Exploiting My A-To-Z Archives: Dedekind Domain

For the Leap Day 2016 A-to-Z — it started around Leap Day; I didn’t write everything in one impossibly massive burst — I threw subjects open to nominations. This was a great choice. People ask me about things I would never have thought of. Or that I did not know about before being asked. So I would learn things just ahead of explaining them.

Dedekind Domains, here, were I think the first example of that. It was also the first time I threw out and rewrote an essay from scratch. The first essay tried to lay out all the rules that made up a Dedekind Domain. Which, for an audience I couldn’t be sure had ever heard of rings before, took paragraph after paragraph of definition. When I realized I wasn’t staying intersted in writing this, I understood I needed a different approach. So this essay taught me several things, one of them truly important.

It turns out I don’t have the Leap Day 2016 A-to-Z essays organized by a convenient tag either. I’ll have to fix that too.

Advertisements

Exploiting My A-To-Z Archives: Characteristic Function

For today I’d like to bring back attention to something from my original, summer 2015, A to Z. The characteristic function is a way of picking out sets of things. Like, there’s a characteristic function that picks out “all the real numbers between 2 and 4”. Or “all the prime integers”. Or “all the space within a set distance of this point”. It’s a handy tool for breaking up a problem into several smaller problems, without requiring that we write it out as more problems. We can use the tools for regular functions to deal with complicated and weird cases.

There are several things called characteristic functions. The other really important one turns up in probability, and the essay linked there isn’t about that one.

Other Summer 2015 A to Z essays are at this link. Also I learn I didn’t tag these essays the way I would come to later on. I’ll have to fix that sometime.

Exploiting My A-To-Z Archives: Benford’s Law

Let me close out the week by bringing back a piece about my car. It was part of the Summer 2017 A to Z. It didn’t by itself determine just when in 2017 I would do an A to Z sequence, but it did set a deadline for me.

Benford’s Law is one of the many mathematics things not quite named for the correct person. Simon Newcomb, one of the United States’s greatest 19th century astronomers, wrote about it first. But Frank Benford in 1938 publicized it, with a paper titled The Law Of Anomalous Numbers. And it’s a great weird phenomenon. Look at sets of data from the real world: street addresses. Lengths of rivers. Heights of tall buildings. The mileage reading on car odometers. The leading digit is more often a 1 than a 2, more often a 2 than a 3, more often a 5 than a 6, and so on. It seems bizarre this should happen. Especially in things that seem to be independent of human construction, like masses of asteroids, it’s bizarre this should happen.

So it’s fun to spend some time looking at this strange business.

My other essays from the Summer 2017 A To Z are at this link.

Exploiting My A-To-Z Archives: Asymptote

One of the little things I like about doing an A To Z sequence is the feeling that I’m writing two paragraphs ahead of catastrophe for several months. It’s thrilling stuff. But another thing I like is that I will put together strings of days with some publication. When I ran these three-a-week this was particularly attainable, since between those posts and Reading The Comics I could often have five or six days of the week covered without making special effort.

I don’t think I can do that much writing just now, especially since I see I’m completely failing to use two-a-week postings to spend less time writing the A-to-Z essays. So I’ll instead bring some attention to older essays that I’m still happy with. Such a thing happens.

So for the letter A, I’d like to bring back up the Fall 2018 essay about Asymptote. This is one of the pieces I’m happiest with. It did something wonderful for me, which is to make me think hard about a concept I’d known for decades and come to a new understanding of it. I’d gone in thinking about asymptotes as, like, the lines that bracket hyperbolas, and how something like that happens with other functions. I came out understanding questions like why sometimes a function will cross its asymptote many times and sometimes it never will. Or, for that matter, why a parabola doesn’t have an asymptote. I don’t know that I’ll ever have a writing moment that good again.

My other Fall 2018 A-To-Z Essays are at this link.

What I Learned Doing The End 2016 Mathematics A To Z

The slightest thing I learned in the most recent set of essays is that I somehow slid from the descriptive “End Of 2016” title to the prescriptive “End 2016” identifier for the series. My unscientific survey suggests that most people would agree that we had too much 2016 and would have been better off doing without it altogether. So it goes.

The most important thing I learned about this is I have to pace things better. The A To Z essays have been creeping up in length. I didn’t keep close track of their lengths but I don’t think any of them came in under a thousand words. 1500 words was more common. And that’s fine enough, but at three per week, plus the Reading the Comics posts, that’s 5500 or 6000 words of mathematics alone. And that before getting to my humor blog, which even on a brief week will be a couple thousand words. I understand in retrospect why November and December felt like I didn’t have any time outside the word mines.

I’m not bothered by writing longer essays, mind. I can apparently go on at any length on any subject. And I like the words I’ve been using. My suspicion is between these A To Zs and the Theorem Thursdays over the summer I’ve found a mode for writing pop mathematics that works for me. It’s just a matter of how to balance workloads. The humor blog has gotten consistently better readership, for the obvious reasons (lately I’ve been trying to explain what the story comics are doing), but the mathematics more satisfying. If I should have to cut back on either it’d be the humor blog that gets the cut first.

Another little discovery is that I can swap out equations and formulas and the like for historical discussion. That’s probably a useful tradeoff for most of my readers. And it plays to my natural tendencies. It is very easy to imagine me having gone into history than into mathematics or science. It makes me aware how mediocre my knowledge of mathematics history is, though. For example, several times in the End 2016 A To Z the Crisis of Foundations came up, directly or in passing. But I’ve never read a proper history, not even a basic essay, about the Crisis. I don’t even know of a good description of this important-to-the-field event. Most mathematics history focuses around biographies of a few figures, often cribbed from Eric Temple Bell’s great but unreliable book, or a couple of famous specific incidents. (Newton versus Leibniz, the bridges of Köningsburg, Cantor’s insanity, Gödel’s citizenship exam.) Plus Bourbaki.

That’s not enough for someone taking the subject seriously, and I do mean to. So if someone has a suggestion for good histories of, for example, how Fourier series affected mathematicians’ understanding of what functions are, I’d love to know it. Maybe I should set that as a standing open request.

In looking over the subjects I wrote about I find a pretty strong mix of group theory and real analysis. Maybe that shouldn’t surprise. Those are two of the maybe three legs that form a mathematics major’s education. So anyone wanting to understand mathematicians would see this stuff and have questions about it. (There are more things mathematics majors learn, but there are a handful of things almost any mathematics major is sure to spend a year being baffled by.)

The third leg, I’d say, is differential equations. That’s a fantastic field, but it’s hard to describe without equations. Also pictures of what the equations imply. I’ve tended towards essays with few equations and pictures. That’s my laziness. Equations are best written in LaTeX, a typesetting tool that might as well be the standard for mathematicians writing papers and books. While WordPress supports a bit of LaTeX it isn’t quite effortless. That comes back around to balancing my workload. I do that a little better and I can explain solving first-order differential equations by integrating factors. (This is a prank. Nobody has ever needed to solve a first-order differential equation by integrating factors except for mathematics majors being taught the method.) But maybe I could make a go of that.

I’m not setting any particular date for the next A-To-Z, or similar, project. I need some time to recuperate. And maybe some time to think of other running projects that would be fun or educational for me. There’ll be something, though.