[FOM] Re: Parallel to Slater on Numbers

Arnon Avron aa at tau.ac.il
Fri Oct 10 02:53:55 EDT 2003

> Arnon Avron writes
> >Do you reject the propositions:
> >"The pair <0,1> isin the exponential function" and "the exponential function
> >is a subset of the upper half-plane" as ungrammatical?
> I think that is enough of 'tangential' issues.  What Avron should 
> concentrate on is not confusing the value of a function with either 
> an argument of that function, or the set of arguments for which the 
> function has that value.  

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!).

> 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 opposed as totally *unmathematical* is to
let every possible equivalence relation induce
new mysterious entities (like cardinalities, directions
and so on). I care about "unmathematical" much, much, much, much more than 
I care about "ungrammatical"!

> There was also the discrepancy between Avron's and 
> Holmes' views on '2={{}, {{}}}', which I documented, which does not 
> inspire confidence in people (like me) who do not  think one can 
> treat or define numbers in several ways making this equation 
> sometimes true and sometimes false.  If that is the case then FOM is 
> really wobbly.  Hello paraconsistency!  (And you might know my views 
> about that!).

This discrepancy is only a trivial, unimportant side-effect of a 
much bigger discrepancy about the foundations
of Mathematics, the importance of NF, the nature of infinite sets
etc. Anyway, if discrepancies like this mean that "FOM is
really wobbly" than the whole of western Philosophy should be 
rejected as paraconsistent (I dont know, by the way, if you mention
paraconsistency because you know that I have personally  done some
work in this area. Let me assure you in any case
that nevertheless, classical
logic is for me the primary logic, the only logic for mathematics,
and the one in which we prove theorems about all non-classical systems).

Arnon Avron

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