# [FOM] Unschooled grammatical intuition v. theoretical identities

Hartley Slater slaterbh at cyllene.uwa.edu.au
Thu Oct 16 19:52:57 EDT 2003

```Now Neil Tennant is getting enraged (FOM Digest Vol 10 Issue 23):

>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 ?

I am not unschooled; I was well taught; my teacher was Steen.  So
have another try, Neil, at interpreting the formula I gave for 'There
are exactly two Fs'.  (Hint: suppose that numbering variables has
implications).

I grant you the latter is a very, *very* interesting issue, but only
for this reason: it reflects on how few people want to get to grips
with substantive issues.  It was quite remarkable how many people
'the theory of number has been mis-represented for 100 years'.  Faced
with the fact that in, for example, '([n+1]x)Fx iff
(Ex)(Fx.(ny)(Fy,-(y=x))', the 'n' is clearly a variable, people seem
to have turned to some trivial pursuit, to avoid the anxiety which
realisation of the dimension of the real issue would bring.

And they are not just my grammatical intuitions, if they are
'intuitions' at all; they are also Neil's, see
http://www.cs.nyu.edu/pipermail/fom/2003-June/006806.html :

>In the singulary case, I proposed (in my 1987 book Anti-Realism and
>Logic) an adequacy condition on a theory of number, using what I called
>Schema N:
>
>Schema N        #xF(x) = _n_ iff there are exactly n Fs

Where, by contrast, in von Neumann's work, does he say anything like
this, or employ formulas of the form '(nx)Fx', where 'n' is one of
his constructs?  Calling something 'a cardinal number' or 'an ordinal
number' does not guarantee they are such.  Where in *any* of
twentieth century FOM are there words with the sense of 'first',
'second', 'third' etc. (remember them?)  Certainly von Neumann's
'ordinals' do not have this sense, so all you need, it seems, is some
ungrounded assertion that von Neumann's chosen sets are ordinals and
everybody thinks they have a foundation for the theory of ordinal
numbers.  Alas: "You can make anybody believe anything, so long as
they are clever enough" (Tom Stoppard).
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html

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