David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Philosophia Mathematica 11 (2):226-241 (2003)
The over-arching theme is that we can redeem Frege's key philosophical insights concerning (natural and real) numbers and our knowledge of them, despite Russell's famous discovery of paradox in Frege's own theory of classes. That paradox notwithstanding, numbers are still logical objects, in some sense created or generated by methods or principles of abstractionÃ¢â¬â which of course cannot be as ambitious as Frege's Basic Law U. These principles not only bring numbers into existence, as it were, but also afford a distinctive form of epistemic access to them. The usual mathematical axioms governing the two kinds of numbers are to be derived as results in (higher-order) logic. These derivations will exploit appropriate definitions of the primitive constants, functions, and predicates of the brand of number theory concerned. (For example: 0, 1; s, +, x; (; N(z); R(z).) No supplementation by intuition or sensory experience will be needed in the derivations of these axioms. The trains of reasoning involved will depend only on our grasp of logical validities, supplemented by appropriate definitions. Result: logicism is vindicated; and the mathematical knowledge derived in this way is revealed to be analytic, not synthetic
|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
Neil Tennant (2013). Parts, Classes and Parts of Classes : An Anti-Realist Reading of Lewisian Mereology. Synthese 190 (4):709-742.
Similar books and articles
Peter Sullivan & Michael Potter (1997). Hale on Caesar. Philosophia Mathematica 5 (2):135--52.
Bob Hale & Crispin Wright, Focus Restored Comment on John MacFarlane's “Double Vision: Two Questions About the Neo-Fregean Programme”.
Edward N. Zalta (1999). Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege"s Grundgesetze in Object Theory. [REVIEW] Journal of Philosophical Logic 28 (6):619-660.
Neil Tennant (1997). On the Necessary Existence of Numbers. Noûs 31 (3):307-336.
Erich H. Reck (2005). Frege on Numbers: Beyond the Platonist Picture. The Harvard Review of Philosophy 13 (2):25-40.
Fraser MacBride (2003). Speaking with Shadows: A Study of Neo-Logicism. British Journal for the Philosophy of Science 54 (1):103-163.
Crispin Wright (1983). Frege's Conception of Numbers as Objects. Aberdeen University Press.
Stewart Shapiro (2000). Frege Meets Dedekind: A Neologicist Treatment of Real Analysis. Notre Dame Journal of Formal Logic 41 (4):335--364.
Bob Hale (1994). Dummett's Critique of Wright's Attempt to Resuscitate Frege. Philosophia Mathematica 2 (2):122-147.
Gregory Currie (1981). Ii. The Origin of Frege's Realism. Inquiry 24 (4):448 – 454.
Bob Hale & Crispin Wright (2009). Focus Restored: Comments on John MacFarlane. Synthese 170 (3):457 - 482.
Bob Hale & Crispin Wright (2001). Introduction. In Bob Hale & Crispin Wrigth (eds.), The Reason's Proper Study. Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford University Press. 1-27.
Added to index2009-01-28
Total downloads23 ( #106,846 of 1,696,463 )
Recent downloads (6 months)4 ( #140,936 of 1,696,463 )
How can I increase my downloads?