They Do Exist! Let me tell you a story about when I was a kid See, I was confused and here's what I did. I said "irrational number, what’s that supposed to mean? Infinite decimal, no pattern? Nah, can't be what it seems." So I dismissed them and called the teacher wrong. Said they can't exist, so let’s move along. The sad thing is that nobody seemed to mind. Or maybe they thought showing me was a waste of time. Then one teacher said "I can prove they exist to you. Let me tell you about my friend, the square root of two." I figured it'd be the same ol' same ol', so I said, "Trying to show me infinity is like making gold from lead" So he replies, "Suppose you're right, what would that imply?" And immediately I thought of calling all my teachers lies. "What if it can be written in lowest terms, say p over q. Then if we square both sides we get a fraction for two." He did a little math and showed that p must be even. Then he asked, "if q is even, will you start believing?" I stood, amazed by what he was about to do. But I responded, "but we don't know anything about q" He says, "but we do know that p squared is a factor of 4. And that is equal to 2 q squared, like we said before." Then he divided by two and suddenly we knew something about q. He had just shown that q must be even too. Knowing now that the fraction couldn't be in lowest terms a rational expression for this number cannot be confirmed. So I shook his hand and called him a good man. Because for once I yould finally understand a concept that I had denied all my life, a concept that had caused me such strife. And as I walked away from the teacher's midst, Excited, I called him an alchemist and exhaled "THEY DO EXIST!"Aside from its lack of poetic content, I think that many mathematicians can relate to this poem, particularly the ones who go into the field for its theoretic principles. For many of us, Set Theory is somewhat of a “back to the basics” course where we learn what math is really about. The focus is no longer on how well you can memorize a formula. Instead, its more of a philosophy course on mathematics – like an introduction to the theory of mathematics, hence the name Set Theory. The poem above focuses on a particular frustration of mine, irrational numbers. Early on, we’re asked to believe that these numbers exist, but we’re not given any answers as to why they should exist. The same could be said for a number of similar concepts though – basically, whenever a new concept is introduced, there is a reasonable question of how do we know this is true. This is not just a matter of practicality, but a necessity of mathematics. I mean I could say “lets now consider the set of all numbers for which X + 1 = X + 2″, but if this is true for any X, then it means that 1 equals 2, which we know is not true. So the set I’d be referring to is the empty set. We can still talk about it, but that’s the set I’d be talking about. So why is this concept of answering the why’s of mathematics ignored, sometimes until a student’s college years? This gives students a false impression of what math really is, which leads to people making statements like “I hate math”, not really knowing what math is about.
Growing up, I never really liked math. I saw it as one of those necessary evils of school. People always told me that if I wanted to do well and get into college, I needed to do well in math. So I took the courses required of a high school student, but I remember feeling utter confusion from being in those classes. My key problem was my inquisitive nature. I really didn’t like being “told” that certain things were true in math (I felt this way in most classes). I hated just memorizing stuff, or memorizing it incorrectly, and getting poor grades because I couldn’t regurgitate information precise enough. If this stuff was in fact “true”, I wanted to understand why. It seemed like so much was told to us without any explanation, that its hard to expect anybody to just buy into it. But that’s what teachers expected. And I was sent to the principal’s office a number of times for what they called “disturbing class”, but I’d just call it asking questions. At the same time, I was taking a debate class. This class was quite the opposite of my math classes, or really any other class I’d ever had. We were introduced to philosophers like Immanuel Kant, John Stuart Mill, Thomas Hobbes, John Rawls, etc. The list goes on and on. We discussed theories, and spoke of how these concepts could be used to support or reject various propositions. Although these philosophies were quite complex, what I loved was the inquiries we were allowed to make into understanding the various positions. Several classmates and I would sit and point out apparent paradoxes in the theories. We’d ask about them and sometimes find that others (more famous than us) had pointed out the same paradoxes and other things that seemed like paradoxes could be resolved with a deeper understanding of the philosophy. Hate is a strong word, but I remember feeling that mathematicians were inferior to computer programmers because “all math could be programmed”. This was based on the number of formulas I had learned through high school and I remember having a similar feeling through my early years of college. But things changed when I took a course called Set Theory. Last year, I wrote a piece that somewhat describes this change: