The Nominalist Theories of Number: A Peano Arithmetic Critique

  • Etorobong Godwin Akpan Department of Philosophy, University of Port Harcourt
Keywords: Nominalism, Peano Arithmetic, Fictionalism, Formalism, Model, Number

Abstract

The paper is concerned with the study of the nominalist theories of number. The objective of the study is to evaluate nominalist theories against the backdrop of their capacity to adequately interpret Peano Arithmetic, having accepted the arithmetic as a standard and a fruitful formalism for general arithmetic. Two strands of nominalism are considered, namely; formalism and fictionalism. Using the method of content analysis of texts and publications on the subject matter, it has been shown that formalism denies symbolic transcendence and the necessity of first order logic in a system model, while fictionalism claims that mathematical statements are vacuous even in the face of the use of bound variables. The claims of the two strands of nominalism, as well as the general claim of nominalism itself, are contrary to the nature of Peano Arithmetic in particular and arithmetic in general. Hence, nominalism cannot provide an adequate theory of number that can satisfy the interpretation of general arithmetic.

References

1. Achebe, C. (1980). Arrow of God. African Writers Series 6. London: Heineman.
2. Adams, M. M. (1977). “Ockham’s Nominalism and Real Entities”. The Philosophical Review. 86(2), 144 – 170. JSTOR. Web. 19 April, 2022.
3. Benacerraf, P & Putnam, H. (1985). Philosophy of Mathematics: Selected Readings. 2nd ed. Cambridge: Cambridge University Press. Web. 19 March, 2022.
4. Burgess, J. (2004). “Mathematics and Bleak House”. Philosophia Mathematica. 12(3), 18 – 36. Web. 19 March, 2022.
5. Ebba, G. (2011). Objective and Objectivity Alternative to Mathematical Realism. Umea: Print and Media. Web. 21 March, 2022.
6. Finedictionary. “Conceptualism”. Dex Form. n.d, n.pag. Web. 19 March, 2022.
7. Gödel, K. (1986). Collected Works Volume 1. Ed. Solomon Feferman: Oxford: Oxford University Press.
8. Hamilton, A. G. (1978). Logic for Mathematics. Cambridge: Cambridge University Press.
9. Hoffman, S. (2004). “Kitcher, Ideal Agent, and Fictionalism”. Philosophia Mathematica. 12(3) 3 – 17. Web. 19 March, 2022.
10. Kant, I. (1965). Critique of Pure Reason. New York: St. Martin’s Press.
11. Korner, S. (1971). The Philosophy of Mathematics: An Introduction. London: Hutchinson and Company.
12. Legg, C. (2011). “Predication and the Problem of Universals”. Philosophical Papers. 30(2) 117 – 143. Web. 19 March, 2022.
13. Peano, G. (1967). “The Principles of Arithmetic Presented by a New Method”. From Frege to Gödel: A Source Book in Mathematical Logic. Ed. Jean van Heijenoort. Cambridge: Harvard University Press, 82 – 114.
14. Quine, V.O. (1971). From a Logical Point of View. London: Oxford University Press.
15. Raatikainen, P. (2003). “Hilbert’s Program Revisited.” Synthese. 137: 157- 177. Web. 5 April, 2022.
16. Raatikainen, P. (2007). “On the Philosophical Relevance of Gödel’s Incompleteness Theorems”. Gödel’s Incompleteness Theorem. N.p Web 5th April, 2022.
17. Resnik, M. (1980). Frege and the Philosophy of Mathematics. London: Cornet University Press.
18. Russell, B. (1998). Introduction to Mathematical Philosophy. London: Rutledge.
19. Russell, B. (1992). The Principles of Mathematics. London: Rutledge.
20. Shapiro, S. (2005). “Philosophy of Mathematics and its Logic”. The Oxford Handbook of Philosophy of Mathematics and Logic. n.d, 6 – 34. Web. Draft. 17 May, 2022.
Published
2023-04-01
How to Cite
Etorobong Godwin Akpan. (2023). The Nominalist Theories of Number: A Peano Arithmetic Critique. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 4(3), 114-123. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/409
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Articles