Friday, January 22, 2010


Have you seen this article by Willfred Hodges about reading people's "refutations" of Cantor's diagonal argument?

It's pretty funny, and he even raises some interesting philosophical issues about logic, how to think about logical mistakes, etc.

Fallacious but psychologically attractive inferences

I like to start with really simple theories and see where they go wrong. Recently this lead to an interesting combination of experiences

When I say:

People are justified in making those a priori inferences which are both necessarily truth preserving and psychologically compelling for normal humans.

People say: But reasoners do have initial justification for accepting certain attractive but ultimately fallacious arguments e.g. tricky arguments for the existence of God.

But when I take out the requirement of being genuinely truth-preserving and say...

People are justified in making those a priori inferences which are psychologically compelling for normal humans.

People say: But what about those bad but psychologically compelling inferences like inferring the consequent?

So which is it (do you think)?

When someone makes a psychologically compelling but invalid inference like the gambler's fallacy or inferences about naive set theory are they:
a) justified, (though presumably thinking about the right questions may later give them justification for changing their mind) or
b) unjustified
c) somehow there's a difference between the gambler's fallacy and naive set theory in this regard

I don't have a dog in this race, or think anything deep is going on here, but I'd really like to know which way normal language intuitions go.

Wednesday, January 20, 2010

Speculation re: "Faculty of Reflection"

This may just be totally ad hominem speculation (though there's no particular person I have in mind), but...

Maybe the reason why some are inclined to think we should always be able (in principle) to reflect and formulate a system that captures any recursively reasoning we do about math and logic has to do with a certain view about reasoning + rule following.

Remember how Wittgenstein (in the Blue and Brown books) criticizes this theory that we manage to obey the command "bring me a red ball" by first imagining a red patch to get an idea of what color "red" is, and then picking the color that matches this imaginary patch. He says: do you first need to imagine another red patch, in order to know what color to make your sample patch? So, this theory leads to a kind of regress. [I'm always tempted to call it the smoke-two-joints-before-you-smoke-two-joints theory of understanding language, after Bob Marely's famous song about regress]

Presumably everyone will agree that in this case we need to just posit (and perhaps scientifically explain) a direct ability to pick red things, when commanded to. Invoking a further layer of person-level thought (where you pick the red balls by first doing something else like imagining a red patch) just leads to regress.

But, an analogous theory with regard to mathematical reasoning would be that when we are asked to answer some mathematical question, what we do is first consider certain rules for how to reason about mathematics, and then do what these rules say. Now, I think this is a very bad theory. But, if you accepted it, you might think there could well be a special process of reflection where you, in effect, remember these rules, or become consciously aware of the rules you were unconsciously appealing to all along. That is, you might think: any recursively enumerable portion of mathematical reasoning you accept, you should be able to formalize (and recognize to be correct) by making explicit all of the (presumably finite number of) rules that you implicitly consulted when doing that reasoning.

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.

Doubting Conceptual Truths

Some things, some Kantians say about the justification for logic suggest the following superficially attractive idea:

a) Certain sentences have the property that anyone who can think about them must thereby be inclined to accept them (e.g. you can't even think thoughts involving `and' if you aren't willing to accept 'If it's raining and it's snowing then its raining')

b) We are justified in accepting such sentences, because we have no other option.

But actually, its plausible that we can reasonably doubt many sentences with this property. This is because sometimes you can turn out to have been working with an 'incoherent concept' like tonk, or bosh, the naive concept of set or perhaps various philosophical concepts. In such a case, you don't count as thinking with the concept unless you are willing to make certain (bad) inferences.

Now, you might argue that someone who was taken in by this kind of incoherent concept doesn't count as thinking anyway (e.g. there's no proposition which ``it's raining tonk its snowing" expresses). So maybe you only count as *thinking* in the good case, where it turns out that your concepts are coherent. But, given that we know that very smart and conscientious people can wind up with bad concepts, it intuitively seems reasonable to not completely dismiss the possibility that various new concepts you are learning are among the bad ones.

Hence*, it would seem that, we can rationally doubt claims which it would not be possible to deny. (what we're concerned about here is not the possibility that the claim is false - which we can't entertain- but that one of the concepts figuring in it is incoherent, so the claim is nonsense)

*if a) is true

Saturday, January 9, 2010

Frege and Obviousness vs. Self-Evidence

In a recent article Shapiro writes"There are obvious propositions, such as 2 + 3 = 5, that are not self-evident. Frege emphasized that to know sums like those, one need not invoke any intuition, Kantian or otherwise, but he insisted that one must reason one’s way to this knowledge."

But in what sense did anyone before Frege "reason their way" to the claim that 2+3=5? From what? The quoted claim seems to have the consequence that either:
a) no one knew any arithmetic before Frege's derivation of arithmetic from logic
b) people before Frege could count as knowing arithmetic because they were unconsciously going through Frege's derivation

Saturday, January 2, 2010

A Sketchy Aesthetics

Currently physics background for another project is devouring most of my mind, but I had this kindof wacky idea for a big-picture theory of aesthetics.

Suppose we say: Naive intuitions about beauty correspond to a folk theory whereby people naturally would all like the same things, except for certain deceptive conditions like:

  • only liking something because you were told it's a great work of art
  • not liking something which you would otherwise like because you've seen
  • it so often
  • liking something only through ignorance of some of its descriptive
  • properties
  • liking something which you would easily get sick of etc.

These are platitudes about which causal influences on aesthetic judgement
are "misleading".

But now, pull a Kantian revolution, and say that it's not that these traits are misleading because they lead us to fail to track some antecedently natural and interesting category of the beautiful, but rather that `beautiful' refers to (roughly) "the kind of thing that people would like if not for the conditions indicated in the platitudes."

So far this sounds like familiar response-dependence theories. BUT there's one more wrinkle.

There might be enough natural variation between people for there to be no interesting class of objects which everyone is (ceterus paribus) disposed to like, when free from the deceptive conditions indicated by the platitudes above. It's a psychological matter whether getting more ideal in these respects would lead to convergence or divergence in aesthetic judgements.

So here's the complete view I want to propose (which, hopefully, handles this objection):

If there is a fairly definite class of things people (in our linguistic community) are inclined to like when free from the platiduinous bad influences then `beautiful' rigidly designates that class of objects.

If there is no such class, `beautiful' is implicitly speaker relative. In the latter case, we can think of arguments about beauty as involving the implicit assumption between both participants in the argument that their dispositions to like or dislike that objects would be the same if free from platitudinous bad influences.