David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Philosophy 25 (95):326 - 330 (1950)
In the most recent edition of Language, Truth and Logic , Professor A. J. Ayer still maintains that pure mathematics is analytic, being in fact merely a vast system of tautology. He is much more confident about this than are most contemporary professional mathematicians who have investigated the foundations of their subject. Following the breakdown of the efforts both of Frege and of Russell and Whitehead to derive pure mathematics from logic, i.e. to prove that the denial of any one proposition of mathematics would necessarily be self-contradictory, Hilbert attempted to prove the more modest thesis that pure mathematics is consistent, i.e. that no two propositions of mathematics can contradict each other; but in 1931 Gödel discovered that even this thesis was undecidable according to the “rules of the game.” As Weyl has recently lamented, “From this history one thing should be clear: we are less certain than ever about the ultimate foundations of mathematics.” The sense in which Ayer uses the terms analytic and tautology implies also that in his view the activities of pure mathematicians lead to nothing new. It is true that he remarks that “there is a sense in which analytic propositions do give us new knowledge. They call attention to linguistic usages of which we might otherwise not be conscious, and they reveal unsuspected implications in our assertions and beliefs.” But, he continues, “we can also see that there is a sense in which they may be said to add nothing to our knowledge. For they may be said to tell us what we know already. ” This denial of novelty in mathematics is as typical of contemporary positivism as the prophecy of Comte that the composition of the stars would never be revealed to us and the objections of Mach to the atomic hypothesis were characteristic of nineteenth century positivism. Indeed, one wonders why the term positivism should have been appropriated by successive philosophers whose common outlook could be so much more fittingly described as negativism
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Dorothy P. Coleman (1979). Is Mathematics for Hume Synthetic a Priori? Southwestern Journal of Philosophy 10 (2):113-126.
G. G. Taylor (1981). The Analytic and Synthetic in Russell's Philosophy of Mathematics. Philosophical Studies 39 (1):51 - 59.
Kevin Meeker (2007). Hume on Knowledge, Certainty and Probability: Anticipating the Disintegration of the Analytic/Synthetic Divide? Pacific Philosophical Quarterly 88 (2):226–242.
A. W. Moore (1997). The Underdetermination/Indeterminacy Distinction and the Analytic/Synthetic Distinction. Erkenntnis 46 (1):5-32.
Kristina Engelhard & Peter Mittelstaedt (2008). Kant's Theory of Arithmetic: A Constructive Approach? [REVIEW] Journal for General Philosophy of Science 39 (2):245 - 271.
Robert A. Holland (1992). Apriority and Applied Mathematics. Synthese 92 (3):349 - 370.
Lucienne Félix (1960). The Modern Aspect of Mathematics. New York, Basic Books.
Carol A. Van Kirk (1986). Synthesis, Sensibility, and Kant's Philosophy of Mathematics. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1986:135 - 144.
John P. Burgess (2004). Quine, Analyticity and Philosophy of Mathematics. Philosophical Quarterly 54 (214):38–55.
Added to index2010-08-10
Total downloads3 ( #340,583 of 1,679,364 )
Recent downloads (6 months)1 ( #183,761 of 1,679,364 )
How can I increase my downloads?