Robert Brooks: The Wizard Of Shape, Part 2.

“A man’s reach should exceed his grasp

or what’s a heaven for?”–Robert Browning

RW:  Perhaps you could tell us something about the origin of topology.

RB:  The founder of topology is considered to be Henri Poincare, who had a very strong philosophical outlook towards mathematics and science.  He was led to topology through differential equations, and, in particular, by the “N-Body Problem”, which depicts gravitation.  This situation raises the following questions:  What’s going to happen to these “N”(number) of bodies in the long run?  Are they going to settle into some definite pattern?  Are they going to collide with each other?  Are they going to fly off?  The prospect of trying to solve such a system of equations is just preposterous.

RW:  Is it because there are too many variables involved?

RB:  Yes.  Poincare also wrote a beautiful essay on probability in which he gave the following example:  Imagine you have some large number of molecules of gas A in a bottle, and you ask, “What’s going to happen to this gas in the bottle?”  Then, you make up some experiment in which you have two different gases in two different parts and you remove the partition.  Now some people believe(Albert Einstein:  “God does not play dice.”) if you could find the location of all the particles and work out the equations, you would know how the system evolved.  But the system is so complicated and the equations so difficult, all you can do is guess.  Poincare wanted to show that you get the right answer precisely because you don’t know all of the factors!  It’s the kind of paradox that he always seems to be looking for in mathematics.

On the one hand, you have a determinist point of view, “Well, if we can just sit down, and calculate all these things, we’ll know the right answer.”  On the other hand, Poincare wrote that just by stepping back from the situation, and relying on insight without getting distracted by all the little things, that you get a very clear picture of what’s going on.  And that’s what topology is all about.

RW:  Now in topology are we talking about the position of points, and the possible arrangement of shapes?

RB:  Well, I tend to think about topology as being certain mental habits;  certain ways of thinking about things.  Let me go to these shapes hanging on my door.Both shapes are built out of the same fundamental pieces, in this case a figure “H”.  So the H’s are the building blocks of these surfaces.The blocks are put together in a fairly clever way, so that on the one hand,they’re the same, and on the other hand, they’re different.  They’re the same in that they make the same sounds, but they’re different enough to make different shapes;  one’s longer, and the other’s fatter.  So we put similar pieces together in different ways, and say. “What can happen so that they’re the same in some ways and different in other ways?”

RW:  But topology has this other characteristic that Michael Guillen points out in his book, Bridges to Infinity, that basically deals with the souls of geometric objects, those aspects of their geometry that don’t change under certain kinds of transformations.

RB:  Yes.  It goes back to the image of Poincare and the molecules of gas.  You step back from the millions and millions of little interactions that you have no control over and look into the soul of these objects.  That’s the beauty of topology.