NAA: Equivalence Relations

New post at Not About Apples.

What do you mean when you say two things are the same? One thing’s for sure, you don’t always mean the same thing. The mathematical idea in the background is called an “equivalence relation”.

Permalink: Equivalence Classes and Equivalence Relations

NAA: “Well-defined”, more examples.

New post at Not About Apples.

Some more examples of how “well-defined” is used and why that concept can be useful, both mathematical and not.

Permalink: “Well-Defined”: More Examples

The Vexing Bumper Sticker

Now, there are a lot of bumper stickers out there, funny, political, all kinds, and I enjoy looking at them when I’m driving. Every once in a while, I see once that makes me think, or gives me a good laugh, but a few years ago, I saw one that has taken up more of my personal processing time than any other. Because I have absolutely no idea what it could possibly mean. Actually, I had come to the conclusion that I must have read it wrong or made the whole thing up, at least until I saw the same bumper sticker just last week.

The bumper sticker: “Just because I’m an atheist doesn’t mean I don’t believe in God.”

So I’m thinking, “But it does, doesn’t it?”

NAA: Math and Poetry

New post at Not About Apples.

Why is it that mathematicians see themselves as cousin to poets? And vice versa?!

Permalink: Math and Poetry

NAA: Well-Defined

New post at Not About Apples.

Another one of my attempts to explain a piece of mathematical jargon that really shouldn’t be specific to mathematicians. The word of the day: well-defined.

Permalink: Mathematician-speak: Well-defined

A Howl at the Moon

I delight in the full moon every month.  And while, despite the large number of objects I own in a wolf motif, I have never literally howled at the moon, I do feel a surge of something when I notice a full moon in the sky.   It’s primal.

I don’t consider myself a superstitious man (a whimsical one, surely, but that’s different), but it often seems that I am more better at what I do when the moon is bright.  It could be a coincidence, it could be my imagination, it could be placebo effect.  Or maybe some part of my mind is on a 28-day cycle of highs and lows, just as I know I am a faster reader in the evening and a faster typist in the morning.  It doesn’t seem so hard to believe that there is a cycle at play.  And if I know anything about how my mind works and where my creativity comes from, it is that I know very little about how my mind works or where my creativity comes from.  So who am I to say?

As I see it, my duty is to to be as open as possible to inspiration and new ideas, and to make my mind a welcome place for sparks of ideas to start a fire, not to try to predict or understand the whimsy of the Muse.

Do I actually believe that there is a Muse for my mathematical ideas?  Sometimes  I do.  The inimitable Ramanujan said that his astonishing (and astonishing really is an understatement) insights and formulas were revealed to him by a goddess in visions.  If you know anything about Ramanujan, it’s as plausible a story as any.

In any case, the last three days have been good for my work; I’ve figured out how to confirm some of my hunches, and taken a chunk out of the next batch of questions I want to answer about number theory.  It’s becoming clearer in my mind how the transition from my current whiteboard and Mathematica explorations into a handful of publishable papers is going to happen.

What I haven’t done much lately is update this blog.  As the full moon winds down, it’s a good time to play a little bit of catch-up and vignette my way to a new blog post.

NAA: The Galois Correspondence

New post at Not About Apples.

At long last, the conclusion of (the first half of) the Galois theory arc. Some payoff for all the ideas and terminology we’ve been acquiring over the last few posts.

Permalink: The Galois Correspondence

NAA: Favorite quotes

New post at Not About Apples.

13 selections from my quote of the day file.

Permalink: Quotations: A Baker’s Dozen

Also, the version of the previous NAA post was an older, incomplete draft. The full version is posted there now.

Permalink: Symmetries of Number Systems

Picasso at the Lapin Agile

Last night I saw Steve Martin’s play, Picasso at the Lapin Agile. It was my first time at a play in years, my first time ever seeing my cousin-in-law act, and a play I’ve specifically wanted to see for ages. And, given my schedule, who knows how long it would be until the next play. So I had a lot riding on that play.

And they did not let me down. I had the best seat in the house (or darn close), and they had me from when I entered the theater. The bar set was spectacularly done: it felt like a place I might go one night for a drink between bouts with the Riemann Hypothesis, and the old-timey music was transporting.

In case you don’t know the play, it’s Steve Martin at the apex of his powers. Filled with comedy of every sort, time travel, and so many fourth wall breaks that it will take a dozen metaphorical carpenters to put that place back together after the show closes on Sunday, and a e-shaped pie, it’s pure entertainment. But the play also deals with very serious themes: the nature of progress, whether it is possible for one man to change history and what that even means, special relativity, the relationships between men and women, the creative process, our attitude toward the future. To see those issues played out at the turn of the LAST century only reinforces their timelessness.

As with any play, a performance can be only as good as its cast, and the cast here was top-notch. Strong performances all, no weak links.

Mouse, my cousin-in-law (without whom I might never have known this play was even happening), was marvelous as Germaine — proud to have married into your family, cuz. Who knew you had such a French accent in you?

I don’t know how old the actor who plays Gaston is in life, but he captures so totally the man’s status on the cusp of old age, so thematically important to the work.

Really, every performance was solid, but special mention should be made of the main protagonists Einstein and Picasso. They skillfully evoked our modern notions of the people without ever being caricatures, and without ever losing sight of the crucial point that these people were still young, proto-Einstein and proto-Picasso, marked not by wporld-changing achievement but by the potential for it.

Actually, the actor playing Einstein reminded me often of Gordon Kaye, a British actor best known for the WWII screwball sitcom ‘Allo ‘Allo. Which means it’s time to get those DVDs from the library and watch ‘em again.

Seeing a play like this energizes me, makes me want to sit down, put pen to paper, because sooner or later someone’s going to crack the Riemann Hypothesis. And who knows, it MIGHT be today, and it MIGHT be me.

Ta-ra-ra-BOOM-de-yay!

NAA: Numerical Symmetries

New post at Not About Apples.

The Galois Theory arc returns from its hiatus. Today we look at what we should mean by symmetry in the context of number systems, connecting the concepts of group and field. This leaves us just one step away from the Fundamental Theorem of Galois Theory, and only a few steps of away from one of the most important impossibility theorems of all time: there does not exist a formula for the solutions of a polynomial degree 5 or higher analogous to the quadratic formula.

Permalink: Symmetries of Number Systems