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Looking for something else, I found this article that I posted on Usenet in 2001. I am repatriating it here. It concerns a particular feature of David Brin's Sundiver universe:
I've spent a lot of time thinking about exactly this problem, whether the real numbers might be a sufficiently artificial construction that the aliens, when they come, won't have them. The conclusion that I've always come to is that the real numbers are unavoidable. The aliens will be unable to avoid the Pythagorean theorem, noting, for example, that the diagonal of a 3×4 rectangle has length exactly 5. The Egyptians knew this three thousand years ago, and the aliens cannot possibly miss out on it and still be able to construct space ships. Of course, they call it the Blorkskug theorem, after the famous Nogulobular Blorksug Urbulgolp. But that's ancient history. The aliens will then observe that if the Blorkskug theorem is to hold, then the diagonal of a square, whose edge is exactly one florzbeem, must have length of !!d!! florzbeems, where !!d·d = 2!!. The alien mathematicians might refuse to admit that the diagonal has any welldefined length, but that's a rather obtuse philosophical position. (No pun intended.) It's quite concrete and specific. Mathematicians get paid to solve engineering problems, not to sit in pubs and insist that you're an idiot for wanting to measure the distance between the west and east corners of your wet glotzpurg field. If an alien mathematician loftily informed you that you can't measure that because there is no such number, you'd hit him in the dewlaps. (Ouch!) Now the aliens take out a florzbeemstick and measure the diagonal of the wet glotzpurg field and find that it seems to be about !!\frac75!! florzbeems. Well, it must be a little more, because the !!\frac75· \frac75 = \frac{49}{25}!!, which is a hair less than 2, and the ancient Blorkskug theorem says it should be 2. It won't take long for the aliens to discover that !!\frac32, \frac{17}{12}, \frac{99}{70}!!, and so on are all a little too large, and that !!\frac75, \frac{41}{29}, \frac{239}{169}!!, and so on are all a little too small. But as the fractions get more complicated, their squares get closer and closer to 2. (Aliens can compare fractions just fine, remember.) The smart aliens soon realize that they can make fractions whose squares are as close to 2 as they like — if !!\frac ab!! is close, then !!\frac{a+2b}{a+b}!! is even closer, and you're still computing with integer fractions. Alien mathematicians will wonder if there is a fraction whose square is exactly 2. Some alien will find the proof that there isn't any such fraction. It's an easy proof, totally watertight. The Greeks knew it about 2500 years ago. It's completely obvious once you know about prime factorizations. The aliens can't possibly be dumb enough to miss this, especially if they are as obsessed with fractions as we have supposed. So far the aliens have followed the same path as human mathematics, and I think it's completely unavoidable. At this point they must do something different. Human mathematics supposes that there is a new kind of number, a nonfraction, whose square is exactly 2. We call such numbers 'irrational' because they aren't ratios. We want the aliens to reject this idea. The aliens might deny that there's any number which corresponds to the length of of the line. Instead, they say, it can only be approximated by fractions, although to any desired degree of accuracy. For example, we can give two sets of fractions, say !!\left\{\frac32, \frac{17}{12}, \frac{99}{70}, \ldots\right\}!! and !!\left\{\frac75, \frac{41}{29}, \frac{239}{169}, \ldots\right\}!! which can be used to pin down the actual value as closely as desired, but we can't pin it down exactly. The width of the wet glotzpurg field is not a number, because all numbers are fractions, but we can get a handle on it by manipulating these collections of rational approximations instead. And in fact this works out pretty well. You can add and multiply these collections of approximations and you can get meaningful results that way. The end result will be two sets of fractions; all the fractions in the first set will be less then the real answer, and all the fractions in the second set will be greater than the real answer. If you want an approximation to any desired degree of accuracy, just pick one of the more complicated fractions from either set. If the result of a computation really is a fraction, you will be able to prove that by looking at the approximations. But the notation is rather cumbersome. Sooner or later, someone is going to invent a shorthand. "My central tentacle is getting tired," she will say. "Instead of writing !!\left\{\frac32, \frac{17}{12}, \frac{99}{70}, \ldots\right\}!! and !!\left\{\frac75, \frac{41}{29}, \frac{239}{169}, \ldots\right\}!! all the time, I will just write !!s_2!! as an abbreviation. But it will mean the same thing; it is only an abbrevation." Oops. This alien has just invented real numbers. We write $$\sqrt2$$ instead of !!s_2!!, but the meaning is exactly the same. The computations that the alien does with its !!s_2!! notation are exactly the same as the computations we do with our square root notation. When the alien wants to know how big the result is, it computes a rational approximation; that's what we do too. (That's how pocket calculators work, for example — everything is a rational approximation.) And the alien computations don't merely arrive at the same results as ours by coincidence. They have the same underlying meaning, in the philosophical sense. The fact is, there's no difference between calculating with 'finite rational approximations' and calculating with 'real numbers', because real numbers are finite rational approximations. If you look in a human mathematics textbook, such as Principles of Mathematical Analysis by Walter Rudin, you'll find a mathematical definition of real numbers by construction, and the text will use one of two techniques: Either the construction by Dedekind cuts, which are paired sets of rational numbers exactly as I showed above, or by Cauchy sequences, which are, guess what, sequences of rational numbers. So perhaps the aliens like to deny the existence of irrational numbers, but guess what? We have a long history of doing the same thing. Consider 'imaginary numbers' for example. We may deny that imaginary numbers exist, but that does not stop us from calculating with them, imaginary or otherwise. The aliens may similarly deny the ontological existence of irrational real numbers, but as we've seen they calculate with them just fine. They probably use base17 decimal fractions, just like we use base10 decimal fractions. (Those are just rational approximations, you know.) They may not even realize that they are denying the existence of the irrational numbers, just as we use imaginary numbers every day without thinking about what the phrase 'imaginary number' actually means. It's tempting to view the statement that "the galactic society doesn't accept the concept of 'real number'" as just an overliteral mistranlsation of some phrase analogous to 'imaginary number'. Probably the aliens are saying the same thing about us. "Those crazy humans don't even accept the concept of the square root of !!1!!!" And since the calculations that the aliens do are identical to our own, using rational approximations, we are exactly like the aliens. Someone pass the roast glotzpurg, please! OK, I have to go now. My central tentacle is getting tired.
