Finitary and Infinitary Mathematics, the Possibility of Possibilities and the Definition of Probabilities

摘要

Some relations between physics and finitary and infinitary mathematics areexplored in the context of a many-minds interpretation of quantum theory. Theanalogy between mathematical ``existence'' and physical ``existence'' isconsidered from the point of view of philosophical idealism. Some of the waysin which infinitary mathematics arises in modern mathematical physics arediscussed. Empirical science has led to the mathematics of quantum theory. Thisin turn can be taken to suggest a picture of reality involving possible mindsand the physical laws which determine their probabilities. In this picture,finitary and infinitary mathematics play separate roles. It is argued thatmind, language, and finitary mathematics have similar prerequisites, in thateach depends on the possibility of possibilities. The infinite, on the otherhand, can be described but never experienced, and yet it seems that sets ofpossibilities and the physical laws which define their probabilities can bedescribed most simply in terms of infinitary mathematics.