[FOM] Intuitionism, predicativism, and ill-defined domains

[Arnon Avron]

As Tolstoy might have put it, all the platonists are platonists
in the same way, but every non-platonist has his/her own way
of being non-platonist (I admit that this fact should be count in favour
of platonism, and is certainly a huge obstacle in trying to convince
mathematicians to adopt  non-platonist foundations). Thus I find
myself arguing here with Nik Weaver, although my views seem to be closer
to his more than to anybody else on FOM (Unfortunately, Sol Feferman
is not on the FOM list). In fact, for some time I am planning to 
send a message explaining why predicativism (or why conceptualism, 
which is the term that Nik prefers. I dont really care, but
I am simply used to "predicativism") - an issue which I find much more
important than the recent discussion concerning intuitionism I got
myself involved in. I simply wanted first to read Nik's three papers
(but up to now I have only found time to read the first. It does 
provide motivation to read the other two!). Since this might take 
time, I'll react now to in brief to Nik's message from Oct 22, leaving
better (I hope)  explanations to a more detailed posting.

> > The upshot is that we cannot avoid thinking of *all* the integers.
> > Neither do we need to be able to actually run through all the integers
> > in order to know that for every integer n we come across it will be the
> > case that n=n, or that either p(n) or its negation will be true,
> > or that if we continue to check integers greater than n, then
> > either we'll encounter a pair of twin primes or else we shall never
> > encounter one. I cant imagine a third possiblity, and if you
> > do than you have a much better imagination than me.
> >
> > In short: Classical logic is valid also for potential infinity!
> 
> That is well put, but I disagree on two grounds.  First, if you
> accept that the term "all the integers" is meaningful, which I
> understand you to say you do, then I don't think you're talking
> about a potential infinity, you are talking about a completed
> infinity which is grasped as a whole.  We must draw a distinction
> between the quantifier "for any integer" (that may appear in some
> indefinite future) and the quantifier "for all integers" (referring
> to elements of a well-defined set, the set of all integers).  Only
> the former can legitimately be used by someone who disbelieves in
> a completed infinity.

I admit that I dont understand the essential difference between
"for any integer" and "for all integers". Maybe it is because
there is no difference of this sort in Hebrew. But it is OK for me
to replace in what I wrote above "for every integer n we come across"
by "for any integer n we come across" etc. The validity of excluded
middle would not be affected.
 
> My second point of disagreement is that I feel that if you are
> willing to assert things like "if we continue to check integers
> greater than n, then either we'll encounter a pair of twin primes
> or else we shall never encounter one" then you are in fact able
> to imagine running through all the integers.  (You seem to blur
> the distinction between this being logically possible and it
> being physically possible in our universe; the latter is not
> at issue.)

No, I dont imagine running through all the integers as 
logically possible (physical possibility is indeed not an issue).
I imagine the logically possibility of running  up to
*any* integer. This is sufficient.

  I believe that the way I understand this is exactly the way
The Greeks understood infinity. Euclide never talks about an
infinite line as a whole: only about finite line segements
that can be extended indefinitely. Yet it was clear to him
that if we continue extending two finite line segments on both sides
then either they will meet at some point (of time and plane),
or else they will never meet.

  Talking about the Greeks, they provide an excellent example
that accepting only potential infinity and being constructive
do not contradict using classical logic. Euclide was an
extreme constructivist in the sense that there is no difference
in his books between "exists" and "constructibe using a ruler 
and a compass". So every existence proof in Euclide is constructive -
but he freely uses excluded middle, double negation elimination
and reduction ad absurdum to show that his constructions
meet their specifications! (By the way, also Gauss rejected
actual infinity, but as far as I know he had no reservations whatsever
about LEM).

> Let's move this debate to an arena where it's easier to distinguish
> between conceiving of something as a well-defined entity versus
> conceiving of it as being necessarily incomplete and always capable
> of extension.  Namely, consider the universe of sets.  I propose
> the following test.  Let a "large cardinal axiom" be any statement
> of the form "there exists a cardinal x such that P" where P is any
> formula in the language of set theory.
> 
> Claim A: every large cardinal axiom has a definite truth-value.

Did you mean "every large cardinal axiom" or "any large cardinal axiom"?
 
> Is it possible to accept this claim without being a platonist?  If
> you disbelieve in the universe of sets as a well-defined complete
> entity, have you any grounds for accepting claim A?  I say no.
> 

The only way I can understad   "having a definite truth-value" is as a
realtion between a statement and a domain - unless there is 
a definite intended domain that the statement is understood to be about.
Now like you I dont think that it makes sense to talk about
one definite domain of sets. Your J2 may be a good domain,
but (like you, if I understand you correctly) I think that
a predicativist should take into account the possibility of 
other acceptable constructions of sets, and what s/he proves should
be *absolute*: true for any domaia of setsn. A given large cardinal axiom P 
might have one truth-value in some domain, and a different one in
another. In this sense it does  not have "a definite truth-value".
However, "P or not P" does have an absolute truth-value:
it is true for any domain. So all classical mathematics is
a part of predicative mathematics.

  Now also according to you classical logic is valid for any specific
"closed" domain. However, you claim that "If we are working in an open-ended
domain that is indefinitely extendible then the law of excluded middle
is not justified". I am not quite sure what you mean by "indefinitely
extendible". If you have in mind something like the potential infinity
of the natural numbers or finite line segments then LEM applies. 
If not then the only way I can understand "indefinitely extendible"
is "extendible to bigger and bigger domains", but then again, in each 
of them classical logic is valid.

Arnon Avron
School of computer science
Tel-Aviv University