The Form and Function of Duality in Modern Mathematics

Philosophia Scientiae 18:95-109 (2014)
  Copy   BIBTEX

Abstract

Phenomena covered by the term duality occur throughout the history of mathematics in all of its branches, from the duality of polyhedra to Langlands duality. By looking to an “internal epistemology” of duality, we try to understand the gains mathematicians have found in exploiting dual situations. We approach these questions by means of a category theoretic understanding. Following Mac Lane and Lawvere-Rosebrugh, we distinguish between “axiomatic” or “formal” (or Gergonne-type) dualities on the one hand and “functional” or “concrete” (or Poncelet-type) dualities on the other. While the former are often used in the pursuit of a “two theorems by one proof”-strategy, the latter often allow the investigation of “spaces” by studying functions defined on them, which in Grothendieck's terms amounts to the strategy of proving a theorem by working in a dually equivalent framework where the corresponding proof is easier to find. We try to show by some examples that in the first case, dual objects tend to be more ideal (epistemologically more remote) than original ones, while this is not necessarily so in the second case.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,593

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Holography and emergence.Nicholas J. Teh - 2013 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (3):300-311.
Whole and part in mathematics.John L. Bell - 2004 - Axiomathes 14 (4):285-294.
Theorizing a contradiction between form and function in architecture.John Hendrix - 2012 - South African Journal of Art History 27 (1):9-28.
Beyond the methodology of mathematics research programmes.Corfield David - 1998 - Philosophia Mathematica 6 (3):272-301.
Unwearables.David Bryson - 2007 - AI and Society 22 (1):25-35.
What the foundationalist filter kept out.Alexander Paseau - 2005 - Studies in History and Philosophy of Science Part A 36 (1):191-201.
Mathematics, Form and Function.Saunders MacLane - 1986 - Journal of Philosophy 84 (1):33-37.
Mathematics: Form and Function.Penelope Maddy - 1988 - Journal of Symbolic Logic 53 (2):643-645.

Analytics

Added to PP
2015-12-01

Downloads
20 (#656,247)

6 months
2 (#668,348)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

David Corfield
University of Kent at Canterbury

Citations of this work

Duality as a category-theoretic concept.David Corfield - 2017 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 59:55-61.
Projective duality and the rise of modern logic.Günther Eder - 2021 - Bulletin of Symbolic Logic 27 (4):351-384.

Add more citations

References found in this work

No references found.

Add more references