Why Study Higher Mathematics?

Just about everyone will tell you that mathematics is important and should be taught in school.  All the arguments are over the type of mathematics that should be taught.  Algebra?  Statistics?  Probability?  Real world problems (“John has 137 erasers to distribute among 17 students…”)?  Math world problems (“How many ways can Ellen arrange six different books on a shelf…”)?

The problem is that you can’t answer the question of “What should we teach in school?” until we answer the question of “Why should we have an educated population?”  There are three reasons for getting an education:

These goals aren’t mutually exclusive:  If you help yourself, for example by getting a better job, then you help the government, because you pay more in taxes (and, because you have a better job, you are more likely to want to keep things the way they are…so you support the government).  Likewise, if you help your community, you’re helping yourself, because you’re making your own living environment better.  At the same time, helping the community might not be helping the government, and vice versa.

This is where things get interesting.  If school mathematics focuses on “real world problems” and “useful mathematics,” then it’s set up to help individuals and the government.  However, there’s no guarantee that this will help the community:  wealthy persons exist, even (and perhaps especially) under repressive regimes.

Here’s why higher mathematics is important.  First, a quick definition (my own):  Higher mathematics is any mathematics that focuses on proofs.  Every branch of mathematics can be taught and learned at a higher level:  3 + 2 = 5 is arithmetic and very basic; higher mathematics occurs when you ask (and answer) why 3 + 2 is 5.

I’ve talked elsewhere about why mathematicians do proofs. Despite my silver-tongued eloquence, not everyone is convinced:  students continue to ask “Why should we have to prove things that everyone knows?”  But consider this:  Throughout history…

• Everyone knew that men were smarter than women.
• Everyone knew the sun went around the Earth.
• Everyone knew that slavery was acceptable.

Progress occurred when people began asking “Well sure, everyone knows these things…but are they really true?”

And this is one of the things that higher mathematics trains you to do:  Proof causes you to ask questions about what “everybody knows.”  Take that “3 + 2 = 5”.  What do we really mean by that?  There’s a few different answers, but one of them is “3 + 2 is the second number after 3.”  (You can go in a few different directions after this point, but that’s another story)

We can go further:  proof requires us to construct a logical argument, with each step based on the step before it.  We can’t make bold leaps, like “Since this happened one time, it must happen all the time.”  Instead, we have to establish a chain of causality, with each step carefully constructed.

We can go further still:  proof forces you to constantly question your own beliefs.  Many people accept $5 \times 3 = 3 \times 5$ without question.  But if you’ve been taught to do proofs, there’s a little voice inside your head saying “Why do you believe that?”  If the answer is “Because someone told me it was true”, then you’re uncomfortable…and try to find evidence.

Now for the punchline:  You can slant history to make sure your country is always right, and other countries are always wrong.  Philosophy can be wrangled to serve the state:  China did this for two thousand years.  Even science can be forced into doctrinal correctness:  physics and chemistry did very well under the Soviets.  The arts can be browbeat into submission; literature can be stifled; engineering can be co-opted.

But questioning belief and demanding evidence is an integral part of higher mathematics.  You cannot remove these components from higher mathematics without destroying higher mathematics.

And if the state chooses not to teach higher mathematics?  Then the mathematics needed to solve new problems will not be developed.  Individuals and society will suffer…and the government will be replaced.    Thus I claim:

The development and continued existence of a free society depends on the teaching of higher mathematics.