FOM: Hume's Principle

[John Mayberry]

In his note of March 23rd Alasdair Urquhart writes

>First, I quite agree with the silliness of naming the
>cardinality principle "Hume's Principle" -- but I've little to
>add to what Bill Tait already said on this.  The passage in
>Hume that is taken to justify this nomenclature (Treatise,
>Book I, Part iii, Section 1) seems in fact to amount to the
>idea that you can determine when two positive integers are 
>the same by writing them out as N = 1 + 1 + 1 + ... + 1 and
>then comparing units.  This has little or nothing to do
>with Cantor's general cardinality principle.  Admittedly, this 
>misreading of Hume goes back to Frege (Grundlagen, Section 63), but 
>I don't see why we should perpetuate this misreading.


This is *not* a misreading. What Hume is employing here is not our 
modern notion of "positive integer", as Uquhart mistakenly supposes, 
but the ancient Greek notion of "number" (arithmos) as a (finite) 
plurality composed of units (See Aristotle, *Metaphysics*, Book Delta 
1020a14 and Euclid, *Elements*, Book VII, Defs 1 and 2, and Heath's 
comments on these Euclidean definitions in his edition of Euclid). 
That this is what Hume has in mind is obvious from the discussion in 
Book I, Part II, Section II of the Treatise (see the paragraph 
beginning "I may subjoin another argument . . . ). 

Unfortunately I have deleted Bill Tate's note by mistake, but in 
Section 63 of the Grundlagen, Frege cites Cantor in a footnote as 
using Hume's notion of numerical equality, and later (Sections 85 and 
86) refers to Cantor's work on transfinite numbers with enthusiastic 
approval. It seems to me that it was only later, in the Grundgesetze, 
that Frege was ungenerous, indeed, unfair, to Cantor.

-----------------------------------------
John Mayberry
School of Mathematics
University of Bristol
J.P.Mayberry at Bristol.ac.uk
-----------------------------------------