[FOM] Platonism, "vagueness" of N, possible inconsistency of PA?

[Vladimir Sazonov]

Randall Holmes wrote:
>      Vladimir Sazonov asks how platonism could survive a proof of
> ~Con(PA)...
> 
>      Platonism doesn't necessarily require a commitment to any
> particular mathematical content; what it involves is a particular kind
> of attitude to mathematics.  A platonist believes that mathematical
> statements have well-determined truth values (independent of our state
> of knowledge) and perhaps that there are special mathematical objects
> to which they refer 

Is it meaningful from platonistic point of view to refer to
"standard model" of PA in such a context where evidently
no formal system like ZFC or the like is assumed?

These kind of beliefs you mention looks to me both meaningless and
useless. What we really need (in addition to mathematical rigour)
is only our intuition whose nature is vague. Platonism pretends on
something more what has no scientific grounds (just beliefs). Alas,
it is almost impossible to reassure those who are inclined to beliefs.

A proof of
> ~Con(PA) would show (among other things) that impredicative
> second-order logic on infinite domains is inconsistent, and so that
> the conditions I state above 

[see the original posting of Holmes and correction in
"[FOM] slip in previous post"]

In fact, as I understand, this may be (essentially) considered
as axioms on the arithmetical successor operation.

> do not hold; 

Just conditions (on successor operation - something which is
rather trivial and I see no reason to *necessarily* deny) or
may be the whole formal system does fail? It seems you intend
to say that second-order logic (which is actually two-or
more-sorted extension of first-order logic by so called second
order objects and Comprehension Axiom) is something non-doubtful
and that contradiction of PA (and hence of Second Order logic
with successor operation) would lead, therefore, only to doubts
on the successor operation.

I do not agree, because second order objects (even in many-sorted
first-order setting - the only one which I understand) have rather
unclear nature what we already know from the situation with the
Problem of Continuum and what can be realized even "from the scratch".
Even in the case of (not so big) finite domains we can feel some
doubts on second order quantification.

But you, seems, have so strong platonistic inclinations that, in
the case of inconsistency of PA would reject the very successor
operation! Do I understand you well? Or impredicative character
of Second Order Logic should be responsible and successor could
be saved some way?

from my standpoint, it would
> show that N does not exist 

I do not know what is that N which would not exist in the case
of inconsistency of PA. Of course, in this case we would have
problems to imagine (informally) *a* model of inconsistent theory
PA. But we will be still able to count "one, two, three, and so on"
and to find another formalization(s) based on this "protointuition".


(and also that the collection of all items
> of syntax (as usually understood) does not exist).  I would not arrive
> that the conclusion that the usual idea of N was "vague", but that it
> was wrong (clearly defined 

clearly???

> but resting on incorrect assumptions).

Assumptions on what? On successor (???) or, say, on Induction Axiom?
Even if we do not have any contradiction of PA, I think, we should be
plausible enough in playing this game "what if...?". You seems intend
to blame the axioms of successor - the most peaceful and modest axioms.
Why not Induction or Comprehension? Or I misunderstood you?

Of course, in the case of a contradiction the usual idea of N
(that is axiomatization PA) will be considered as wrong. Then
it becomes difficult to discuss whether N *was* vague or not
before the contradiction was found, but, nevertheless, possible.
The naive, contradictory set theory has an evident axiomatization
and intuition behind it. Even if we already know on paradoxes,
this intuition is still with us. Is not it clear now that it is
vague (and wrong to some degree; there is, anyway, something
rational in it)? In the case of a contradiction in PA - the
heart of mathematics - it would become clear that we should be
careful with assertions on existence and uniqueness of any version
of N to which we will come after reconsideration of PA. There is
always potential possibility that we again may do something wrong.
What is then the status of any current version of N, even if we
really have no contradiction, but can, in principle, expect it?
Just a vague idea (may be even wrong in some aspects) - what else?

Again, we have an idea which may be, *in principle*, wrong.
No guarantees, except beliefs. Is not it the most appropriate
term "vague idea" (as any idea whatever it could be).


[...]

> Such systems are certainly worth investigating for their own sake, and
> so are systems for implementing strictly "feasible" mathematics.

OK!

it appears
> likely that if one were doing metamathematics the syntactical objects
> to which one would be _referring_ would actually have to be different
> from the syntactical objects one was _using_, 

This is quite right note in any case (whether PA is consistent or not).
It is meaningless to identify reality with our abstractions.


which would require
> extraordinary care.  

I see nothing extraordinary here. Just a "honest" scientific approach.

It seems it makes sense here to illustrate which way Induction
Axiom is related with mixing syntax and semantics. (This seems
well-known.)

It is sometimes explained why A(0) and A(n) -> A(n+1) imply
forall n A(n) by reference to iterated application of modus ponens:

A(0) and A(0) -> A(1) imply A(1),
A(1) and A(1) -> A(2) imply A(2),
...
A(n) and A(n) -> A(n+1) imply A(n+1),
...
therefore forall n A(n).

What is n here, syntactic or semantic object? It should be
syntactic (numeral). But then how is it possible to
conclude forall n? Asserting that any number (an imaginary
object in the imaginary model for PA) may be denoted
by a numeral is again a mixing of syntax and semantics.
Also, syntactic objects have only feasible size whereas
Induction Axiom proves existence of non-feasible numbers.
What is then the real ground for Induction Axiom, if not
just a reference to the fact that it is very useful formal
tool, although based on a vague (even doubtful) intuition
which mix things in so bad way. Is not the idea of N based
on such a big mess just vague (even wrong to some degree)?

I think, it is better and "honestly" to realize this fact (even
while continuing application of Induction in mathematical practice)
than just to ignore it.


Kind regards,

Vladimir