Few things truly blow my mind. It’s not that I’m so intelligent and perceptive that any surprising fact seems unsurprising to me: it’s more that I like to reserve hyperbole for special occasions, and not every mildly surprising fact is a special occasion. One thing that truly blows my mind, though, is math. Not math in general, but the orgasmic parade of mindblowing facts and theories that exist within it. I’m meta-mindblown by the sheer amount of mind-blowing that goes on in mathematics. It’s such a shame that this isn’t in mathematics education: instead, math is taught variously as a practical, but boring toolkit, or as a series of highly theoretical, but uninteresting abstractions. The real joy of math is nowhere to be found in mandatory schooling anywhere in the world, as far as I know.
I feel confident that this is not simply frustration of a burnout—I don’t have any mathematical education beyond high school and scattered outside reading—because many professional mathematicians have lamented the same thing. However, clearly the solution to this problem isn’t obvious. In the 1960s, there was an attempt to bring this abstract but interesting stuff into elementary math education. It was called New Math, and it was an abject failure.
A while ago I linked to Edge’s question for 2012: what is your favorite deep, elegant or beautiful explanation? I didn’t really consider the question myself, content to revel in the explanations of others smarter than me. But then today, I stumbled on a link, and suddenly it hit me. My favorite explanation that is both deep, elegant and beautiful is Georg Cantor’s infinite set theory in general, and his diagonal argument in particular.
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Representing a wide range of real numbers in finite space is tricky business. Computers use floating point numbers, which behave like real numbers in most ways, but sometimes lead to surprising results. Around zero, not all numbers are representable, and some are only representable in a special form that sacrifices precision and processing speed: the denormalized floating-point numbers.
It shouldn’t come as a surprise that not all real numbers in a given range can be represented in finite space. That’s because between any two reals, or indeed any two rational numbers, there’s infinitely many more. When we start talking about infinite sets, we’ve abstracted so far away from the puny origins of numbers used for counting that our intuitions break apart.
I keep returning to this self-contained little crash course in infinite set theory. It’s really quite readable and assumes no prior knowledge of set theory, but the learning curve is steep. It’s also a guaranteed mind-blower. For a little ape brain used to counting things one by one, the idea of a set of numbers that can’t be counted is pretty hard to grasp in itself. But it gets better, or perhaps worse: there are no more rational numbers than there are natural numbers, even though the natural numbers are clearly a subset of the rationals.
Georg Cantor is the founder of set theory. Cantor’s big insight was that when comparing the sizes of groups, counting isn’t the essential thing. When we want to know if we have more apples than oranges, or if Scrooge McDuck has more money than the bum on the street corner, we count them separately and arrive at neat numbers that we can compare in the usual way. But this clearly doesn’t work on infinite sets. However, there is another way to go about this. We could put the apples in one bag and the oranges in another, and then keep making pairs, one from each bag, and then see which bag empties first. This notion of one-to-one correspondence does generalize to uncountable sets, and it forms the basis of Cantor’s extension of the concept of size to infinite sets: cardinality. Two sets have the same cardinality (which reduces to “size” in finite sets) when they can be put into a one-to-one correspondence with each other.
There is some recent evidence that correspondence, or matching things up with each other to estimate magnitude, is a more basic cognitive strategy for humans than is counting. The widely studied Pirahã tribe of the Amazon appear not to have any words to express exact quantities, not even “one.” However, as experiment has shown, they perform well on one-to-one matching tasks. This also refutes, or at least strongly suggests the falsehood of previous claims that the Pirahã’s lack of words for number has left them completely without the concept of exact quantity. Of course, Cantor died long before the Pirahã came to scientific attention.
Cantor demonstrates a clever way to put the rationals in a one-to-one correspondence with the natural numbers, thus showing that in an important sense—perhaps the only sense that makes sense—they are of equal magnitude. From this result, we may be tempted to think that all infinite sets are the same cardinality. But this is a mistake. The crash course states it succinctly: “infinity is not synonymous with ‘totality’, a clarification which alone dispells many of the ancient conundrums and paradoxes surrounding the infinite.” Cantor has an ingenious method called the diagonal argument that shows that you cannot put the real numbers in a one-to-one correspondence with the natural numbers, and thus that the reals have a larger cardinality.
It’s a proof by negative. Assume we have some pairing of each real number to each natural number. Express the real numbers as infinite decimal expansions. Put them in a table, so that the naturals (1, 2, 3, …) run in the left column, and the corresponding real numbers in the right column. Then start with the first digit of the first real, and change that. Walk diagonally down to the second digit of the second real, and change that. Continue diagonally down through all the real numbers. Here is an illustration from Wikipedia:
The resulting infinite string of digits is demonstrably not in the list, since it differs from the first real in the first digit, from the second real in the second, and so on. Thus the real cannot be in the list, and the one-to-one correspondence can’t be complete. Therefore, the infinity of real numbers is larger than the infinity of natural numbers. Quod erat demonstrandum.
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The most mindblowing sciences are those that are in some sense foundational. There is physics, which explains the most fundamental nature of reality. There is math (and some technical philosophy), which explains the most fundamental nature of rational thought. There’s the cross-section of neurology and psychology, which explains the most fundamental nature of subjective experience. And finally, there’s linguistics, which explains the most fundamental nature of human communication. (Which is to say, they allattempt to explain these things.) Everything else in some way builds on these, and while there’s plenty of interesting stuff in other sciences, there’s nothing quite like turning the basic elements of reality and of experiencing reality as a human being upside-down, or blowing to pieces our most cherished intuitions about the world. Or, for that matter, trying on for size entirely novel categories of thought.
I have a natural attraction to explanations and stories. All my interests beyond those that are purely emotional and physical (martial arts, sex) in some way reduce to stories and explanations. They are what I want to do with my life, in some sense or fashion. Fundamental explanations are especially attractive, since they tend to be the deepest, most elegant and beautiful. It makes me wonder, why didn’t I go into science? It’s still not too late: why don’t I? But then I remember my attraction to smaller stories that aren’t particularly deep or elegant, but are frequently beautiful: the human stories that are best told in literature and in photographs and films. And it saddens me that it seems impossible for me to seriously pursue both kinds.
