Hajime Ishihara Professor
School of Information Science、Intelligent Robotics Area
B.S., M.S. and Ph.D.from Tokyo Institute of Technology (1980,1987,1990)
Researcher at Mitsubishi Research Institute, Inc. (1980), Associate at Hiroshima University (1988), Associate Professor at Japan Advanced Institute of Science and Technology (1992)
Constructive Mathematics, Mathematical Logic, and Foundations of Mathematics
constructive mathematics, intuitionistic logic, reverse mathematics, constructive set theory
We have been exploring constructive mathematics as mathematics with intuitionistic logic which originated in Brouwer’s intuitionistic mathematics and formalized by Heyting and Kolmogorov. We have been dealing with constructive functional analysis such as theory of Hilbert and Banach spaces and theory of distributions, and with constructive topological spaces such as neighbourhood spaces, formal topology, and basic pair. As a foundation of constructive mathematics, we have also been studying constructive set theory (CZF) which is a predicative system and has a quite natural interpretation in Martin-Loef type theory. Furthermore, we have been advocating, and leading a research on, constructive reverse mathematics which aims at classifying, arranging and systematizing mathematical theorems, in classical mathematics, Brouwer’s intuitionistic mathematics and constructive recursive mathematics developed under different philosophies of mathematics, by logical principles and/or function existence axioms from a uniform point of view.
We have been studying proof theory and semantics of intuitionisitc logic. Since there is a natural correspondence, called the Curry-Howard correspondence, between proofs in intuitionistic logic and programs (terms of lambda-calculus), we are able to extract programs from proofs in intuitionistic logic, and program extraction systems, based on this fact, such as the Minlog system at University of Munich, has been developed. We have been dealing with relationship between classical logic and intuitionistic logic, and doing a research on extracting programs from constructive contents of classical proofs. Also we have been studying proof theory and semantics of substructural logics such as linear logic.
Theory of Computation
We have been studying computability theory and computational complexity theory, and relationship between them and constructive mathematics. We have been characterizing classes of computable functions, such as the class of polynomial time computable functions, as function algebras, and trying to explore relationship between computational complexity and degrees of unsolvability, and logical principles and function existence axioms in constructive reverse mathematics. Furthermore, we have been dealing with lambda-calculus and type theory such as simple types, intersection types and union types.
- The binary expansion and the intermediate value theorem in constructive reverse mathematics，Josef Berger, Hajime Ishihara, Takayuki Kihara and Takako Nemoto，Archive for Mathematical Logic，published online 10 May 2018
- On Brouwer's continuity principle，Hajime Ishihara，Indagationes Mathematicae，29，1511-1524，2018
- Consistency of the intensional level of the Minimalist Foundation with Church’s thesis and axiom of choice，Hajime Ishihara, Maria Emilia Maietti, Samuele Maschio and Thomas Streicher，Archive for Mathematical Logic，57，873-888，2018
◇Lectures and Presentations
- A monad on the combinatory algebras，Hajime Ishihara，Operations, Sets, and Types，Muenchenwiler, Switzerland，April 18 - 20, 2016
- Constructive reverse mathematics: an introduction and recent results，Hajime Ishihara，Frontiers of Non-Classicality: Logic, Mathematics, Philosophy，Auckland, New Zealand，January 26-29, 2016
- Constructive reverse mathematics and omniscience pringiples，Hajime Ishihara，14th Asian Logic Conference，Munbai, India，January 5-8, 2015
◇Academic Society Affiliations
- Association for Symbolic Logic，1992-
- American Mathematical Society，1989-
- Mathematical Society of Japan，1988-
- Computability and Complexity in Analysis 2011，Scientific Programme Committee，2011/01/31 - 2011/02/04
- Constructive Mathematics: Proof and Computation，Programme Committee，2010/06/07 - 2010/06/11
- Computability and Complexity in Analysis (CCA) 2009，Scientific Program Committee，2009/08/18 - 2009/08/22