Making Those Distinctions: 2 is not 3  

By Bobby Neal Winters
As I’ve mentioned before, I am a low-dimensional topologist.  At least I was before I began spending my time shuffling papers.  Doe the quality of being a low-dimensional topologist persist over time?  Perhaps it can be proven or perhaps I am an example that it does not.  
But I digress.
My time away from doing mathematics has given me time to think about mathematics.  Does a eunuch think more about women than other men?  I am not sure I want to research this. In any case, when I was doing mathematical research, I didn’t take the time to reflect on it. Perhaps that’s why I wasn’t more successful.  Now that I am viewing it from a distance there are things I ponder.  One of the things I ponder is the methods by which mathematicians make distinctions.  
In low dimensional topology this comes to the forefront because so much of the subject deals with pictures and mathematicians distrust pictures.  Pictures can be deceiving either purposefully or accidentally.  As a result of this, we make a practice of translating things from pictures to something that is closer to words.  We create artificial languages in which we can carry on our arcane conversations.  Sometimes differences which are easy to see (but difficult to be sure of)  in the picture are difficult to see (but absolutely certain) in our symbolic language.
Consider the following two objects:

2-holed torus

3-holed torus

These are called the two-holed torus and the three-holed torus, respectively.  They are generalizations of the torus.  A torus is a surface that looks like a donut and I will omit the picture, assuming everyone has seen one.
These two surfaces are different.  I mean, look at them.  One has two holes and the other has three.  As an old professor of mine used to say, “A blind cow could tell them apart.”  Indeed, once it has been proven, we can say they are different, but such are the standards of the subject that we can’t just assume this a priori; it has to be proven.
Okay, how?
There are a number of ways.  Consider something called homology theory. This is complex (and that is a great pun for the in-group, but unless you want to spend a couple of semesters working at it, let it go). You begin by creating models of your surfaces using triangles.  One can then obtain abelian groups from these triangles.
Okay, I’ve just slipped in the concept of “group” along with the notion that there are special groups that are referred to as “abelian.”  Let it go.  Let it go, I say.
I can’t.  
A group is a mathematical object that has a binary operation that is modeled on multiplication.  The positive real numbers with multiplication are a group.  Those of you who have been abused by being taught about matrices will take comfort in the fact that two by two (n by n, actually) matrices with a nonzero determinant form a group.
Now, those of you who have studied two by two matrices may recall that they differ from real numbers because A times B is not necessarily B times A.   Groups where A times B is always equal to B times A are called abelian.  If you didn’t know that before, now you do.  Among those abelian groups there are free abelian groups.  To describe them would make this too technical and you might rightly fear that prospect.
One can use homology theory to associate a different abelian group to each of the surfaces shown above.  The two-holed torus can be associated to a free abelian group of order two and the three-holed torus can be associated to a free abelian group of order three.  
And, yes, the number of holes of the torus will always be the same as the order of the free abelian group that it is associated with.  It’s kind of happy (or ironic depending on your mood) that it turns out that way.  We are allowed to know that two does not equal three when we are in the world of free abelian groups, but we are not when we are just looking at the pictures.