Math Thoughts from Jake Marcus

Jake Marcus landed on this blog because his aunt Margy Rochlin told him about it.

Margy is my first cousin and a full time writer. She and I were talking about my algebra class and my blog and writing. Jake graduated from Yale in spring 2008 with a degree in math so it was natural that Margy mention my experiment.

Jake thinks about the arts and philosophy and other countries and cultures, and he thinks a lot about math.

So Jake visited here and wrote a long letter to me that I want to share parts of with you. There was a lot there, so I'll try to share only some of the highlights in bite size chunks.

Jake's words are in purple.
---------------------------------------------------------------

Math is the product of making ideas as precise as possible.

If you take any argument and try to clarify its assumptions, define all of its terms, strengthen each step of the argument so that any one step indisputably leads to the next, then you will find yourself doing math.

What really got my attention was that Jake was thinking about "ideas and precision" rather than "counting and precision." I'm sure I can't yet appreciate what he is thinking, but I want to and I think that those couple of sentences alone might sum up a good part of my attraction to math as a subject.

One other thing Jake wrote summed up something else I've been thinking about:


A lot of the creativity in math is picking the right definition for what you are interested in exploring. Once you decide on that definition, then there is one right answer for what that definition implies. Of course, there are many ways to pick that definition in the first place, some much more reasonable than others, but no one right definition


Jake went on to make a point about precision and the importance of definitions using "volume" as an example. (Jake's writing is fairly clear, but it is still easy to get lost. I did.)

The field tends to focus on simple and idealized concepts like triangles, distance or whole numbers, because it takes so much effort to make even these deceptively simple ideas precise.

Take the concept of volume.

We all have a good idea of what volume means and sure we toss the word around idly in conversation. But what if we want to make volume precise?

We might start by looking it up in the dictionary. Merriam-Webster defines volume as the amount of space occupied by a three-dimensional object. This definition suffices for daily living, but not if we’re sticklers about it. We need a definition that will allow us to make unassailable arguments about volume and its properties.

If we use Merriam-Webster’s definition, someone will inevitably come along and question what we mean by “object” or “space” and use that ambiguity to poke holes in the arguments we try to make about, say, the volume of spheres or donuts or any manner of exotic and absurd three-dimensional shapes.

Let’s use some mathematics to think about volume.

First, decide on some reference point in your house, for example the rightmost atom on your living room sofa and assign it the numbers 0, 0, 0. Now, assign to every point in the entire universe three numbers.

The distance from your reference point walking in the horizontal direction, the distance from your reference point walking in the vertical direction and the distance from your reference point walking upwards towards the sky (imagining that you can walk that way).

Some of these numbers will be very small. You might assign the atom just to the right of your reference point, the numbers .00000001, 0, 0 and the atom just to the left of your reference point, the numbers 0, .000000001, 0 and the atom just above your reference point, 0, 0, .000000001, but you might assign the numbers 1000000, 2000000, 1000000 to some atom in your bedroom and assign some really very big numbers to the points off in far corners of the milky way.

Now we have an x-axis, a y-axis and a z-axis for the universe and can start thinking about collections of those points, or x, y and z coordinates.

The collection of all the points a distance of one or less away from the rightmost atom on your living room sofa is a sphere with a radius of one centered on that rightmost atom.

You can think about any collection of points you want. You can cut out cubes, pyramids, cylinders and cones from the earth as if you were cutting shapes out of fabric.

The question then becomes, how do we assign to each collection of points a number, called volume?

Whatever process we decide on for assigning a number (volume) to different collections of points will in effect define the concept of volume for us.

Suppose you collect some points together and they make a cube of length 2, width 3 and height 4. Then we want our process to assign the number 2x3x4 or 24 to this particular cube and to assign the product of the length, width and height to cubes in general. We can probably agree on a few other properties that our volume-assigning process must have.

There are four properties in particular that seem reasonable for our process to preserve no matter the collection of points:

1) Volume should never be negative, it should be 0, infinity or some number in between.

2) The volume of two distinct objects put together should be the sum of the volume of those two objects separated.

3) A sphere with a radius bigger than 0 should have a volume somewhere between 0 and infinity.

4) If you can make one object exactly the same as the other by rotating it or moving it around, then the volume of those two objects should be the same.

It seems like whatever process we decide on for assigning a volume to different collections of points should observe these four rules.

But here’s the punch line: no process exists that observes those four rules!

No one will ever discover one either. Mathematicians have proved that no such process can exist.

As soon as you imagine a way of assigning volume to objects that always observes one of those rules, it contradicts another. If you assume a process exists that sticks to rules 1-3, then the Banach-Tarski Paradox shows that this process does not observe rule 4, that is a ball can be taken apart into five pieces, each of these pieces can be rotated and moved around, put back together and the ball will be bigger than it was. Granted these 5 pieces have to be really weird shapes that couldn’t be practically constructed (in the real world, we know how to cut a ball in half, but not how to cut a ball in 1/Ö2).

There is no one way to define volume and the task of making whatever definition you decide on precise is a subtle one.

Jake's last point he wanted to make to me was motivating, if not daunting.

Calculus gives some very good definitions for concepts that seem ethereal and unclear. It is not any less precise than other forms of mathematics; rather its strength lies in making precise concepts like infinity, continuous, smooth and infinitesimal.

Jake gave an example from Zeno’s paradoxes. If you're interested in reading about the paradoxes, here's a link.

I'll finish this entry by letting Jake finish:

In mathematics beyond a certain point, how you approach a problem does, I think, matter the most.

Some very famous mathematicians made their discoveries by making connections between fields thought to be completely unrelated.

Evariste Galois saw the connection between mathematical objects called fields, groups and polynomials and with this insight solved geometrical problems that had gone unsolved for thousands of years.

Since the Greeks, mathematicians had wondered with it was possible to trisect an angle (split an angle up into three equal angles) using only a compass and a ruler. With Galois theory, you can prove that it’s impossible.

I remember taking Galois theory and being amazed that the professor showed us how to solve three problems in one lecture that had taken humanity thousands of years to figure out.

Jake, thanks for your thoughts!