What is Non-Classical Mathematics
The 20th century has witnessed several attempts to build (parts of) mathematics on grounds other than those provided by classical logic. The original intuitionist and constructivist renderings of set theory, arithmetic, analysis, etc. were later accompanied by those based on relevant, paraconsistent, contraction-free, modal, and other non-classical logical frameworks. The subject studying such theories can be called non-classical mathematics and formally understood as a study of (any part of) mathematics that is, or can in principle be, formalized in some logic other than classical logic.
The scope of non-classical mathematics includes any mathematical discipline that can be formalized in a non-classical logic or in an alternative foundational theory over classical logic, and topics closely related to such non-classical or alternative theories. Particular topics of interest include (but are not limited to) the following:
- Intuitionistic, constructive, and predicative mathematics: Heyting arithmetic, intuitionistic set theory, topos-theoretical foundations of mathematics, constructive or predicative set and type theories, pointfree topology, etc.
- Substructural and fuzzy mathematics: relevant arithmetic, contraction-free naïve set theories, axiomatic fuzzy set theories, fuzzy arithmetic, etc.
- Inconsistent mathematics: calculi of infinitesimals, inconsistent set theories, etc.
- Modal mathematics: arithmetic or set theory with epistemic, alethic, or other modalities, modal comprehension principles, modal treatments of vague objects, modal structuralism, etc.
- Non-monotonic mathematics: non-monotonic solutions to set-theoretical paradoxes, adaptive set theory, etc.
- Alternative approaches to classical mathematics: alternative foundational theories over classical logic, categorial foundations of mathematics, non-standard analysis, etc.
- Topics related to non-classical mathematics: metamathematics of non-classical or alternative mathematical theories, their relative interpretability, first- or higher-order non-classical logics, etc.
This webpage gathers the activities in non-classical mathematics following the first conference on non-classical mathematics held in Hejnice (Czech Republic), June 2009.