FOM: Neo-Fregean reverse mathematics

Alasdair Urquhart urquhart at
Mon Mar 26 11:18:26 EST 2001

On the subject of Hume's notion of number,
John Mayberry may be right in saying that Hume's
notion of number is Euclid's.  But so what?
The middle books of Euclid's elements are universally
agreed to be on the subject of number theory.
Or should we say that Euclid was writing about some
other topic, say "Euclid-numbers"?   Just before
the passage in dispute (Treatise, Book I, Part III,
Section 1), Hume explicitly refers to "algebra and arithmetic."
So, it's obvious that he has in mind numerical calculations
such as 2 + 2 = (1+1)+(1+1) = 1+1+1+1 = 4.  The last equation
is an algebraic depiction of Hume's notion of 4 as composed
of 4 units.  So I see no inconsistency in talking of Hume's numbers
as positive integers.

In any case, these historical questions are irrelevant.  The point is,
Hume's remarks in no way justify the general cardinality principle,
applying to finite and infinite sets, 
rightly attributable to Cantor, as Bill Tait correctly said.

My colleague Jim Brown doesn't get my position on definitions 
quite right.  Let me say what it is.  My belief is that in
discussions on logic and foundations, we should stick pretty
closely to the strict logical notion of a definition.
Roughly speaking, a definition gives the meaning of a symbol
in terms of earlier meanings.  In particular, that should
be ALL that a definition should contain, and in particular
a definition should not have as a consequence the *existence*
of certain defined entities.  These requirements are
formalized as the requirements of eliminability and conservativity
(non-creativity).  For details, see Chapter 8 of Suppes's 
Introduction to Logic.  

The notion of "contextual definition" has a fairly well-defined
meaning in the older logical literature.  It refers to definitions
that are not given by explicit equivalences (as in Suppes), but
by a set of recursive rules for elimination.  The key examples
are Russell's theory of descriptions and no-classes theory
(Whitehead and Russell describe these as part of their doctrine of "incomplete 
symbols").  This is the standard usage of the term, as can
be seen by looking at Quine's "Ways of Paradox" and "From
a Logical Point of View."  Now I have no objection at all 
to such definitions, and so I don't agree with Jim's remark
that I think "contextual definitions in general are nonsense."

What I object to is the new usage of the phrase that has appeared
in the "Neo-Fregean" literature.  George Boolos explicitly 
defines a "contextual definition" to be a formula of the form:

	!U = !V <--> U ~ V,

where U and V are second-order entities, !U and !V are first-order
entities, and ~ an equivalence relation (LL&L, p. 137).  Now of course,
people are entitled to call things what they want.  But it's not unfair
to point out that this new notion of "contextual definition" is only 
very loosely related to the old notion.  In particular, when
added to higher-order logic, the cardinality principle is simply a new axiom, and not a 
definition (in the logical sense, that is), since the requirement 
of conservativity is violated.

--Alasdair Urquhart

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