Derivative of the Natural Number Power Function

The leaves are changing. Fog settles in the morning and lifts as day breaks, allowing sunlight to scatter across gold and orange leaves. The Hudson Valley is so beautiful this time of year. It makes me happy to be alive, to be thinking and working and experiencing all of this.

Days blur into one another. I feel like the year has just begun, right as it’s starting to come to a close. My sessions are a little light some days, and yet I’m busier than ever. About a month ago I was taking a walk and remembered a question I had a few years back, when I was taking calculus. I was never given evidence or proof for why the derivative of sin is cosine, and why for cosine it is -sin. In attempting to write a little proof for this, I realized that I was making many assumptions. Certain limit properties I just take for granted, without having any sort of basis for those assumptions. While we know these properties hold true, if I was going to spend the time to prove something that we also know, namely the derivative of sin or cosine, I might as well find the justification for these limit properties as well. Mathematical truth cannot be assumed; it must be found definitively and worked out. This endless need for proof is the beauty in all things. It’s the reason I love mathematics. The fact that you can start atomically, from singular basic principles, and build outwards is amazing.

With this need in mind, I took a break from what was becoming a rather long document investigating these derivatives and began reading Terence Tao’s lovely, if mathematically intensive book Analysis 1. I relearned proofs by induction, which is currently one of my favorite things, and grappled with a variety of set theoretic ideas. Realizing that there was a lot of set theory I did not know, and that would be beneficial to understand, I paused around page 60 and started on Enderton’s Elements of Set Theory. This book is gentler than Tao’s, but no less confusing in nature. Like machine code in computer science, set theory lays the groundwork for all of higher mathematics. Sets, functions, and even classes appear to be everywhere in analysis and algebra.

Despite the difficulty of this material, certain axioms and ideas are beginning to make sense. The Axiom of Replacement, Axiom of Regularity, Axiom of Extensionality, and so on, are all starting to feel like second nature to me. Just like understanding machine code as a software engineer isn’t generally a necessity, but may make you better at your job, I feel that knowing the ins and outs of set theory may result in overall higher mathematical ability for myself.

There’s something horrifying but strikingly beautiful about set theory. Because it underpins the rest of mathematics, there are many ideas within it that seem to run counter to intuition. Different kinds and levels of infinity, the Banach-Tarski theorem/paradox (of which VSauce made a fantastic video), Russel’s Paradox within the context of naive set theory, and more come to mind. Even the idea that there is no set containing all sets, which lends itself to some mathematicians working in a type of set theory known as von Neumann–Bernays–Gödel set theory(or NBG for short), which insists on the existence of these seemingly impossible to define containers for sets known as “classes.” Why can’t we have a set containing all other sets? Because while you can put as many objects as you want in a box, the only thing you can’t put in the box is the box itself. You can’t have a set containing all sets, because that would imply that it contains itself, which leads to all sorts of logical issues.

The truth is, beneath all the shiny luster of theorems and proofs and “objective reality,” mathematics is an ontological nightmare. In reality, we can’t really know anything with certainty, because the very framework that all of this is built on seems rocky at best. From a theoretical perspective, this is troubling, though from the viewpoint of practical application, it’s all just good enough.

Anyways, I’m having a good time and learning a lot. Most likely some or most of the material in this little post is wrong, and maybe I’ll come back and correct it later if I find issues. Or maybe not; perhaps it’s better to leave as an artifact of my ignorance at this point, as evidence of growth to my future self.

In the spirit of showing what I’m currently learning, I’ve included a handful of proofs from the early sections of Enderton’s short book. Though they’re just a few of the many problems he poses (or variations of them at least), these are a few I felt were the most interesting so far.