Moffett On An Interesting Thought Experiment

On Close Range, Marc Moffett sketches the case of a mathematician who, only by stretching the limits of his abilities of ratiocination and concentration, can comprehend a very complex proof. At t1 he does so, and thus believes the conclusion of the proof to be true. At t2 his concentration flags a bit, and he thus can no longer see how the proof supports its conclusion. At t3 he drinks an espresso, and once again sees how it is that the proof supports the conclusion. And so on, until tn, when, regardless of whether his powers of concentration are up to the task of following the proof, he remembers enough of his earlier moments of concentrated lucidity to be confident of the proof -- even when he doesn't engage in the cognitive calisthenics that would be required to follow out the way in which the proof supports its conclusion.

Moffett suggests that there are three ways to understand when it is that the mathematician knows the conclusion of the proof. (i) He knows the proof at t1, t3, etc., but not at t2, t4, etc. Moffett finds this option problematic. (ii) He doesn't know until tn, because he isn't justified until tn. Moffett -- justifiedly -- finds this option problematic as well. (iii) He has justified true beliefs at t1, t3, etc., but doesn't know until tn. Moffett likes this option, and thinks that it implies that "the JTB analysis can fail, not only because of instability in justification, but also because of instability in belief".

First, a minor point. If Moffett thinks that option (iii) involves "instabilities" of justification or of belief, isn't he in fact denying that the mathematician has justified true beliefs at t1, t3, etc. -- either because, at those times, he doesn't in fact believe or because he isn't in fact justified? If this isn't what "instability" means, then it would be nice to know what it does mean.

My major goal here is to provide a bit more support for option (i) -- the mathematician knows at t1, t3, etc. Consider the following case. Bill has extreme short term memory loss and attention deficit disorder. This unusual combination means that Bill's attention flits from place to place within his visual field, and, once he attends to something new, Bill forgets what it was he was attending to a moment before. Bill's eyesight, however, is perfectly good; we can even imagine (if it'll help) that Bill knows his eyesight is very good -- remember, Bill's long-term memory is fine. Suppose that, at t1, Bill attends to a white spot on a black background in the extreme left of his visual field. At t2, he turns to the right, so that he is no longer attending to the white spot, and thus forgets all about it. At t3, Bill attends again to the white spot. And so on. It seems to me that Bill knows there's a white spot there at t1, t3, etc., and doesn't know at t2, t4, etc.

So the problem with option (i) isn't that such situations are impossible. If there is a problem, it's going to have to be a problem having something to do with mathematical knowledge or ratiocination. I don't think I see what it could be, however. If, as Williamson argues to great effect, there can be unsafe knowledge, I don't see why there can't be unstable knowledge -- even extremely unstable knowledge, as the case of Bill (and, I would suggest, the mathematician) attests.

Comments

[marc]

Hi Joe--

Thanks for the comments. I've tried, perhaps somewhat Quixotically, to defend the claim that knowledge requires stability--though considerations such as the ones you raise seem to require some fancy footwork. I've sketched, all too vaguely, how this might go (http://rationalhunter.typepad.com/close_range/). I suspect that if Schaffer's right about knowledge-wh, then something along the lines I've suggested is defensible.

[j.s.]

Thanks for the response, Marc. I've put an initial reaction to your response in a comment on your blog.

[Matt Weiner]

On the minor point: I thought that instability in justification or belief means that at t1, t3, etc., Kurt is justified and believes; but those conditions are unstable because he's close to not believing or not being justified. Is that the kind of answer you're looking for?

[j.s.]

Thanks, Matt. Yes, having seen Marc's response to me, I agree with you that that's what he means with "instability". I was prevented from seeing this reading due to the fact that it seemed (and still seems) obvious to me that being close to not believing or not being justified is not an obstacle to knowing.