[FOM] inconsistency of P

Studtmann, Paul pastudtmann at davidson.edu
Tue Oct 4 17:13:03 EDT 2011

There is an epistemic variant of the position that Chow is advocating on behalf of Nelson.  According to the view, whatever the case is with respect to truth, our knowledge only extends to finite aspects of mathematics.  Such a view would be quite a natural one for an empiricist to adopt, though there are of course other ways of articulating an empiricist view of mathematics.  The view would draw a boundary around our knowledge of mathematical truth at the places that Chow has already suggested extend to mathematical truth itself.  

And why would an empiricist draw such a line?  For the simple reason that such facts are accessible through ordinary experience, while experience of the infinite is beyond the ken of finite creatures like us who are embedded in a small portion of space and time.  

If this view is correct, we do know that the consistency of various parts of PA is provable from various sets of axioms.  But we fail to know the truth of the axioms themselves, and so we couldn’t really claim to know that PA is consistent on the basis of those proofs.  This is not to say that some axioms don’t strike us as intuitively compelling.  But it is one thing for us to have an intuition about structures on the basis of our empirical grasp of small finite portions of those structures and quite another for those intuitions to give us an indication into the nature of the infinite.  

A proponent of this view might suppose that the axioms are hypotheses that serve to organize finitely given portions of mathematics and that they can in some sense be justified on the basis of their playing that role.  This would be similar at least in spirit to Hilbert’s view of axioms.  But notice that such a view does make a principled distinction between the types of justification available to the different portions of mathematics, so much so that one might very well be tempted to accord a much higher epistemic status to the finite portions of mathematics. 

A proponent of this view might also take axioms to be something like enlightened decrees that serve to guide those conducting their mathematical inquiries.  According to such a view, axioms, like decrees, are not truth-evaluable, though they can of course have any number of interesting pragmatic features like: organizing existing theory, seeming to be intuitively true, etc.  

One should note how extreme such a view is.  According to it, we don’t even know that zero has no natural number predecessor, since such a sentence is PI_1.  But such an implication, some might argue, puts an epistemic line in exactly the right spot.  Why?  Because skeptical scenarios can be given that would show that we don’t in fact know that zero has no natural number predecessor. 

Scenario: Suppose there is a highest natural number that is also zero’s predecessor.  Suppose that the number is so large that it cannot be represented (except indirectly via description) by a physical creature.  In other words, suppose that all ‘true’ arithmetic is modulo that number.  Moreover, in the vein of Chow’s thought experiment, suppose that such a number places an upper bound on some physical constant, perhaps the number of fundamental particles.  

How do we know that such a scenario does not hold?  

It is not clear that we do, at least not on the basis of ordinary experience.  (Admittedly, physical science complicates the picture.)  And so the epistemic line appears to be in exactly the right place.

Of course, it is one thing to articulate a view of this sort and quite another for one to provide some sort of justification for it.  Interestingly enough, given the view’s explicitly empiricist leanings, any such justification would have to be empirical.  And that is not an easy thing to imagine: an empiricist justification of an empiricist view of mathematical knowledge.  But such is the lot of empiricists.

For what it’s worth, I suggest a way empiricists might provide some such justification in my book 'Empiricism and the Problem of Metaphysics'.

Paul Studtmann
From: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] On Behalf Of Timothy Y. Chow [tchow at alum.mit.edu]
Sent: Tuesday, October 04, 2011 11:22 AM
To: fom at cs.nyu.edu
Subject: Re: [FOM] inconsistency of P

Harvey Friedman wrote:

>I am reasonably aware of Nelson on this, but my main point is that
>mathematicians (including Nelson, by the way) seem to regularly use - and
>teach - facts which logically imply the consistency of at least
>significant fragments of PA, including PRA, or single quantifier PA.  At
>least, the former Nelson - and what about Nelson's current math teaching?

I think I indirectly answered this already in my post where I pretended to
be Nelson (or someone with similar views), but to be explicit: When we
teach, we play by the rules of the game that society pressures us to play.
In my soul I am free to believe the truth.  According to the party line,
PA is consistent, as Nelson well knows, but if you permit him to speak
freely, he will dissent.

As an aside, lying in calculus class is something we all do in one form or
another so I don't think Nelson can be ethically faulted for doing so.

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