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Is calculus necessary?

Oliver Knill
Recently, there were a few articles dealing with this topic. Here are a few thoughts which I plan to expand more in the future.
  1. Calculus is essential for many other fields and sciences.
  2. It is a prototype of a though construction and part of culture.
  3. Teaching calculus has long tradition. Its teaching can be learned.
  4. If we wanted to teach something else, what would replace it?
  5. Calculus develops thinking and problem solving skills.
Added August 30, 2017:

Why calculus is valuable

Calculus is lucrative business. Calculus is made to gold in many industries. Today!
Geography (google earth), computer vision (autonomous driving of cars), Photography (panoramas), artificial intelligence (optical character recognition), robots (mars bots), computer games (world of warcraft), Movies (avatar), network visualization (social networks). We live in a time where calculus is made to gold. If linear algebra is added to calculus (page rank example), then applications get even bigger. There is a gold rush on calculus. It currently makes is responsible for billions of profit.

Why calculus is a prototype

Calculus helped to understand of what we are and to plan where we go.
No other field of mathematics is so rich in history and culture than calculus. From fundamental geometry like Pythagoras as part of vector calculus, measuring volumes with ideas of Archimedes, to deal with velocities and forces which was essential in the development of astronomy and to answer questions about our place in the universe. The motion of planets, the development of galaxies and phenomena like black holes all need calculus and the quest to understand our role in the universe has led to calculus. Calculus could be essential for our survival since we need to develop and understand climate or population growth models, spread of diseases or mechanisms to resolve conflicts or deal with economic and financial crisis.

Why calculus is easy to teach

Curriculum developers: Teach it yourself first and look whether it is teachable.
There are other subjects which are important. Some of them are more difficult to teach. Probability theory is an example where the subject can only be taught well if some mathematical maturity is reached in which calculus plays an important part. It needs also mathematical maturity from the teacher. A core problem in curriculum development is that pioneers developing the material are often not teaching it themselves or do not see the difficulties a typical teacher has. Curriculum developers are by nature optimistic and want things to work. For calculus, the task is relatively easy. No other field of mathematics is so well developed pedagogically.

What should replace calculus?

Calculus is inherent in every other subject, even discrete structures.
Discrete mathematics comes in mind. But calculus is already inherent in discrete mathematics. Combinatorics, set theory or graph theory are usually core elements in a discrete math course. Unfortunately these are rather different beasts. Combinatorics belongs to probability theory, set theory to the foundations of mathematics and graph theory to topology. Newer models of calculus see discrete structures as special cases of a more general calculus. We need to continue to look out for modifications and alternatives and also for new ways to teach the subject.

Is it needed in life?

Calculus develops the ability to think and solve problems.
Is the ability to move fast necessary? In fact, we do no more need to hunt our own food any more and the ability to run fast is not essential anymore. Still, all sports have walking components as part of the exercise plan, for swimmers, rowers, boxers, tennis players or car racers. Running in particular is a simple sport, you only need minimal equipment and can be done anywhere. In principle, it can be replaced by biking, swimming, golf, or walking. But not everything works universally. I myself would love to climb mountains and fly down with a paraglider (as I did in graduate school) but it needs money and the right geography. I can do a workout in 30 minutes which includes walking to the gym and showering doing something for 10 minutes can already be effective. Like walking or running stay part of the athletic mix, calculus will always be part of mathematics education. Calculus has proven to help in any other field, like graph theory, game theory or statistical or data visualization. Here is a readers note (March 9, 2016):

"Your short article on why we teach calculus is marred with flaws. Why should we continue to teach something just because it has a long tradition? And not having a thing to replace it is your most valid excuse? Also, since when is calculus a part of everyday culture? It sounds like you just want a reason to defend your profession."

Here is my response:

Its a valid point. As a teacher one is of course biased and in general, everybody overestimates and naturally hypes his or her own work or profession.
There is a general principle however: if one wants to replace something, one has to constructively build an alternative which works and demonstrate that it can work on a large scale. Just calling for a replacement is cheap. In the case of calculus, it is not only the results which have an excellent track record (major industries mine calculus today), but also calculus as a "tool to sharpen thinking and problem solving skills" and prepare for other fields.
As replacement of calculus, both discrete math or statistics comes in mind. I myself know that both need solid calculus skills to be used effectively. Whoever calls for using stats as a replacement of calculus does not know stats. Whoever calls for discrete math as a replacement does not understand discrete math. Both fields really only shine if one knows calculus. I love discrete math, (work on it) and statistics (example [PDF]) and even discrete or alternate versions of calculus like here. Unfortunately, in many of todays implementations of discrete math or statistics, the replacement is used as an alternative, the implementation is an excuse to stop practicing harder problems or acquire more sophisticated problem solving skills or to learn harder subjects. Many discrete math courses are a race to the bottom, the reason being the lack of a clear goal to reach. Calculus has the fortune to have a clear goal: the fundamental theorem of calculus (both in single and multivariable calculus), as well as established levels of sophistication like integration skills, knowledge about series and the ability to solve differential equations. Yes, these skills can be hard to reach, but it is worth it. If acquired, the usual discrete math or stats curricula are more rewarding.

Maybe we have to look at history also to see what worked and where things were successful. The biggest advancements in discrete math, physics, statistics,philosophy or computer science were done by people who knew calculus well: Euler invented graph theory and was a master in calculus, Leibniz invented determinants and a computing device and a master in calculus, Newton figured out the laws of gravity, and was a master in calculus, Kepler figured out the laws with which the planets move and was a master in calculus, von Neumann invented modern computers, game theory and was a master in calculus, Archimedes invented countless of machines and was a master in calculus. Kolmogorov wrote the first textbook in probability and was the first put the subject on a solid foundation and was a master in calculus. Riemann dove into the deep mysteries of the prime numbers and was a master of calculus.