Calculus without Calculus

Two-thousand years before Isaac Newton was born, an unknown Greek reflects upon the world convinced there must be some order to the chaos. We imagine our hero come beard-to-beard with Aristotle's crude physics -- spurred on, we imagine, by recent tragedy: a brother's trireme lost off the coast of Attica in a storm.

Aristotle consoles. The sailor drowned because the bottom of the ocean is the natural place for him.

This vexes our hero (let us call him/her Kittycules), for s/he observes some things do not seek the bottom of the ocean. Ducks, exempli gratia. Witches. Very small rocks.

Because that, too, is their natural place, Aristotle retorts undeterred. Very tremendously natural place. Lots of people are saying that.

It cannot be denied Aristotle's physics leaves much to be desired. Talk of "things seeking their natural place" looks nice on a bumper sticker, but it don't provide a useful tool. However, what form would a useful tool assume? It would take two millennia (and a plague) for humans to work that out.

But it need not have.

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We place prediction at the apex of nous (where Aristotle placed poetry, which always sounded to Kittycules like so much dog-ate-my-homework). What it do? That is the question one should ask. And demand in response no mere vagary but numbers attached to the doing. A what at a where and a when.

Here the narrator steps, Our Town-like, from the wings.

We speak of Dynamics, the iron handmaiden that supplanted Aristotle's mushheadedness. Heraclitus, too. Yes, yes. Panta rhei. Everything flows. But flows where and when and how? An answer requires not only Newton's calculus, but also Newton's laws. Isaac's glimpse behind the Pantagruelian curtain; the gear workings of the universe revealed. Heutzutage, any competent physics undergrad can apply F = ma and calculus and obtain a time series of motion -- a prediction -- but those days lie thousands of years in the future. What can Kittycules hope to accomplish without Newton's golems?

Kittycules is a student of Archimedes. One must observe carefully, Archimedes always said. (Also: Give me all the grant money. Some sayings in academics are eternal.)

So observe we do. But what? The choice of the right problem is paramount.

Fire? Wind? The ocean sea? All far too complex.

Instead, and to move the story along, Kittycules settles on the lowly pendulum.

Fire burns, wind blows. Oceans wave. Pendulums pendulate. Pull the bob to one side. Release. It swings for a bit then stops.

Aristotle would explain thusly: The pendulum seeks out its natural state which is the ground, but is repeatedly prevented from doing so by the gronkulation of the fluxozoids.

Yes, professor, but can you predict the shape of the motion? The details? The period or when the bob will come to rest?

Aristotle shrugs. Beats me. (Give me all the grant money.)

In a storied back room of his/her hovel, far away from the Lyceum, Kittycules establishes a modest laboratory. A pendulum -- say, a bit of ostraka suspended on a lyre string -- hangs before a wall which Kittycules has marked in a careful exquisite grid so that the vertical position of the pendulum can be measured as it oscillates. A regular measure of time is also required. For this, Kittycules uses his/her pulse (hey, it worked for Galileo).

Archeologists have since recovered a fresco of the experiment:

Kittycules asks: Given a snapshot of "now," can we predict the snapshot of "next" -- that is, the position of the pendulum bob on the next heartbeat?

The extremes are instructive, in science as in life. When the bob passes through the bottom of travel, if moving leftward it continues leftward, if moving rightward it continues rightward. The now tells us the next. However, at the far right and far left of travel, the bob pauses to reverse direction. The next changes character.

Apparently there is more to this problem than simply now predicting next. How to proceed?

Moons pass. Then Kittycules hits upon an idea.

Everything flows, that one guy at the Academy keeps saying. Inspired, Kittycules invents a new philosophy: everything remembers. If you know the state of things now, that may not give you the next. But include in your knowledge how things were, then a much more accurate prediction of the next becomes possible.

Yet, memory also fades, so the contributions of "now" and "were" must be combined in proper proportion. Kittycules writes:

  y[n+1] = α y[n] + β y[n-1]

Here, y is the vertical position of the bob (measured relative to its lowest position as it passes through the center line) so y[n+1], y[n], y[n-1] are the next, is, and was values, respectively. Finally, α and β are numbers that control the contribution of the now and the were to the next.

At last. A system which can produce numbers. Numbers tabulated from experiment for comparison.

All that is required are the two values α and β.

How are these obtained?

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Intermission -- S/he proves by algebra that Hamlet's grandson is Shakespeare's grandfather

We, with the benefit of hindsight, can demonstrate Kittycules has invented calculus. Recall the governing equation for the free motion of an undamped pendulum can be written:

  y'' + ky = 0

Here y is the vertical position relative to rest, k is a constant involving mass and the acceleration of gravity, and primes indicate derivative.

Now, approximate derivatives using finite differences:

  y' = (y[n] − y[n-1]) / Δt

  y'' = ( y'[n] − y'[n-1] ) / Δt
   = ( (y[n] − y[n-1]) − (y[n-1] − y[n-2]) ) / Δt²
   = ( y[n] − 2 y[n-1] + y[n-2] ) / Δt²

Substituting into the governing equation we obtain:

  y[n] − 2 y[n-1] + y[n-2] + k Δt² y[n] = 0
   => (1+kΔt²) y[n] − 2 y[n-1] + y[n-2] = 0

Rearrange:

  y[n] = 2/(1+kΔt²) y[n-1] − 1/(1+kΔt²) y[n-2]

This is true for all n, so it is true if we replace n with n+1:

  y[n+1] = 2/(1+kΔt²) y[n] − 1/(1+kΔt²) y[n-1]

Defining α = 2/(1+kΔt²) and β = -1/(1+kΔt²), we arrive at:

  y[n+1] = α y[n] + β y[n-1]

which is the model Kittycules has proposed. Calculus without calculus. For is modeling nature not the entire raison d'être of calculus? A question we shall return to.

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All that remains is two numbers α and β. These come from expressions involving k, and k involves the pendulum mass and the acceleration of gravity.

Kittycules knows nothing of such quantities.

How to proceed?

The oracle of course! How hard can it be for the gods to look up two numbers in their etherial ledger?

Kittycules sends to Delphi and burns the right incense and greases the right palms and waits for a reply.

Alas, oracles being oracles, the answer returned is a disappointment. "The banjo is angry at midnight" or some such. Kittycules begins to suspect the business is just a bunch of hooey designed to golden fleece the yokels. ("Bingo," comes the reply.)

No matter. If the gods will not deliver, then labor must. Brute force trial-and-error will reveal the correct constants.

There was no Matlab back then, but Kittycules could call upon the universal fount of cheap computation: graduate students. S/he lures them away from Aristotle, who -- let's face it -- hadn't produced anything notable since the Macedonians invaded and he took that tutoring job.

Each student is assigned a value of α and a value of β and told to get crackin'. Archeologists have since recovered a few frescos of the results, including this historic specimen:

The dots are the data and the line shows the values generated by the model. As you can see, the Kittycules model is able to predict the behavior of a pendulum with rather impressive accuracy. [1]

So why did two millennia pass before the stranglehold of Aristotelian thinking was overthrown?

Giddy with success, Kittycules takes to the street, demonstrating his/her protocalculus to any passer-by with hasty diagrams sketched in the dust of the agora. One day, whilst Kittycules was thusly preoccupied, a pair of Roman soldiers appear. What's all this, then? one asks in an inexplicable cockney accent. Calculus, innit? Kittycules waves them off. Weg! There is a scuffle and Kittycules is slain. it would not be the last promising academic career ended by ignorance in a ham-fisted metaphor.

The protocalculus is lost to the mists of time. The rest, as they say, is history.

Epilogue

In the end the children screamed, the lovers cried, and the poets dreamed.

Such a calculus alternative was formulated not by a hypothetical ancient Greek, but by the statistician George Yule in 1927 [2]. All the more astonishing in that Newton's laws had been triumphant for some 300 years at the time of Yule's work (and the assault by Heisenberg et al. had not yet torn them down). It was almost unthinkably brazen. Step aside ya Liverpool wanker, I gots a NEW idea! Skating uphill, as Blade would say.

No, Yule was not proposing to replace calculus with simple calculations every schoolboy and soccer mom could understand. What Yule was driving at was --- well, to know with certainty you'd have to ask Yule and that opportunity was lost to a Cambridge grave in 1951 -- but we will say Yule was driving at the difference between phenomenology and theory. His famous paper took up the problem of periodic variation in sunspots for which there was no underlying understanding. Yet, he produced a no-calculus sunspot model that reproduced the data just as well as a no-calculus pendulum model could.

Hence the quandary: If arithmetic can produce results that otherwise require four semesters of calculus and a few of physics, who gets the last laugh? Fast-forward 2000 years and in countless applications we are now asked to accept differential equations that cannot be solved to be a solution.

What are we to make of such an attitude? Is prediction no longer our gold standard? A theory should predict, surely. But failing that, which is worse: A theory that cannot predict or prediction without understanding?

Bah! Mere curve fitting, the physicists scoff. Nothing but useless scribbling, the engineers counter. A slap fight ensues, a flurry of dainty garter snake arms flailing fecklessly.

All you need is love, the Beatles said. Or, in our case, arithmetic, which the ancient Greeks understood at least a little. Do that, and we get calculus two millennia before Newton. What might have been. What might be. Would we now be colonizing the stars, as Carl Sagan posited? Or would the new tool have been turned to war and the Earth to ash?

Newton Silver Plated Necklace
by LabKitty

Footnotes

1. Those of you scrutinizing closely will note an inconsistency here. In the intermission we derived the Kittycules model for an undamped pendulum. This places constraints on the values of α and β (to wit: α = -2 β and β ≤ -1). Violate these constraints and you're no longer modeling an undamped pendulum. What are you modeling? Short answer: Who knows. But choose wisely (say, by choosing α = 0.9 and β = -0.9) and you may obtain the model for a damped pendulum, which the fresco suggests is the case.

2. Yule, G.U., On a Method of Investigating Periodicties in Disturbed Series, with Reference to Wolfer's Sunspot Numbers. Phil. Trans. Royal Soc. London Series A. 226: 267-298. (1927).