Fake insights

Fire walk without me, thanks

Today I was helping a friend with a math homework problem. I spent half an hour trying to solve it several different ways, and got nowhere. Every approach generated pages of ugly equations.

“This problem is really hard,” I said, as I gave up. “You should probably just skip it and go on. They’ve made it artificially difficult. I mean, just look at the problem statement—that’s a massively complicated formula!”

Right then, I had a flash of insight—an aha moment! One of the messy formulae could be reinterpreted as an equation that made a substitution possible—and once you did that, the whole problem collapsed in on itself. Various terms cancelled each other, and the solution popped out in a few lines.

This felt good—as flashes of insight always do—and it rescued my reputation as someone who once could do basic mathematics. However:

It was a fake insight.

It was fake because the problem had been set up to be artificially easy, not artificially difficult. Most similar problems could not be solved the same way. It was possible only because exact details made that substitution possible.

My friend and I had been trying to solve it using the general methods that should work for any problem of that class. I think we were right to do so. In the real world, hard problems rarely collapse into trivial ones because constants happen to fit together.

When I was a math major, the hotshots in the department spent most of their time preparing for the Putnam Exam, a national undergraduate math competition. As far as I could tell, most of the problems on the exam were of this sort: impossibly difficult if approached using a general method, but trivial if you noticed some artificial feature that made a special-purpose method applicable.

I wanted no part of that. It is bad pedagogy, I think. It reduces math to puzzle-solving: a fancy version of Sudoku. The process is quite unlike mathematical research, which is open-ended and creative, and (in my limited experience) relies on a quite different mind-set.1

It seems that this pattern applies in other areas of life.

Some personal development seminars are carefully crafted to produce fake insights of this sort. They deliver an aha moment that feels really good at the time and makes you think you accomplished something significant. You believe then that the insight will totally transform your life—but it doesn’t. The actual scope of the insight was quite limited—perhaps even only to the context of the seminar.

An extreme example is the fire-walking trick. The guru explains that, if you just are sufficiently confident in yourself, and repeat this magical mantra, you can accomplish anything—even walking on coals that he shows objectively measure at a thousand degrees! And, if you are sufficiently confident, you try it—and you do it! But this has nothing to with confidence or the magical mantra, and everything to do with thermal conductivity and specific heat capacity.

In less extreme examples, the insights are accurate—like the solution to today’s math problem—but too narrow to have much value.

And, often, they are set up by the instructor rather than discovered by the student. Only the last step in a chain of reasoning is missing. You do get the “aha!” from finding it—but you don’t learn the tools that would enable you to find the whole chain yourself.

Many of the insights generated in psychotherapy also seem to fit these patterns. It may be true that something your mother said when you were seven affects how you view relationships today—but does recognizing this lead to significant changes in your actions next week? Less often than one would hope. You will need a great many insights like that—and a way of generating them yourself.2

To contradict everything I’ve said so far…

Recognizing special cases, when general methods fail, is an important meta-rational skill. Often it does lead to breakthroughs. Persisting with a general method when it’s clearly failing is characteristic of organizational dysfunction (as well as scientific stagnation).

So, what makes an insight “fake”?

This may addict you to getting fed fake insights by teachers who refuse to reveal their tricks.

That is common both in spiritual circles and in mathematics education!

  1. 1.To be fair, many Putnam winners have gone on to do significant mathematical research. I should also say that my impression of the Putnam problems was pretty superficial, and may have been inaccurate.
  2. 2.And, to be fair, some psychotherapeutic methods do teach you to do that.