A math odyssey
Today I read more of Infinite Powers. Intrigued by a quote contained there, I also did a little extra reading about Johannes Kepler, the mathematician and scientist who discovered the elliptical orbits of the planets. It seems that for him also, doing mathematics may have had spiritual dimension.
Despite that and what I wrote of myself yesterday, I’ve decided to take the next three days off study and blogging for religious observances. Expect me back on Monday.
Today I read a little of The Mathematical Tourist and also a little of Infinite Powers, Steven Strogatz’s book on the history of calculus. The Mathematical Tourist skims over its material very quickly and I was finding it hard to follow in my current state of mind.
I am feeling very detached from my review project just now. Nevertheless, tonight, at the midpoint of Western Christian Holy Week, I’ve been reflecting that, when I am able to apply myself to it, doing mathematics becomes a spiritual practice for me. There is a certain attitude of mind that engages with the grandeur of the universe with both humility and joy. I’ve always called it play, but there may be a better word. At any rate, this play can be a response to any natural or artistic beauty—is there really a distinction?—and the impulse behind any creative work. I think what evokes it is very different for different people. For some people it may well be basketball or cake decorating. For me it is certain natural phenomena, music, and more than anything else, higher mathematics. And for me, this play is also a form of prayer. Engaging with the wonder of creation with humility and joy is also singing a hymn to the Creator. To play is to pray, and nothing inspires play in me like mathematics.
I’m feeling better today, but still dragging around. I watched more math videos on YouTube, notably a 3blue1brown video about symmetry and group theory. It was very interesting, though I wouldn’t recommend it to readers who are not math enthusiasts.
I’m hoping to be up to some more active study tomorrow. For one thing, I’ve run out of intelligible permutations of the words “not”, “much”, and “doing”.
Today was quite bad on the mental health front. Spring has long been the worst season for me. I was able to make myself do something, though. I spent about an hour watching math videos, including a pleasing intro to the concept of compactness by a YouTuber called Morphocular.
My mental health was better today, but I haven’t yet bounced back physically. I took a long nap after church, and I was only up for some light math reading after. The section of The Mathematical Tourist that I read was about tests for primality, which is appropriate given my current interest in primes. It made me wonder if SageMath tests numbers directly or looks them up in a table. There might be a forum where I could ask.1
Today I got a haircut, which has improved my mental outlook. Otherwise, though, I have not had it together at all, and the only study I managed was a little reading from The Mathematical Tourist. Better luck tomorrow, I hope.
Today I continued to play with the problem from my post Puzzling Primes. Among other things, I installed the computer algebra system SageMath, which I can use to test quickly whether numbers are prime. With that tool, I might be able to solve the problem by brute force. I can’t imagine that this would be chosen as a problem of the week if there were not a better way, though.
So far I have been able to prove (a) that a superprime may contain at most two occurrences of the digits 1 or 7, (b) that all the other digits must be 3 or 9, except the first digit, which may also be 2 or 5, and (c) that if the first digit is 2 or 5, there will be no occurrences of 1 or 7.
All this means that, if there were some factor limiting the number of trailing 3s and 9s a prime could have, I would at least be able to prove a limit on the length of superprimes, and therefore that there must be a largest one. So far I’ve had no luck with that avenue of inquiry, though.
I will share the actual proofs of these claims in the future, but I’m still feeling unwell today. In the meantime, here is a chart of the ways that potential superprimes can be constructed. Red represents potential starting digits while blue represents digits that may be added.
Today was another bad mental health day, but I did read some of The Mathematical Tourist in the evening. Something written there reminded me of the problem I shared in my post Puzzling Primes, and I ended up working on that for a while, as well. I didn’t make any progress, though.
Today I read more of The Mathematical Tourist. I’ve acquired an ebook version, which I can read using text-to-speech software, my preferred reading method for text-heavy books such as this. (As I think most current readers know, I’m dyslexic.) It’s a big improvement on my attempts to read my hard copy in the ordinary way.
I’ve decided to take a three-day break from my regular study schedule and from updating the blog. There is a lot going on for me during the first half of the coming week. Expect me back on Thursday.
Today I read the first section of the first chapter of my calculus book and worked on the associated exercises. (Yippee!) My concentration, which was such a problem on Thursday, was fine today, and I’m feeling encouraged.
The section I read was about functions and discussed the vertical line test. According to the test, a graph represents a function of $x$ if and only if no vertical line intersects the graph more than once. The way I remember first being taught the test, though, the “of $x$” condition was not included. It always bothered me that $y=x^2$, an upward facing parabola, should be a function, while $x=y^2$, a rightward facing parabola, should not. It is, of course. It’s just a function of $y$ rather than a function of $x$. This has me wondering whether, say, the graph of a diagonally facing parabola could be interpreted as representing a function, and what it would be a function of.1