The Absolute Nature of Mathematics

I received this email questioning a statement made in a book about black holes. The question and the concerns the of writer suggest a closer look is necessary before I answer. But I will respond to the tone of the statement until I verify the conditions of the actual problem.

The Email:

My plea for help: The only way I have the remotest way of learning about what's going on in the real sciences - mathematics, physics, chemistry, biology, biochemistry, and so on, is to read what their practitioners write for lay people. Fortunately, there is a lot of such writing produced. As a result, I don't always feel entirely in the dark. But one of the things about reading their work that irritates me is the way they often write about absolutes, about certainties, as if they "know" and that's the way it is. Period. Of course, I have no way of challenging them, because I can't do the mathematics that underlie their bald-face assertions. I have to accept their declarations and move on. But one of the things that brings me pause is when I run into such assertions in situations where common-sense applications don't square with what they say. Sometimes, when that happens, they explain why the common-sense solution doesn't apply. But other times they don't explain it at all. They just say, in effect, that's the way it is - without any explanation - and move on.

These things usually happen with very simple statements that a lay person like me can understand. What bothers me, is if they don't explain, and I can't understand why what they say is such an irrefutable truth - and it doesn't seem to me that it is - if they are wrong ( they don't explain why they aren't) - how can I be sure of their more esoteric explanations?

Well, anyway, a book I'm reading now about Black holes, written by Leonard Susskind, made an absolute declaration that irritated me because he didn't explain why it's true.

He was talking about the ways you can divide a rectangle in half. He said there are only four possible ways to do it, and he illustrated them. I've included them in the attachment (by using shading, since that's the only way I could figure out how to do it on my computer). Basically, he showed that you could only do it vertically or horizontally. What I want to know is why you can't do it diagonally, which would give you four more ways - doubling the possibilities? It's a minor point in his argument, but I would hate to think he's careless about such a simple point. Can you help me out on this.

Season's Greetings!

The writer has all the kudos I can muster for his continuance to involve himself in the world of the hard sciences. I do detect a sniff of smugness in his communication about the way scientist/mathematician communicate. Perhaps this attitude is a barrier to understanding how thoughts are communicated with certainty. Without addressing his question about the number of ways to divide rectangle in half until I verify that it is the statement made by the author, I will comment on his reference about absolute statements in the hard sciences.

First there are no absolute statements in mathematics and the hard sciences. Mathematicians generate mathematics is in a world called a space. Assumptions describing the nature of the space determine whether a statement is true or false in this mathematical world. If a statement is true, then mathematicians assert with certainty its truth, only within the discussed mathematical world.

As an example, a triangle is an object that lives in a plane, a flat surface. In a plane, the sum of the angles of a triangle always total 180 degrees. Furthermore, a triangle can live on the surface of a sphere. On the surface of a sphere there are triangles whose angle sum total 270 degrees. So the angle sum total is absolute only within the plane. As in this case when one moves from one mathematical landscape to another, the truth of a statement may change.

For more on this topic, click this link.

Thank you for the question. Merry Christmas!