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infinitesimal calculus

«Beautiful moment, do not pass away!»: these words are borrowed from Goethe’s Faust, to remind us of the flow of time, its inescapable advance that slowly and patiently corrupts everything. But why is the moment, the instant, so important? In itself? Nothing, because although it is a fraction of time, it is a fraction that has no duration nor extension. Just like in mathematics the difference between the number “0,9 periodic” (0,9999999… with infinite 9) and the number 1 is null. In fact, for every number that we choose smaller than 1, “0,9 periodic” is larger than it. The conclusion is that there is no number that can come between “0.9 periodic” and 1, and so the two numbers are actually the same. But then it is necessary to say that if we take an exact instant from our life, we have taken nothing. And therefore we have not caused any damage. Then how can that instant be so important?

Art, among all human disciplines, is the one that mostly demands an explanation to this same question. What is more, among the founding purposes of art is the search for a solution to the corruption of time, depicting something that can imprint an instant for eternity.

An example of this are the paintings of the great twentieth-century artist Balthus. Influenced by the lessons of the poet Rainer Maria Rilke – his mother’s lover and his *maître à penser* since he was a boy. In his paintings, he depicts those moments of suspended existence through which the path of life abruptly deviates, leaving a scar at the exact moment in which the age of innocence is shattered.

In one of the many letters Rilke sent to young Balthus for his birthday (which fell on February 29th), he wrote: «At midnight there is always a tiny crack between the day that ends and the new day that begins. A highly adroit person who could slip through it would emerge from time and find themselves in a realm separate from all the changes we undergo; held in this place are all the things we have lost. It is there, my dear Balthus, that you should sneak in on the night of February 28th, to take possession of your own hidden party».

It is also precisely there that Balthus was able to enter, in those moments of time that separate before from after on the vertices of a broken line where destiny turns unexpectedly, with his art. Balthus’ characters are captured in their own intimacy, often in the passage between childhood and adolescence, in the instant in which the first lascivious thought breaks through their innocence. The greatness of Balthus lies in his capability of showing the exact moment in which this thought will change the life of the portrayed person forever: a moment before he was one person, the next moment he would be another; we see him eternally imprisoned in the canvas while the instant between the before and after is prolonged forever.

*Avatars of the Tortoise* (*Other Inquisitions*), Achilles runs faster than a tortoise, but the tortoise has an advantage, let’s say, of one meter. In the time it takes Achilles to run that meter, the tortoise will have moved, say ten centimeters, and therefore will be ten centimeters ahead of Achilles. Achilles then runs those ten centimeters, but in the meantime the tortoise will have advanced one centimeter, remaining ahead. Achilles runs that centimeter, but the tortoise will have already run one millimeter. And this can go on forever without Achilles ever reaching the tortoise.

The paradox is so far from our own experience and from our day to day life that it appears insane, but a formal solution requires some mathematical knowledge of what are the “series”, or sums of infinite numbers. Zeno, in his paradox, implicitly took for granted that by adding infinite numbers, no matter how small, we would obtain infinity as a result. This is an incorrect idea, although instinctively it may seem likely. In fact, by adding up infinite numbers, one might not always get an infinite result, it depends on which numbers are added. An example, just to understand what we are talking about, is a particular case of the so-called *harmonic series*, in which we add all the numbers of the form 1/n², so 1+1/4+1/9+1/16+1/25+1/36+… and so on to infinity. The result of this sum, far from being infinite, is π²/6.

Something similar happens for Achilles: the infinite intervals of time used by Achilles to cover the distances that separate him from the tortoise, if added together, give a non-infinite result, and therefore Achilles reaches the tortoise in a finite time.

But this mathematical treatment – summarised here in words – had to wait many centuries and the input of mathematicians of the caliber of Carl Friedrich Gauss, Augustin-Louis Cauchy, Bernard Bolzano, to get at a coherent and well-defined formalism.

Let’s take a fundamental concept in this field: the so-called accumulation point. We can intuitively define it as a point that has around itself infinitely many points, infinitely close. To understand this definition, let us imagine the interval of numbers ranging from 0 to 1. We want to show that 0 is an accumulation point because it has infinitely many other points close to it. Let’s take any point between 0 and 1 and call it A (for example, A could be the number 0,5). The number A/2 is always a number belonging to our interval and is between 0 and A. We repeat what we just did but starting from A/2, taking A/4 (which is between 0 and A/2). We can repeat the operation as many times as we want: whatever number we arrive at, by dividing it by two we will get closer to 0, without ever reaching it. This implies that 0 has infinite numbers at an infinitesimal distance, which makes it an accumulation point. The same happens for an instant, which is an accumulation point of time, having other infinite instants next to it at an infinitesimal distance.

In addition to pure mathematicians, two great scientists like Gottfried Wilhelm von Leibniz and Isaac Newton used concepts similar to those just described to invent the infinitesimal calculus, in which we introduce infinitesimally small quantities to describe the reality that surrounds us. Let’s make a practical example of what infinitesimal calculus allows to do: we are going on a trip, so we leave by car from Trieste at ten in the morning and we arrive in Genoa at four in the afternoon. Knowing the distance between Trieste and Genoa and measuring the time we spent to cover it, we can calculate the speed at which we traveled. However, this is an average speed, it is not at all certain that during the journey we were constantly maintaining that same speed. For example, we might have slowed down due to traffic near Venice and accelerated elsewhere. If we had measured the exact time we passed through Verona, we would be able to provide a more correct estimate of our speed during the trip. The more cities we add to the list of measurements, the more correct the speed description becomes. However, in order to have the exact speed at each instant, we would have to divide the itinerary into infinitely many segments of infinitesimal lengths, traveled in infinitely many times of infinitesimal lengths. This is what infinitesimal calculus does: calculating speeds, accelerations, forces, and other quantities, using infinitesimal quantities.

Dividing a space of null length by a time of null duration properly, yields the velocity of an object at a given point and time. In jargon we say that velocity is the *derivative* of space with respect to time. You can also do the opposite operation: the speed that an object has at a certain instant, multiplied by the infinitesimal duration of that instant gives an infinitesimal length. Adding up infinite infinitesimal distances we obtain exactly the space covered by the object and its exact trajectory. This process of summation of infinite infinitesimal quantities is called *integral*. Carlo Emilio Gadda who, not by chance before consecrating himself as a writer had a past as an engineer, wrote: «the integral of fleeting moments is the hour».

Our lives are a long integral of infinitesimal instants that at each moment continues to extend by an infinitesimal amount. Balthus was able to extract an infinitesimal interval from the integral to extend it eternally on the canvas. Newton and Leibniz built the system of modern calculus, based entirely on infinitesimal quantities. Hence, as much as we can rationally understand the mathematical language, we will continue to wonder about the real meaning that an instant has in our lives, about what would happen if we were able to extract a specific one. What consequences would lead to change it, how a single instant could be reflected in the course of a lifetime. While, in the meantime, Achilles will reach the tortoise.

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