Wednesday, January 20, 2010

Reflection and Limits on Mathematical Knowledge

It's a classic question whether there are mathematical truths which are unknowable by creatures like us. And, as Bill Clinton might have said, the answer to this question naturally depends on what you mean by "creatures like us".

A smart philosopher recently suggested that I should take the following possibility seriously: even if particular human's brains were well well approximated by a Turing machine, the faculties by which humans access mathematics include (once suitably idealized) a faculty of reflection, whereby one could transcend the possibilities of any system to which Incompleteness applies.

So, I'm taking that possibility seriously, here's why I reject it :)

If by `a faculty of reflection' you just mean something that lets you say, of any particular formal system which you believe to be sound, that it is consistent, then this is not enough to get around incompletness for familiar Putam vs. Penrose reasons.

If by `a faculty of reflection' you mean something which lets you produce a system which formalizes all your current reasoning about mathematics, and then recognize that the system does this (so that then then you can deduce this system is consistent and arrive at its con sentence) then I don't buy that humans can be plausibly idealized as having anything like this kind of faculty.

Behaving in a way that matches a given algorithm is one thing, coming to know that this is what you are doing is quite another! The issue here is essentially the same as with the Kant Puzzle I poster earlier. Certainly we can work out that particular examples of conclusions that the formal system proves, and check that yes we accept that conclusion. But to arrive at the con sentence you would need to know that everything the formal system proved was correct not just some finite number of instances.

Now, admittedly, after trying enough cases, (if the system was simple and elegant enough) you might be willing to accept that yes everything the formal system proved was something you accepted, and hence infer the con sentence.

a) its somewhat controversial whether beliefs formed in this way would count as knowledge

b) this process might dead end at a point where all the reasoning you accepted could only be summarized by an alogorithm/formal system that looked ugly and gerrymandered to you, and hence was not a plausible candidate for induction.

c) if this is the sense in which we could always get access to con statements, no faculty of 'reflection' in particular would be involved, just a general ability to apply something like scientific induction to mathematics. The same kind of reasoning that gets you from 'the first million things proved in this system are ones I accept as true, so all of them are true (so the system is consistent)' would also get you from 'the first million numbers have property p, so all numbers have property p'. In both cases we're accepting a simple general principle on the basis of seeing that it holds true in finitely many cases.

Thus I think that if there's any sense in which idealized human mathematical reasoning transcends the limits imposed by incompleteness, it's not because we have a specific faculty of reflection.

Instead, it's because of something much less glamorous: because we're willing to apply not-always-truth-preserving methods like scientific induction to mathematics, and hence disposed to accept certain claims that are actually inconsistent.

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