Solvable Paradoxes

Nothing wrinkles the brain quite like a paradox. It goes like this: We encounter two opposed truths, we try to reconcile them, and our train of thought is caught in a nauseating loop-the-loop. Sometimes we beat the paradox eventually; other times, we submit to its cosmic impossibility. We’re intrigued by paradoxes because they challenge our minds so formidably, and they drive us mad for the same reason. We love them and we hate them, paradoxically.

Perhaps that’s why paradoxes are so ubiquitous in language. They’re even represented by their own stylistic device, the oxymoron. How many times haven’t you heard “silent scream,” “one-man band,” “holy hell,” or “Thinking SJW?” Entire sentences can be paradoxical as well, like this poignant bit of advice my brother once shared on Facebook:

“The moment you embrace imperfection, everything becomes perfect.”

Source unknown

Imperfection and perfection are each other’s opposites, so how could embracing one bring about the other? It’s a good quote, but clearly ambiguous. Paradoxes also manifest in the visual arts. Consider Ascending and Descending, a lithograph by master mindbender M.C. Escher:

“Bathrooms are on the top floor.”

You’ve no doubt seen it before. The staircase that feeds into itself is a celebrity among illusions, even landing a minor role in Inception. The stairs can’t be going upwards, or downwards, because they end at the same elevation as where they start (wherever you consider the end point to be). But they can’t be flat either; the angles and shading make that all too clear. It’s a delightfully infuriating paradox that rouses the philosopher in all of us.

And then there are evil paradoxes like this:

“This statement is false.”

Satan, again

This isn’t a stylistic device, but a harrowing mind game. If we accept that the statement is false, then it must be true, thus making the statement false. Insane yet?

Okay, I’ve listed a number of paradoxes; let’s try to solve them. The perfection advice is the easiest, so we’ll tackle it first. Perfection and imperfection are opposites, yes, but they’re also nebulous words with potential to overlap. I doubt you took many seconds to discern the message: Allow your idea of perfection to encompass the little imperfections of life, and those will cease to bother you. The paradox isn’t truly a paradox – it just plays on the semantic flexibility of two abstract words, delivering an important message with a pseudo-paradoxical punch.

Seriously though, perfectionism easily gets toxic – you’re already good enough 🙂

Escher’s stairs are trickier, but we’ll beat them by examining the properties of perception. We don’t really see in three dimensions, but two; the brain displays the third dimension of depth as diagonals and curves on a plane. That’s why this is instantly recognizable as a cube, despite the image being flatter than the one plastic glass of soda that went untouched at the party. Escher used this visual shorthand to his advantage for Ascending and Descending. He angled the lines to resemble our perception of three dimensions, alleging to represent reality. Then he went on to connect the lines in ways that are only possible in a flat drawing, swiftly abandoning reality. Once again, the paradox isn’t truly a paradox. Escher isn’t illustrating some glitch in existence; he’s only pulling out the representational rug from under us. We can solve the paradox by denying the artwork its claim on the real world.

But the last example, the perplexing false statement, is impossible. There’s no way to have those four words make sense without a referential cheat. This time, it seems like we’ve come upon a glitch in existence. A loop-the-loop on the fringe of our understanding. Contrast it with the perfection advice and the lithograph, which were paradoxes only at a glance. When we inspected the possibilities and limitations of their respective media – language and 2D art – we could beat them. They’re solvable. With this example, though, there’s no medium to consult, only the bedrock of our minds. It’s a bona fide contradiction. The infinity of space is another example. Or how about this classic: If God is all-powerful, could he create a boulder that’s too heavy for even him to lift? If he can’t, well, he’s not all-powerful, because we just discovered something he can’t do. But if he could, that would mean there’s now a boulder that God can’t lift – in which case you’re also not all-powerful, are you God?

(Sorry, I don’t know why I keep channeling Satan in this post.)

Let’s distinguish, then, between two types of paradox: one that arises only in representations of reality, and one that arises in true reality and proceeds to entangle our thoughts. We’ll call them representational paradoxes and cognitive paradoxes, respectively. (I haven’t done much research to find out if there exist better terms already, but that’s old news to those of you who know me.) Oxymorons, contradictory quotes, and Escher’s impossible buildings are representational paradoxes. They can be fun and stylistically potent. But they’re only paradoxical because their media permit it. Step outside those media and into the world, and you’ll render the contradiction moot. Cognitive paradoxes, however, are utterly inescapable. They’re the true paradoxes.

Or… (and here comes a surprise that you all saw coming) could it be that cognitive paradoxes are also representational paradoxes?

Perhaps human cognition is just another representation of reality – not truly reality? Maybe thoughts are a medium of their own, their possibilities and limitations artificial? What if true reality is unparadoxical, and impossibilities only emerge in its interplay with the human brain? In that case, paradoxes aren’t cosmic adversaries on our philosophical escapades. They’re just the result of flawed representation. Art, language, human thought, they’re all approximations, imperfect, inviting contradiction. Consider again the mind game with God and the boulder. Perhaps it’s only impossible because the human concept “all-powerful” is a faulty operation. Perhaps our thoughts are like lines in a lithograph – able to combine in ways that reality can’t support. But if we could somehow escape our minds, as easily as we can choose to look away from a drawing, we might understand the world without hitches. “All-powerful” might make beautiful sense. Going even further, perhaps all paradoxes are solvable. Not by us humans, but solvable in the greater reality that eludes us. In that case, true paradoxes don’t truly exist.

I have more to say, but I’ll stop here before both our flawed brains simultaneously short-circuit. Not only from the subject matter, but from the countless little paradoxes I’ve slipped into the text (including its title, arguably). I’ve underlined them all, actually, to maximize both our existential headaches. Sorry, not sorry!


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