The Hidden Art of Resolving Paradoxes

Anyone who’s read Atlas Shrugged surely remembers the recurring motif:

“Contradictions do not exist. Whenever you think that you are facing a contradiction, check your premises. You will find that one of them is wrong.”

This is true. The universe, ultimately, compiles. Everything must come together into a coherent whole, and perhaps one way to look at science is as a project that seeks to resolve contradictions. The Douglas Adams cut on this is that the universe is the answer, and our task is to figure out what are the right questions to ask, in order to unlock its mysteries.

If you find that you can’t resolve a contradiction, it means that your frame of reference is too narrow. You need to pop out of the system in order to see it in its entirety. A certain metaphor for this notion really stuck with me, so let me reiterate it here...

Imagine that you exist in flatland, and your whole world is a two-dimensional plane. How would you visualize a cylinder — a strange kind of object that has no place in the world in which you live? Would you stand it upright and say that it’s a circle? Or would you lay it flat and declare it a rectangle? Maybe some kind of two-dimensional hybrid of the two?

Clearly not. What you actually need to do here is step out of your 2D frame of reference. Welcome to the third dimension. Here, you find that you’re able to reconcile the facts that you know about this paradoxical object of yours. It’s both a circle and a rectangle, fit together in a way only possible if you’re working in 3D space. Use this metaphor as you see fit.

However, there’s a problem you can run into, if you’re not careful. Say you knew that your measly 2D world just doesn’t have the capacity to visualize your cylinder. But, it turns out you don’t have the information necessary to make the synthesis. Of course you’re not aware of this. Say you’ve only observed the shape as rectangle, for example. You’ve even rotated it 90 degrees on its side to see if the shape changes. Nope, still a rectangle. Well a rectangle in 3D is a… prism!

Obviously not what we want. And here’s the rub: you shouldn’t be too quick to resolve paradoxes, even if you know that ultimately, there is a paradox. You must learn to hold contradictions in your head — hold them in tension, like a Mexican standoff — until you’re truly ready for a synthesis. Sometimes you’ll never be ready. Some mysteries take multiple lifetimes to crack. But trying to resolve a paradox early will leave you with the wrong model. It will make a prism of your cylinder. Better to have an incomplete model, or several incomplete, contradictory models, than a wrong, but unified and all-encompassing one.

This approach to resolving paradoxes is but one of the skills you’ll need if you wish to think from first principles (or in other words, to think like a scientist). But not the only one. Others heuristics I’ve written about include faceting and the pre/trans fallacy (which is basically the bell curve meme before it was cool).

To end, let me quote the great Benoit B. Mandelbrot, from his book The (Mis)Behavior of Markets:

Let us mull the promises that science makes to society to win its support. The grand promise is to endeavor solving the great mysteries—to the list of which I have added one. But there is also a more practical promise. It consists in helping society to improve, to prevent it from acting on the basis of theories that sound nice but are not true to the facts, and to help it act on the basis of facts—even if those facts have yet to find a theory that fully explains them.

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