Are Minds Like Numbers?

Before you either embrace or dismiss the possibility that souls could be number-like, you might want to give some more thought to the question

Here are some readings to help you get started.

PAUL BENACERRAF, "WHAT NUMBERS COULD NOT BE." Philosophical Review. 1965; 74,47-73
IN REVIEWING THE ESSENTIALS OF A LOGICIST ANALYSIS OF NUMBER IT IS NOTED THAT NO ANALYSIS IDENTIFYING NUMBERS WITH PARTICULAR SETS IS "CORRECT" TO THE EXCLUSION OF OTHER ANALYSES, WHICH IDENTIFY THE NUMBERS WITH DIFFERENT SETS. BUT IF THE SENSE OF, E.G., "THREE" DETERMINES ITS REFERENCE, AND AT LEAST TWO ANALYSES OF "THREE" ARE EQUALLY "CORRECT" BUT ASSIGN IT TWO DIFFERENT SETS AS ITS REFERENT, THEN THE CONDITION IN THE ANALYSES THAT STATES THAT THREE IS A SET IS A SUPERFLUOUS ONE, AND NUMBERS COULDN'T BE SETS AT ALL. IN A FINAL SECTION IT IS SUGGESTED THAT BY SUBSTITUTING THE WORD "OBJECT" FOR THE WORD "SET" A SIMILAR ARGUMENT ARISES WHICH CAN THEN BE USED TO REFUTE THE IDENTIFICATION OF NUMBERS WITH ANY GIVEN SYSTEM OF OBJECTS: TO CHARACTERIZE THE NUMBERS IS TO CHARACTERIZE NOT A SYSTEM OF OBJECTS BUT AN ABSTRACT STRUCTURE WHICH MANY SYSTEMS OF OBJECTS MIGHT EXHIBIT.

PAUL BENACERRAF, "MATHEMATICAL TRUTH." Journal of Philosophy. 1973; 70,661-679
THIS PAPER SERVES MERELY TO POSE THE FOLLOWING PROBLEM: TWO QUITE DISTINCT KINDS OF CONCERNS HAVE SEPARATELY MOTIVATED ACCOUNTS OF MATHEMATICAL TRUTH: (1) THE DESIRE FOR A HOMOGENEOUS SEMANTICAL THEORY IN WHICH SEMANTICS FOR THE PROPOSITIONS OF MATHEMATICS PARALLEL THE SEMANTICS FOR THE REST OF THE LANGUAGE, AND (2) THE CONCERN THAT THE ACCOUNT OF MATHEMATICAL TRUTH MESH WITH A REASONABLE EPISTEMOLOGY. IT IS ARGUED THAT ACCOUNTS OF TRUTH THAT TREAT MATHEMATICAL AND NON-MATHEMATICAL DISCOURSE SEMANTICALLY IN RELEVANTLY SIMILAR WAYS DO SO AT THE COST OF LEAVING IT UNINTELLIGIBLE HOW WE CAN HAVE ANY MATHEMATICAL KNOWLEDGE WHATSOEVER; WHEREAS THOSE ACCOUNTS WHICH ATTRIBUTE TO MATHEMATICAL PROPOSITIONS THE KINDS OF TRUTH CONDITIONS WE CAN CLEARLY KNOW TO OBTAIN, DO SO AT THE EXPENSE OF FAILING TO CONNECT THESE CONDITIONS WITH ANY ANALYSIS OF THE SENTENCES WHICH SHOWS HOW THE ASSIGNED CONDITIONS ARE CONDITIONS OF THEIR 'TRUTH'. THEREFORE, NO EXISTING ACCOUNT SEEMS TO MEET BOTH CONCERNS. YET BOTH MUST BE MET BY ANY ADEQUATE ACCOUNT.

Paul Benacerraf, Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings Cambridge University Press, 1983. Edition: 2nd ed. Material: viii, 600 p. Notes: Bibliography: p. 571-600. DH Hill Library QA8.4 .P48 1983 c.1

LINDA WETZEL, "THAT NUMBERS COULD BE OBJECTS." Philosophical Studies. 1989; 56,273-292

Paul Benacerraf, "What Mathematical Truth Could Not Be -I" in Benacerraf and His Critics, Morton, Adam (ed) Blackwell, 1996 [see below]
A review of certain developments on issues treated in the author's "What Numbers Could Not Be" (WNCNB), the paper discusses 1) the philosophical atmosphere in which (and in response to which) it was written; 2) the argument of WNCNB, identifying several "lacunae", and suggesting how one might address them; and 3) philosophical uses of metamathematical results, especially recent appeals to the Lowenheim-Skolem theorems to establish limits on our conceptual abilities. In these, as in the better understood cases regarding the Godel incompleteness results, the desired philosophical results are claimed to depend on a significant prior injection of philosophical presuppositions.

Adam Morton and Stephen P. Stich, eds. Benacerraf and His Critics Blackwell Publishers, 1996. Material: xi, 271 p.: ill. Notes: Includes bibliographical references and index.
1. What Mathematical Truth Could Not Be - I / Paul Benacerraf. 2. The Legacy of 'Mathematical Truth' / Penelope Maddy. 3. Prospects for Platonism / Steven J. Wagner. 4. On What Possible Worlds Could Not Be / Robert Stalnaker. 5. Skepticism about Numbers and Indeterminacy Arguments / Jerrold J. Katz. 6. On the Proof of Frege's Theorem / George Boolos. 7. Logicism 2000: A Mini-manifesto / Richard Jeffrey. 8. Shadows of Remembered Ancestors: Mathematics as the Epitome of Story-telling / Richard E. Grandy. 9. Wittgenstein: Mathematics, Regularities, and Rules / Mark Steiner. 10. Mathematics as Language / Adam Morton. 11. Infinite Pains: The Trouble with Supertasks / John Earman and John D. Norton. Bibliography of Paul Benacerraf to 1995. Bibliography of Works Discussing Paul Benacerraf's Work. DH Hill Library QA29.B515 B46 1996 c.1

Charles S. Chihara. Constructibility and Mathematical Existence Clarendon Press, 1990. Material: xv, 282 p.: ill. Notes: Includes bibliographical references (p. [273]-278) and index. System ID No: ACM-1953 DH Hill Library QA8.4 .C45 1990 c.1

Jody Azzouni, Metaphysical Myths, Mathematical Practice: the Ontology and Epistemology of the Exact Sciences (Cambridge University Press, 1994) Includes bibliographical references (p. 235-244) and index.
Pt. I. Mathematical Practice and Its Puzzles. 1. Metaphysical Inertness. 2. Metaphysical Inertness and Reference. 3. The Virtues of (Second-Order) Theft. 4. Intuitions about Reference and Axiom Systems. 5. Comparing Mathematical Terms and Empirical Terms I. 6. Comparing Mathematical Terms and Empirical Terms II. 7. The Epistemic Role Puzzle. 8. Benacerraf's Puzzle. 9. Comparing Puzzles. 10. Quine's Approach I. 11. Quine's Approach II -- Pt. II. The Stuff of Mathematics: Posits and Algorithms. 2. An Initial Picture. 3. Application and Truth. 4. Systems, Application, and Truth. 5. Quine's Objections to Truth by Convention. 6. Grades of Ontological Commitment. 7. Multiply Interpreting Systems. 8. Intuitions about Reference Revisited -- Pt. III. The Geography of the a Priori. 2. Algorithms Again. 3. Some Observations on Metamathematics. 4. Incorrigible Co-empiricalness. 5. Why There Are No Incorrigible Co-empirical Truths. 6. Normative Considerations, the Success of Applied Mathematics, Concluding Thoughts.

Michael D. Resnik. Mathematics as a Science of Patterns Oxford University Press, 1997. Material: xiii, 285 p.: ill. Notes: Includes bibliographical references (p. [275]-281) and index.
1. Introduction. 2. What is Mathematical Realism?. 3. The Case for Mathematical Realism. 4. Recent Attempts at Blunting the Indispensability Thesis. 5. Doubts about Realism. 6. The Elusive Distinction between Mathematics and Natural Science. 7. Holism: Evidence in Science and Mathematics. 8. The Local Conception of Mathematical Evidence: Proof, Computation, and Logic. 9. Positing Mathematical Objects. 10. Mathematical Objects as Positions in Patterns. 11. Patterns and Mathematical Knowledge. 12. What is Structuralism? And Other Questions. DH Hill Library QA8.4 .R473 1997 c.1

Michael D. Resnik, ed. Mathematical Objects and Mathematical Knowledge Aldershot, England ; Brookfield, Vt.: Dartmouth, 1995. Material: xxi, 647 p. Notes: Includes bibliographical references and index.
Pt. I. Field's Response. There Are No Mathematical Objects. Mathematics is Theoretically Dispensable. 1. Realism and Anti-Realism about Mathematics / Hartry Field. 2. Is Mathematical Knowledge Just Logical Knowledge? / Hartry Field. 3. Review of Science without Numbers: A Defense of Nominalism / David Malament. 4. Conservativeness and Incompleteness / Stewart Shapiro. 5. Synthetic Mechanics / John P. Burgess. Pt. II. Another Anti-Realist Response: Mathematics is Really Talk About Possibilities Involving Concrete Objects. 6. A Simple Type Theory Without Platonic Domains / Charles S. Chihara. 7. Arithmetic for the Millian / Philip Kitcher. Pt. III. Maddy's Realism: Mathematical Objects are Like Ordinary Concrete Objects and Known by Similar Means. 8. Physicalistic Platonism / Penelope Maddy. 9. Perception and Mathematical Intuition / Penelope Maddy. 10. A Godelian Thesis Regarding Mathematical Objects: Do They Exist? And Can We Perceive Them? / Charles S. Chihara. Pt. IV. Other Approaches to Realism. 11. Truth and Proof: The Platonism of Mathematics / W. W. Tait. 12. Why Numbers Can Believably Be: A Reply to Hartry Field / Crispin Wright. Pt. V. Second-Order Logic: A New Tool for Logicists and Other Mathematical Reductionists? 13. To Be is To Be a Value of a Variable (Or To Be Some Values of Some Variables) / George Boolos. 14. Nominalist Platonism / George Boolos. 15. Saving Frege from Contradiction / George Boolos. 16. Second-Order Logic Still Wild / Michael D. Resnik. 17. Second-Order Logic, Foundations, and Rules / Stewart Shapiro. Pt. VI. Structuralism: Mathematics Studies Structures. 18. Mathematics as a Science of Patterns: Ontology and Reference / Michael D. Resnik. 19. A Naturalized Epistemology for a Platonist Mathematical Ontology / Michael D. Resnik. 20. Structure and Ontology / Stewart Shapiro. 21. Modal-Structural Mathematics / Geoffrey Hellman. 22. The Structuralist View of Mathematical Objects / Charles Parsons. Pt. VII. New Approaches to Mathematical Intuition. 23. Mathematical Intuition / Charles Parsons. 24. Phenomenology and Mathematical Knowledge / Richard Tieszen. System ID No: AIB-9689 DH Hill Library QA8.6 .M377 1995 c.1

Thomas Tymoczko, ed. New Directions in the Philosophy of Mathematics: an Anthology Princeton University Press, 1998. Edition: Rev. and expanded ed. Material: xvii, 436 p. Notes: "First Princeton paperback printing 1998"--T.p. verso. Includes bibliographical references (p. 399-436).
Pt. I. Challenging Foundations. Some Proposals for Reviving the Philosophy of Mathematics / Reuben Hersh. A Renaissance of Empiricism in the Recent Philosophy of Mathematics? / Imre Lakatos. What Is Mathematical Truth? / Hilary Putnam. "Modern" Mathematics: An Educational and Philosophic Error? / Rene Thom. Mathematics as an Objective Science / Nicholas D. Goodman. From the Preface of Induction and Analogy in Mathematics / George Polya. Generalization, Specialization, Analogy / George Polya. Pt. II. Mathematical Practice. Theory and Practice in Mathematics / Hao Wang. What Does a Mathematical Proof Prove? / Imre Lakatos. Fidelity in Mathematical Discourse: Is One and One Really Two? / Philip J. Davis. The Ideal Mathematician / Philip J. Davis and Reuben Hersh. The Cultural Basis of Mathematics / Raymond L. Wilder. Is Mathematical Truth Time-Dependent? / Judith V. Grabiner. Mathematical Change and Scientific Change / Philip Kitcher. The Four-Color Problem and Its Philosophical Significance / Thomas Tymoczko. Social Processes and Proofs of Theorems and Programs / Richard A. De Milio, Richard J. Lipton and Alan J. Perlis. Information-Theoretic Computational Complexity and Godel's Theorem and Information / Gregory Chaitin. Pt. III. Current Concerns. Proof as a Source of Truth / Michael D. Resnik. On Proof and Progress in Mathematics / William P. Thurston. Does V Equal L? / Penelope Maddy. DH Hill Library QA8.6 .N48 1998 c.1

2000 David F. Austin