David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
The paper outlines a project in the philosophy of mathematics based on a proposed view of the nature of mathematical reasoning. It also contains a brief evaluative overview of the discipline and some historical observations; here it points out and illustrates the division between the philosophical dimension, where questions of realism and the status of mathematics are treated, and the more descriptive and looser dimension of epistemic efficiency, which has to do with ways of organizing the mathematical material. The paper’s concern is with the first. The grand tradition in the philosophy of mathematics goes back to the foundational debates at the end of the 19th and the first decades of the 20th century. Logicism went together with a realistic view of actual infinities; rejection of, or skepticism about actual infinities derived from conceptions that were Kantian in spirit. Yet questions about the nature of mathematical reasoning should be distinguished from questions about realism (the extent of objective knowledge– independent mathematical truth). Logicism is now dead. Recent attempts to revive it are based on a redefinition of “logic”, which exploits the flexibility of the concept; they yield no interesting insight into the nature of mathematics. A conception of mathematical reasoning, broadly speaking along Kantian lines, need not imply anti–realism and can be pursued and investigated, leaving questions of realism open. Using some concrete examples of non–formal mathematical proofs, the paper proposes that mathematics is the study of forms of organization—-a concept that should be taken as primitive, rather than interpreted in terms of set–theoretic structures. For set theory itself is a study of a particular form of organization, albeit one that provides a modeling for the other known mathematical systems. In a nutshell: “We come to know mathematical truths through becoming aware of the properties of some of the organizational forms that underlie our world. This is possible, due to a capacity we have: to reflect on some of our own practices and the ways of organizing our world, and to realize what they imply..
|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
Davide Rizza (2011). Magicicada, Mathematical Explanation and Mathematical Realism. Erkenntnis 74 (1):101-114.
Erich H. Reck (2009). Dedekind, Structural Reasoning, and Mathematical Understanding. In Bart Van Kerkhove (ed.), New Perspectives on Mathematical Practices: Essays in Philosophy and History of Mathematics. World Scientific. 150--173.
Charles Sayward (2005). A Wittgensteinian Philosophy of Mathematics. Logic and Logical Philosophy 15 (2):55-69.
Janet Folina (1994). Poincaré's Conception of the Objectivity of Mathematics. Philosophia Mathematica 2 (3):202-227.
David Corfield (2003). Towards a Philosophy of Real Mathematics. Cambridge University Press.
Mark Colyvan (2011). Fictionalism in the Philosophy of Mathematics. In E. J. Craig (ed.), Routledge Encyclopedia of Philosophy.
O. Linnebo (2003). Stewart Shapiro. Philosophy of Mathematics: Structure and Ontology. Philosophia Mathematica 11 (1):92-103.
Penelope Maddy (1990). Realism in Mathematics. Oxford University Prress.
Added to index2009-01-28
Total downloads29 ( #61,192 of 1,102,742 )
Recent downloads (6 months)2 ( #182,643 of 1,102,742 )
How can I increase my downloads?