Note: Dealing with a subject that I almost worshipped, this post along with being a book review is also a toast to the endless hours of Calculus practiced through-out my high-school and tries to explain its importance in shaping the modern world. Do comment out with similar memories of school.
School has just ended for another batch of students. Results are slowly seeping out of the alcove, as the students are enjoying the aftermath of their 12 years of school lives. Although this marks the beginning of a new frontier, it’s always quite painful to forget the golden years at school.
Through middle school and high school, a student develops special affinities towards few subjects, which affect in someway their decisions for higher studies. Though in no way had it played a decisive role in opting for my higher studies, Mathematics was a subject quite close to me. It almost had a melody and rhythm to it which dragged me towards deeply. Directed love for Calculus drove me towards Mathematics even more.
The first class of Calculus still lingers in my memory. The rattling sound of Mr. Raghavan’s chalk coming in contact with the jet black-boards. The swooshing sounds while the boards being changed. A single problem covering all the four boards. Mr. Rhaghavan being from South India, had a funny way of uttering the words ‘Laaamiiit X taends to jzero’ (Limit X tends to Zero).
The actual meaning for his frequent used term, took some time to set in. But finally when it did, it felt quite empowering. The term ‘x’ is a being of unknown, collecting ‘x’ with other terms gives birth to things like 2×2 + 3x + 5. Along with the slightest change in ‘x’ its values chance. Like if x=4, then 2×2 + 3x + 5 becomes, 2(4*4) + 3(4) + 5 = 32 + 12 + 5 = 49. Now that it has been established that with the change in value of ‘x’, its value changes, so imagine if the value of x becomes unimaginably small, not zero (0) but still ‘very close’ to zero. Remember here the keyword is ‘very close’, ‘x’ is not completely zero (0), but its difference from zero (0) is so small that it’s almost negligible.
Slowly due to the subject’s attraction combined with Mr. Rhaghavan’s teaching skills, it started setting in that Calculus exceeds far more than just being a general subject in school. It is characteristically different than all other Mathematics. It is almost like worshipping the unknown.
To understand the concept better, let’s take up an example. We all are aware that due to the earth’s gravity, an apple will fall with an acceleration of 9.8m/s when it is dropped from a tree. But we also know that acceleration also depends upon the distance between the two object’s centers from the general law of gravitation. So the apple will have a starting acceleration when it falls from 4 meters. But it will certainly increase when the apple reaches 3.9999……99m and will again increase at 3.9999……98m, 3.9999……97m and so on. But what if we are to find out the exact speed of the apple at any given time? Can we? That is where lies the beauty of Calculus.
The word Calculus originates from calculi, which literally means pebbles. The basics of counting Greek counting started with pebbles hence Calculus in Greek means to count. Words like calculation and calculate also have the same origin. But whatever be the origin, it’s needless to say that Calculus is not simply a method to count. It is a specialized math to fulfill a specialized need.
How did this specialized way of Mathematics came into being, surely it didn’t in one day! Like every other thing, this also has a place in history. Its not causeless. History gives us two names related to Calculus. One of Sir Isaac Newton and the other of William Gottfried Leibnitz. There have been a massive quarrel in history between England and Germany questioning which of these two scientists invented the way, which is familiar to almost all students of the subject. But is that all that is to the history of Calculus.
To understand this genesis of Calculus better, I had read The History of the Calculus and its Conceptual Development by Carl Benjamin Boyer. In its short foreword given by Richard Courant pointed out a very important topic. He said that the teachers, students and scholars of a particular scientific subject should always be aware of the long time and change it took to attain today’s knowledge. Its unfortunate that Calculus taught today doesn’t include the long struggle spanning over two millenniums.
What Courant had said in the foreword spanning few paragraphs was clearly portrayed in a book I read recently, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World by Amir Alexander. Through his diligent investigation, Alexander in his book points out an immense struggle in history which is both religious and political. ‘Laaamiiit X taends to jzero. The idea of the infinitesimal, debates surrounding it through which he recounts the birth of Calculus. Through-out history, people have tried to come to terms with this concept by offering their explanations.
One such man was Zeno. He was a was a pre-Socratic Greek philosopher, who devised a set of three paradoxes, to support his teacher Parmenides’s doctrine that contrary to the evidence of one’s senses, the belief in plurality and change in the universe is mistaken, and in particular that motion is nothing but an illusion.
The first paradox presents Achilles, in the paradox Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise’s starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.
Similar is his second paradox, an arrow left towards a target can never reach there as first it has to travel half of the total distance, then half of that, then again half of that and so on. And it’s impossible for the arrow to reach its target in a limited time.
His third paradox presents a different scenario. He poses that an apple can never fall from the tree, as it is still in every frame pictured if time is divided into infinitesimal parts.
Needless to say, these paradoxes completely contradict our factual experiences. However large be the head start, anybody can beat a tortoise in a race. A bow leaving an arrow will surely reach its target and apples fall from the tree everyday. But the artifice of the Greek philosopher is commendable. Claiming some impossible clauses, he presents some not so easily breakable logic.
Alexander further elaborates, that modern civilization is unthinkable without Calculus. All the theories and formulas explaining the universe, is arranged with the help of Calculus. Not only the theories, but modern technologies are also dependent fully on Calculus. Aeronautics, Electronics, Finance & Economics. Calculus is omnipresent.