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NUMBERS, SYMBOLS, AND INDUCTION:
THE GENETIC AND FINITE CONCEPTS OF NUMBER
by
Ashton Betancourt
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHILOSOPHY)
December 2006
Copyright 2006 Ashton Betancourt
Object Description
| Title | Numbers, symbols, and induction: the genetic and finite concepts of number |
| Author | Betancourt, Ashton |
| Author email | jambi14@hotmail.com |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Philosophy |
| Date defended/completed | 2006-06-14 |
| Date submitted | 2006 |
| Restricted until | Unrestricted |
| Date published | 2006-11-15 |
| Advisor (committee chair) | Damnjanovic, Zlatan |
| Advisor (committee member) |
Higginbotham, James Penner, Robert College of Letters, Arts and Sciences |
| Abstract | In this essay I present and defend a formalist conception of arithmetic. I begin with an analysis of Hilbert's instrumentalist formalism. Hilbert's program stands out as the best known attempt to develop an arithmetic of number signs that does not presuppose the finite number concept. He and his assistant Bernays laid the groundwork for a theory of number as number sign, number as symbolic construction. Yet this theory was not developed, as the demands of Hilbert's program led him to abandon his original goals and to appeal to semantic reflection and the concept of finite number in order to justify his 'finitist' reasoning.; Chapter Two explores the connection between semantic reflection, the concept of finite number, primitive recursion, and recursive definitions more generally. I conclude that Primitive Recursive Arithmetic is inadequate as a theory of symbolic constructions, that the finitist standpoint cannot be expressed by any formal theory, and moreover that the essence of Hilbert's original program can best be captured by a theory of number as symbolic construction.; In Chapter Three I build on Charles Parsons's work on the concept of mathematical intuition to develop an intuitive arithmetic of number as symbolic construction. The potential infinity of a sequence of symbolic constructions is a major question left to Chapter Four. There I defend the notion of an indefinitely extensible sequence of possible constructions. I contrast the potentially infinite class of symbolic constructions with the infinite domain of natural numbers.; Chapter Five begins with an analysis of the concept of predicativity. I then argue that the concept of finite number cannot be predicatively justified. The final section of the chapter centers around Edward Nelson's use of the Hilbert-Ackermann consistency theorem to establish limits on any formula that adequately expresses the concept of number within a formal theory. The result here reinforces the argument of the previous chapters that the arithmetic of number as symbolic construction diverges in important ways from ordinary natural number arithmetic. |
| Keyword | arithmetic; induction; finitism; Hilbert; Nelson |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Type | texts |
| Legacy record ID | usctheses-m142 |
| Rights | Betancourt, Ashton |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Betancourt-20061115 |
| Archival file | uscthesesreloadpub_Volume40/etd-Betancourt-20061115.pdf |
Description
| Title | Page 1 |
| Full text | NUMBERS, SYMBOLS, AND INDUCTION: THE GENETIC AND FINITE CONCEPTS OF NUMBER by Ashton Betancourt A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHILOSOPHY) December 2006 Copyright 2006 Ashton Betancourt |
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