A friend asked me a wonderful question recently.
> I feel like math is so wonderful and useful to a certain point. (Like sixth grade.)
> Then it seems abstract... I can't see the purpose.
She was very much right that mathematics becomes abstract at that point.
And that is very much the point of it. It allows us to think abstractly. It allows us to find patterns in many different things that work the same way. That allows us to think a problem through in mathematically form and then recognize that form again and again.
One good example is modular arithmetic. We could call it clock arithmetic and talk about integral hours that wrap around the 12 hour clock. We could build an addition table that tells us everything there is to know about that kind of arithmetic. We could extend that addition table to be multiplication by repeated multiplication.
That is all well and good. But it only tells us about clock arithmetic on a 12 hour clock. It doesn't tell us as it stands about clock arithmetic on an 11 or 13 or 24 hour clock. It doesn't tell us about arithmetic on a clock where the time isn't just an integer, but can be between the hours. Maybe we don't care about those things because we don't have to tell time on a 17 hour clock very often.
But what about the fact that the original arithmetic that we started with on our 12 hour clock also works just fine for .... music on a western scale.
Should we spend the same amount of time reinventing all of what we learned about the 12 hour clock when it is exactly the same as for musical scales? Or should we re-use that knowledge?
Well, to re-use that knowledge, we have to stop a moment and erase all the places where we originally said "clock" or "hour" or "one day later" and replace them with the integers from 0 to 11 inclusive and replace "move clockwise one hour" with "add one, reduce by removing 12's". This abstracts the original system which can make it harder to learn, but it also makes it much more useful because we don't have to learn it again and again.
But what if somebody asks about pentatonic scales? Or about microtones? Or the 35 note scales that were talked about in the sixties?
Well, if we had started by abstracting away the number of hours on the clock and abstracting away whether the numbers were integers or real numbers, then we would be able to answer those questions instantly.
And if we had abstracted the idea of rotation a bit more, we would see that rotation around a circle can be generalized into rotation around a sphere or even more complex things. Now our clock arithmetic can solve navigational problems and get us home on a dark night.
But with just a bit more extension, that same clock arithmetic (now with 4 hours and 9 dimensions) could solve a Rubik's cube with about 5 minutes of thought.
None of the details here matter. For instance, the fact that 12 is not prime plays a big role in the nature of the arithmetic that we get on clocks, but that isn't the point of all of this.
Now, this abstraction is really frustrating because you can't always point to something and say that it is what you are talking about. In fact, the point of abstraction is that you aren't talking about something specific. This is exactly what makes it hard to teach abstraction using manipulatives. Manipulatives are all about grounding intuition in naive physics. Abstraction is all about UNgrounding your intuitions.
The point is that mathematics is all about abstraction. It is just a mechanism so that we can repeat what once appeared the work of genius without having ourselves to be genii.
So is it useful for most people to understand these wheels within wheels that turn behind the world we see?
To me, yes, it is useful and beautiful and wondrous. But I can't speak for others.
I do think that anybody with a spiritual tendency is turning away from the hand of god if they choose to not see all the kinds of order in the world.