Observation and Intuition

In Carolin Antos, Neil Barton & Venturi Giorgio (eds.), Palgrave Companion to the Philosophy of Set Theory (forthcoming)
  Copy   BIBTEX

Abstract

The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated -- realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate is true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones.

Similar books and articles

Analytics

Added to PP
2023-06-11

Downloads
506 (#39,209)

6 months
269 (#8,932)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Justin Clarke-Doane
Columbia University

Citations of this work

No citations found.

Add more citations

References found in this work

Two Dogmas of Empiricism.W. Quine - 1951 - [Longmans, Green].
Fact, Fiction, and Forecast.Nelson Goodman - 1973 - Cambridge: Harvard University Press.
Modal Logic as Metaphysics.Timothy Williamson - 2013 - Oxford, England: Oxford University Press.
New work for a theory of universals.David K. Lewis - 1983 - Australasian Journal of Philosophy 61 (4):343-377.
The logical basis of metaphysics.Michael Dummett - 1991 - Cambridge: Harvard University Press.

View all 98 references / Add more references