Leon Horsten

In his Gibbs lecture, Gödel argued for the thesis that either the human mind is not a Turing machine, or there exist absolutely undecidable mathematical propositions. He believed that this disjunction can be deduced with mathematical certainty from certain results in mathematical logic. He thought that his disjunctive thesis is of great philosophical importance. First, Gödel's argument for his disjunctive thesis is discussed. It is argued that thisargument contains an ambiguity. But when it is made precise in a certain way, the argument is seen to be ultimately inescapable. Second, the disjuncts of the thesis are considered in their own right. It is argued that there are today no convincing arguments in favor of the first disjunct, nor are there at present convincing arguments against it. But there do exist today strong reasons to believe the second disjunct to be true, although these reasons do not establish the disjunct with anything like mathematical certainty

In this contribution, we focus on probabilistic problems with a denumerably or non-denumerably infinite number of possible outcomes. Kolmogorov (1933) provided an axiomatic basis for probability theory, presented as a part of measure theory, which is a branch of standard analysis or calculus. Since standard analysis does not allow for non-Archimedean quantities (i.e. infinitesimals), we may call Kolmogorov's approach "Archimedean probability theory". We show that allowing non-Archimedean probability values may have considerable epistemological advantages in the infinite case. The current paper focuses on the motivation for our new axiomatization.

This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact mathematical constructions. Therefore, they are legitimate components of the theory and the pictures of Cayley graphs are a part of practical mathematical procedures.