These Fellows Keep Squabbling


I. Fourier & Chronos

Tartarus. As far beneath Hades as heaven is above the earth: a bronze anvil dropped from the one would fall nine days to reach the other, and nine more to reach here.1 There is no light, no floor worth the name, and nothing whatsoever to do. Two very old figures have been not speaking to each other for some time. A third arrives, wrapped to the chin in wool.

FOURIER. It's cold.

CRONUS. Is it a child?

CHRONOS. It is not a child.

CRONUS. It's small.

FOURIER. I am sixty-two and I am a Baron of the Empire, and I would like to register that it is cold down here. There is no source. Heat requires a source and a gradient. This is simply an absence with opinions.

CHRONOS. Welcome. I am Chronos.

FOURIER. Cronus. Yes. The sickle, the father, the — the business with the children.

CHRONOS. No.

CRONUS. (delighted) Here we go.

CHRONOS. I am Chronos. Khrónos. Time itself, unbegotten, older than the sky, older than him. He is Cronus. Krónos. A Titan. A son. He ate six babies and got beaten by the seventh.2

CRONUS. Five. I ate five. Zeus was substituted with a rock, which I would like the record to reflect was very convincing.

CHRONOS. Some Roman put us in the same coat and now I carry a farming implement.

CRONUS. My farming implement.

CHRONOS. Your mother's, technically.

CRONUS. It was adamant and it unmade the sky.3 Now it mows. Now it appears on greeting cards at the turning of the year, in the hands of him, with sand running out of a little glass, and everyone says ah, Father Time, and I am not thanked, and I am not feared, and I am not even correctly spelled.

FOURIER. (after a pause) Then I have come to the right hole.

§

CHRONOS. You know why you're here.

FOURIER. I was told there had been a complaint.

CHRONOS. There has been a complaint. You removed me.

FOURIER. I did no such thing.

CHRONOS. You took a thing that happened — a thing that happened in me, at particular moments, in an order, my order — and you turned it into a list of how often. And the list has no moments in it. I have read your list. I have read it very carefully, because I have had a great deal of me to spend on it, and there is no when anywhere in it.

FOURIER. That is correct.

CHRONOS. Where did you put it?

FOURIER. I integrated over it.4

CRONUS. (genuine admiration) You ate it.

FOURIER. I summed it. Each number in my list is made by walking the entire history of the thing, from its beginning to its end, multiplying as I go, and arriving at a single quantity. The walk is necessary. The walk is, in fact, all I do. But when I arrive, the walk is not in my hand. Only what it gathered.

CHRONOS. So you did eat me.

FOURIER. I spent you. There is a distinction and I'd like it noted.

CRONUS. There isn't. I promise you there isn't. I have been on both ends of this.

§

CHRONOS. Then tell me the price. There's always a price and men like you always know it and never lead with it.

FOURIER. (reluctantly) You cannot have both sharply.

CHRONOS. Both.

FOURIER. When, and how often. If you want to know precisely which tones are in a thing, you must take a long look, and a long look smears the moment — you will not be able to say when they arrived. If you want the moment exactly, you must look briefly, and a brief look cannot tell one tone from its neighbor. It is not a defect of my method. It is a wall. Push at one and the other swells. There are men after me who put numbers on the wall, and a man named Gabor who wrote it down properly, and a physicist who found the same wall standing between position and momentum and got rather more famous for it.5

CHRONOS. (slowly) So I am not abolished.

FOURIER. No.

CHRONOS. I am blurred.

FOURIER. Yes.

CHRONOS. That's worse. You understand that's worse. To be denied is a kind of respect. To be approximately present —

CRONUS. It's worse. It's much worse. I'd take the rock.

§

CRONUS. Tell him the other thing.

FOURIER. There's no other thing.

CRONUS. There's an other thing. You've got the face. I had that face for about four hundred years.

FOURIER. (stiffly) My series requires that the thing repeat.

CHRONOS. Repeat.

FOURIER. Forever. In both directions. My tones are eternal — they have no beginning, they will have no end, they simply are, at their frequencies, and what I call a curve is the sum of them. So for the method to be exact, the thing it describes must never have started and must never stop. If it started, there is an edge. Edges leak. I spent a great deal of my life pretending I had not noticed this.6

CRONUS. (long pause) So it can't be eaten.

CHRONOS. So it never happened. There's no moment in it. Nothing began. It's not a life, it's a — it's a standing hum.

FOURIER. It's a very good hum. You can build a corner out of it. Lagrange said you couldn't.

CHRONOS. I ruled over a race of men who did not toil and did not age and I was thrown into a hole for it, and I am now told the mathematics of the world prefers a thing with no birthday.7

CRONUS. You didn't rule that. I ruled that. The Golden Age was mine.

CHRONOS. It is genuinely astonishing how little I care.

CRONUS. There's a German upstairs been shouting for a century that everything comes round again exactly the same, forever, and it's supposed to be the most terrible thing he can think of, and he says it like a gift.8 I'd have him down. He'd hate it here. He'd love hating it here.

§

Nothing continues to happen for an unmeasurable interval.

FOURIER. May I ask something.

CHRONOS. Ask.

FOURIER. How long have I been here?

CHRONOS. (pleasantly) That is not a question you get to ask me anymore.

Footnotes

  1. Hesiod, Theogony, lines ~720–725: a bronze anvil falling from heaven would take nine days to reach the earth, and nine more from the earth to reach Tartarus. He gives the pit a measurement, which is remarkable — the oldest surviving Greek cosmology hands you a distance in units of falling-anvil-days. ↩

  2. They are two figures. Cronus (Κρόνος) is the Titan: son of Uranus and Gaia, father of Zeus, who devoured his children to forestall a prophecy until Rhea substituted a stone for the youngest. Chronos (Χρόνος) is time personified, a primordial being of the Orphic tradition, unrelated by descent. Ancient writers enjoyed the near-homophony and etymologized one into the other, the Romans folded both into Saturn, and the result is the modern Father Time — who carries a scythe belonging to the Titan and an hourglass belonging to the abstraction. An identity destroyed by a pun is a fair grievance. ↩

  3. Theogony, ~160–180. Gaia makes a sickle of adamant and Cronus uses it on his father Uranus, separating sky from earth. The implement that appears in January cartoons performing agricultural metaphors is that one. ↩

  4. Accurate, and it is the whole complaint. Each Fourier coefficient is produced by sweeping the entire signal — every instant of it — and accumulating a single number. The time variable is the thing being summed over; it is not preserved anywhere in the result. You can recover the signal from the complete set of coefficients, so nothing is destroyed, but no individual coefficient knows when anything happened. ↩

  5. Dennis Gabor, "Theory of Communication," Journal of the Institution of Electrical Engineers 93 (1946), which formalized the trade: the product of resolution in time and resolution in frequency has a floor. The mathematics is the same as Heisenberg's 1927 uncertainty relation between position and momentum — one relation, two applications, wildly unequal fame. This is why a spectrogram must choose a window length and can never be sharp in both axes at once. ↩

  6. The Fourier series proper represents a periodic function — one with no beginning and no end. Real signals start and stop, and treating a finite stretch as though it repeated produces artificial discontinuities at the seam, whose energy smears across the whole spectrum: spectral leakage. The working fix is windowing, tapering the ends toward zero, which trades a little resolution for a lot of honesty. Every practitioner meets this on their first day and it is never mentioned in the story of the discovery. ↩

  7. Hesiod, Works and Days: the golden race lived under Cronus's reign without toil or age. Rome kept the association in the Saturnalia. Cronus's reward for presiding over paradise was to be overthrown by his own child and imprisoned, which is a structure worth noticing. ↩

  8. Nietzsche's eternal recurrence — The Gay Science §341, where a demon proposes that you will live this same life again innumerable times, and asks whether that would crush you or whether you would call it divine. He meant it as the hardest possible test of a life rather than as a consolation. He has not been down yet. ↩