Tamar Lando: The topology of gunk

by Ignacio Ojea

The topology of gunk
Tamar Lando (Columbia University)
Monday, May 13th, 4 to 6 PM
2nd floor seminar room, Philosophy Department, NYU (5 Washington Place).

Abstract. Space as we typically conceive of it in mathematics and physics is composed of dimensionless points. Over the years, however, some have denied that points, or point-sized parts are genuine parts of space. Space, on an alternative view, is ‘gunky’: every part of space has a strictly smaller subpart. If this thesis is true, how should we model space mathematically? The traditional answer to this question is most famously associated with A.N. Whitehead, who developed a mathematics of pointless geometry that Tarski later modeled in regular open algebras. More recently, however, Whiteheadian space has come under attack, because it does not allow us to talk about the size or measure of regions in a nice way. A newer approach to the mathematics of gunk, advanced by F. Arntzenius, J. Hawthorne, and J.S. Russell, models space via the Lebesgue measure algebra, or algebra of (Lebesgue) measurable subsets of Euclidean space modulo sets of measure zero. But problems arise on this approach when it comes to doing topology. According to Arntzenius, the standard topological distinction between ‘open’ and ‘closed’ regions “is exactly the kind of distinction that we do not believe exists if reality is pointless.” I argue that the turn to non-standard topology in the measure-theoretic setting rests on a mistake. Once this is pointed out, the newer approach to gunk can claim two important advantages: it allows the gunk lover to talk about size and topology—both in perfectly standard ways.