[FOM] inconsistency of P and extreme formalism

Brian White white at math.stanford.edu
Sat Oct 8 22:17:31 EDT 2011

On Oct 6, 2011, at 3:47 PM, Arnon Avron wrote:
>> On Mon, 3 Oct 2011, aa at post.tau.ac.il wrote:
>>> Can Nelson understand what is Pi^0_0 statement? How?
>>> Can he  understand what is a formula? How?
>>> Can he  understand what is a proof? How?
>>> I cannot understand how he can use all these
>>> concept and yet claim to doubt the consistency of P
>>> (how does he understand the notion of consistency of P
>>> at the first place?)

The notions of formula, proof, consistency can be defined
in systems that are much weaker than P.  

So belief in consistency of P is not necessary to talk about (and
to reason about) notions such as formulas, proofs, consistency, etc.

> Note, by the way, that Nelson was *not* claiming that he has shown
> anything of the form "there is a proof of `0=1' in PA using at most 
> one million symbols." At least in his message to FOM he gave no bound
> on the alleged proof of `0=1'.

Implicitly he did give such a bound.  He said he was in the process of
writing a proof with every detail supplied.   Presumably he believed
he could finish that within his lifetime (for example).  So he certainly
wasn't talking about a proof of length 10^10^100.

> Note also that what he did was to
> give an *outline* of a complicated proof of `0=1'. Obviously, like
> anyone of us he was confident that if the outline is correct then an 
> actual proof can be constructed out of it. It is beyond me
> on what this confidence was based, given that he pretends not to understand 
> the collection of natural numbers, or the collection of formulas of P,
> or the collection of formal proofs in P.

What is mysterious about it?
Let's suppose someone is an extreme formalist.
He believes that when we're doing mathematics, we're just playing
a game with certain formal rules.   He doesn't believe that the strings
of symbols that occur in proofs are assertions about anything, just as
when we play chess, we don't believe that configurations of chess
pieces are assertions.
Nevertheless, he might enjoy the game and get good at it.
He develops (from experience) instincts about what things
can be proved and what can't.
If shown something that is possibly a proof, but with certain steps
omitted, he may have a good instinct about whether the missing
steps can be supplied.

It's analogous to a very good chess player.
If you show him a certain configuration, he may feel quite sure
that white can force a win.

Or maybe it's not obvious to him at first, but after some thought, he
figures out an outline of a solution: he sees that white can
take black's queen and he's confident (based on that) that
white can force a win, even though he hasn't worked out the

> It is beyond me
> on what this confidence was based, given that he pretends not to understand 
> the collection of natural numbers, or the collection of formulas of P,
> or the collection of formal proofs in P.

Is it beyond you to believe that a very good chess player
might have (for very good reasons) confidence that in a certain configuration
white can guarantee victory, even if that player does not pretend to understand
the collection of all possible configurations in chess?  In fact, perhaps the player believes
that the set of all possible chess configurations is so astronomically large
that for all practical purposes it is a meaningless concept.

> To sum up: I am conviced that a lot practice in doublethought
> is needed for someone like Nelson in order to really believe
> that he believes his official views.

Why are you so confident that P is consistent?
I would guess it's because you think you "have" a model of it.
(That's why I feel confident that P is consistent!)
But for all we know, there are only finitely many particles in the
universe.   In any case, even if there are infinitely many particles, we can't see
them all, so I think it's more accurate to say that
we are confident of P's consistency because we can imagine a model
for P: we have a clear (to us!) mental picture of something we are convinced is a model for P.

Now, as I said, I personally find my model convincing.
But is it really so surprising that some people (e.g. Nelson)
are a bit skeptical?

-Brian White

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