Some thoughts and a proposal in the philosophy of mathematics

Abstract
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)
Options
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
 
Download options
PhilPapers Archive


Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 11,817
External links
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
Analytics

Monthly downloads

Added to index

2009-01-28

Total downloads

29 ( #63,332 of 1,099,909 )

Recent downloads (6 months)

1 ( #303,846 of 1,099,909 )

How can I increase my downloads?

My notes
Sign in to use this feature


Discussion
Start a new thread
Order:
There  are no threads in this forum
Nothing in this forum yet.