In the animal kingdom, the math does you!
One of the most comical denials of evolution that I have ever seen was when one leftist literary intellectual argued against kin selection by laughing that animals certainly can't multiply and compare fractions in their heads.
This mistake, in a more general form, is to believe that an agent must consciously understand why the strategy that it follows is good in order for that agent to reap the benefits of that strategy. Of course this is totally false. The result of using some strategy is determined by the environment that surrounds the agent, not by the beliefs that the agent has (or doesn't have) about the strategy and how it works. For example, a chess computer doesn't have any real understanding of the game, but as long as it makes moves that are objectively better than the moves that its opponents makes, it will win.
Having accurate conscious beliefs about strategies can, of course, help choosing between the available strategies and explaining why they are good or bad, but simply following a good strategy is enough to reap the benefits from it, no matter whether you follow this good strategy after conscious analysis or because whatever process it was that built you has forged your behaviour to be concordant with this strategy under the circumstances that you typically happen to find yourself in.
NPR's "Math Guy" Keith Devlin's (column archive) new book "The Math Instinct" examines the innate mathematical abilities of various animals and humans. Of course, as the writer notes, we should be careful not to proclaim that a human "solves" a calculus problem over the Newtonian gravity field by jumping off a tall building. In this case, the math, insofar as there is any, is done on the human by the universe. Similarly, the author explains, beaver shouldn't get the credit for building its dam, since a beaver merely collects sticks and mud and piles them up with no coherent plan, and the flow of water shapes the dam to a stable state.
After discussing the mathematically complex and computationally extremely demanding (but totally nonconscious) feats of ants, bats, birds and other lovable animals in echolocation, migration, balancing their bodies or retracing their steps, the book moves on to discuss conscious understanding and manipulation of numbers. Of course, here it is mostly humans who can do anything at all, even though some animals are clearly able to count in ways that they must have some kind of abstract notion of a number. (According to the book, even very small human babies can count.) In the chapter about what the author dubs "street math", we meet a bunch of people who have had no formal education in math, but who can perform complex price calculations as street vendors, even though they can't repeat these feats when the same problems are posed to them out of their usual context.
By the way, here is one general gripe that I have had for a long time towards books, and I will mention it now since the Devlin's book reminded me of it. You see, when I am reading some book, I very rarely forget what book it is that I am currently reading. And if I ever did, it would take only a second to flip the book over and look at its front cover. Hence, it is unnecessary to keep repeating the name of the book in the header of each page. It would be far more useful to put the chapter title in these page headers, thus making it easier for the reader to keep track how far he has been reading and search for a particular topic in the book. Thank you.
This mistake, in a more general form, is to believe that an agent must consciously understand why the strategy that it follows is good in order for that agent to reap the benefits of that strategy. Of course this is totally false. The result of using some strategy is determined by the environment that surrounds the agent, not by the beliefs that the agent has (or doesn't have) about the strategy and how it works. For example, a chess computer doesn't have any real understanding of the game, but as long as it makes moves that are objectively better than the moves that its opponents makes, it will win.
Having accurate conscious beliefs about strategies can, of course, help choosing between the available strategies and explaining why they are good or bad, but simply following a good strategy is enough to reap the benefits from it, no matter whether you follow this good strategy after conscious analysis or because whatever process it was that built you has forged your behaviour to be concordant with this strategy under the circumstances that you typically happen to find yourself in.
NPR's "Math Guy" Keith Devlin's (column archive) new book "The Math Instinct" examines the innate mathematical abilities of various animals and humans. Of course, as the writer notes, we should be careful not to proclaim that a human "solves" a calculus problem over the Newtonian gravity field by jumping off a tall building. In this case, the math, insofar as there is any, is done on the human by the universe. Similarly, the author explains, beaver shouldn't get the credit for building its dam, since a beaver merely collects sticks and mud and piles them up with no coherent plan, and the flow of water shapes the dam to a stable state.
After discussing the mathematically complex and computationally extremely demanding (but totally nonconscious) feats of ants, bats, birds and other lovable animals in echolocation, migration, balancing their bodies or retracing their steps, the book moves on to discuss conscious understanding and manipulation of numbers. Of course, here it is mostly humans who can do anything at all, even though some animals are clearly able to count in ways that they must have some kind of abstract notion of a number. (According to the book, even very small human babies can count.) In the chapter about what the author dubs "street math", we meet a bunch of people who have had no formal education in math, but who can perform complex price calculations as street vendors, even though they can't repeat these feats when the same problems are posed to them out of their usual context.
By the way, here is one general gripe that I have had for a long time towards books, and I will mention it now since the Devlin's book reminded me of it. You see, when I am reading some book, I very rarely forget what book it is that I am currently reading. And if I ever did, it would take only a second to flip the book over and look at its front cover. Hence, it is unnecessary to keep repeating the name of the book in the header of each page. It would be far more useful to put the chapter title in these page headers, thus making it easier for the reader to keep track how far he has been reading and search for a particular topic in the book. Thank you.
Of course, old books actually used the header space to say things like the chapter or whatever relevant organization there was. It made naming the chapters worthwhile.
And then for some reason they changed it. I remember when I stopped looking at the headers altogether, because they only told you two very obvious things.
I also used to be able to skip chapters I decided were boring, but now you have to find the exact page the next chapter starts on to know you've skipped it.
Posted by Alan | 11:46 PM