FOM: Hume's Principle

[William Tait]

At 4:47 PM +0100 3/25/01, John Mayberry wrote:
>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 . . . ).

Insofar as this is connected with my posting on `Hume's principle', 
let me say that I agree entirely with this: it is fairly fully 
discussed in my (1996) ``Frege versus Cantor and Dedekind: on the 
concept of number'', section IX.

But notice that Frege, too, makes Alasdair's mistake about Hume: he 
mistakes Hume's definition (in effect certainly in Euclid Book VII) 
of two finite sets being equal (in power) for a definition of when 
two cardinal numbers are identical. E.g. Frege ( Section 63 of 
*Grundlagen*) remarks in  connection with Hume's definition is that 
the *identity* relation is, after all, given in general. [There is a 
lot of obscurity in Frege's book which turns on the translation of 
``gleich''. I recall that, in some but not all cases, it is the 
translator's fault.]

>Unfortunately I have deleted Bill Tate's note by mistake,

I would send you a copy, John; but at the moment I don't have it available.

>  but in
>Section 63 of the Grundlagen, Frege cites Cantor in a footnote as
>using Hume's notion of numerical equality,

Frege mentions Cantor, but makes no distinction between Hume's 
definition of when two `numbers', i.e. finite sets, are equal (in 
power) and Cantor's adopting that definition for infinite sets. 
Frege, which was revolutionary. By implication, because of priority, 
Frege attributes Cantor's definition to Hume. In my posting, I 
pointed out that it was Cantor's definition, for arbitrary sets, that 
Frege employed, not Hume's---since he defined the cardinal numbers in 
general and then picked out the finite ones from them.

>  and later (Sections 85 and
>86) refers to Cantor's work on transfinite numbers with enthusiastic
>approval.

Frege indeed mentions Cantor's 1883 paper in his Grundlagen; but he 
obviously did not absorb its content well: Cantor warns in that 
paper---and in his 1885 review of Frege's book---that there are 
totalities (extensions of concepts), e.g. the totality of transfinite 
numbers and that of powers, which have no power, i.e. are not sets. 
(As I recall, Frege did not really understand Cantor's notion of 
well-ordered set, either: but I may be misremembering. I'm not able 
to check right now.)

>It seems to me that it was only later, in the Grundgesetze,
that Frege was ungenerous, indeed, unfair, to Cantor.

And also in later reviews of Cantor's work. But I don't agree with 
the view of Frege turning ungenerous only in his later work. The 
Grundlagen is filled with ungenerous readings of his contemporaries. 
I feel that even his misunderstanding of Hume, indeed of Euclid, was 
a result of an ungenerous impulse to criticize rather than to search 
for a meaning of what they wrote which makes sense. Its not too hard 
to find that meaning: after all, John, you and I did.

Bill Tait
-- 
William W. Tait
Professor Emeritus of Philosophy
University of Chicago
wwtx at midway.uchicago.edu
Home:
5522 S. Everett Ave
Chicago, IL 60637
(773) 241-7288