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Friday, August 8, 2008

Counting: science we can all count on

I'm browsing blogs regularly now to steal ideas, and this one caught my eye with a funny line from an actual scientific paper quoted in the New York Times:
While attending lectures on dementia, the doctors, Kenneth Rockwood, David B. Hogan and Christopher J. Patterson, kept track of the number of attendees who nodded off during the talks. They found that in an hourlong lecture attended by about 100 doctors, an average of 16 audience members nodded off. “We chose this method because counting is scientific,” the authors wrote in their seminal 2004 article in The Canadian Medical Association Journal. (emphasis added)
Is this the funniest thing ever in a scientific journal? There's the old tale of the paper with a one-sentence abstract, "No, it does not," but I think this is in a league of its own.

Now to over-analyze it with two related questions: is counting scientific, and is it science we can all agree on?

In the mathematics of set theory, we begin to answer that question. In logic, counting numbers is an extension of the propositional logic that is so familiar to lawyers in philosophers (hence, I don't like it much). Also, I've never formally studied logic, am not a logician, and can't do it justice. But being outside my specialty hasn't stopped me so far on this blog.
The counting numbers (integers) are called a mathematical group, while the real numbers as a whole are a field. The main property that defines these sets is closure under some operation, like addition or multiplication. This means that any elements of the set that are operated on will generate other elements of the set: add two numbers and get a number, multiply them and get another number, etc. You cannot add apples and oranges to get a pear is the point.

What is interesting about this subject is that our numbers are only one type of field. There's the complex field, the quaternions, the rationals, vectors, matrices, and all sorts of stuff that do not behave like "counting" should. In matrix multiplication, for example, A*B does not equal B*A: it is noncommutative (in the real world, this is why quantum mechanics works the way it does: the universe is not commutative).

But the concept of a field can be too restrictive. A while back some crackpot math teacher proposed adding a new number to math classes called "nullity." While I could analyze it briefly but incorrectly with my limited topology background, the idea is cleanly shot down by Cale. I would make a simple analogy: we count on a number line that extends to infinity in both directions. But what if we added an extra point to that line where infinity should be? Then we have a closed ring (reminds me of the temperature scale that includes negative temperature, actually). We can even add two more points above and below and get a sphere, or something even more exotic, leaving our simple linear counting structures far behind.

At the same time as we destroy the infallibility of counting, set theory reminds us that it's all good: through homomorphism, we can still understand systems in the real world by ordinary counting. Physics, even though it sees the world in exotic groups and complex fields, still works in a counting framework.

I will address briefly whether or not counting is a universal science. It is not. See an article on the Pirahã tribe for an example of people whose concept of integers fades at "3 or more" (see also the original paper, Nevins et. al.'s response, and Everett's response). And before anybody brings it up, Sapir and Whorf are still idiots. So is Stephen Hawking.

Several Amazonian and Austronesian tribesmen have settled in the cities, many with similarly exotic languages. The human mind can adapt - it's not a slave to language or culture - but you'd have to start learning by counting on your fingers by going back to kindergarten. I wonder again what logic is universal. Obviously it's not first-order or second-order, and I would still argue that it's not even basic propositional: according to me, all logic is learned from the environment.

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