Non-Classical Foundations of Analysis

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Supported by the New Zealand Marsden Fund.

"If between any two points in space there is always a third point, can anything touch anything else?" - Le Poidevin, in Travels in Four Dimensions, Oxford University Press, 2003.


Project Outline

Foundations for real analysis traditionally comes in two kinds: classical (as per Cauchy, Weierstraß, Dedekind, etc.) and constructive (Brouwer, Weyl etc.). These are underpinned by, respectively, classical and intuitionistic logics. We seek to establish a solid non-classical foundation for real analysis relying on paraconsistent logics.
Paraconsistent logics are characterized by rejection of the universal validity of the principle ex contradictione quodlibet. It is this principle which, in classical and intuitionistic logics, causes global absurdity in the presence of local contradiction. We build on recent developments in set theory, geometry and arithmetic to investigate models of the continuum capable of supporting both an interesting structure and interesting substructure; inconsistent phenomena that arise need not lead to disaster. A 3D-printed version of the Penrose Triangle illusion, created with partial inverted cubes. The 3D model is available under the Public Domain here: http://www.thingiverse.com/thing:6513

A detailed description of the project can be found here.

Location

The project will be based in the Department of Mathematics & Statistics at the University of Canterbury, Christchurch, New Zealand.

Research Participants

Principal Investigator: Dr. Maarten McKubre-Jordens, University of Canterbury.

Associate Investigator: Dr. Zach Weber, University of Otago.

Ph.D. Researchers: Mr. Erik Istre (USA) & Mr. Anggha Nugraha (Indonesia).

Project Co-ordinator:

Maarten McKubre-Jordens
Mathematics and Statistics Department
University of Canterbury
Private Bag 4800
Christchurch 8041
Phone: +64 3 364 2987 extn 8878
Fax: +64 3 364 2587
Email: maarten.jordens@canterbury.ac.nz

Some Links (in no particular order)

McKubre-Jordens, M. (2011) This is not a carrot: Paraconsistent Mathematics, +PLUS Magazine, August 2011.

McKubre-Jordens, M. and Weber, Z. (2012) Real Analysis in Paraconsistent Logic, Journal of Philosophical Logic 41(5):901-922. (Unofficial Preprint)

Mortensen, C. (1995) Inconsistent Mathematics. Kluwer Mathematics and Its Applications Series, Dordrecht: Kluwer.

Weber, Z. (2009) Inconsistent Mathematics, in Fieser, J. and Bowden, B. (eds.) The Internet Encyclopedia of Philosophy, http://www.iep.utm.edu/.


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