# Intuitionistic Analysis, Forward and Backward

## Joan Rand Moschovakis, Emerita Professor of Mathematics at Occidental College, Los Angeles, California

In the early 20th century the Dutch mathematician L. E. J. Brouwer questioned the universal applicability of the Aristotelian law of excluded middle and proposed basing mathematical analysis on informal intuitionistic logic, with natural numbers and choice sequences (infinitely proceeding sequences of freely chosen natural numbers) as objects. For Brouwer, numbers and choice sequences were mental constructions which by their nature satisfied mathematical induction, countable and dependent choice, bar induction, and a continuity principle contradicting classical logic. Half a century later, S. C. Kleene and R. E. Vesley developed a significant part of Brouwer’s intuitionistic analysis in a formal system FIM. Kleene’s function-realizability interpretation proved FIM consistent relative to its classically correct subsystem B, facilitating a detailed comparison of intuitionistic with classical analysis C. Continuing Brouwer’s work into the 21st century, Wim Veldman and others are now developing an intuitionistic reverse mathematics parallel to, but diverging significantly from, both classical reverse mathematics as established by H. Friedman and S. Simpson, and constructive reverse mathematics in the style of E. Bishop. This lecture provides the basics of intuitionistic analysis and a sketch of its reverse development.