Sophomore Year.

First semester Sophomore year, I took my first real math class. I'm not saying Calculus isn't real math. It's just the ball pin at Chuckie Cheese. This class was like the first wooden roller coaster you ever go on. It was called Foundations of Mathematics. It's goal was to teach you how to prove mathematical statements. It was a peep hole into how complex, deep, and beautiful mathematics can really be. For the final, the first question was this:

What is a proof?

This is what I wrote:

A proof is a special kind of explanation of some mathematical idea or observation. It is an argument that some statement is true. The difference between a proof and a regular explanation is the ability to say that you know something to be true instead of you think something to be true. In order to be a good proof, it must leave the reader with no doubt that the given statement is true in all cases. It must also be clear and concise, and every step should be explained. This is where mistakes can occur in proof writing. Some common errors occur by using circular reasoning or logic, or by assuming too much about the statement. However, when done correctly, proofs are beautiful and useful mathematical necessities.


There are many reasons why we prove things. One reason is to show 100% certainty about something. This lets us know if a statement is simply plausible, or if it is something we can rely on. Another reason is to correct mistakes. If you make assumptions about a statement just by looking at patterns, then you can’t say that you know it’s always true. Mathematics is a pure science, so experiments may fall short for high numbers. Therefore, proofs are necessary. Another reason is to gain insight. If you force yourself to look at a statement more closely, it forces you to look at it differently and learn more about it. Proofs help you understand not only the given statement but the process to get to the given statement.


While everything above is true, there is a deeper meaning to proofs, at least there is to me. Proofs allow mathematicians to separate themselves from every other science. We don’t just rely on patterns or experiments, we need 100% certainty to believe something. It also connects us to other sciences. Once mathematicians prove a statement, other scientists can apply them to real-life applications. Mathematicians do the dirty work. Proofs are awesome explanations to why something is true. Actually not just something, but something we are interested in, something we love. That’s the real reason we prove things, because we love math!

Interesting fact: I didn't type this answer. I hand wrote it on loose leaf. My professor liked my answer so much that he typed it. Then he sent it to the other professors, and it soon spread through the department.

Can you believe it?

I didn't actually find this out until over a year later. I just answered the question honestly and wholeheartedly. Apparently it's still the best answer they've ever seen. And I just got it in my possession, for the first time in two years, tonight.



When I read this, I'm surprised at how insightful I was two years ago. But I'm not surprised at how much I've grown as a student of mathematics.


I think this answer was the beginning of a beautiful love affair...