Paraconsistent mathematics, sometimes called inconsistent mathematics, represents an attempt to develop the classical infrastructure of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
(e.g.
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
) based on a foundation of
paraconsistent logic
A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syst ...
instead of
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
. A number of 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. The theory of differentiation turned out to have these advantages, while the theory of integration did not. (2)
References
* McKubre-Jordens, M. and Weber, Z. (2012). "Real analysis in paraconsistent logic". ''Journal of Philosophical Logic'' 41 (5):901–922.
doi:
10.1017/S1755020309990281
* Mortensen, C. (1995). ''Inconsistent Mathematics.'' Dordrecht: Kluwer.
* Weber, Z. (2010). "Transfinite numbers in paraconsistent set theory". ''Review of Symbolic Logic'' 3 (1):71–92.
doi:
10.1017/S1755020309990281
External links
* Entry in the ''Internet Encyclopedia of Philosophy'
* Entry in the ''Stanford Encyclopedia of Philosophy'
* Lectures by Manuel Bremer of the University of Düsseldor
Philosophy of mathematics
Proof theory
Paraconsistent logic
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