Calculus in Oppositeland

In Oppositeland, birds swim and fish fly. You drive on the driveway and park in the parkway, Donald Trump is a stolid civil servant, Hillary Clinton is a paragon of transparency, and people find LabKitty's political humor to be insightful.

Also, in the OL (as I like to call it) schoolchildren love calculus. Mostly because they aren't squirming sacks of hormones, but also because in Oppositeland calculators were invented before Isaac Newton. (Aside: Is it appropriate to say Newton was invented? The more you get into his biography, the more he starts to read like a piece of performance art, like Bansky, 4chan, or the Tea Party. But I digress.)

As everybody knows, before Newton came along calculus was shoulder-shrug obvious. You'd say "calculus" and the guidos hanging out at the local pizza joint would go fuggitdaboudit. (Aside: Is that term culturally insensitive? Perhaps I should have written "pizza establishment." Again, digress.)

Yes, have a seat, for it's time for another charming LabKitty math fable.

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Add up All the Things

The first problem Olandian mathematicians took up when inventing calculus was computing the area of a circle. Verily, Olandian Socrates challenged his assembled protocalculeans, we seeketh the area enclosed by a circle, the most perfect of shapes. Yet we know not how, for the boundary is curveth, which vexes even the most superior mind. (This is how you spoke at the Academy if you expected to get tenure.)

There things stood for many years, for upon issuing this challenge Socrates would immediately turn to more pressing matters, such as where they should build another Socrates statue on campus. But one day, this day, a graduate student speaks out of turn. Dude, just cover the circle with a bunch of little square tiles and add them up. You'll at least get an approximate answer, and an approximate answer is better than no answer at all.

Socrates puffs out his cheeks and scratches his beardy beard. He considers punishing the upstart for this insolence, perhaps by having him grade the midterms or milking some undergraduates. But it cannot be denied the lad has a point. Socrates seizes the idea and adds a critical modification. Instead of obtaining one approximation, he obtains a series of approximations using more and more smaller and smaller tiles. The process would go something like so:

Count the tiles that lie inside the circle; exclude the tiles that do not. Boom! Area. But a problem remains. Clearly, the true answer lies somewhere at the end of this process, but we have not the time nor energy to continue forever. When, then, to cease refinement?

Back in those days, mathematics was still the handmaiden of reality and so the answer was easy: You stopped refining when further refinement would no longer change the outcome at hand. If the calculation was, say, determining the area of an annulus of land for the purposes of levying a property tax, further refinement that did not change the amount of the tax charged was pointless. If the calculation was determining the area of a regulation discus and thus its weight (by multiplying by its thickness and density), refinement beyond the smallest difference the scales of the day could measure was pointless. Practicality ruled in Oppositeland, or at least it did until the Academy began to admit topologists and art history majors.

This is how area calculation came to be approached in calculus. When calculating devices became commonplace, refinement was standardized to eight significant figures. For eight was the number of digits displayed in the standard Olandian calculator. Once improvement is beyond the display there's no point in continuing. As such, the Olandians came to use the symbol ∞ to indicate how long to continue the process, which looks like an eight lying on its side. Also, they used the letter sigma to indicate summation and delta to indicate a chunk of area and attached an ∞ subscript to indicate "continue to add up bits of area until the answer to eight places no longer changes" so they didn't have to keep writing lots of words. That is, Σ_∞ ΔA

Cool cats started omitting the subscripts and smoothing off the sharp corners and just wrote ∫ dA. For an area bounded below by the x-axis -- the most commonly encountered calculoid problem -- the notation becomes ∫ f dx, where f denotes the function that bounds the area on top and the calculation is now divided into a collection of rectangular tiles having width dx.

This idea wasn't just for area; it worked for any problem that could be characterized as "add up all the things." In one dimension it worked for finding lengths. In three dimensions it worked for finding volumes. It even worked for adding up abstract stuff. Olandian physicists used it to add up chunks of energy to find total energy, or to sum power across a spectrum divided into frequencies or forces distributed around a path to find work.

This technique brought together all sorts of problems that heretofore hardly seemed related at all, and so the Olandians called it integration.

The Derivative is Derivative

The divide-and-conquer approach of integration can be applied to problems that have nothing to do with finding areas. Another major class of problem facing the Olandians was computing a rate of change. Suppose you seek to construct a speedometer for your Tesla electric car (which are free in Oppositeland, by the way). This is a simple matter of applying the definition: speed = distance / time, which we can write compactly as Δf / Δt -- "t" for "time" and "f" for "how far." Compute distance, say, by counting how many times the tires go around. Note the elapsed time on a stopwatch. Divide the former by the latter. Viola! You have computed your approximate speed. A rate of change.

A version of refinement applies here: As you use shorter and shorter time intervals in your calculation, the result becomes more accurate because there is less opportunity for the speed to vary during the calculation. If you want your speed right at this very instant, that's seems unpossible because Δt is zero and you can't divide by zero, not even in Oppositeland. However, as in integration, refinement comes to the rescue. We make the same argument that once refinement no longer changes the answer beyond eight calculator places, we can stop.

Someone suggested the notation df/dt to indicate the refinement limit of Δf/Δt, the "d"s kinda looking like a sideways 8 if you squint, which doesn't make total sense but at least the notation spares future historians from looking up some pain-in-the-ass HTML unicode. The quantity came to be called the derivative, because the concept was derived from the idea of using refinement that had been established by integration.

The Plague Arrives

Things hummed along happily in Oppositeland for many years, until one summer a terrible power outage swept the country. With no Internet available, some took to amusing themselves in perverse ways. A young upstart named Isaac Newton discovered calculus results could be produced without the use of a calculator. No, not simply by playing calculator -- tallying up columns of sums long into the night by hand that a computer would sum in a jiffy -- but rather by devising clever algebraic schemes that leapt right to the answer as if by magic.

Describing all of Newton's parlor tricks would likely exhaust your patience, so let's just look at one example. Here's how Isaac computed the derivative of t^2, in other words f = t^2, or in still other words: how far you've gone is equal to the square of the elapsed time. Instead of selecting a specific Δt and finding a number, we leave the quantity in symbolic form and find a general result. In this scheme, Δf becomes f(in a little bit from now) – f(now) = (t + Δt )^2 – t^2. Expanding, we have:

     Δf / Δt = [ (t + Δt )^2 – t^2 ] / Δt

     = [ t^2 + 2t Δt + (Δt)^2 – t^2 ] / Δt
     = [ 2t Δt + (Δt)^2 ] / Δt
     = 2t Δt / Δt + (Δt)^2 / Δt
     = 2t + Δt

Newton then argued since we're going to make Δt small anyway, the answer is just 2t. That is to say, if you punch 2t into your calculator, you'll get the answer for d(t^2)/dt for any value of t. You need not bother with a bunch of refinement argle-bargle.

Let's test the idea for f(t) = t^2 at t = 1 using refinement argle-bargle. The following table shows the progression of Δf/Δt (right column) using ever-decreasing values of Δt (left column):

1.00000000000000 3.00000000000000
0.50000000000000 2.50000000000000
0.25000000000000 2.25000000000000
0.12500000000000 2.12500000000000
0.06250000000000 2.06250000000000
0.03125000000000 2.03125000000000
0.01562500000000 2.01562500000000
0.00781250000000 2.00781250000000
0.00390625000000 2.00390625000000
0.00195312500000 2.00195312500000
0.00097656250000 2.00097656250000
0.00048828125000 2.00048828125000
0.00024414062500 2.00024414062500
0.00012207031250 2.00012207031250
0.00006103515625 2.00006103515625
0.00003051757812 2.00003051757812
0.00001525878906 2.00001525878906
0.00000762939453 2.00000762939453
0.00000381469727 2.00000381469727
0.00000190734863 2.00000190734863
0.00000095367432 2.00000095367432
0.00000047683716 2.00000047683716
0.00000023841858 2.00000023841858
0.00000011920929 2.00000011920929
0.00000005960464 2.00000005960464
0.00000002980232 2.00000002980232
0.00000001490116 2.00000001490116

Newton says the result should be 2⋅1 = 2 which, alas, appears to be true.

We could test other values and other functions but, long story short, Newton had hit the jackpot. Once he discovered this chink in the armor he proceeded to burn down all of the simple and beloved calculator calculus the Olandians had so belovingly constructed. Integrals fell next; Newton computing areas and arc lengths and volumes and much else besides, all with more mad scribbling. He then invaded other disciplines -- mechanics, optics, eye forking -- demonstrating his algebraic approached worked in computing the motion of objects and how light is bent in prisms, the attraction of apples to the earth and men to cows.

Yet, the most outrageous claim of Mr. Newton was still to come. In a stroke of penultimate outrageousness, he outrageously claimed integrals and derivatives were mirror images. Two sides of the same coin. That is to say, if you took the derivative of any function and then took the integral of that result, you would arrive back at the function you began with. Or, if you compute the integral of a function, the derivative of that result will produce the original function.

This was, of course, bark at the moon crazy, for what on Earth did areas and rates of change have to do with one another? Injecting this sort of talk into the classroom would incite a riot, the polite and well-groomed Olandian schoolchildren turning to jazz and heroin. Women would demand pantaloons and the vote. Dogs and cats living together. Mass hysteria.

If civilization were to survive, it was necessary to put a stop to Newton's blasphemy. Fortunately, as often happens given the vicissitudes of history, or when a post is running long and I'm running low on whiskey, convenient salvation was at hand.

One day, the National Academy of Oppositeland received a bulgy parcel containing all of the mathematical output of one Isaac Newton. For in Oppositeland, Newton was just an unknown bumpkin jockeying for recognition and a faculty position and he needed the Academy's seal of approval (not to mention the primacy of his ideas was being challenged by a continental rival named Gottfried Leibniz). Here was the Academy's opportunity to crush Newton's spirit, and if there's anything academics live for it's crushing spirits.

Newton's calculus had its own chink in the armor. A skeleton in its closet, a roofie in its Appletini. To wit: It contained functions that had no integral. This was very strange, given all of the hoopla about integrals and derivatives being two sides of the same coin. What are we to make of a coin with only one side? Picture these poor functions without a proper pedigree -- bastards in every sense of the word -- wandering the mathematical landscape, threatening to appear at any moment in the solution process like a rabid dog appearing in a scenario involving dogs.

The Academy had their casus belli. In replying to Newton's application, they wrote:

Dear Mr. Newton,

We have received all of your mathematical output. How quaint your claim of producing the results of the Calculus without use of a calculator, doing so with what appears to be scribbles drawn on parchment using a feather plucked from a duck's bottom. However, all your method does is introduce unsolvable problems into what is currently a satisfactory and useful system. The prospect of functions that have no integral is especially troubling. They're not even hard to find. Consider finding the arc length of a sine wave. In our Calculus the problem is trivial. In yours, the problem is unsolvable.

Should you come up with something better in the future, feel free to drop us a line.

Warmest regards,
The Establishment

PS: We use the term "derivative" not "fluxion," and "Gottfried Leibniz" not "bum snacker." Also, we call "chips" "french fries."

Thus dejected, the following evening Newton arrives at le field d'honneur to defend the honor of Linda Lovelace against the inappropriate advances of Gottfried Leibniz. A crisp Parisian midnight, a prime number of steps, and one musket ball later, Isaac lies dead.

The rest is future, as they say in Oppositeland.