Truth, Beauty and Practical Math

In an earlier post, I discussed whether math was really real, or just made up.  I came to the conclusion that there’s a difference between the math that is implicit in the laws of logic, and the math that people happen to study and learn. The difference is that the latter is much much smaller than the former.

The Math We Know is a small subset of the totality of All Possible Math. It includes just the stuff that

• We find useful, OR
• We find interesting, AND
• We don’t find too difficult.

However, there’s a lot more mathematics than just this. Most of the math we find useless, boring or too difficult, well, it never sees the light of day. Nobody ever hears about it.

The way something enters the Math We Know is like this.

• First, some Jane (or John) Doe studies as much mathematics as in known in some particular speciality.
• Then, she gets motivated to solve some particular problem that occurs to her, or that someone else poses. She asks other mathematicians, but it seems that nobody has solved the problem before.

Her motivation might come from a number of different places. Perhaps the problem is a very practical one, so some business, government department or royal court is ready to pay Jane a huge amount of money for the solution. Perhaps a lot of other mathematicians – or Jane herself – need the solution to solve their own problems. These other problems might be in math, or might be in other fields – astrology, economics, ecology, bioinformatics or whatever. Or perhaps Jane just finds the problem fascinating – a challenging and beautiful puzzle that captures her heart and mind.

When Jane is motivated to solve a difficult mathematical problem at the forefront of human knowledge, she takes her quill and parchment, or biro and notebook, or phone and stylus and gets to work. Let’s suppose that after a few weeks or months (or even years) of hard thinking and false starts, she finally solves the problem! The details of what happens next depend on exactly when in history Jane has been born. However, in broad brush strokes, it goes like this.

• Jane is the first person ever to solve this mathematical problem. Maybe she applies her solution to do something practical, maybe not.
• She tells some other mathematicians about what she has done. If the problem is interesting or useful, they listen, learn, pass it on and build on it.
• If Jane’s work turns out to be very useful or very interesting, it eventually starts being taught in institutes of higher learning.
• If it’s really really really useful, it eventually ends up boring the skulls off millions of first year college students who will never quite understand why there are calculus or statistics units in their engineering or psychology course.

Note that if the problem was not useful or interesting, Jane would never have tried to solve it, and the solution would not have become part of the Math We Know. Likewise, if the problem was too difficult for Jane or anyone else, it would again fail to become part of the Math We Know.

Therefore, the Math We Know consists of just a tiny bit of what we might call All Possible Math. To enter into the realm of the Math We Know, a mathematical truth must pass through one of two doors. the first door is to appeal to Homo Sapiens‘ sense of wonder and intellectual challenge. The second door is to solve some practical problem that (by an accident of history) we encounter. Once the truth has passed through one of these two doors, there’s a third that all must pass through – the problem must be sufficiently simple that our brightest and best can tackle it with the tools they have been given.

The good news in all this for educators is twofold.

• Any piece of math you teach – any known piece of math – was found, once upon a time, to be beautiful or useful to somebody, somewhere. It is therefore possible to motivate students to learn it, by helping them see the beauty or the use.
• Any piece of known math is, in fact, possible for the human mind to grasp. It is therefore possible to enable students to learn it. After all, someone, somewhere, at some time in history, managed to grasp it without the help of any teacher. True, they were probably brighter than some of your students – but they didn’t have you to help them, either!