[FOM] Voevodsky

[Richard Heck]

On 05/18/2011 03:01 AM, Robert Black wrote:
>
> Now what I take Voevodsky to be saying: we can coherently doubt the
> consistency of PA. This seems obviously true, since if it weren’t, the
> Hilbert programme for PA would have been incoherent. Further, although
> there are proofs of the consistency of PA, for each such proof we can
> doubt its cogency. (The examples Voevodsky considers are firstly the
> obvious proof that the axioms of PA are true and the proof rules
> preserve truth (formalized, this is the proof of Con(PA) in
> second-order arithmetic) where it is possible to doubt whether we
> really understand induction over arithmetically definable but
> non-recursive properties, or Gentzen-style proofs where it is possible
> to doubt whether the ordinals less than epsilon_0 (modulo some
> suitable notation) are really well-ordered. This seems to me correct.
> (I don’t mean I *do* doubt these things - I don’t - but one *can*
> coherently doubt them.) The consistency of PA is not absolutely
> indubitable.
>
> But of course if PA is inconsistent then anything is provable from it,
> so it follows that proving something from PA doesn’t make it
> absolutely indubitable.
>
Personally, I doubt that very much is "absolutely indubitable".
Descartes made that pretty clear. But what is supposed to follow?

This entire style of argument looks suspicious. It looks to me like
classical skeptical reasoning. In some sense it isn't absolutely
indubitable that I'm not being deceived by an evil demon, or a brain in
a vat whose perceptions are produced by a computer, or whatever.
Skeptics have long argued that this fact, by itself, shows that what I
take myself to know with certainty---and, frankly, I'm more certain that
I have hands than I am of much mathematics---I do not really know at
all. But the argument is a bad one, even if people still disagree about
why.

Yes, if PA is inconsistent, then not all its axioms are true, and if
we've proven stuff using some of the false ones, then we don't really
know what we think we do. But the mere (very abstract) possibility that
PA *might* be inconsistent doesn't show that we don't really know what
we think we do, if PA isn't inconsistent (as we are agreed it isn't,
though of course we could be wrong). To think otherwise is simply to
repeat the skeptic's mistake.

That doesn't mean, by the way, that Voevodsky's program isn't coherent.
It just means it needs better motivation. If one really thought, for
example, that the axioms of PA might not really be true, even if they
are consistent, then that might be a reason to pursue such a program.

Richard Heck