Thursday, November 3, 2011

Can Jellyfish Do Math?

I suspect we'd all agree that jellyfish as we know them on Earth cannot do mathematics. But don't go away, because what I actually meant to talk about are jellyfish-like aliens that may exist on a distant planet. That's cool, right?

Yesterday I went to a talk at SETI. Yes, the Search for Extra-Terrestrial Intelligence is still on, and they have free public lectures on Wednesdays, with complimentary Nutella sandwiches. True! And delicious! This week's lecturer, John Stillwell, talked about ET Math.

Really.

The idea that math is some kind of universal language we could use to communicate with sentient life elsewhere in the universe has percolated out of nerd-dom into popular culture--and who knows? It could be true. But, says Stillwell, it may not be a trivial task to recognize alien mathematics. He illustrated this point with a couple of examples of curious, non-intuitive, roundabout bits of mathematics developed by our fellow humans right here on Earth.

ANCIENT GREEKS AND THE FOURTH DIMENSION

Start with an elementary principle of arithmetic: that multiplying numbers together doesn't depend on the order in which you multiply them. To wit:

2 x 3 = 6
3 x 2 = 6

This rule can be generalized by considering any numbers a and b. Using the algebraic tradition of snuggling two numbers close together to indicate multiplication, we get the commutative law of multiplication:

ab = ba

It may seem preposterous to even try to prove this basic rule. After all, it's obviously true.

But it actually matters how you prove it to be true. Euclid in ancient Greece thought of a and b as the two sides of a rectangle, and the product ab to be the area of the rectangle. Obviously, it doesn't matter which number is which side of the rectangle--the area will still be the same.




You can multiply three numbers together by moving into three dimensions. With three sides, the product is the volume of the cube.


But it stops there. Humans live in three dimensions and have a very hard time visualizing anything more than that, so the ancient Greeks simply did not believe you could multiply more than three numbers together.

That's what Stillwell said, anyway, and he wrote a book about the history of math so I guess I'll take it.

Over the centuries since Euclid, humans have come up with at least two additional ways to prove that ab = ba, without needing rectangles or cubes. Both alternative techniques are extremely weird, and, although I was able to grasp the basics while Stillwell was talking (he's a good lecturer!), I've decided not to try to explain them here. Instead, I'm only going to babble about their relevance to aliens. And jellyfish.

ALIEN PROOF #1: PROOF BY INDUCTION

Humans are highly visual creatures, and we're really good at visualizing things in our heads even if we don't see them with our eyes. But there's no reason aliens would be like that. What if aliens found our ideas about rectangle math just awfully difficult to comprehend?

Well, there's an utterly non-visual way those aliens might still be able to prove that ab = ba.

It starts with a German high school teacher named Hermann Grassmann who moonlighted as a Sanskrit scholar--he translated the Bhagavad Gita, among other things. He came up with a method of thinking about arithmetic that required no visualization at all. It was later rediscovered by this guy Peano (pronounced like the instrument!) and named for him, so poor Grassmann got short shrift.

I don't think his high school students really appreciated him, either. Peano (Grassmann) arithmetic is rigorous, brilliant, and totally non-intuitive. You can use it to prove that ab = ab, but first you have to redefine addition and multiplication in terms of successor functions. At that point, I suspect most fourteen-year-olds would throw up their hands in despair. And probably most jellyfish, too. So let's try something else.

ALIEN PROOF #2: PROOF BY PROJECTIVE GEOMETRY

Stillwell argues that one of the reasons our human concept of geometry is rigid is that we have rigid bones in our bodies. A squishy, soft-bodied creature might not think in such angular terms as rectangles and cubes.

But any creature that knows about light will be able--indeed, forced--to think in straight lines. Light travels in straight lines, which is the whole basis of perspective, including our remarkable ability to look at a square tile at the far edge of a floor, where it's all squashed and thin, and still see it as a square.


The trick to perspective is that parallel lines always meet on the horizon. When drawing straight lines in perspective, based on this rule, you come across some interesting coincidences. Coincidences, as Stillwell explained them, where three points just happen to fall on a single straight line, for no apparent reason.

But there is a reason; in fact, there's a rule. Pappus' hexagon theorem claims that for any six points, three of which lie on one line and the other three on a different line, you'll always have one of these coincidences.

It's kind of hard to explain with just words, so check out these two visual explanations.

What's even more difficult to explain is that you can actually use this theorem and these points and lines to do arithmetic--to add and multiply numbers together. And once you're doing that, you can prove that ab = ba through purely perspective-based geometry.

Thus, this image:



is a sort of proof that ab = ba, and as friendly a way for ET to say hello as anything else we could imagine. So keep your telescopes peeled!

WELL MAYBE NOT JELLYFISH BUT HOW ABOUT OCTOPUS MATH?

Given Stillwell's idea that Alien Proof #2 might be good for boneless jellyfish-style ET, I had to approach him after the lecture and introduce myself as a marine biologist. Here's what we talked about, and what I've been thinking about ever since:

In looking for all the diverse ways that aliens might do math, Stillwell has restricted himself to humans. But exobiologists are always using the weirdest Earth creature they can find to guide their search, whether it's a primate or a scallop or a bacterium. Why shouldn't exomathematicians do the same?

Now, I'm not suggesting than any non-human animals are doing calculus, but they do have to solve problems, and some of them are extremely good at it. So wouldn't it be interesting to look at something boneless here on Earth, something that is also highly visual, like, say, an octopus? How does it solve problems? How does it calculate where to put an arm in order to grab a rock, how does it determine whether it can fit through a given hole?

Stillwell seemed quite receptive to this idea, and a short discussion with him and another questioner delved briefly into fluid dynamics and topology. After all, many of the problems jellyfish or octopuses have to solve are related to calculating or predicting water flow, both in their environment and in their bodies. And problems like the "how do I get in that box" and "can I fit through that crevice" are quite topological in nature.

Finally, Stillwell mentioned homotopy theory. While I was pleased to note that my mathematician brother actually works on homotopy theory and even tried to explain it to me once, I confess I didn't quite understand the connection. What do you think, Mike: homotopy theory? topology? alien math? ehhh?

And then, of course, there are the SETI goofballs who just had to bring up aliens living in regions of space-time where the very fabric of mathematics might be different. Stillman graciously acknowledged the point:

"I didn't talk about the possibility of ba not equal to ab, but that's very important as well."


Major hat tip to Terry S. for alerting me to the lectures and the associated abundance of chocolate hazelnut spread.

4 comments:

  1. Well, people are now just starting to think about basing math on homotopy theory rather than on sets or numbers. See here and here for instance. I don't think anyone has written anything about it for a popular audience, yet, though, so those pages probably won't be comprehensible at all.

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  2. Hah! I started reading the n-Category Cafe post, and after a little bit I thought, "I wonder who wrote this," and scrolled up. =)

    "Thus, the ∞-categorical revolution, if carried out in the language of homotopy type theory, will support and be supported by the inevitable advent of better computer-aided tools for doing mathematics."

    I find this exciting in an I-don't-quite-understand-it-kind of way, but yay!

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  3. Danna, thanks for a very nice writeup of my talk. I can see now that I should have made it clear that the Greeks wanted to multiply *lengths*, rather than numbers, because they did not believe in irrational numbers, and hence they thought that length was a more general concept. This led them to the idea of
    multiplying lengths by placing them at right angles to each other,
    hence the difficulty in multiplying four lengths.

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  4. Wow! Many thanks to John Stillwell for finding this post and clarifying my confusion over Greek multiplication. Yay, the internet.

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