Today has been very bad, but I read a little of Infinite Powers.
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Frustrating Days
Today was frustrating. I had more to do than usual, but I was also tired and, late in the day, fairly depressed. I pushed myself to do a small amount of light math reading, and I am hoping for better conditions tomorrow.
My review project hasn’t been moving forward very quickly over the past few weeks. That, too, is frustrating. Early spring is frequently very rocky for me, though. At the moment, I’m just trying to make it through without completely losing my momentum.
Discontinuous
Things are a bit disrupted today. I am out of one of my medications and am feeling the withdrawal symptoms more and more. Nevertheless, I read more of Infinite Powers. The section I read contained several interesting ideas. One was that, while calculus is very useful in modeling the natural world, the assumptions that space and time are continuous that are fundamental to calculus may not be accurate. At the smallest scales, space and time may be broken into indivisible segments that span units called Planck length and Planck time. I will have to think more about this and how it relates to the idea that mathematics is inherent in nature.
Infinitesimal
Today I had a lot of things to do that were not math, and they wore me out so that I spent most of the rest of the available time sleeping. I did manage to read a section of Infinite Powers, though. It concerned Archimedes’ early use of calculus concepts to find the area of a circle. I would name my owl after him, if I had an owl.
Another Quiet Day
Today I was tired and lay low. It’s been a lot of effort to keep at least some of my plates spinning over the past several weeks. For study time, I made more electronic flashcards based on the review I have been doing. It’s useful to make the flashcards a while after I finish reviewing a topic. That way I can tell which pieces of information have taken root in my mind and which ones will need more cultivation.
Flash News
Today I spent a long time testing flash card apps and making electronic flash cards about various math information I’ve encountered during my review, such as details about conics and their significant points. I’m still not convinced I’d rather have these than physical flash cards, though they are more portable and easier to keep organized. Perhaps tomorrow I’ll make some physical flash cards as well and compare them.
(The app I eventually settled on is called Mochi.)
Proofs That Go Poof
Today I read another section of Analysis with an Introduction to Proof. This one was dedicated to proof techniques, and one of those made use of the tautology $[p\implies (q\lor r)]\equiv [(p\land \lnot q)\implies r]$. This is intuitive enough, but proofs that use it feel like cheating. For instance, to prove that if $x^2=2x$, then $x=0$ or $x=2$, you can assume that $x^2=2x$ and $x\neq0$, then prove that $x=2$. That’s it. Poof! Done! No showing that $0^2=2(0)$ required. I do not find this intuitive.
A Wheel on a Wheel
For study today, I watched the three-video series The Wonderful World of Weird Wheels by Morphocular, which is a fun look at the mathematics of curves rolling on other curves. I think it would probably appeal to any readers who have a basic understanding of what a derivative is, and maybe to others as well, since there are a lot of engaging visuals.
The video series motivated me to read a bit about hyperbolic functions, which I never really encountered in my mathematical education. $\sinh$ and $\cosh$ were always just mysterious markings on certain calculators. I can’t say they are that much less mysterious now, but I do know their definitions and have some sense of their relationship to hyperbolas.
Shiny New Week
Today I spent two hours reading a section of Analysis with an Introduction to Proof and doing the associated exercises. I’m feeling better than I was for most of last week, though the mental weather is still not entirely clear.
Part of the section I read explained the inadequacy of examples, even quite a few examples, in proving general statements. As an illustration, it offered the statement that $n^2+n+17$ is prime for every natural number $n$. This statement is true for values of $n$ up to 15, but is not true for 16 or 17. The latter, in particular, could be predicted without trying each value of $n$, since $17^2$, $17$, and $17$ are clearly all divisible by a common factor (meaning that their sum must also be divisible by that factor).
Later, an exercise asked me to find a counterexample to the statement that $3^n+2$ is prime for every natural number $n$. This had me wondering whether there was a way to predict a counterexample, as there had been in the earlier case.1 I could not think of one, however, so I answered the question by trying each $n$. It turns out that I didn’t have to go far, as $n=5$ breaks the pattern. I still wonder if there is a better way, though.
Thank you to all of you who have commented on my posts recently. It really helps me stay committed to my project. I will try to reply soon.
- To do. ↩︎
Easter Break
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.