Monthly Archives: May 2013

“Let’s talk about group isomorphism!” Said no one ever.

Group isomorphism is obscure, complicated, and technical. It’s also one of the coolest ideas I’ve ever encountered, one of those real wow! moments that changed the way I think about math, the universe and everything. And, I believe, it’s very relevant to AI.

So here’s the deal: I’ll do my best to lead you through the swamp, and in return, you’ll get a new perspective on mathematics – and maybe even the nature of intelligence. Agreed?

We’ll start with something simple. Or rather, two somethings: addition and multiplication, + and ×. Our old arithmetic friends. Can’t get much simpler than these guys, right?

6 + 8 = 14

3 × 7 = 21

Have you ever thought about how similar addition and multiplication are? Both of them take two numbers, do something mathy to them, and spit out another number.

In fact, the similarities don’t stop there. Both groups are also associative, meaning we can throw in parenthesis and it doesn’t change the answer:

(1 + 2) + 3 = 1 + (2 + 3) = 6

(2 × 3) × 5 = 2 × (3 × 5) = 30

That may not seem like much, but it’s a property subtraction doesn’t have.

Addition and multiplication also both have an identity element. For addition it’s zero, for multiplication it’s one. Either way, it doesn’t change the identity of whatever you pair it with.

5 + 0 = 5

7 × 1 = 7

What’s more, addition and multiplication both have the idea of an inverse. For any number, you can pair it with its inverse, and get the identity element.

18 + (-18) = 0

37 × (1/37) = 1

Addition and multiplication really are a lot alike, aren’t they? So much alike, in fact, that you almost get the feeling that at some deeper level, they’re doing the same thing, just with different symbols. Like they’re two different languages, and all we need is a translator…

In fact, you can translate (or “map”) between multiplication and addition. The key is the natural log function, ln(x).

For those who aren’t familiar with it, the natural log function is a button on any scientific calculator, and a function in Excel too (just type “=LN(5)” for example). If you want more details about what it actually means, Google is your friend. For now, what we care about is that it will translate between multiplication and addition.

What do I mean by that? Let’s see an example.

3 × 5 = 15

Well, ln(3) = 1.098612, and ln(5) = 1.609438, and ln(15) = 2.70805. These numbers may look like random garbage, but they have a remarkable property:

1.098612 + 1.609438 = 2.70805

(Ignoring rounding errors, of course.)

In other words:

ln(3) + ln(5) = ln(15)

There’s our translation.

Say we wanted to calculate 6 × 7, but we’re allergic to multiplication. We just can’t do it, for whatever reason. But we can do addition, and that’s all we need.

All I have to do is “translate” our multiplication problem into the language of addition, using the natural log function:

ln(6) = 1.791759

ln(7) = 1.94591

Now we’re in the world of addition, and I can do addition!

1.791759 + 1.94591 = 3.737669

We’ve got our answer! We just need to “translate” it back into multiplication world.

We know that ln(answer) = 3.737669. What we need is the ln function in reverse, a sort of un-natural-log, to go back to multiplication world. In mathematical terms, we’re looking for the inverse function, the function that will undo ln(x). As it happens, the inverse function for ln(x) is e^x, so we take e^3.737669 and get…

(drum roll please)

42, the answer to our original multiplication problem, 6 × 7.

Roughly speaking, addition and multiplication are known as groups in the mathematical sense. And because they are fundamentally the same, they are called isomorphic (“iso” means same, “morph” means form). Group isomorphism.

To me, it’s fascinating that addition and multiplication – which seem totally different on the surface – are somehow the exact same thing “under the covers.” Any result you derive for one operation can be “translated” to the other. Our world hides deeper symmetries than we suspect.

Tomorrow, I’ll show how some of those deeper symmetries apply to AI.

Questions?

For a long time, I’ve thought it would be cool to know how to unicycle. Probably for the same reason I learned how to juggle: chicks dig it. (NOTE. This is not accurate.)

Last week, I was at the park (outside!) with a group of friends, and one of them brought a unicycle. He rode it around a little, and the rest of us tried to. To get an idea of our success rate, imagine a herd of antelope learning to touch-type, and that should get you pretty close. Still, I had fun, and was fascinated by this one-wheeled monstrosity.

I had to have one.

After doing a little research, I ordered this bad boy…and then waited. Yesterday, it arrived.

Not pictured: my neighbors pointing at me, saying “Why the hell is he photographing a cardboard box?”

Bubbly as a can of Faygo, I carted my prize to the back of the house to assemble it.

Carrot-flavored, my favorite.

The instructions said it could be assembled in 15 minutes. Knowing my own technical expertise, I mentally adjusted this to “four days,” but in the end I was surprisingly successful. Using a couple of wrenches from my wife’s toolbox, and making a quick run to the local Speedway to inflate the tire, I got it together in about an hour.

And here she is:

Shown in the vertical orientation. In my experience, horizontal is more realistic.

If the seat seems pretty high up, it’s because I am freakishly tall.

Sadly I don’t have any pictures (yet) of me trying to ride it. Betsy was away at the time, and the list of Ways To Make Unicycle-Riding Easier does not include Trying To Take a Picture of Yourself While Doing It. But my method’s pretty simple: find something to lean against for stability (like a tree, or a bench) and get on the seat, pushing down on one pedal to steady yourself. Then put your other foot on the other pedal, find your balance, release your support, and go.

Experiments thus far have shown the “and go” step to be significantly more difficult than the others. I do in fact go, but the direction is mostly earthward.

It’s pretty clear, when you’re sitting on a unicycle, that this is not something God intended humans to ride. A little-known fact about unicycles is that they only have one wheel. When you’re watching someone else unicycling, it looks cool; when you’re trying to do it yourself, you wonder if you might be mildly retarded.

My friend said he never got hurt learning to unicycle. You fall a lot, he said, but since your instinct is to put out your feet, you generally don’t fall down.

This was, of course, a transparent lie. I had my first legit fall after about two dozen tries:

I admit this is not a very badass scratch. But give it time.

I love it anyway. Can’t wait to try it again tonight!

My About Page claims: “I have personally seen the largest living thing in the world.”

Here’s the proof:

Click to enlarge. Not that it needs to get any bigger.

I took this photo of the General Sherman Tree in Sequoia National Park, California, on June 17, 2005. It was part of a two-week road trip out west with two of my best friends, both of whom are probably reading this right now.

Sherman’s status as the largest living thing is debatable, but all other claimants to the throne are superorganisms of some kind – at least from what I’ve seen. If anyone knows a specific organism that’s bigger, I’d be interested to hear about it.

Even if there is something bigger, it’d be tough to look much more impressive.