Mathematical Reflections by Christopher Havens
Last night, during my studies of category theory, I found a passage in my textbook which brought me into some neat thoughts. It reads: “Mathematical discovery is by no means a matter of systematic deductive procedure. It involves insights, imagination, and long explorations along paths that sometimes lead nowhere.” by Robert Goldblatt
I love this passage because I’ve experienced this intimately. My first discovery was born from tying to solve the continued fraction whose partial denominators make the sequence of natural numbers. If you’re familiar with number theory, it looks like [1:2,3,4,…] . Also my favorite irrational number! But it has been solved using things we call Bessel functions. Really, solving it in terms of integrals and infinite sums is not hard. But I wanted to find an explicit expression in finite terms.. like e or pi. I tried for a year! Failing and failing at every avenue I took. I even tried to invent nonsensical methods hoping to manipulate the fraction into giving me what I wanted. I kept notebooks o my attempts at solving this problem. I then tried to solve a general case.. I figured if I looked at ALL solutions, I could find THE solution. Here was where things fell apart.
I built closed forms for classes of these fractions. But what I learned in the process was that there WAS no solution in the terms that I was looking for. It really did seem like a long exploration on a path that led nowhere.. well. That’s not entirely true. Although there was no solution to my year long problem, it marked my most beautiful failure to date. The closed form formulas I made had never previously been found. It was all new math. the next step was to prove what they were. That was also a long time! But at this point, I had entered into a collaboration to mature my results so that they were ready for publication. When all was said and done, this research gave several new theorems to the beauty that is number theory. It was all because of my exploration into the secrets of the constant [1:2,3,4,…]. I’ve wanted to name this constant ever since.. because it inspired and led to new discoveries in mathematics. Let’s worry about the name later.. but denote the irrational number [1:2,3,4,…] by the lowercase Greek letter Xi (sorry guys, I don’t have the symbol).