Mathematics based on incremental learning—Excluded middle and inductive inference

Susumu Hayashi
2006 Theoretical Computer Science  
Learning theoretic aspects of mathematics and logic have been studied by many authors. They study how mathematical and logical objects are algorithmically \learned" (inferred) from nite data. Although they study mathematical objects, the objective of the studies is learning. In this paper, a mathematics of which foundation itself is learning theoretic will be introduced. It is called Limit-Computable Mathematics. I t w as originally introduced as a means for \Proof Animation," which is expected
more » ... to make i n teractive formal proof development easier. Although the original objective w as not learning theoretic at all, learning theory is indispensable for our research. It suggests that logic and learning theory are related in a still unknown but deep new way. 1 Mathematics based on Learning? Mathematical or logical concepts seem to be one of the main research targets of learning theory and its applications. Shapiro 25] investigated how axioms systems are inductively inferred by ideas of learning theory. W e m a y s a y t h a t Shapiro studied how logical systems (axiom systems) are learned. Stephan and Ventsov 27] investigated how algebraic structures are learned and have given some interesting learning theoretic characterizations of fundamental algebraic notions. We m a y s a y that they investigated learnability of the mathematical concepts. Contrary to them, we a r e n o w d e v eloping a mathematics whose semantics and reasoning system are in uenced by ideas from computational learning theory. Let us compare these two lines of research. Email address: shayashi@kobe-u.ac.jp (Susumu H a yashi).
doi:10.1016/j.tcs.2005.10.019 fatcat:bjcpur2pxzeihdscoer2fz6ty4