Infinity is the bipolar form of finitude

It comes across as either a theorem or the beginning of your 19th nervous breakdown.

“Infinity is the bipolar form of finitude” is just that kind of sentence. It arrives wearing the tie of mathematics but the slightly frantic smile of metaphysics.

Crazy-ass wackadoodle. Flip city. Oh-they-used-to-laugh-at-me-when-I-refused-to-ride-on-all-those-double-decker-buses-all-because-there-was-no-driver-on-the-top certifiable.

You suspect a sentence like that knows something you don’t, but also that it might, at any moment, start pacing the room and explaining itself with diagrams on napkins, doodles on Möbius strips or increasingly baroque grocery lists.

First, though, we need to talk about finitude, which is a word philosophers use when they want to say “things end” but in a tone suggesting that the ending has deep structural significance and is not merely the universe forgetting to order more mayonnaise. A finite thing has edges. It stops. It runs out. A pizza slice is finite. A Thursday afternoon is finite. The amount of patience you have for some idiot mansplaining finitude in a blog is also, as a rule, quite finite.

The standard contrast, which you learned sometime between arithmetic and existential dread, is that infinity is what happens when the edges go away. Numbers keep going. Time keeps extending like Hey Jude or Layla or Comfortably Numb.

The line has arrows on both ends and the teacher draws it with the serene confidence of someone who does not personally intend to travel all the way along it.

But here’s the small conceptual hitch that mathematicians handle with admirable professionalism and philosophers handle by inventing longer, unecht German-sounding words: infinity is not actually a thing you ever encounter.

You never open a cupboard and discover an infinite number of mugs cascading out like a ceramic apocalypse. Infinity shows up only as a gesture. A direction. A refusal to stop counting.

And this, to my customary Donnie Darko way of thinking, is where the “bipolar” idea starts to earn its keep. Hello? (Hello, hello, hello.) Is there anybody in there?

Imagine finitude not as a prison cell but as a corridor with two doors: one labelled “O” and the other labelled “Enough Already.”

Every finite quantity lives somewhere between these two doors. Three apples: closer to zero than to “Enough Already,” unless you are extremely hungry. A thousand apples: drifting toward the latter door, depending on the size of the apples and the size of your refrigerator crisper.

Infinity, in this picture, is not outside the corridor. It is what happens when the corridor has a manic episode.

Instead of quietly existing between its endpoints, finitude begins sprinting toward them with increasing enthusiasm. It charges toward the right-hand door — More! More! Blacken my eye! Set fire to my tie! I’m still not satisfied! — but the door keeps receding like a treadmill set by a malicious gym instructor named Helga. Every time you reach a larger number, there is another number politely waiting behind it, wearing a Fitbit and saying: “Good work, but have you considered the next one?”

The manic pole of finitude is therefore the runaway escalation: add one more, add one more, add one more forever and a day. Just nod if you can hear me. Is there anyone home? Mathematicians call this “diverging to infinity,” which sounds dignified but is basically the numeric equivalent of someone who has had six espressos and decided to alphabetize the stars.

The depressive pole is the opposite sprint: the collapse toward zero. Divide again. And again. Halve the distance. Shrink the interval. You can keep cutting forever and never quite arrive at nothing, like a Zeno-style carrot being sliced into ever more philosophical coins. There is always a smaller fraction, a thinner sliver, a quieter whisper of quantity. There is no pain. You are receding. A distant ship. Smoke on the horizon ...

So infinity appears at both ends of finitude’s emotional spectrum. At the manic pole, things blow up without bound. At the depressive pole, they dissolve toward an infinitely fine granularity. The corridor remains finite in any moment you can actually stand in, but its dynamics — the directions it can run — are pathological in a very particular way.

This is why calculus, that famously stern branch of mathematics, often behaves like the clinical psychologist of numbers. It studies what happens when finite processes become extreme enough that their limits start acting strange. You let the step size go smaller and smaller. You let the count go larger and larger. Eventually the mathematics begins to speak in the language of infinity — not because infinity is present, but because finitude has begun oscillating toward its unreachable moods.

You could say that infinity is what finitude looks like when it refuses moderation and refuses to take its quetiapine.

There is also a slightly unsettling human analogy here, which you can safely ignore if you prefer your mathematics unsullied by anthropology but which nevertheless lurks around the edges of the idea like Nurse Ratched’s ward guards armed with clipboards. Clipboards and broom handles.

Human attention, for instance, is finite. You can read only so many words, scroll so many posts, watch so many videos before your brain begins to feel like a browser with forty-seven tabs open and one of them playing music you cannot locate but is blowin’ through the jasmine of your mind like a summer breeze.

Yet the systems built around that attention — feeds, queues, recommendation engines, Leonard Cohen’s “flabby liars of the Aquarian Age” — simulate infinity by exploiting the same manic logic as the number line: there is always another item. The corridor extends into Andy Warhol’s Exploding Plastic Inevitable.

Conversely, there is the depressive pole: the sense that the individual unit of meaning keeps shrinking.

Posts become shorter, messages briefer, thoughts compressed into emojis or reaction icons until communication begins to resemble subatomic physics, where the particles are tiny but the interpretive apparatus around them is vast. Think Leonard Cohen’s “upturned bellies of fallen sparrows.”

Both directions feel infinite. Neither actually is. Still. Wasn’t it a strange way up? Wasn’t it a long way down?

Which suggests that the sentence we started with — “Infinity is the bipolar form of finitude” — is less a theorem than a diagnosis. Infinity is what happens when the idea of “more” or “less” is pursued so relentlessly that the endpoints stop behaving like endpoints and start behaving like asymptotes: destinations you approach forever but never quite arrive at. And not to put too fine a point on it: Not in utter nakedness but trailing clouds of glory do we come. Don’t quote me, but I’m pretty sure Bob Guccione said that.

The weird consolation in all this is that the world you actually inhabit remains stubbornly finite.

Your afternoon contains a precise number of minutes, many of which will be spent trying to remember why you opened a particular browser tab. This one, for instance. The universe, according to several cosmologists who are disturbingly cheerful about the matter, may have a finite amount of stuff arranged in an inconveniently large configuration.

Even your coffee cup contains a specific, disappointingly countable amount of coffee. When properly J. Alfred Prufrocked, you can measure out your life with coffee spoons. But who wants to wind up pinned and wriggling on the wall? And how should you presume?

Infinity, meanwhile, continues to hover at the edges like a dramatic relative — seen through the small end of a telescope — who never quite moves into the house but is forever sending postcards from the horizon at sunrise — a crack, a crack in everything where the light gets in.

The peril here is that if you’re not mindful about where you’re looking, the crack in the tea-cup will open a lane to the land of the dead. O look, look in the mirror, O look in your distress: Life remains a blessing, but the dead just deliquesce.

Finitude provides the room you can actually live in. It’s where the sausage gets made.

Overhung by endless forests of contemporary tubular lamps, infinity provides the two doors at the ends of the corridor — one flung open for your reception, promising endless expansion; the other endless subdivision — through which thought can throw itself into the convulsion of the world, restlessly, forever.

Thing is, the lines blur, the loop opens; the sorcerer’s apprentice knows how to initiate the spell but not how to end it. The dish runs away with the spoon. Infinity sulks into finitude; finitude erupts into infinity. The poles are less destinations than moods, less places than ways of experiencing the same stubborn, uncooperative reality. We invent symbols like ∞, which looks reassuringly like a closed loop, as if infinity could be contained within a shape that begins and ends in the same place.

What’s interesting — and faintly disturbing — is how readily the mind toggles between these modes. And the end of all our exploring will be to arrive where we started and know the place for the first time.

This is where the “bipolar” metaphor earns its keep, not as a clinical diagnosis (which would be both inaccurate and in dubious taste) but as a way of capturing the instability inherent in our concepts of limit and boundlessness.

Infinity is not simply the absence of limits; it is the experience of limits dissolving and reconstituting themselves in rapid succession.

Finitude is not simply the presence of limits; it is the experience of those limits asserting themselves with a kind of desperate finality that, paradoxically, invites their own transcendence.

Which is how we get to Baudelaire’s famous formula encompassing absolute beauty, spiritual eternity and overwhelming melancholy in one mind-blowing Electric Ladyland of an aporia (typed in this toked-up sentence by the expiring winter’s long expressive fingers and manicured nails at the butt-ends of her days and ways and crinkly, age-spotted, vellum arms slouching toward spring):

C’est infini dans le fini.

And you know, as the late, great Annie Ross was sent here to remind us, two heads are better than one.