Blog on hiatus

Not feeling well right now. Hopefully I’ll be back before too long.

Friday Link

Just this :-)

Starting abstract algebra

In school, math progresses more or less in a straight line: arithmetic, algebra, geometry, trigonometry, calculus. In college you may do some more advanced calculus, or a few other things like discrete math or differential equations, but that’s generally it.

But suppose you want to dive deeper. What comes after calculus?

I’m hardly an expert, but from what I can tell, modern mathematics – the stuff that real mathematicians work on – consists of three parts: analysis, topology, and abstract algebra.

I’ve started learning about abstract algebra.

In elementary algebra, you take numbers and add/subtract/multiply/divide them together. In abstract algebra, you take a step back. You say “Instead of numbers, let’s use any elements from a set,” and “Instead of adding/subtracting/multiplying/dividing, let’s do any operation that takes in two and spits out one.”

With those and a few other simple rules, you’re on your way.

The key insight here is that elementary algebra isn’t the only algebra, it’s just an algebra. There are others axiom-based systems that turn out to be just as good.

For instance, in matrix algebra, the things you operate on are matrices instead of just numbers. And multiplication is non-commutative (that is, AB doesn’t necessarily equal BA). How do we handle such a strange situation?

And it isn’t just matrices. Strange new algebras pop up everywhere you look, operating on anything you can think of. Rather than trying to figure out each one individually, abstract algebra asks: what can we say about the structure of mathematics in general?

I got this book from Amazon, and I’m working through it now. Good stuff so far.

What kind of math interests you?

Postmortem: In the Land of Invented Languages


Ever heard of a language called Esperanto? Hundreds of thousands of people speak it worldwide, yet it’s not the official language of any country. That’s because it’s a constructed language, something invented by Ludwig Zamenhof in 1887. He wanted an international auxiliary language, easy to learn, belonging to everybody, owned by nobody, to promote world peace.

Or perhaps you know about Lojban, a more recent language created to be unambiguous and grammatically precise.

Arika Okrent’s In the Land of Invented Languages is a whirlwind tour of these and many others, from Hildegard of Bingen’s Lingua Ignota in the twelfth century, down to Star Trek‘s Klingon in the modern day.

And it’s utterly fascinating.

Part of it is the sheer variety of languages themselves, each trying to fill a different niche in the vast sphere of human activity – like John Wilkins’ philosophical language, where the structure of each word describes the meaning of the word itself.

Part of it is the personalities involved – like Charles Bliss, who was so controlling and erratic that he made it almost impossible for anyone to actually use his “Blissymbols.” He demanded (and received) $160,000 from a center for disabled children as part of a settlement involving his language.

And a big part of it is Okrent herself. Her style is light, quick, and full of vivid detail, which makes her a delight to read. Even better, she leaps into her research, going to Klingon-speaking conventions to see firsthand what it’s all about. You couldn’t ask for a better guide.

If you’ve ever wondered about made-up languages, this is the book for you.

So flying snakes are a real thing

Chrysopelea, better known as the flying snake, is a real thing that actually exists. Technically it glides rather than flies – it can’t gain altitude – but it does in fact move through the air. The video above is legit.

So. How did I not know about these? How does one get through twenty-eight years of life without finding out about flying snakes?

Doesn’t that seem like something they should teach you in schools? First day of biology: Hey kids, there are snakes that can fly. You’d better believe that would’ve gotten my attention.

Perhaps, you argue, it’s irrelevant because they don’t live in the U.S. Even if I grant you that, people need to be warned before they travel to these places. Southern India, southern China, Sri Lanka, the Philippines, Vietnam, Cambodia, Laos, Indonesia. If you buy a ticket to any of these areas, it should come stamped in giant letters HELLO DO YOU KNOW THERE ARE FLYING SNAKES WHERE YOU ARE GOING.

People. I have been to the Philippines. I did not know about this.

It’s not that I’m even especially terrified. They’re only mildly venomous. They’re not going to hurt you. I get that. I’m just saying, that is some necessary, up-front information right there.

Flying. Snakes.

Tell your children!

Sick today


On suffering

Indeed, the truth that many people never understand, until it is too late, is that the more you try to avoid suffering, the more you suffer, because smaller and more insignificant things begin to torture you, in proportion to your fear of being hurt.
-Thomas Merton, “The Seven Storey Mountain”

I’m still reading Merton’s fascinating and insightful autobiography, and recently I came across the quote above. It’s striking, in part, because it mirrors the First Noble Truth of Buddhism: the truth of suffering.

You are going to suffer. To a greater or lesser extent, you are going to suffer every day of your life. You will be nervous, anxious, angry, bored, impatient. You will experience pain. There is no getting around it. Even if you had all the power in the world and could summon every conceivable pleasure and comfort, something would still be missing. Humans suffer; it is in our nature.

But once you understand that – once you truly and finally accept that suffering is not going away – you begin to find a new kind of power. A new space opens up, a space in which you need not constantly run toward pleasure and away from pain, a space in which real peace is possible. Far from being pessimistic or depressing, the truth of suffering is the root of enormous joy, a joy so deep that it doesn’t depend on external circumstances.

Or so I am told. I am trying to apply this to my own life now.

Of course, accepting the truth of suffering doesn’t mean I put my finger in a buzz saw for the fun of it. It also doesn’t mean I should turn a blind eye to the deep suffering of much of the world, or my own rather privileged position in it. It doesn’t mean we should stop trying to improve things, or stop taking medicine. Far from it. The world desperately needs us, and we should be active and engaged within it.

What it means, though, is that the constant barrage of ads, proclaiming you can be happy if only you buy X, can be seen for the nonsense they are. It means that you can stop seeing your dissatisfaction as something broken that needs to be fixed. It means accepting that there will always be someone smarter, someone richer, someone better. That’s okay. That’s life.

Or so it seems to me on this Monday morning.