Building a Different Box of Tools

  Experimental hairy ball

The hardest part of learning is our desire to be curious about something, our ability to think through what we find, and then engage with the data to make it ours. Even when we have access to the infinite repository of the Internet, the actual reading and processing takes time and effort. That is if we want to form our own opinion about something, or take things further.

We know we're onto something when we read actively, looking things up in context, referencing material outside our discipline. Admitting, I don't know, and looking it up or investigating a question, a word, an expression, technical jargon, takes us further than nodding and pretending we know.

A story in Surely You're Joking, Mr. Feynman! illustrates this well. The story comes directly from theoretical physicist Richard Feynman and is part of a collection of recorded conversations between Feynman and Ralph Leighton over years.

A different box of tools

It starts with a typical university environment, if not more posh than the regular campus, at Princeton University, where Feynman made a home for a a few years. The graduate school had the physics and mathematics departments share a lounge where tea at 4 pm was an opportunity to play games like Go, or discuss theorems.

Feynman says topology was the big thing at the time.

In mathematics, topology is the study of the properties of geometric figures or solids that are not changed by homeomorphisms, such as stretching or bending. Donuts and picture frames are topologically equivalent, for example.

or from the Princeton University WordNet:

the branch of pure mathematics that deals only with the properties of a figure X that hold for every figure into which X can be transformed with a one-to-one correspondence that is continuous in both directions.

As with different groups when they come together, there is some kind of playful rivalry, each pulling for its own domain and body of knowledge. For example, physicists laughed at the abundance of “it's trivial” expressions to describe when something had been proven. Occasionally laughing became “mathematicians can prove things that are obvious” teasing.

Which led to a dare of sorts:

Topology was not at all obvious to the mathematicians. There were all kinds of possibilities that were “counterintuitive.” Then I got an idea. I challenged them: “I bet there isn't a single theorem that you can tell me — what the assumptions are and what the theorem is in terms I can understand — where I can't tell you right away whether it's true or false.”

It often went like this: They would explain to me, “You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?”

“No holes?”

“No holes.”

“Impossible! There ain't no such a thing.”

“Ha! We got him! Everybody gather around! It's So-and-so's theorem of immesurable measure!”

Just when they think they got me, I remind them, “But you said an orange! You can't cut the orange peel any thinner than the atoms.”

“But we have the condition of continuity. We can keep on cutting!”

“No, you said an orange, so I assumed that you meant a 'real orange.'”

So I always won. If I guessed it right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.

Elaborating further about his guesses, Feynman gets to the system he uses to think through what someone is explaining something he's trying to understand. It goes like this:

I had a scheme, which I still use today when somebody is explaining something that I'm trying to understand: I keep making up examples.

For instance, the mathematicians would come up with a terrific theorem, and they're all excited. As they're telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball)—disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as I put more conditions on.

Finally they state the theorem, which is some dumb thing about the balls which isn't true for my hairy green ball thing, so I say, “False!”

If it's true, they get all excited, and I let them go on for a while. Then I point out my counterexample.

The reason why Feynman guessed right most  of the time was that the mathematicians thought their topology theorems were counterintuitive. They weren't as difficult to figure out. By getting used to the “ultra-fine cutting business” he did a pretty good job of guessing.

But there was one thing he did not learn — countour integration, a method of complex analysis. It wasn't in the books. His physics teacher, seeing him restless “you talk too  much and make too much noise,” he said, and thinking boredom was the cause, lent him a book that had the answer. The book demonstrated how to differentiate parameters under the integral sign#.

This gave Feynman a leg up on trying it out and having it work most of the time:

So I got a great reputation for doing integrals, only because my box of tools was different than everyone else's, and they had tried all their tools on it before giving the problem to me.

When we broaden our box of tools by building our knowledge in multiple domains and disciplines, we learn to go beyond the obvious answer to the more nuanced understanding and increase our ability to be effective at solving problems and finding opportunities. We can start by being curious about what we don't know well or at all and engaging in the “process of discovery.” 

Feynman's many interests and the lengths to which he went to pursue learning, including testing whether a  gifted person is gifted in anything, as the saying goes, demonstrate how seriously he took his responsibility as a scientist and as a human being.