Monday, December 8, 2008

RtBotS, Part 5 of 9: Epissedemology


(7) Some religious believers accuse atheists of being not just interpersonally arrogant, but intellectually arrogant, and it's usually because of the very first point in the Rationalist Credo: the idea that everything can eventually be known and understood. At the end of the day, this is the most theologically radical notion of the whole Credo, which is why it appears first in the list. Religion not only takes it for granted that there are some truths that are forever beyond the grasp of man, it exalts it as an explicit doctrine. "The Lord works in mysterious ways, his wonders to perform" is a baroquely constructed admission that humanity should, in principle, find ignorance satisfying.

The reason I gave (5a) first place in the Rationalist Credo is that it requires the nakedest of leaps of faith; the only way to arrive at a position about it is to appeal to one's own intuition and personality. The final sentence of the previous paragraph either leaves one filled with revulsion and distaste, or it doesn't molest one's mood at all. I personally find the idea abhorrent, but I must concede that there is some evidence for it, however uncompelling I may find it. The fact is that brains are like penises: size matters. A concept simple enough for an 8-year-old human to grasp could well be beyond the cognitive reach of a dog, or a bear, or a walrus, simply because an 8-year-old human has a bigger brain than those other mammals. Chimpanzees and dolphins are comparatively much closer to our level because the size difference between their brains and ours is smaller. So, presumably, an animal with a brain twice the size of ours could be vastly more sophisticated, and find it trivial to understand things that would forever remain as beyond us as calculus will forever remain beyond a platypus.

Having said that, I must point out that there is a significant difference between saying that there are things we'll never understand because we fail to cross a certain threshold in neocortex size versus saying that we'll never understand them because they are divine. The religious assertion that some things are eternally unknowable can be interpreted in one of two ways: it could simply be an allusion to the aforementioned cognitive limits imposed on us by our brain size, or it could be a claim that there are really two kinds of knowledge, divine and mundane. At the core of point (5a) is the assertion that there is only one kind of knowledge; knowledge is knowledge. On that basis, I reluctantly accept the possibility of the first interpretation, but flatly reject the second.

Allow me to make a technical detour to explain why I find the brain-size problem uncompelling evidence for a human cognitive barrier.

While it is true that cognitive capacity is proportional to brain size, that proportionality is exponential, not linear. Doubling brain size doesn't double cognitive capacity, it squares it, at minimum; we don't have a precise way to measure "cognitive capacity," so we can't work out what the exponent in that equation would actually be, but we know it would indeed be an exponent. (Think of the Richter scale: a 7.0 earthquake isn't twice as bad as a 6.0 earthquake, it's 10 times as bad. The decibel scale of auditory volume works the same way.) If you look at graphs of exponential curves, you'll find that they start out with a very gradual slope upward, and then hit a point of explosive growth; that "elbow" is the inflection point, or "tipping point" to put it in Malcolm Gladwellese.

The reason this matters is that, ultimately, one's position on whether there are things we can never understand depends on what we mean by "understanding." At root, understanding is synonymous with functional decomposition: we understand something when we know what components make up that thing, and how those components interrelate with each other (which usually involves enumerating the components of the components, and so on). The "and so on" is the hint to what is really going on here; it is code for recursion. Understanding relies on functional decomposition, and functional decomposition relies on recursion. One way to evaluate where brains sit on the exponential size/capacity curve -- whether they are on the low side or the high side of the tipping point -- is to see whether they are capable of recursion.

According to Steven Pinker, language is what you get when you harness ears and vocal cords and put them in the service of recursion. Language is the simplest possible form of behavior that can prove recursive cognition. The capacity to think recursively is at the heart of what he calls "the language instinct" -- recursion is the cognitive technology through which Noam Chomsky's Language Acquisition Device does its job, and converts pidgins into languages through the injection of recursive grammars.

There is lots of auditory communication in the animal world -- the barking of dogs, the eerie singing of whales, the clicking and squealing of dolphins, the chirping of birds -- but there is no evidence that any of it is structured according to a recursive grammar. That seems to be the province of humans alone.

This gives me hope in the face of the brain-size problem. We can't say for sure whether there is a human cognitive barrier, but we can say for sure that there is a universal cognitive barrier, i.e. the tipping point on the exponential size/capacity curve. Recursion is what a brain has to be capable of to pass through that barrier. Since language requires recursion, and since humans have language, our brains can support recursion, so we must be on the far side of that barrier already. The prospect of yet another barrier beyond that, one we have yet to encounter, is unlikely, since exponential curves have only one tipping point.

Since we're on the subject of science and human understanding, in the larger context of atheism, we might as well pay a brief visit to quantum theory, which is something that shows up often in YouTube debates between atheists and religious believers. Many famous scholars of quantum theory have gone on the record to lament how little they've learned about it. "Quantum mechanics is magic," said Daniel Greenberger. "Everything we call real is made of things that cannot be regarded as real," said Niels Bohr, who also said, "Those who are not shocked when they first come across quantum theory cannot possibly have understood it." John Wheeler echoed this with, "If you are not completely confused by quantum mechanics, you do not understand it." Richard Feynman -- one of the few physicists, along with Hawking and Einstein, to be not only universally respected but universally liked -- summed it up: "It is safe to say that nobody understands quantum mechanics."

Some religious believers gleefully present these admissions of bewilderment as proof that there is indeed a human cognitive barrier, and that quantum theory rests on the far side of it. But this conclusion belies a corrupt understanding of what it means to understand. I fear now that I may have furthered that corruption with my recursion discussion above.

I stand by the idea that recursion is the universal cognitive barrier, but by using the universality of human language to prove that we're beyond it, I may have implied that this makes recursive thinking easy. Children are born wanting to learn languages and invent them when there aren't any nearby, but just because that application of recursion is subconscious and effortless doesn't mean that recursion can be subconsciously and effortlessly applied to other cognitive domains. Nothing proves that more than everybody's favorite subject, math.

We seem to be born with a "number instinct," too, but it is far less powerful than our language instinct. It allows us to conceive of the idea of "number" and to apply that concept to the problem of counting, but anything else, even basic arithmetic, has to be taught and practiced. This is far from subconscious and effortless; indeed, most people find it painful, humiliating even, especially when they get beyond arithmetic into algebra and calculus.

Mathematics is often poetically described as "the universal language," but if that's true, it's the only language in human history not to have a concept of time built into its structure. (Notice that even music, the other candidate for a universal language, has an innate dependence upon time built into its structure.) Mathematics does share with other human languages a reliance on symbols and recursion, but it differs from them in its timelessness. In language, useful statements are always narrative; they start at one point in time and end in another, and something happens in between. In mathematics, useful statements are equations, which don't start or end, but simply are. Mathematical statements have no relationship to time unless time is explicitly referenced as one of the variables. Linguistic statements are inextricably embedded in time, whether they explicitly reference it or not.

Fluent speakers of multiple languages occasionally run into trouble when they encounter a word in one tongue that has no direct equivalent in the other; they have to rely on some kind of "linguistic intuition" to convert the word into a roughly equivalent phrase in the other tongue. Douglas Hofstadter has written quite a bit about how thorny such problems of language translation can be, but they are merely potholes in the road compared to the difficulty of translating from mathematical statements into narrative statements or vice versa. Such a translation requires a "mathematical intuition" that can not only compensate for mismatches in vocabulary, but mismatches in structural assumptions. It's a cognitive problem that requires an effort just shy of, as the saying goes, "dancing about architecture."

But it turns out that, in the real world, math-to-language translation is much more useful than architecture-to-dancing translation, so there is an incentive to forge ahead and develop one's mathematical intuition. This is the rationale behind word problems (and also the reason why so many people hate them) -- you have to take a narrative representation of a problem, translate it into a mathematical representation, solve it, and translate it back. The point of word problems is not so much to get the right answers as to provide a workout for your mathematical intuition. This is also why so many math teachers suck; they get the math fine, but don't have a sufficiently well developed mathematical intuition to translate that understanding into a narrative form that their students can easily digest.

The relevance of all this to the mysteriousness of quantum theory is that, when Feynman said "nobody understands quantum mechanics," what he meant is, "there are many people who have a thorough mathematical understanding of quantum mechanics, but we have yet to find someone with a sufficiently well developed mathematical intuition to translate that understanding into a narrative form that others can easily digest." The ultimate goal of science is to make accurate predictions about real-world phenomena, and by that standard, physicists in fact have quite a formidable understanding of quantum theory. But the mathematical statements that express it are so strange that they defy translation into sensible narratives.

That's why I don't view the mysteriousness of quantum theory as proof, or even evidence, of a human cognitive barrier. There is a big difference between saying "we have yet to find someone with a sufficiently well developed mathematical intuition" and saying "we know in principle that there cannot be a sufficiently well developed mathematical intuition." Mathematical intuition is a skill, like playing a musical instrument or piloting an airplane. It can be improved through effort and dedication. To concede that there are things that cannot, in principle, be known or understood would eliminate our motivation to keep practicing.

Ultimately, it comes down to optimism versus pessimism. Only the pessimist claims there is no point in even trying. Only the pessimist would be satisfied with eternal unknowability.


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