The Local Conception of Mathematical Evidence: Proof, Computation, and Logic

In Michael David Resnik (ed.), Mathematics as a science of patterns. New York ;: Oxford University Press (1997)
  Copy   BIBTEX

Abstract

The fact that mathematics is ordinarily practised as an autonomous science with its own, peculiar type of evidence constituted mainly by deductive reasoning is often taken as evidence that mathematics and science have specifically different evidential supports and specifically different subject matters. I argue against this conclusion by first analysing deductive proofs, and the type of evidence that is usually required for axioms, and claiming that most of the evidence for the most elementary and fundamental parts of mathematics is empirical. I then appeal to the role of computation to argue that non‐deductive inference from empirical premises is part of the contemporary methodology of mathematics, and so some of our proofs turn out not to be purely logical deductions. Finally, I discuss the relation between mathematics and logic and argue against logical realism by denying that statements attributing logical properties or relations are true independently of our holding them to be true, our psychology, our linguistic and inferential conventions, or other facts about human beings. In the end, both mathematics and logic turn out to be a priori only in the sense that some mathematical and logical truths are obtained through deductive proofs, and for pragmatic reasons, are insulated from experience; but neither mathematics nor logic are a priori in the sense of being immune to empirical revision.

Other Versions

No versions found

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 101,795

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Non-deductive Logic in Mathematics: The Probability of Conjectures.James Franklin - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Dordrecht, Netherland: Springer. pp. 11--29.
The Philosophical Problems of Applied Mathematics.Robert Anderson Holland - 1989 - Dissertation, University of Illinois at Chicago
Non-deductive logic in mathematics.James Franklin - 1987 - British Journal for the Philosophy of Science 38 (1):1-18.
Holism: Evidence in Science and Mathematics.Michael D. Resnik - 1997 - In Michael David Resnik (ed.), Mathematics as a science of patterns. New York ;: Oxford University Press.
Problemas para a Explicação Matemática.Eduardo Castro - 2017 - Revista Portuguesa de Filosofia 73 (3-4):1437-1462.
The Case for Mathematical Realism.Michael D. Resnik - 1997 - In Michael David Resnik (ed.), Mathematics as a science of patterns. New York ;: Oxford University Press.
Why Is Proof the Only Way to Acquire Mathematical Knowledge?Marc Lange - 2024 - Australasian Journal of Philosophy 102 (2):333-353.
Elements of Logical Reasoning.Jan von Plato - 2013 - Cambridge and New York: Cambridge University Press.
Towards a theory of mathematical argument.Ian J. Dove - 2009 - Foundations of Science 14 (1-2):136-152.

Analytics

Added to PP
2016-10-25

Downloads
7 (#1,644,695)

6 months
7 (#740,041)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Michael Resnik
University of North Carolina, Chapel Hill

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references