Transcript – Philosophical Foundations of Probability and Mathematical Reasoning

This paper is about philosophical issues for the applicability of the mathematics of the infinite, ever since the identification of the axiom of choice as an important principle about infinite sets in the early 20th century. There has been some mathematical and philosophical uneasiness about its meaningfulness as the only central mathematical principle that claims the existence of an object without providing any means of identifying the object. In this paper, we specifically investigate the worldly applicability of the hyperreal, a structure extending the standard real numbers with non-standard, infinitely large and infinitesimally small elements. Various philosophers and mathematicians have made creative use of the hyperreal, both in mathematical proofs and in theories that purport to describe part of the world. We note that the way mathematical objects are used in describing parts of the world always involves some number of arbitrary choices to describe some convention, which then gives meaning to the rest of the mathematical structure. When we decide that the concatenation of two linear distances will be represented by the sum of the numbers representing the individual distances and then decide that the number one will represent the length of a particular iridium rather than Paris, we have then specified the meaning of every real number as a potential distance. What is essential for this meaning to be defined uniquely is that the algebraic structure of the real numbers is rigid. Once we have specified a unit and the operation of addition, if a distance represented by the number X under one application is represented by the number Y under another, then either the distance unit must be different in the two applications.

Say one uses the meter and the other uses the model or the concatenation. Operation must be different in the two applications. Perhaps one of them is a slide rule where multiplication represents concatenation. However, the hyperreal don’t have this property of rigidity for every finite or even accountably infinite collection of units and operations. There is an auto morph ism that moves some X to a distinct Y, but preserves all of these units and operations. Thus, it is impossible to give definite empirical meaning to all the elements of this structure in any world. The application this causes no problem for the mathematical use of the hyperreal. They are perfectly consistent mathematical structure. The proofs of certain mathematical results and standard real analysis are much easier and more intuitive when we use the hyperreal as a notational device to keep track of the dependencies among multiple quantifiers. Abraham Rubinson even developed an introductory calculus class that takes advantage of this use to simplify the understanding of concepts, have limits and continuity in ways that are similar to the intuitive methods of Newton and Leibniz in the 17th century. But we claim that the individual hyperreal have no empirical meaning, and we suggest that this may carry over to all mathematical structures that essentially depend on the action of choice for the proof of their existence. This doesn’t vindicate the skepticism of the mathematicians and philosophers of the early 20th century, but it does suggest a way to reinterpret the skepticism.