[FOM] Parallel to Slater on Numbers

Hartley Slater slaterbh at cyllene.uwa.edu.au
Fri Oct 10 21:06:40 EDT 2003

Arnon Avron writes:

>If I understand you correctly, your answer to my question is "yes": you
>do reject these two sentences as ungrammatical. But you have not said so
>explicitly, and English is not my mother's tongue. I might therefore have
>misunderstood you. So can I humbly ask you to give an explicit "yes" or "no"
>answer to the question above (calling it 'tangential' is just refusing to
>draw the obvious conclusions from your claims!).

I have not given any answer to the question, nor do I propose to do 
so.  I have been making a point about grammaticality in connection 
with basic numerical statements, and that is all I am concerned with. 
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.

Avron goes on
>  > Avron believes that cardinalities themselves have
>>  cardinalities, indeed he must say that Card{{}, {{}}} = {{}, {{}}},
>>  and likewise for all the finite von Neumann ordinals.
>Avron believes that this is so according to the best and most natural
>definition of cardinality he knows, but he would not mind much
>if some other finititary definition is adopted, provided it does
>the job. What I strongly oppose as totally *unmathematical* is to
>let every possible equivalence relation induce
>new mysterious entities (like cardinalities, directions
>and so on).

He 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'.  And if you find 
directions mysterious, how do you find your way about?  More to the 
point, how do you give other people directions?  Someone pointing to 
a paradigm road 'Main Street' in the direction of North and asking 
where it heads gets the answer from you: 'Main Street'.

Avron also says, about the discrpeancy between himself and Holmes:
>  Anyway, if discrepancies like this mean that "FOM is
>really wobbly" then the whole of western Philosophy should be
>rejected as paraconsistent

If Arithmetic could be reduced to the status of Philosophy that might 
be so.  But here is one thing to cling on to: 2+2=4!
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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