Calculus: A Light Refresher

The term “calculus,” coined by the Enlightenment genius Gottfried Leibniz, is not as frightening or off-putting as it may seem. It derives from the Latin word “calx,” little stone(s) once used for calculation. The term itself and its “derivatives” have acquired a certain currency in colloquial lingo and popular culture. One uses the word in many different contexts as a multiple signifier that has nothing to do with the science, as when we say, for example, “moral calculus” in political or ethical matters.

We find it used as a popular name, as in The Adventures of Tintin comic-book series and 2011 movie. Professor Cuthbert Calculus is described in the official site as absent-minded and intuitive, “capable of the most unexpected, and sometimes really weird, connections with reality, by simply using his pendulum. Professor Calculus is intrigued by everything, including botany, physics, electronics and dowsing. He has all the traits of a scientist who is determined to make his ideas work.” He is a dreamer and brings “an element of freshness and fantasy to the series.” He is also the inventor of motorized roller skates, the precursor of EV skateboards. For those unfamiliar with calculus, Professor Calculus is a fun place to start.

The central mathematical concept of “the limit” is press-ganged from Infinity to serve the formulaic vehicle. Limits describe how a function behaves infinitely near rather than precisely at a given point, calculating how a rate of change develops as the function approaches zero or infinity. These functions or derivatives almost magically turn approximations into exact quantities, the very secret of calculus, a scheme of improbable limits that work both in equations and real-world results. It puts me in mind of the hit Eagles song "Take It to the Limit," co-authored and sung by Randy Meisner, which also appeared in two movies, Space Cowboys and the aptly named Vertical Limit. The key stanza and line are repeated many times:

So put me on a highway
And show me a sign
And take it to the limit one more time

Meisner saw the sign and took it to the limit one more time too often. He left the Eagles after experiencing severe anxiety hitting the high notes in the song, departing the group after an intense quarrel with band leadership. But in calculus, we can take it to the limit as often as we wish. There is no end to it, since the limit is infinite.

Let’s look at our subject a bit more seriously, since the concept is for many people quite inhibiting—though it need not be. Calculus works, for example, with the essential datum of speed that can only occur, obviously, over a time interval. But instantaneous speed is an average speed over incrementally shorter time intervals approaching but never reaching zero, that is, approaching the limit of an infinity but plainly never getting there. The interstice to or from the actual limit is so tiny that it has no practical effect. 

Calculus conquers the discrepancy, the reason being, as noted, that infinitesimal decimal expansions are so small they can be subsumed in the major power series of simpler areas. It’s a kind of trick, but close enough to count as a perfect semblance. The solution to a computation is infinitely close, not infinitely far—so close it may as well be actually present. 

Think of movies where pixels blend into a single trajectory toward a tangible result, as a consummated act that can be discretely observed and measured as one entity though endlessly separate. The fact is that the mathematical model works beautifully, even though it stays approximate. It’s all a bit freaky, but then, the world is infinitely surprising.

Some familiarity with the language of calculus is certainly an aid to understanding. A “derivative” is the rate of change found in a process known as differentiation. This is relatively straightforward. An “integral” is regarded as the sum of infinitesimal strips or fragments or gaps, i.e., as unit-accumulations of infinitesimal change at a changing rate.  As we’ve seen, a “limit” is the value that a function approaches as it approaches infinity. “Infinity,” as Steven Strogatz says in Infinite Powers, “is the original sin of calculus.” Another way of saying this is that we live in a world of approximations. Nothing is ever 100%. But in the real world, approximately enough is good enough.

Original sin or civilizational bonus, calculus is everywhere, in language, in music and fiction, in the vast technological apparatus of an advanced civilization—such as atomic clocks, the global positioning system, electrical and aerospace engineering, actuarial tables, microwave ovens, quantum physics, computer algorithms, statistics, market decisions, modifying drugs and therapies for infectious diseases like AIDS and hepatitis C, etc. etc.—and, of course, it is now securely lodged in the scientific almanac as it is in the array of modern experience. 

It is important, then, to recognize the influence of calculus for daily life since we depend upon it constantly, if unbeknownst. It is more than a mere elective. We need to know something about calculus as we do about politics, news, sport, and popular issues. Indeed, it has been remarked that Thomas Jefferson adopted his hero Isaac Newton’s calculus method in framing the Declaration of Independence. He begins with an axiom (“We hold these truths to be self-evident”), deduces from it a series of propositions (including the right to secede from the Crown), and leads to a practical, real-world result (separation).

     Also read: Rage Against the Machine: Mathematicians Declare War on AI — While Taxpayers Foot the Bill!

So we keep taking it to the limit one more time. Calculus is an illustration and a pivot of mankind’s foundational need for order, as expressed in the formula for summing up increments into integrals y(t)=1/2at2 to solve the fundamental problems of curve, area, volume, slope and distance and their relation to one another. It integrates local steps to give the global whole and organizes the world into comprehensible structures we map, interpret, understand and control. That is the money question. It is an imperative that the poet Wallace Stevens has memorialized in a gorgeous and much-acclaimed poem, “The Idea of Order at Key West,” the concluding portion of which reads:   

The lights in the fishing boats at anchor there,   

As the night descended, tilting in the air,   

Mastered the night and portioned out the sea,   

Fixing emblazoned zones and fiery poles,   

Arranging, deepening, enchanting night.

Oh! Blessed rage for order, pale Ramon,   

The maker’s rage to order words of the sea,   

Words of the fragrant portals, dimly-starred,   

And of ourselves and of our origins,

In ghostlier demarcations, keener sounds.

The ghostlier demarcations of the blessed rage for order form a splendid definition of calculus.

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