10 December, 2018

Slow Growing Functions

I'm a terrible amateur mathematician.  Sure, I watch Grant Sanderson's 3Blue1Brown videos for fun, but I never pause them to work out the math on my own. I participate in math forums occasionally, and every once in a while something I play around with gets some press (e.g., when I helped in a thread with Ed Pegg, Jr., and Laura Taalman with determining that the scutoid shape always has non-planar faces, for which Taalman's 3d print model was later popularized in a Matt Parker video), but to be honest, these are just nothing more than weird flexes. Beyond the thesis I wrote back in my school days about applying GĂ¶del numbering to Aristotelian logic (which had no discernable practical applications), I haven't added anything novel to the field of mathematics at all.

Nevertheless, I love math. There's something about the way you can navigate its simple rules and come up with surprising results that makes me feel excited and full of genuine wonder. I enjoy board games and video games for much the same reason: I like to play around with rulesets and see what comes out. But mathematics has an unreasonable effectiveness when it comes to reality that few other invented systems have, so it occupies a special place in my heart.

Six years ago, I found myself talking with my friend Dale about extraordinarily large numbers. The conversation prompted me to write a short blog post on the topic. It was written just for my own enjoyment, but a number of better mathematicians than I got their hand on it and wrote a few discouraging words. One commenter in particular pointed out a few errors in the last few paragraphs of my post, and then, after I replied and edited my post, they wrote: "I'm sorry if I came off a little brusk and harsh. It's good that you're interested in this stuff and trying to learn more!" As a layperson, it felt simultaneously good and bad to read their comment. Good, because they're right: it is good that I'm trying to improve on this stuff. But also: Bad, because they're right: I'm just a nonmathematician writing another poorly written post on mathematics.

Anyway, the part of that past blog post where I was most confused was on fast growing functions. I not only explained what I knew poorly, but I also didn't fully understand the concepts behind those ideas. I really should not have included fast growing functions in that post, since it was not something I fully understood at the time, but it fit thematically and I really wanted to make the post thorough.

Now, I realize how much more important it is that all portions of a blog post are researched well enough to pass for at least acceptable to experts in whatever field it is. I've striven to ensure that even reddit posts I make in specialized subreddits are suitable enough so that experts in those fields wouldn't downvote me. It's a weird goal to have, not wanting experts to downvote me, but it's the best a layperson can strive for, I think. My contributions to r/philosophyofscience, r/boardgames, r/startrek, r/philosophy, amongst others, are examplars of what I aim to do in my everyday life: to know enough in each facet of life to not be a total idiot in it. My eventual aim of competence starts with a desire to function adequately, and slowly grow to more knowledge in each field as I can.

It sounds a bit silly when I put it this way. There are areas where I have a great deal of competence: effective animal advocacy, communications data analysis for organizations, knowing every nook and cranny of the worlds of balance and ruin in Final Fantasy VI. But for everything else, I just want to do well enough so that an expert in that field wouldn't laugh at me, and then I want to slowly build from there.

It's in that vein that I'd like to make up for the mistake I made six years ago. And so I present a short essay on fastly fast growing functions, written for a lay audience that's moderately comfortable with high school level mathematics.