Non-classical analysis

In mathematics, non-classical analysis is any system of analysis, other than classical real analysis, and complex, vector, tensor, etc., analysis based upon it.

Such systems include:

• Abstract Stone duality, a programme to re-axiomatise general topology directly, instead of using set theory. It is formulated in the style of type theory and is in principle computable. It is currently able to characterise the category of (not necessarily Hausdorff) computably based locally compact spaces. It allows the development of a form of constructive real analysis using topological rather than metrical arguments.
• Chainlet geometry, a recent development of geometric integration theory which incorporates infinitesimals and allows the resulting calculus to be applied to continuous domains without local Euclidean structure as well as discrete domains.
• Constructivist analysis, which is built upon a foundation of constructivist, rather than classical, logic and set theory.
• Intuitionistic analysis, which is developed from constructivist logic like constructivist analysis but also incorporates choice sequences.
• Non-Newtonian calculus, which provides alternatives to the classical calculus of Newton and Leibniz.
• Non-standard analysis, develops rigorous infinitesmals within a new number system along with a transfer principle allowing them to be applied back to the real numbers.
• p-adic analysis
• Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory.
• Smooth infinitesimal analysis, which is developed in a smooth topos.