The Journey through Terra Incognita

By Bobby Neal Winters

The subject of mathematics offers us a wonderful mixture.  It teaches us down to earth useful skills, but it also teaches us the disciplined use of imagination.

I love trigonometry. I took a class in it in high school back in 1978.  That would be 45 years ago.  I have good friends who are adults of some level of achievement that have not been alive as long.

Trigonometry has two parts to it.  One part is its use for measuring physical objects.  If you know how far you are from a tower, you can measure the angle from the ground to the top and use trigonometry to tell you how tall the tower is.

That’s just cool.  Students of a practical mind can get their teeth into that and chew even if they don’t think it’s cool.

The other part consists of functional identities.  Let me explain what I mean by that. You see, you’ve got these trigonometric functions sine, cosine, tangent, secant, and cosecant, and there are all sorts of formulas about how they relate to each other.  A large part of a well-taught course consists of proving those formulas are true.  

You can go on for hours and hours filling page after page with formulas that would make a hawk dizzy.

While I think the first part is cool, I think the second part is really cool.  But that sets me apart as a person with a peculiar turn of mind.

The thing is: When you learn the first part in your initial course, that is just about all there is to it.  What you see is what you get. You’ve got something that is quite useful in the practical world, but you are done. It’s over. Fini!

The functional identities are a doorway to a whole new world. (Key music from Aladdin: “A whole new world...’)  The functional identities are used in integral calculus, Fourier series, vector analysis, complex analysis, differential equations, quantum mechanics, et cetera, et cetera, et cetera.

A freshman in trigonometry will look at these functional identities and say that they are useless. They will cling to the measurement formulas as being the only part that should be taught. But the stone that was rejected by the builders has become the chief of the corner.  Where did I hear that?  I think another teacher said it.

Mathematics puts things in order.  It finds patterns.  While this statement will put quite a few chins on the floor, I will say it anyway: It seeks to make things easy.

Every once in a while we find a pattern we can extend and extend and extend.  It will go on like Michael Pena’s stories in the first Ant Man movie.  It goes all over town, all around the world, but then it comes back to the here and now and points out a truth.

In mathematics, we create a language in which we talk about things that we can’t see, things that no one can see.  But we are careful.  We mark our paths with every step that we take.  We map out territory through terra incognita indeed through terra incomprehensibilis.  However wild our maps might be, we are vindicated because when we come back to good ol’ terra firma we are right.

We are living in an odd age.  I wrote that and immediately thought maybe not. Maybe it’s always been this way.  What I mean is that we don’t trust experts, but we don’t have patience. 

On one hand, we praise--and perhaps rightly so--those who won’t just take someone’s word that something is true.  Trust has been lost.  On the other hand,  these same people don’t have the skills to ferret out the truth themselves.  Not only do they lack the skills, but they won’t take the time.

Mathematics, by its nature, teaches skills that allow us to ferret out certain kinds of truth.  But this takes patience. Patience is one of those gifts of the Holy Spirit that is mentioned in the Bible.  I’ve been accused of having too much of it, and I listened to that accusation...patiently.  It might be true; I will figure that out eventually.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.