaw, James;Gupta, Anil.

Dissertation Note  

Thesis (Ph.D.)--University of Pittsburgh, 2024.

Summary, Etc.  

요약Arithmetic and logic seem to enjoy an especially close relationship. Frege once wrote that arithmetic is reason's nearest kin. To deny any of the basic laws of arithmetic seems tantamount to denying a basic law of logic. My dissertation is concerned with two great attempts to make something more of this informal idea. In one direction, Frege tried to reduce arithmetic to nothing but quantificational logic and definitions. Neo-logicists continue to follow in Frege's footsteps, pursuing a version of this program today. In the other direction, Godel tried to reduce certain applications of quantificational logic to nothing but arithmetic and definitions, by means of his Dialectica translation.In the first half of my dissertation, I prove new theorems (with Jeremy Avigad) that shed a surprising light on the prospects for neo-logicism. An important objection against neo-logicism is that it makes use of allegedly stipulative definitions that are not conservativeover pure logic, i.e., definitions that settle open questions that we could not have settled before. This violates a basic requirement on stipulative definitions. I argue that by passing to a richer logical and definitional framework, it is possible to overcome the conservativeness objection. However, there is a subtlety: the strategy succeeds only if conservativeness is understood semantically rather than deductively. This suggests that the viability of neo-logicism is highly sensitive to the way in which epistemic commitments are represented in formal theories.In the second half of my dissertation, I argue that Godel's Dialectica translation succeeds in assigning a constructive meaning to quantificational theories of arithmetic. Virtually all commentators have objected that Godel's translation makes use of definitions which presuppose the very quantificational logic that Godel was trying to eliminate. This, of course, would render the translation philosophically circular. Godel was adamant that there was no circularity here, but no one has been able to understand his defense of this claim. I vindicate Godel, showing that there is no circularity and answering a longstanding exegetical question in Godel scholarship.

Subject Added Entry-Topical Term  

Mathematics.

Subject Added Entry-Topical Term  

Set theory.

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Mathematicians.

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Theorems.

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20th century.

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Philosophy.

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Logic.

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Theoretical mathematics.

Added Entry-Corporate Name  

University of Pittsburgh.

Host Item Entry  

Dissertations Abstracts International. 86-05B.

Electronic Location and Access  

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Control Number  

joongbu:657254