[FOM] Re: numbers and sets

[A.P. Hazen]

   Hartley Slater has repeatedly emphasized that "numbers are categorically
different from sets."  I think he may be referring to a strong (and widely
shared) intuition, stated in a way that (indirectly) goes back to Russell.
I would ask Slater to consider, however, whether the non-identity-standard
interpretation of arithmetic in set theory (interpretation 4 in my post of
a couple of days ago) doesn't go some distance toward meeting the objection.

   A "category mistake" is a statement that "isn't right," but in such a
way that it "isn't even wrong."  It isn't right to say that the number 3 is
red, but it also isn't right to say it is green: numbers aren't the KIND
(category) of thing that can be colored.  Much of the time I'm happy to say
that 3 **isn't** red, or green, but some philosophers and linguists have
felt that this is inadequate: because of the category-mismatch of the
numerical subject and the color predicate, the statement "3 is red" is
NEITHER true NOR false: numbers are not the sort of thing that can,
meaningfully, be said either to have or to lack a color.  The term
"category mistake" was introduced by the English philosopher Gilbert Ryle
about fifty years ago; I believe he thought of himself as generalizing, and
applying to non-mathematical subject matter, what Russell had said about
type errors in the context of the type-theoretic system of "Principia
Mathematica".  (Ryle's discussion inspired a bit of technical work in
logic: his notion provided one of the motivations for studying 3-valued or
partial logics in the 1950s and 1960s; Merrie Bergmann's "Logic and Sortal
Incorrectness" ("Review of Metaphysics" vol 31: 1977) discusses how to
handle category mistakes in the framework of, roughly, Montague grammar.)

    There is a certain feeling that saying that a number either does or
does not have something as a member is a category mistake: sets have
members, but numbers aren't the KIND of thing that CAN have members.
Identifying numbers with, e.g., the Von Neumann "numbers" gets the truth
values of arithmètic sentences right (it gives an interpretation of PA in
ZF, so every theorem of PA gets proven in ZF), but it also assigns truth
values to some statements that seem to be category mistakes.  (And not just
the blanket, default, assignment of falsity to all categoreally odd atomic
statements: {1} is not a member of any Von Neumann number, but 1 itself IS
a member of 2, 3 and all larger numbers.)

   Now, this has tended not to bother mathematicians, and it is easy to see
why: the identification doesn't give the WRONG truth value to any statement
they antecedently thought HAD a truth value; the objection is rather that
it wrongly gives truth values to statements no one in their right mind had
ever asserted or denied!  Mathematicians have thought of these categoreally
odd statements as (to use a bit of philosophers' slang) "don't cares," and
haven't cared.

   Some philosophers-- Slater, for one-- DO care, however.  And maybe, for
at least some philosophical purposes, it would be nice to have a theory of
sets and numbers that recorded the intuitive concpetual distinctions....  I
claim that such a theory is available.  A natural way to formalize the
three-way distinction of true/false/categoreally-odd is to use a
MANY-SORTED logic, with notationally distinguished alphabets of variables
for numbers and for sets, and with predicates that can only form
well-formed atomic formulas with terms of the appropriate sort. (Category
mistakes, in such a formalization, become syntax errors.)  When we
formulate an intuitive theory of sets and numbers in this two-sorted way,
however, the "reduction" of arithmetic to set theory becomes very neat.  By
using the non-identity-standard relative interpretation I described, the
whole theory is interpreted in its purely set-theoretic sub-theory,
arithmètic theorems going to set-theoretic theorems, arithmètic refutables
to set-theoretic refutables, and category mistakes not being translated!

    (Since, for higher mathematics, we will want sets with numbers as
members, the details will get complicated: we may want to interpret a
theory including ZFU (ZF with ur-elements) with numbers taken as
individuals in ZF.  At least in the presence of the axiom of foundation,
however, this is more tedious than problematic.)

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Allen Hazen
Philosophy Department
University of Melbourne