Where Creativity Comes From

When I was a freshman studying math at Princeton, I spent lots of time at the tutoring center with my problem sets, groveling at the feet of very smart upperclassmen math majors. Math is the major that by far is dubbed the most difficult at Princeton (as much as I hate to admit it, since Computer Science majors usually have some weird inferiority complex when it comes to math), but in my experience, the math majors really were some of the smartest kids on campus. But in particular, they were the most creative, where:

Creativity is the phenomenon whereby two previously unrelated concepts are united in a useful way.

This is a particular flavor of creativity, one which is exemplified well through Einstein’s discovery of special relativity. Einstein came upon the kernel of his theory of special relativity when he was sixteen, and asked himself what he would see if he chased a beam of light at the speed of light itself. Knowledge of the time would say that to an observer moving at the speed of light, light itself should appear stationary, but this conclusion was inconsistent with observation. So Einstein’s thought experiment, which employed only rudimentary physics, led him square into a paradox. What makes this so creative is that all of Einstein’s contemporaries had enough information to come to the same conclusion, but none of them did. They were all obliviously living a lie, sporting inconsistent beliefs about reality without ever realizing it. So Einstein was able to uniquely see a paradoxical relationship between two seemingly irrelevant concepts.

But back to the math majors. They impressed me because I would ask them questions they’d never seen before that appeared totally inscrutable to me, and they would solve them instantly by understanding how my problems were just questions about simple principles in disguise. They would take a math problem I had and recognize that the problem could be rephrased in such a way that the answer became obvious . They were creative because they took seemingly disparate concepts and understood how they were one in the same. When difficult but solvable problems exist in the world, they are solved in one of two ways: by the discovery of new information (from particle colliders, biology labs, etc), or by some creative human realizing that a solution exists that hasn’t yet been recognized as such. Since there is no such thing as observational mathematics, all discoveries in math must be exclusively of the second kind, the creative kind. So we can learn a lot about creative thinking by studying clever math majors.

One trademark of math students is that they tend to think very abstractly, i.e. about generalizations or concepts rather than about particular problems or ideas, which is necessary since higher math itself is inherently abstract. But the difference I found between the good and great math majors was that great math majors were the best at taking mathematical ideas and distilling them into their most essential forms, which is to say, they were the best at abstracting.

But what does abstraction have to do with creativity? Imagine the creative engines of our brains, mashing together random concepts in our subconscious minds. We experience a creative flash of insight when our brains throw together relevant concepts and realize the results are new and useful. Now the mind of an average person will contain lots of little bits of random information sorted and stored in no particular way. But the brain of a highly abstract person will contain less information, because abstraction is a method of information compression. For example, my brain doesn’t understand the concept of a chair by storing snapshots of every picture of a chair I’ve ever seen in my life — it just stores a single piece of information under the “definition of a chair” label, which is that a chair is a platform designed for human sitting. An abstracting brain takes data points and tolerates a degree of loss of detail in exchange for a highly efficient encoding of information — it does a lossy compression. But as a result, when the mind of an abstract person mashes together concepts, it has many fewer concepts to sort through. The search space of the brain of an abstract thinker is more compact than the search space of just any brain, while still containing almost all of the important information. A great math major doesn’t ask herself whether a particular mathematical method bears on a problem, she asks whether an entire class of methods does.

Additionally, the mind of abstract thinkers are in theory highly organized. When my math major tutor reads my Analysis problem set problems, he (and it always was a he, sadly…), takes my problems and maps them into abstract concept space. Using the abstract version of my problem, he finds other problems whose abstracted versions are the same, paying little regard to the fact that one problem might have come from a Probability class or a Geometry class or a Philosophy of Nietzsche class. Said again, abstraction naturally provides hierarchical structure to information. As a result, abstract thinkers are more likely to see hidden connections amongst ideas because the very way by which they store information forces like concepts together.

So math majors are abstract thinkers, abstraction illuminates hidden connections between data, and this is what we call creativity. Nice.

But there is some sort of a difference between having a creative way of thinking and actually making creative discoveries. I might be a great, creative problem solver, but that doesn’t necessarily mean that I’ll make great discoveries. I might pick the wrong problems to work on, or not pick enough problems, or I might simply be unlucky. But if we were to look at the greatest scientists of the past century, I think we would find something beyond good problem solving that separates the goods from the greats, and that something is a sort of obsession or tenacity with respect to important problems. The famous Computer Scientist/Mathematican Richard Hamming said it best in his talk You and Your Research:

Most great scientists know many important problems. They have something between 10 and 20 important problems for which they are looking for an attack. And when they see a new idea come up, one hears them say “Well that bears on this problem.’’ They drop all the other things and get after it.

That is, it’s not enough to be good at solving problems. We must also constantly have on hand a set of important problems that we tenaciously care about. Scientists who are deeply obsessed with a set of problems will find themselves daily in contact with new ideas unrelated to their own problem set, but they will jam every conceptual key they learn of into their set of problems until one clicks. And this paired with abstract thinking allows great scientists to better triage this onslaught of incoming information. So where discovery-making is concerned, it is the combination of creative thinking and constant, dogged commitment to problems that makes for the best discoverers.

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