A theory of learnability

A new theory of learning is presented. Learning has been discussed by philosophers as a problem of acquiring knowledge. Historically, we can distinguish empiricism and rationalism. The rationalist insists that the essential part of our knowledge is innately built in ourselves, while the empiricist asserts that all knowledge derives from experience. The statistical learning theory originating in Perceptron is conceptually based on the association psychology which is one of the schools of empiricism. A recent study of language learning, however, informs us that, only from the rationalist standpoint of view, we can interprete the evident fact that one knows a remarkable amount of matters which he has never learned before. From this standpoint, the present paper deals with the mystery: done can know all from hearing one, i.e., a word to the wise is enough, and mathematically demonstrates that such a possibility of knowledge acquisition manifests highly topological characteristics. This will give some new suggestions to the study of pattern recognition and learning.