047. Natural

tldr: obvious, trivial, clear

Every few weeks or so, ‘Math’ is trending on Twitter. Though, on occasion, this is due to someone claiming they solved the Riemann Hypothesis, more often than not it’s some PEMDAS problem that won’t ever have a fully right answer given that the question itself is purposefully misleading. To this, there seems to be three common responses. 1) Someone arguing why they are right and others are wrong, 2) those who choose to ignore the problem, and 3) those who take it upon themselves to explain why the question itself is flawed. Then, inevitably, ‘Math’ drops further and further down the trending tab as people finally decide that it doesn’t really matter; or at the very least, forget about it entirely.

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I have two shirts related to math. One that says “Zero is in the natural numbers”, and another that says “tau is greater than pi”. On occasion, this will lead to a mini-debate about the naturality of zero, or why pi is actually better than tau. But in the end, of course, this debate doesn’t particularly matter. In the end, whether or not 0 is in the natural numbers won’t affect the cardinality of the set of natural numbers; and it will always be true that tau>pi, given that tau is defined (in the context of this shirt) as 2pi.

To me, these debates (the one on Twitter, and the other regarding my shirts) are one and the same. Arguments that to some cause annoyance, to others confusion, and eventually, to everyone, negligence. The issue with arguments like this though [and the reason I am blogging about this], comes with those who simply dismiss the question entirely. Not those who simply don’t care; those who see the problem, realize its flaw, and feel a sense of superiority when they choose not to discuss “such frivolities”.

At most, including zero in the natural numbers or using tau instead of pi will simply change one’s mathematical notation in a problem/book/writing. And yet, there are deeper questions at play. We use the Peano Axioms to construct the natural numbers, and Peano originally assumed that 1 was in N. But why couldn’t we have constructed the Peano axioms such that the natural numbers start at 0? Why does 2pi (or, tau), show up everywhere in mathematics? So much so, that there is an entire manifesto arguing for the use of tau over pi?

While in the end these questions don’t matter, I think the conversations do. At the root of all of these questions, and the conversations that can occur because of them, we start to get a deeper appreciation of the fact that our mathematics was created. Maybe the natural numbers existed before we wrote them down, and perhaps calculus was waiting to be discovered. But when we rigorously defined these concepts, we inherently created our own system to understand the world around us. Something about that, to me, feels beautiful.

Why does mathematics work the way it does? Because we created it do. In mathematics, we are the Gods who decide the answer to unanswerable questions. We control this tiny aspect of our uncertain world. And yet, there are those who think they are they are better for dismissing these questions as trivial. In my opinion, this belief is exemplary of a deeper issue in the mathematics community: we need to talk about how we talk about math.

At this point, it is almost a meme for a mathematics textbook to state, at one point or another, “This proof is left as an exercise to the reader”. This can feel extremely frustrating when you’re just trying to learn the material. But if you ask a professor about this ‘meme’, they may likely tell you that “The proof is important for mathematicians to work out on their own”.

So. Why. Not. Simply. Say. This.

Why not have a footnote, or a sidebar in the preface, that says “When I say a proof is left as an exercise to the reader, it is because I believe the reader can solve the proof, and that there is something to gain from doing so.” To me, it feels like these questions are dismissed as obvious, trivial, or inherently clear when often times they aren’t. Which hurts as a mathematician.

Last semester, I got through 18.155: Differential Analysis I with Professor Dyatlov. Professor Dyatlov is an amazing teacher if you can ever take a class with him I highly encourage it. And on occasion, he would leave a lemma or corollary to us to prove, and there were these two graduate students who would laugh. They would hear that this exercise was left to us, and would laugh at the idea that this proof would even be considered an exercise. They had seen the material before, and had to make sure everyone knew. I just wanted to *yell* at them. I wish I had told them that what they were doing was wrong.

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But I didn’t; I dismissed this argument as unimportant. Why have the conversation, when in the end it doesn’t matter.

Right?