Paraconsistent mathematics, sometimes called inconsistent mathematics, represents an attempt to develop the classical infrastructure of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

(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 (38 ...

) based on a foundation of

paraconsistent logic Paraconsistent logic is a type of non-classical logic that allows for the coexistence of contradictory statements without leading to a logical explosion where anything can be proven true. Specifically, paraconsistent logic is the subfield of log ...

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 c ...

. 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 {{mathlogic-stub