Calculus and the Power of Infinity

Mathematics emerged from the practical needs of humanity. The shepherds required to count their flocks. The famer needed to weigh their harvest. Traders needed to set their exchange standards etc. In the beginning we tallied with our fingers and toes and later we invented numbers. Along with numbers, shapes also mattered a lot to ancient humans. Geometry become very central in the West. It meant measurement of the earth (Geo- Earth and mete- measure). We could see geometry everywhere on the earth. There are imperfect triangles, rectangles, squares, circles, curves etc. We can see circles in the eye of our loved one. Maybe because of that our wedding rings are round. Circles point to eternal return and hence could represent reincarnation, cycles of seasons, eternal life or never ending love. But as time went, calculus began to out grow geometry.

We may describe 250 C as the origin of Calculus . Using our terminology, today we call it mathematical start up centered on the mystery of the curve. The ambitious plan of these startup was to reduce curves to more familiar straights. This means the curve could be thought of as a line but to do this we needed to push the curve to infinity. Maybe a broad example might help us understand how we harnessed the power of infinity. The surface of the earth is curve but the earth is big enough ( roughly ‘pushed to infinity’) and, therefore, we cannot feel the curve as we move on the surface of the earth the way we can feel the curves of the tennis ball or football. In the ancient days, humanity had the enigma to work out the area of the circle. To do this , the circle could be sliced into small parts that roughly appear like the triangles. We can push such triangles that are appearing as bits of circle to infinity and it will look like rectangle. We now know how to calculate the area of the rectangle. Thus, we arrive at the area of a circle using infinity. The area of circle was first calculated by using similar reasoning by Greek mathematician Archimedes.

Infinity is some what dubious. Nobody seems to know what it is. Is it a number, concept or place? Nevertheless, infinity became a God sent gift to humanity for solving difficult problems of geometry. We have another important tool to consider in this regard. It is the idea of limits. Limit is unattainable goal. We can come close to it but never to it. The slices of the circle can come close to a triangle, when we push it to an infinitesimal size. But it can never really become a triangle. It has been observed in calculus that unattainability of the limit does not really matters. Limit is subtle but is a core concept to calculus. But t is not easily practically demonstrable. Walking to a wall, for instance, by beginning to walk half way to the wall, then walking half way from that half and continuing walking half way of that half will take us to the limit of going to the wall. We will never really reach the wall. We will always remain half way short. This is so because every time we walk, we walk just half way to the wall. This is called the riddle of the wall. It demonstrates the subtleties of the notion of limit.

Infinity does not behave like ordinary number. It does not grow big when we add one to it. Even adding infinity to it does not help. But one of the craziest thing is when we push something to infinity, we come to deal with in terms of the finite. Maybe we can think of an infinite polygon. We can take a circle and put some dots by spacing them evenly on the circumference and then joining them. When we put three dots and join them, we will have triangle, joining four dots will give us rectangle or square, five will give a polygon , running through these polygons, we will see that the distance between the points go on decreasing as we go forwards with the increasing sides of the polygon. As the sides of the polygons approach infinity, we come close to the figure of a circle. The circle, therefore, becomes the limit of a polygon that approaches infinity. This means at the finite stage the polygon is a polygon and will never become a circle. It come close to a circle as it approaches infinity. Now a circle is a simpler geometrical figure and easy to deal with rather than a thorny polygon. Hence, the paradox is as we push something to infinity, we have to deal it with something simple and finite.

We have to be always humble and accept that we cannot reach infinity. Trying to go beyond infinity is to treat infinity like a number. None other than Aristotle had warned us about it bring us all sorts of logical troubles. I am trying to put his argument in our terms. He did not have the luxury of the number zero. If we divide a line to a length that is approaching zero, we can do so. But when we divide a line by zero or by a length zero, we reach infinity. This becomes meaningless as any line of any size divided by zero reach infinity. Hence, to avoid such weirdness, we can divide by any number and that number may approach zero but cannot be zero. Dealing with infinity is like dealing with God. It is also a kind of forbidden fruit. It lies at the heart of many of our big and unanswerable questions. But when we consider a limit that is approaching infinity, it works marvelously and solves several of our puzzling problems through the power of calculus.

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Hypocrisy is the tribute that vice pays to virtue.

- Fr Victor Ferrao