Hume's Mathematical Points and Physical Points

by Corry Shores
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In his Treatise of Human Nature, Hume distinguishes mathematical points from physical points in order to make his case against the infinite divisibility of time and space.

Mathematical points do not extend in space. They thus do not have parts. Hume says that our idea of an extension is made-up of simpler ideas. The simplest of these ideas are not made-up of any more ideas. But things that extend consist of parts. So these simple ideas of extension are ideas of mathematical points, which do not extend in space. [As inextensive, they might be intensive, in the sense of calculus differentials being made of intensive magnitudes.]

Physical points do extend in space. And thus they are made-up of parts.

Hume argues that:

1) extensions are made-up of parts,

2) something that is made-up of parts can still be divided,

3) if extension were only made-up of extensive parts, then it would be made-up of things that could still be divided,

4) if all of extension's parts were divisible, then we may divide it infinitely,

5) but if we may divide it infinitely, then extension would not be made-up of any fundamental physical parts. For, any given fundamental physical part would still be divisible, and hence not fundamental.

6) if extension is not fundamentally made-up of any physical parts, then it is not ultimately divisible into physical parts. [it would be a sort-of indivisible continuum]

7) Thus, extension can only be made-up of mathematical points. We may have an idea of one of these mathematical points. If we continue adding these simple ideas, we will obtain the complex idea of an extension.

(Hume's Treatise page 40, and prior.)

Hume, A Treatise of Human Nature

PDF available at:

http://ia341013.us.archive.org/3/items/treatiseofhumann01humeuoft/treatiseofhumann01humeuoft.pdf

Definitions of mathematical and physical points from:

J. M. M. H. Thijssen Journal of the History of Ideas, Vol. 53, No. 2 (Apr. - Jun., 1992), pp. 271-286. Published by: University of Pennsylvania Press.

Available online at:

http://www.jstor.org/stable/2709874

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