I’m A Mathematician and I Hate Numbers
By: Daniel Goldman
Tags: Education, Learning, Philosophy, Science, Mathematics

A Rendering of a Calabi Yau Manifold

Oh how wonderful numbers are. You can add them, subtract them, multiply them, divide them, blah blah blah. Okay, so I can’t stand numbers, and I hate accounting. But I’m a mathematician and I love mathematics. Lucky for me, dealing with numbers isn’t really that important in mathematics. Sadly, I’ve run across so many people that comment about mathematics and how mathematics is all about numbers and calculations. I blame the education system, but I digress.

I really want to get people to understand what mathematics is, and why I love it so much, even though I’m not a fan of number crunching. And maybe after reading this article, some people who say that they don’t like mathematics will find that it’s accounting, not mathematics that they dislike!

It’s true that mathematics has much of its origins in calculation. Abstract algebra, group theory, topology, and so much more comes from abstraction of everyday operations that we do with numbers. But mathematics has evolved into a field of study all its own.

Mathematics is really an investigation. Mathematicians investigate the very fabric of logical consequence. We take basic assumptions and definitions, and forge conclusions and proofs. But we’re also artists. And every mathematician has a unique way of going about doing mathematical proofs. It’s like a fingerprint. The specific way that we construct proofs and reach our conclusion varies based on our instruction, as well as our own personal way of thinking. So, contrary to popular belief, there are very often numerous correct ways of doing things.

There are three components that exist at the heart of mathematical inquiry: axioms (things taken as true as a matter of fact), definitions, and systems of logical consequence. By using logical consequence we can move from our axioms and definitions towards various theorems and proofs. Through this method, we are able to construct our entire system of numbers that we use every day. We’re able to describe probability and explain how it seems to work. We can define various geometries, including the standard euclidean geometry — geometry on a plane — spherical geometry, and many even more complex geometries.

But what if our axioms are wrong? The funky thing about mathematics is that there really isn’t such a thing. “Wrong axioms” don’t exist. All of our truths in mathematics are dependent on what axioms we select. And there are alternatives in many cases, which makes mathematics even more interesting. Because mathematics started as a study of our own reality, the axioms that we generally use “make sense.” But they don’t have to. We can have very bizarre constructions which seem to defy any sense of reason.

It’s really this area of study that interests me the most, especially if the axioms that we thought properly explained reality don’t really hold true. For more detail on what I mean, I would suggest reading Ramblings on a Paraconsistent Reality.

Numbers are studied in mathematics, but even when they are, it’s generally from a more abstract point of view. For instance, no actual number is used in proving that there are infinitely many primes. It’s all a matter of logical reasoning and abstraction of concepts.

One of the common ways of showing that there are infinitely many primes is to assume otherwise, and show that this assumption reaches a contradiction. This proof was developed by Euclid over 2,000 years ago! (UTM)

Proof: If we assume that there aren’t infinitely many primes, then there must be a largest prime. So we can take all the prime numbers, up to the largest one, and multiply them. If we add one to that product, it must be a composite number (a product of prime numbers) since it’s larger than the largest prime! So there’s some prime that divides this new number. But it cannot be any of the primes used in the product, since it does not divide one. But that means there’s a prime that we missed, and we constructed this number so that we don’t miss any prime number. So that can’t be right, so there must be infinitely many primes.

Sure, I used “one” in this proof, but aside from that, everything in the proof is abstract. I didn’t even really do any actual calculations either. But this kind of proof, which is technically a number theory proof, is a basic type of proof that a mathematician would learn how to produce. So the next time someone tells you that mathematics is all just numbers, or that they don’t like math because it’s number crunching, you’ll know better.