Monday, August 25, 2008

An intuitive thinker

One of the students at the math camp is very clearly an intuitive thinker. I'm not exactly sure how I know this; I just know. Listening to him speak--and he speaks a lot--is like listening to a stream of consciousness. A lot of it is nonsense. As if he is thinking out loud, not really expecting anyone to understand.

But more tellingly is the way he solves math puzzles. It is so very hard to describe, especially to those who haven't had much experience with math or puzzles themselves. I will try.

So some of our lectures were about number theory. The following warm-up problem was given:

Prove that any common divisor of x and y is also a divisor of the greatest common divisor of x and y.

If you're like me, you first wonder why this needs proof. It's so obvious. At least to me. It has to do with the prime factorization, and the gcd function, and the... but darn it, this is no proof! Such is the problem with intuitive thought. You can look at many problems and think they're so obvious, but that doesn't always help you with proof. Who knows what underlying assumptions our intuition makes. It could be using circular reasoning, for all we know. Sometimes a flash of intuition lights the way, but other times it's so bright that it blinds us.

The above question is an example of very basic number theory. This kind of question gives even non-intuitive thinkers problems, because it asks us to prove something we are so used to assuming. For a highly intuitive thinker, this occurs all the time, even for math that is less basic. Everything seems so obvious, but it is difficult to say why.

And so it was, intuition was often an obstacle for this student. And sometimes, his intuition not only prevented him from finding a proper proof, but also gave him the wrong final answer. Such is the peril of intuition; it cannot always be trusted.

Not to say that the student was the worse because of it. He was a star student, one of the youngest and brightest.

I, too am a bit of an intuitive thinker, and I intend to defend intuition, even praise it.

Intuition is not some innate knowledge, gained at birth. It's not encoded into our ancient genes. At least, not always (I think phobias are pretty amazing myself). Carl Jung would have you believe that intuition is opposed to sensation, that intuition comes from within and sensation comes from without. But I don't put much stock into the MBTI, so I don't see why "within" and "without" should be necessarily opposed. I think that intuition is a skill that can be acquired from the outside. It can practiced and improved. It can be a learning tool, a way of internalizing new knowledge.

If I'm learning something, say special relativity, I find it useful to visualize. As we boost spacetime, the axes scissor together such that the determinant is conserved. If we go near the speed of light, objects will flatten. Distances will shorten. Clocks will slow down, but the plane of simultaneity tilts as you accelerate. Sorry if none of this makes any sense--but this is how I do it. I can easily visualize the resolution to the twin paradox, as if there were no paradox at all.

I use special relativity as an example, because it's so often thought to be counterintuitive. But it really can be intuitive, if you work on it hard enough. I worked on it for a long time myself, looking at relativity from all sorts of viewpoints. I try to do the same for other "counterintuitive" concepts (quantum mechanics, anyone?). It's possible, but it takes work. Intuition doesn't start out as a reliable way of knowing, after all. (Spiders aren't that dangerous.)

Equally importantly, you have to work on other skills to support intuition. You must be able to translate intuition into reasoning. You must be able to break down your intuition into its constituent parts, and examine them if needed. Without these supporting skills, intuition is impossible to verify. Some intuition is bad, and some is good. If we have the skills to distinguish between the two, it becomes a powerful tool.


Yoo said...

Hear hear! I consider intuition to be something we get out of the internal programming of our brains (being programmed either innately or by repetition), so it's a great shortcut, but I often want to augment it by rigorous justifications so I can consciously see why the result is justified (not to mention intuition being wrong all too often).

Larry Hamelin said...

Prove that any common divisor of x and y is a divisor of the greatest common divisor of a and b.

Am I missing some jargon? What relationship do x and y have to a and b?

miller said...

Oh, my mistake! I switched from x and y to a and b midsentence. It's fixed now.

Yoo said...

That's a relief. I was secretly wondering if something was wrong with my intuition. ;)

Larry Hamelin said...
This comment has been removed by the author.
Larry Hamelin said...

OK. I haven't done math for a very long time, but the truth of the statement seems intuitive, and the proof seems obvious.

Larry Hamelin said...

During my all-too-brief stint in college, I overheard one professor remarking to the other that he simply could not relate to a student who didn't find it intuitively obvious that the square numbers were the the sum of consecutive odd numbers starting from 1.

Since I found the statement profoundly counter-intuitive and the proof refractory, I was dissuaded from a career in mathematics.

Of course, once I saw the geometric proof, the whole thing gelled in a heartbeat.

Anonymous said...

I've found that what mathematicians find intuitively obvious and what their students do, are not one and the same. It leads me to think that intuition is learned - or rather you learn a pattern of thinking that supports intuitive understanding of subject