### Things Vs. Stuff : A Satyrical Philosophy Discussion

Many years ago, Dabney House held a extra super official philosophy discussion on "The Distinction Between Things and Stuff". I recently recalled this discussion, and wondered if the following notions are relevant :

Things are members of finite or countable sets, while stuff is a subset of an uncountable set. I'm not sure what a single element (or other measure-zero subset) of an uncountable set would be... perhaps "nothing".

Confusing theorems like the Banach-Tarski paradox suggest that, if you are working with uncountable sets, notions of volume are poorly defined. This paradox says that if you define a sphere, there is a way to cut up the sphere into parts that can be re-assembled into two spheres. At first glance, this would seem to be an amazing violation of common sense. The reconciliation of this paradox, however, is simply that the human notion of "volume" works poorly on uncountable sets ( and maybe even countably infinite sets). As far as I know, the Banach-Tarski paradox can not be phrased on finite sets of points.

This would imply that "stuff" can not have a definite volume or area, and that some of what we call "stuff" is in fact a very large collection of "things".

Anyway, I'm not an expert or even passably competent in most areas of math, so if F reads this he's probably going to thing "this is dumb, and also wrong, and dumb".

... that is all.

I like it, actually. But the Banach-Tarski paradox involves cutting stuff (spaces) up in ways that you can really only do with things (sets). So rather than saying that stuff cannot have definite volume, you can say that sufficiently large collections of things can't.

ReplyDeleteThere's a kind of back-and-forth between discrete and continuous things in math. For various reasons you often want to change one into the other.

OK, time to hectically prep my class.

To display more obviously the counterintuitive nature of the Banach-Tarski paradox, it should be precised that it allows to cut a ball into pieces and reassemble them (through translations and rotations only) into two balls OF THE SAME VOLUME as the original ball (so the total volume has doubled).

ReplyDeleteThis paradox comes from the fact that there is no measure on a ball. A measure is a way to assign a volume to every subset of the ball, in a way consistent with the usual intuition (so that for instance the volume of the union of two disjoint sets is the sum of the volume of each components). It is however possible to restrict oneself to so-called "measurable" subsets of the ball, and if we do so, a measure exists (the Lesbegue measure). The Lesbegue measure coincides with the measure one would intuitively associate to simple subsets of the ball. Of course the subsets considered in the Banach-Tarski paradox are not Lesbegue-measurable.

I guess it just means that the mathematical ball is an abstraction which has no exact equivalent in our world. In some sense that's the case for any mathematical object (like a point for instance).

So are you saying... that the problem is not that the concept of volume is undefined for a ball, but that the concept of volume is undefined for the particular subsets of the ball that the Banach-Tarski paradox uses ?

ReplyDelete"So are you saying... that the problem is not that the concept of volume is undefined for a ball, but that the concept of volume is undefined for the particular subsets of the ball that the Banach-Tarski paradox uses ?"

ReplyDeleteYes, exactly.