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Book Joy

Olly / / Updates

Today I finished my relisten of A Brief History of Mathematics and also read more of Infinite Powers. The final section of A Brief History discussed algebraic geometry in very general terms, which piqued my interest. So I went and read part of the summary of that topic in The Princeton Companion to Mathematics.

That book is really cool, with outlines of both the content and history of the subject. I am pleased finally to be getting some use out of it. It was a gift from my father soon after I left college and before I realized that math was no longer something I could do. I nearly got rid of it, along with all my other math books, at numerous points over fifteen years. I always held on, though, lugging a several shelf feet of math books through two interstate moves, and I am so glad now that I still have my library to draw on. The Internet is also a good resource, but most of the time it can’t match the organization, depth, or reliability of a book.

That said, I am still feeling mentally weary and struggling with this project. I am not sure whether to keep pushing or take time away. (I hesitate to do the latter, since I don’t know what the problem is and therefore have no evidence that things would be better after a break.) Possibly the change in medication that I started today will help.

Collections of Things

Olly / / Math Posts

Today I listened to more of A Brief History of Mathematics. Most of what I heard was about Cantor. For non-enthusiasts, Cantor is famous for discovering that there are multiple sizes of infinity. The set of natural numbers (i.e. 1, 2, 3, 4, and so on) is infinite, and so is the set of real numbers (i.e. 1, $-\frac{1}{3}$, 2.5837, $\pi$, and all the other numbers on a number line). Cantor discovered, however, that the cardinality of the set of natural numbers—how many things are in it, essentially—is less than the cardinality of the set of real numbers. The infinity of natural numbers is a smaller infinity than the infinity of real numbers.

Not only that, but Cantor discovered that there are infinitely many sizes of infinity, called infinite cardinals. Listening to that idea being discussed, I was struck by a question I’d never thought of before, which was “what size of infinity describes the infinitude of sizes of infinity?” or, more intelligibly, “what is the cardinality of the set of all infinite cardinals?”

I don’t have an exciting answer to share with you, unfortunately. Basically, the answer is that the question is not intelligible. I was able to find a thread on Mathematics Stack Exchange about the topic, and the answers given there explained that there is no set of all infinite cardinals, because the collection of infinite cardinals is in fact a class. Unlike sets, classes do not have cardinality, so there is no answer even to an adjusted form of the question. I was and still am a bit confused about what a class is, but another thread on Mathematics Stack Exchange suggests that it’s any collection of things that can be described but can’t be a set for logical reasons founded in the set theory axioms.

(I also spent a bit of time today reading about categories, which are another mathematical collection of things. They were something I’d heard of but never actually encountered, and my other researches brought them to mind.)

Baby Steps

Olly / / Math Posts

Today I relistened to more of A Brief History of Mathematics. That might seem like it hardly qualifies as study, but I think it is helping revive my desire to continue this project. It comprises mathematical bedtime stories, told by an enthusiast, with a very light sprinkling of actual mathematical ideas—just enough to be tantalizing without being overwhelming.

I also thought a bit about one of those ideas, a theorem in number theory discovered by Gauss: that every natural number is the sum of three triangular numbers. I didn’t put my full focus into it, though, and I didn’t come up with any insights.

[Later note: The theorem only holds as stated if we let 0 be a triangular number. Definitions appear to differ on that point.]

What’s Up

Olly / / Updates

Having had a good day on Monday, I didn’t manage to do any study at all on Tuesday. My mind was on another project and time got away from me.

Today I tried to continue working on the proposition about trapezoids, but found I didn’t have enough mental energy. I relistened to more of A Brief History of Mathematics instead, hearing about Fourier and Galois. (Galois is one of math’s romantic heroes. Political radical, briefly imprisoned for his republican activities. Killed in a duel at 21, but leaving highly original work behind.) Made curious about Galois’s work, I left off listening and went to look him up in a lovely reference book I have called The Princeton Companion to Mathematics. I read the article on Galois himself and part of the article covering the branch of math called Galois theory. I did lose the thread of that article after a while.

In other news, I’m currently working through the involved application process for a volunteer position I hope to get. There are many applicants, so there is a good chance I won’t be selected. If I am, though, I’m not sure what it will mean for my math review project and this blog. I’m not sure what the future of those things should be, in any case. Despite my good day on Monday, I still feel as if I’ve lost something I need to sustain them.

Happy Day

Olly / / Math Posts

Today I had fun working on a geometric proposition about trapezoids and reading some mathematical correspondence. I may do more math activities later, as well.

The proposition I worked on, which comes from Elementary College Geometry, is that the base angles of an isosceles trapezoid are equal. (That book defines a trapezoid as a quadrilateral with exactly two parallel sides. An isosceles trapezoid is a trapezoid that has legs of equal length. The legs are the non-parallel opposite sides.)

I drew a number of pictures trying to find a good way to prove this proposition.

Trapezoid One Small

The first one shows the approach I came up with when I first looked at the problem some time ago: construct isosceles triangles on either end of the trapezoid, show those are congruent (likely requiring some lemmas), use alternate interior angles. I think this will probably work, but it will need some details ironed out. For instance, what if $\angle ABC$ is acute? (It can’t actually be with Elementary College Geometry’s definition of a trapezoid and $AD$ equal to $BC$, but I may need to prove that.)

This method is not very elegant, even ignoring the niggly details, so before working those out, I decided to see if I could find something better. In the process, I came up with the figures below. None of these approaches has born fruit so far, though.

Trapezoid Two Small

Trapezoid Three Small

Trapezoid Four Small

In the first drawing, I tried the most basic quadrilateral proof move, but just went around in circles. The second drawing didn’t really go anywhere at all. Something resembling the third drawing will be useful for proving the converse of the proposition, I think, but didn’t yield a proof in this direction.

I’m feeling much encouraged and less inclined to simply drop this project, which I had been resisting the urge to do.

Squares Yet Again

Olly / / Math Posts

Today I watched this fun video from Numberphile. I watched it earlier in the month—it’s the one I stopped so I could investigate its topic a bit myself—but I had since forgotten the details. I’d recommend it to all of my readers. It is very accessible but has connections to more advanced topics. When I am feeling better, I would like to work on proving the theorem presented at the end of the video.