Why Do We Study Calculus?

The question students ask most often is, "why do we study this?" (or its variant, "will this be on the exam?").

Indeed, it`s not always immediately obvious how some of the things we study will be of any use in life. For example, few people will ever need to prove anything in calculus. We are willing to trust the work of pure mathematicians whose job it is to certify the reliability of such theorems. Why, then, should we study epsilons and deltas, and all these other abstract concepts in mathematics?

Calculus is not a just vocational training course. In part, students study calculus for the same reasons that they study Darwin, Marx, Voltaire, or Dostoyevsky: These ideas are a basic part of our culture; these ideas have shaped how we perceive the world and how we perceive our place in the world.

To understand how that is true of calculus, we must put calculus into a historical perspective; we must examine and contrast the world before calculus with the world after calculus. (Indeed, there is a growing movement among mathematics teachers to do precisely this i their lessons.

The earliest mathematics was perhaps the arithmetic of commerce: If 1 cow is worth 3 goats, how much does 4 cows cost? Geometry grew from the surveying of real estate.

The ancient Greeks and Egyptians did a great deal of clever thinking, but very few experiments; this led to some errors. For instance, Aristotle observed that a rock falls faster than a feather, and concluded that heavier objects fall faster than lighter objects. Aristotle`s views persisted for centuries, until the discovery of air resistance.

Perhaps the most dramatic part of the story of calculus appears in the history of astronomy. People studied and tried to predict things that were out of human reach and apparently beyond human control.

They theorised that the earth was the center of the universe. Each day, the sun rose in the east and set in the west. Each night, the constellations of stars rose in the east and set in the west. The stars were fixed in position, relative to each other, except for a handful of "wanderers,"; later referred to as "planets".

The motions of these planets were extremely erratic and complicated. Astrologers kept careful records of the motions of the planets, so as to predict their future motions and (hopefully) their effects on humans.

In 1543 Copernicus published his observations that the motions of the planets could be explained more simply by assuming that the planets move around the sun, rather than around the earth. He concluded that the earth was also just another planet moving around sun too, rather than at the centre of the universe. He explained that these planet movements or orbits were nearly circular. Unfortunately, the church did not like this idea. This made the earth appear less important, detracted from the idea of humans as God`s central to creation and trouble ensued.

During the years 1580-1597, Brahe and his assistant Kepler made more accurate observations of the planets. Based on these observations, in 1596 Kepler published his refinement of Copernicus`s ideas. Kepler showed that the movements of the planets are described more accurately by ellipses, rather than circles. Kepler gave three "laws" that described, very simply and accurately, many aspects of planetary motion.

In 1609 Galileo took a "spyglass", a popular toy of the time, and used it as a telescope to observe the heavens. He discovered many celestial bodies that could not be seen with the naked eye.

The moons of Jupiter clearly went around Jupiter; this gave very clear and simple evidence supporting Copernicus`s idea that not everything goes around the earth. unfortunately, the church punished Galileo, but his ideas, once released to the world, could not be halted.

in this period Galileo also began experiments to measure the effects of gravity; his ideas on this subject would later influence astronomy. He realized that Aristotle was wrong; heavier objects do not fall faster than light ones.

He established this by making careful measurements of the times that it took balls of different sizes to roll down ramps. There is a story that Galileo dropped objects of different sizes off the Leaning Tower of Pisa, but it is not clear that this really happened.

We can easily run a "thought-experiment" to see what would happen. Drop 3 identical 10-pound weights off the tower; all three will hit the ground simultaneously. Now try it again, but first connect two of the weights with a short piece of thread; this has no effect, and the three weights still hit the ground simultaneously.

Now try it again, but instead of thread, use superglue to stick two weights together; the three weights will still hit the ground simultaneously. However, we no longer have three 10-pound weights; rather, we have a 10-pound weight and a 20-pound weight. Some of the most rudimentary ideas of calculus had been around for centuries, but it took Newton and Leibniz to put the ideas together. Independently of each other, around the same time, those two men discovered the Fundamental Theorem of Calculus. Though Newton and Leibniz generally share credit for "inventing" calculus, Newton went much further in its applications.

In 1687 Newton published his "three laws of motion," now known as "Newtonian mechanics"; these laws became the basis of physics.

1. If no forces (not even gravity or friction) are acting on an object, it will continue to move with constant velocity

2. The force acting on an object is equal to its mass times its acceleration. 3. The forces that two objects exert on each other must be equal in magnitude and opposite in direction.

Newton`s universe is sometimes described as a "clockwork universe," predictable and perhaps even deterministic. With support from Keplar, these laws revealed that the complicated movements of the heavens were consequences of very simple mathematical principles.

This gave humans new confidence in their ability to understand and ultimately, to control the world around them. No longer were they subject to incomprehensible forces.

The works of Kepler and Newton changed not just astronomy, but the way people viewed their relation to the universe. A new age began, commonly known as the "Age of Enlightenment"; philosophers such as Voltaire and Rousseau subsequently wrote about the power of reason and the dignity of humans.

Browne and Schechter