FOM: Neo-Fregean reverse mathematics

[Richard Heck]

At 11:47 AM 3/26/2001 -0500, Charles Parsons wrote:
>At 6:03 PM -0500 3/23/01, Alasdair Urquhart wrote:>
> >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.
>
>The name "Hume's Principle" has become quite entrenched in spite of the
>strong objections to it. But perhaps some effort should still be made to
>get writers on the subject to change it. I have at various times suggested
>"Cantor's principle". I used that in my class this semester. The difficulty
>is shown by the fact that the papers they recently wrote mostly spoke of
>Hume's principle, although the authors are undergraduates who couldn't have
>much familiarity with the literature on this subject.

Charles is right that the name has become "entrenched", and of course there 
have been some quite strong objections to its use. Many of these criticisms 
rest, though, upon dubious assumptions about the semantics of the 
possessive. To call HP (as Boolos suggested we call it, citing Chomsky's 
use of "LF") "Hume's Principle" need not be to suggest Hume ever possessed 
it, say, in the sense of having formulated it. It may be a principle 
related to Hume in some weaker way, and it is an open question whether 
there is some such weaker way. Last I checked, Bill Tait wasn't (and never 
claimed to be) an authority on Hume. I have serious questions myself about 
the details of his interpretation, as well as about claims he tacitly makes 
about the semantics of the term "number" (which it is not at all clear ever 
means or has meant "group" or "collection"). But I'll not pursue either of 
those points here. On the latter, though, see some of the recent work on 
plurals.

Second, no one, so far as I know, has ever suggested that any passage in 
Hume justifies the use of the term "Hume's Principle". Its use emerged in 
the context of Frege studies, where its meaning and motivation were quite 
clear. In that context, "Hume's Principle" obviously means: The princple 
Frege (mis)ascribes to, or (thinks he) takes from, Hume, and that, note, is 
one of many things such a term can mean in context. If various people 
outside the community using the term are so badly confused as to think 
Boolos, who was the first to use it, meant to be suggesting Hume scooped 
Cantor, that is unfortunate, but not Boolos's fault.

That said, many writers on the subject now agree that, in part because the 
term has started to be used outside the context of Frege studies, and 
because it obviously continues to mislead some people, that some other term 
might better be used. As I mentioned above, Boolos suggested, and at the 
end of his life started using, "HP", which preserves something of the 
history of the development of neo-Fregean approaches to arithmetic.

> >All this is just terminology, and hardly serious.  But a more
> >serious problem in the "Neo-Fregean" literature is the persistence
> >with which people refer to "Hume's Principle" as a
> >"contextual definition."  Boolos (who certainly knew better)
> >refers repeatedly to the principle as a contextual definition
> >in the papers reprinted in "Logic, Logic, and Logic."
> >The Arche web site also refers to the axiom in this way.

Hardly anyone speaks this way any more, except when they slip up. All of 
this got worked out quite a few years ago, largely as a result of Boolos's 
prodding: A check of publication dates would show as much. In any event, no 
one ever really thought that HP was a "definition" of any kind. Wright is 
quite clear about this way back in his book, though he did continue to use 
the nomenclature for a while, until suggesting the term "abstraction 
principle", which is now commonly used.

Re the Arche web site: Where is this usage? I've just scanned most of the 
site and can't find any such thing.

> >But how can it be a "contextual definition" when it has
> >the axiom of infinity as a direct logical consequence?
> >How can it be a definition of any kind at all?

As I said, it's not, and no one ever thought it was. If, on the other hand, 
the question is: How can HP be any sort of analytic or conceptual truth, 
given that it is (or implies) an axiom of infinity, then it's a familiar 
one. Obviously, however, those interested in the thought that it might be 
some sort of conceptual truth don't find it so obvious that no axiom of 
infinity can be such a truth. Indeed, what reason is there exactly to think 
that the modes of cognition that underlie (deductive) reasoning in general 
can not provide us with knowledge of the axioms arithmetic? (That's the 
epistemological content of logicism, or of logicism naturalized.) Kant had 
some reasons, but it is controversial (to say the least) whether they are 
good ones.

Richard Heck


----------------------------------------------------------
Richard Heck
Professor of Philosophy
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Harvard University
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E-mail: heck at fas.harvard.edu
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