[FOM] Unschooled grammatical intuition v. theoretical identities

Neil Tennant neilt at mercutio.cohums.ohio-state.edu
Tue Oct 14 16:20:56 EDT 2003

On Sat, 11 Oct 2003, Hartley Slater wrote:

> All functions have a domain and a range, and specifically the 
> functional expression 'the number of members in...' (#) takes a set 
> abstract to form an expression with the same grammatical place as a 
> numeral.  So '#{{}, {{}}}=2' is grammatical, but '#2=2', '#2={{}, 
> {{}}}', and '#{{}, {{}}}={{}, {{}}}' are not.

Let's get clear ... Someone who cannot write down a correct formula of
first-order logic to capture "there are exactly two Fs" is now telling us
that his grammatical intuitions should prevail over the enormously
fruitful theoretical identification of the natural numbers with the finite 
von Neumann ordinals ?

Would that all foundational questions were so easily settled! This is 
the method, then:

1.  Regard introspective grammatical intuition as the highest
tribunal of experience; and
2. adopt the atittude of dogmatic defeatism---the late Jonathan Bennett's
lovely phrase---about the value of theoretical systematization.

Hey presto!---your problems are solved. Alas, it is also the method that
has made Wittgensteinian philosophers of language completely irrelevant to
(and indeed, obstructive of) foundational advances. 

In this connection, I would like to share with the list a quote from James
Gleick's book "Isaac Newton", at p.132:

	Newton himself said that he had considered composing a "popular"
	version [of the Principia] but chose instead to narrow his
	readership, to avoid disputations---or, as he put it privately,
	"to avoid being baited by little smatterers in mathematicks."

At least Newton would not have had to worry about smatterers in
linguisticks. That anxiety was apparently to be reserved for
foundationalists from the middle of the twentieth century onwards.

> [Avron] should check the large volume of material now coming out of the 
> Arch'e project at St Andrews, for instance, and the similar work 
> previously done by Crispin Wright.  There is also plenty of relevant 
> material in Boolos' 'Logic, Logic and Logic'.

Please do us the favor of explaining how this "large volume of
material" represents any worthwhile theoretical advance over the system of
foundations provided by first-order logic and the axioms of set theory. 
Perhaps you could supply a metatheorem about the consistency-strength of
Wright's system? Or show that it furnishes a better account of the
real numbers than, say, Dedekind? ... And while you are about it, can we
please have your considered judgment as to whether the Arch'e project's
aims are either advanced or put in limitative perspective by Kit Fine's
monograph on abstraction?

Neil Tennant

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