Thursday, March 20, 2008

Two kinds of mathematical intuitions

Mathematicians have two kinds of intuition. A speculative intuition occurs when they think about a problem, perhaps think quite a lot, and conclude that the problem has answer A, even though they how no idea how to prove this. "It just looks like A is the answer." I do not know how reliable speculative mathematical intuitions are. I suspect that they are not very reliable. In particular, I think they rarely if ever justify belief. Certainly, I did not acquire belief in an answer on the basis of speculative intuitions when I was a practicing mathematician.

However, there is also such a thing as pedestrian intuition. This tells the mathematician: "Clearly, p." The "clearly" is not speculative. The content of the intuition is not just that p is true but that p can be easily proved from what preceded. John Fournier, my mathematics thesis director, once gave me the following advice on papers submitted for publication: when there are two obvious steps in a row in a proof, you can omit one, but not both.[note 1] When a mathematician sees that something follows, even if she does not actually go through the proof of the fact that it follows, that pedestrian intuition is, I think, very reliable. It may even be that had the mathematician written down the proof, the proof would have contained some minor mistakes. For this intuition does not seem to be based on having the proof in one's mind. Rather, it seems to be a direct non-inferential grasp of the easy provability of p.

One small piece of evidence for the reliability of pedestrian intuition is the incredible reliability of mathematical publications. Errata are extremely rare in mathematical journals.[note 2] I suspect this is not just because of the refereeing process, but because this highly reliable intuition was guiding the mathematician in writing the proof. In fact, I think the epistemic weight of the result proved in a mathematics paper goes beyond the validity of the published proof. The published proof may indeed contain a minor slip here or there. But what makes these slips be minor is precisely that one can intuitively see what should be in their place. My last mathematics paper was published when I was significantly out of practice. It went back and forth between me and the referee several times, and the referee was rightly exasperated by the amount of mistakes in the proofs. However, all the mistakes were easily fixable: the intuition was exactly right, in a pedestrian way, despite the logical gaps in the proofs.

This is surprising. One might think that a proof with logical holes has no value at all—it is like tracing your ancestry to Charlesmagne with only two gaps in the chain (this isn't my comparison). But somehow the reliability of the pedestrian intuition goes beyond the proof written down.

What explains the extremely high reliability of pedestrian intuition in a well-trained mathematician? One possibility is that it is a highly developed pattern-matching skill. In the past we've seen p-type claims following from q-type claims, and we can see that the present case fits into the pattern, and so p follows from q. This explanation fits well with the fact that experience seems important for this kind of intuition. But I am not sure this would be sufficient to give the intuition the kind of reliability it has. Pattern-matching would, I doubt, have the right kind of reliability. In typical cases of writing down a proof of a new result, the case at hand is unlikely to be exactly like past cases.

Or could it be that there is a process involving a mental representation of a proof, but a representation not directly available to consciousness? If so, what is interesting is that this is just as reliable as, or even more reliable than, consciously going through the steps of a proof (in fact, I suspect that the reliability of consciously going through the steps often or always depends on the non-conscious process occuring side-by-side). This is kind of neat and reminds me of the speculations central to Peter Watts' novel Blindsight. Moreover, if this is right, then I think it should challenge internalist epistemologies that require justifications to be conscious. In these mathematical cases, the justification can be made conscious, but the making-conscious does not seem central, since the non-conscious reasoning is more reliable than the conscious reasoning.

It is an interesting question how the two kinds of mathematical intuition connect up with kinds of philosophical intuition. I do find myself with a quite reliable intuition in philosophy akin to the pedestrian sort of mathematical intuition—an intuition as to what conclusions can be made to follow from what kinds of assumptions. In fact, this is probably just the same intuition at work, though I find it is a bit less reliable in philosophy than in mathematics. (I think I have at least three times been significantly deceived by such an intuition, and in a number of other cases have needed to add plausible ancillary assumptions to make an argument go—though on reflection that probably can happen in mathematical cases, too, which slightly weakens what I said in previous paragraphs.)

There is, however, a second kind of intuition: an intuition that pointless torture is wrong, or that we are not identical with our left big toes, or that identity is non-relative, or that the good is to be pursued and the bad avoided, that nothing can be causally prior to itself, or that every contingent truth has an explanation. I am inclined to class this intuition as different from both the pedestrian and the speculative mathematical intuitions. This intuition is of variable strength, unlike pedestrian mathematical intuition which is pretty uniformly very strong. Sometimes, this kind of philosophical intuition gives us certainty, as in the case of the good being to be pursued, and sometimes it merely inclines us in favor of a proposition. The range of strengths here makes it different from speculative mathematical intuition which, I think, never justifies belief, while this kind of philosophical intuition does justify belief.

Or maybe we need to split this second kind of philosophical intuition into two kinds. One kind is speculative, and this is akin to speculative mathematical intuition. I am, let us suppose, inclined to think electrons are not conscious, but this intuition is not sufficient to compel or justify belief. Another kind is self-evidential which presses belief on us, and I suspect justifies it as well. This kind is more like the highly reliable pedestrian mathematical intuition in respect of the way it compels belief (the reliability question is a different matter on which I want to remain silent), but is unlike the mathematical case in that it is substantive and not merely logical in nature.


Paul M. Rodriguez said...

What do you mean by belief in this connection? Do you mean a kind of belief which is available only to mathematicians, and only subsequent to proof?

Otherwise, I think that there are thing which many mathematicians believe, in the common sense, without being able to prove. For example: I gather that most mathematicians believe that P is not equal to NP, though that cannot now be proved.

Alexander R Pruss said...

This was a bit of an autobiographical post. I suspect that P is not equal to NP, but I don't believe it. :-)

I guess I take to heart Aristotle's remark that we shouldn't accept plausibilistic arguments in mathematics.

But there may well be mathematicians who are less fastidious, and maybe they are right to be less fastidious.

Actually, that P not equal to NP may be a yet third kind of mathematical intuition, something like a strong philosophical seeming...

Anonymous said...

If they were less fastidious, would it amount to something like "having faith" in some theory or mode of analysis? For instance, thinking that "categorification" or n-category theory lends deep insights into the nature of topology, physics, etc., clearly requires some faith at this stage, especially if one decides to take it on board as one's primary research project...

Alexander R Pruss said...

That seems to me as more of a metamathematical intuition--an intuition that a line of investigation would be insightful mathematics. What do you think?