David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Notre Dame Journal of Formal Logic 41 (4):317--334 (2000)
We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term Frege's Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis
|Keywords||abstraction principles analysis Frege structuralism|
|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
Sean Walsh (2014). Logicism, Interpretability, and Knowledge of Arithmetic. Review of Symbolic Logic 7 (1):84-119.
Philip A. Ebert & Stewart Shapiro (2009). The Good, the Bad and the Ugly. Synthese 170 (3):415 - 441.
Stewart Shapiro (2004). Foundations of Mathematics: Metaphysics, Epistemology, Structure. Philosophical Quarterly 54 (214):16 - 37.
Similar books and articles
João Branquinho (1990). Are Salmon's 'Guises' Disguised Fregean Senses? Analysis 50 (1):19 - 24.
Stewart Shapiro (2000). Frege Meets Dedekind: A Neologicist Treatment of Real Analysis. Notre Dame Journal of Formal Logic 41 (4):335--364.
Bob Hale (2000). Reals by Abstractiont. Philosophia Mathematica 8 (2):100--123.
Bob Hale (ed.) (2001). The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford University Press.
Matti Eklund (2009). Bad Company and Neo-Fregean Philosophy. Synthese 170 (3):393 - 414.
Ignacio Angelelli (2012). Frege's Ancestral and Its Circularities. Logica Universalis 6 (3-4):477-483.
Marco Ruffino (2003). Why Frege Would Not Be a Neo-Fregean. Mind 112 (445):51-78.
Nikolaj Jang Lee Linding Pedersen (2009). Solving the Caesar Problem Without Categorical Sortals. Erkenntnis 71 (2):141 - 155.
Richard Heck (2011). The Existence (and Non-Existence) of Abstract Objects. In Frege's Theorem. Oxford University Press.
Michael A. E. Dummett (1991). Frege: Philosophy of Mathematics. Harvard University Press.
Bill Wringe (2008). Making the Lightness of Being Bearable: Arithmetical Platonism, Fictional Realism and Cognitive Command. Canadian Journal of Philosophy 38 (3):pp. 453-487.
Wilfrid Sellars (1964). The Paradox of Analysis: A Neo-Fregean Approach. Analysis 24 (Suppl-2):84 - 98.
Bob Hale (2005). Real Numbers and Set Theory – Extending the Neo-Fregean Programme Beyond Arithmetic. Synthese 147 (1):21 - 41.
Added to index2010-08-24
Total downloads19 ( #94,245 of 1,102,135 )
Recent downloads (6 months)3 ( #128,850 of 1,102,135 )
How can I increase my downloads?