Logic Colloquium: Realism in Mathematics: The Case of the Hyperreals

Colloquium | November 15 | 4-5 p.m. | 60 Evans Hall

 Kenny Easwaran, Texas A&M University

 Department of Mathematics

Although there was controversy about the role of the Axiom of Choice in mathematics in the early 20th century, it has become an accepted part of mathematical practice. One interesting consequence of its acceptance has been the development of non-standard analysis, which studies systems that behave like the real numbers, with the addition of infinitely large and infinitely small elements. These have been used to develop various models of physical, biological, and philosophical phenomena. We argue that these models can be used to develop understanding, and to make predictions and perform some calculations. However, we claim that these models can't correspond to reality in ways that some other mathematical models can. We claim that no physical meaning can be given to the claim that a specific non-standard number represents some quantity. Some uses of the standard real numbers have this feature (like their use in differential equations representing predator-prey systems), but there is no in-principle barrier to their correspondence with physical reality, the way we claim there is for non-standard reals. This feature derives from their dependence on the Axiom of Choice, rather than from the presence of infinitely large and infinitely small elements. (joint work with Henry Towsner)