Mackereth, Stephen Gary
(2024)
Logic, Arithmetic, and Definitions.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
Arithmetic 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, Gödel 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 conservative over 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 Gödel’s Dialectica translation succeeds in assigning a constructive meaning to quantificational theories of arithmetic. Virtually all commentators have objected that Gödel’s translation makes use of definitions which presuppose the very quantificational logic that Gödel was trying to eliminate. This, of course, would render the translation philosophically circular. Gödel was adamant that there was no circularity here, but no one has been able to understand his defense of this claim. I vindicate Gödel, showing that there is no circularity and answering a longstanding exegetical question in Gödel scholarship.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
27 August 2024 |
| Date Type: |
Publication |
| Defense Date: |
11 June 2024 |
| Approval Date: |
27 August 2024 |
| Submission Date: |
18 June 2024 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
215 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Philosophy |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
logic,
arithmetic,
definitions,
philosophy of mathematics,
foundations of mathematics,
Frege,
logicism,
neologicism,
neo-Fregeanism,
conservativeness,
conservativity,
second-order logic,
Gödel,
Dialectica translation,
proof theory,
constructivism,
intuitionism,
finitism,
higher type computability,
reductive provability |
| Related URLs: |
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| Date Deposited: |
27 Aug 2024 13:46 |
| Last Modified: |
27 Aug 2024 13:46 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/46578 |
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