Paraconsistent mathematics
Encyclopedia
Paraconsistent mathematics (sometimes called inconsistent mathematics) represents an attempt to develop the classical infrastructure of mathematics
(e.g. analysis
) based on a foundation of paraconsistent logic
instead of classical logic
. A number of interesting reformulations of analysis can be developed, for example functions which both do and do not have a given value simultaneously.
Chris Mortensen claims (see references):
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
(e.g. analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) based on a foundation of paraconsistent logic
Paraconsistent logic
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
instead of classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
. A number of interesting reformulations of analysis can be developed, for example functions which both do and do not have a given value simultaneously.
Chris Mortensen claims (see references):
- One could hardly ignore the examples of analysis and its special case, the calculus. There prove to be many places where there are distinctive inconsistent insights; see Mortensen (1995) for example. (1) Robinson's non-standard analysis was based on infinitesimals, quantities smaller than any real number, as well as their reciprocals, the infinite numbers. This has an inconsistent version, which has some advantages for calculation in being able to discard higher-order infinitesimals. Interestingly, the theory of differentiation turned out to have these advantages, while the theory of integration did not. (2)
External links
- Entry in the Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/mathematics-inconsistent/
- Lectures by Manuel Bremer of the University of Düsseldorf http://www.mbph.homepage.t-online.de/Logic/ParaLec.htm