Thursday, July 19, 2007

Geometry and intuition

Here is a story I've heard from mathematicians in a few different contexts. I will quote it from Stilwell's text Four Pillars of Geometry.
"The first theorems of hyperbolic geometry were derived by the Italian Jesuit Girolamo Saccheri in an attempt to prove the parallel axiom. In his 1773 book, Euclides ab omni naevo vindicatus (Euclid cleared of every flaw), Saccheri assumed the non-Euclidean parallel hypothesis, and sought a contradiction. What he found were asymptotic lines: lines that do not meet but approach each other arbitrarily closely. This discovery was curious, and more curious at infinity, where Saccheri claimed the asymptotic lines would meet and have a common perpendicular. Finding this 'repugnant to the nature of a straight line,' he declared victory for Euclid."
I've heard this story a few times, and each time it was to illustrate the importance of deriving a contradiction in a reductio, not just stopping when things get unintuitive or "repugnant". I haven't seen the original book, so I don't know if Saccheri says that his conclusions contradict the nature of a straight line, in which case he would have misunderstood what was happening to the notion of line under the new assumption. However, if he thinks that the repugnance is enough for a reductio, he is wrong.
(Apparently there is a lot of literature on intuitions. I've read none of it. It could very well discuss this story in excruciating detail.)

It seems, just from my own reading, that the charge of having unintuitive consequences is often leveled against various theses. I don't have a nice example off-hand, but I'm doubtful that it will be terribly hard to track one down. (I think I used the phrase in the paper I read at UT Austin, but I am pretty sure I have seen real philosophers use it.) It is never clear to me what the force of that charge is. If it is meant in the sense that Saccheri supposedly used it, then that seems to be bad. If it is meant ad hominem, then it is fairly easy to ignore. There are certainly at least some areas of philosophy, logic at least, where the charge of unintuitive consequences is fairly meaningless. Are there any areas in which saying that something has unintuitive consequences is particularly damning? There probably are, but I'm not sure what exactly.

6 comments:

Justin said...

I typed a long post, then got booted from the net as I tried to submit it.

A good philosophical example is mereology, where people will take the nihilist or universalist positions, then say things like "If that's true, then I don't exist. Absurd!" or "if that's true, then there's something made up of Cal Ripken, my ties, and moon dust. Absurd!"

As for philosophical literature on intuitions, it's beset by the trouble that people often either are unspecific which sort of intuitions they're discussing or (even worse) assuming that intuitions are a natural kind, which can be discussed without making any further distinctions. I can probably deliver you reasonable citations if you'd like.

Finally, you're still linking to my old blog. That should be fixed.

Shen-yi said...

i've always thought of it like this: in a lot of purely analytic areas, intuitions are the data we have to test our models. of course the model could still work, but faced with unintuitive consequences, one at least has the burden of why it is acceptable.

and sometimes the explanation is simple, like in logic. i don't think it's true that unintuitiveness is not a charge leveled in that domain. intuitions have different strengths; some are more epistemically secure than others. if a theory is built on pretty secure intuitions, and works well until higher-level, when it runs up against less-secure intuitions, then one might just say the theory rules in this case. i think that's what happens in logic.

Shawn said...

Justin,
I thought I had changed the URL right after you switched to wordpress. I must have not hit save or something... The link has been fixed though.

Shawn said...

I'm doubtful that intuitions play any sort of dialectical role in logic. There might be some instances from the history of it that prove it wrong. Intuitions certainly lead to new ideas, like the unintuitiveness of material implication leading Belnap and Anderson to relevance logic. But, that doesn't falsify classical logic in any sense. It is certainly motivational, but I feel like the intuitions play a more central dialectical role in, e.g., mereology. Justin or other more knowledgeable people may want to correct me here. The little story about the parallel axiom is an instance of using intuitions as argument and coming up short. If that characterization is right, then there are some things from Mark Wilson's Wandering Significance that are relevant that I'll try to write up in the near future.

Shen-yi said...

"Intuitions certainly lead to new ideas, like the unintuitiveness of material implication leading Belnap and Anderson to relevance logic. But, that doesn't falsify classical logic in any sense."

is that true? doesn't it falsify classical logic AS a way to capture our inference patterns (or a part of our epistemic norms or something like that)? similarly, you might think that the challenges in mereologies doesn't falsify, say, four-dimensionalism as a formal system; but it falsifies four-dimensionalism AS a way to capture what the world is like.

Shawn said...

You are probably right that that sort of unintuitiveness charge would disqualify some classical inferences as capturing our inference patterns. I'm hesitant to say that formal logic is in the business of accurately portraying human inferential practice. I think old Aristotelian logic was in the business of trying to classify which of the inferences people regularly engaged were good and which were bad, but there is a bit of a disconnect between that project and more contemporary aims.

"similarly, you might think that the challenges in mereologies doesn't falsify, say, four-dimensionalism as a formal system; but it falsifies four-dimensionalism AS a way to capture what the world is like."
I don't understand what makes that transition okay. Our mereologists were arguing using intuitions, and then they draw conclusions about the nature of the world. That seems a little sketchy, but I know next to no mereology.