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Mathematicians use both self-reference and proof by contradiction all the time
Mathematicians use both self-reference and proof by contradiction all the time

Make a case that numbers are interesting

In a set of uninteresting numbers, the smallest one is interesting for being smallest

After many days of non-stop numbers generated by one election or another, I’m actually mildly burnt out. There’s only so many times you can read about leads being whittled down or up, or what it actually means to say one candidate has a 10% chance of winning, or why a popular vote majority does not necessarily mean a majority in either an electoral college or a state assembly. I say this even though I generally find numbers, and the things they can do for you, interesting.

So for this column I’m going to turn my attention to some less-portentous aspects of numbers.

Start from the top: are numbers indeed interesting at all? (Restricting ourselves for the time being to the positive integers—5, 89, 4402 and the like). Or let’s put it like this: Is there something interesting you can say about any given number? For example, 125 is interesting because you can express it as the sum of two squares in two different ways: 52 + 102 = 25 + 100 = 125; and 22 + 112 = 4 + 121 = 125. Similarly, and as the great S. Ramanujan famously told his mentor G.H. Hardy, the number 1729 is interesting because you can express it as the sum of two cubes in two different ways: 13 + 123 = 1 + 1728 = 1729; and 93 + 103 = 729 + 1000 = 1729. Or take 28, a so-called “perfect" number because it is the sum of its divisors: 1 + 2 + 4 + 7 + 14 = 28. If you exclude 1 itself, the smallest such perfect number is 6.

If these are interesting facts about those respective numbers, is it true that there’s something interesting to say about every single number?

I once listened to a mathematician answer that in a semi-facetious way. Let’s assume, he said, that it’s not true. That is, let’s assume there is a pool of numbers about each of which there is simply nothing interesting to say. Now naturally when you have a pool of numbers, one must be the smallest. So it is with these deadly boring numbers—one of them has to be the smallest of the lot. Well, well, well, what do we have here? That it is the smallest of the uninteresting numbers is clearly an interesting factoid about that particular number. Which means that we have a contradiction right there: in this pool of uninteresting numbers, one turns out to be interesting indeed. Since we’ve reached a contradiction, our assumption that there are some totally uninteresting numbers must be false, and every number is indeed interesting.

You might scoff at this analysis, and I wouldn’t blame you. Yet note that I said “semi-facetious" above. That’s because there is something essentially mathematical—logical, really—in how this reasoning goes. Here we used it to explain the so-called “interesting number paradox", but it can and has been used in many more serious mathematical situations too. There’s the idea of self-reference, in that the smallest uninteresting number becomes interesting precisely because it is the smallest uninteresting number.

There’s the technique of proof by contradiction, or reductio ad absurdum, when an assumption you start with leads logically and inexorably to an absurdity or a contradiction. This can only mean the assumption itself was mistaken: in this case, the assumption that there are uninteresting numbers.

Mathematicians use both self-reference and proof by contradiction all the time. For example, consider one of the most famous and elegant proofs in all mathematics, one that the remarkable Greek mathematician Euclid dreamed up in about 300 BC.

The hypothesis: There are infinitely many prime numbers.

Proof: Start by assuming the hypothesis is false. That is, let’s say there is a pool of all the prime numbers. Of those, clearly one must be the largest. Call it M. Now we can list all the primes in this pool in order from smallest to largest: call them m1 (which is actually 2), m2 (3), m3 (5) and so on, ending with M.

Multiply all these primes together, add 1 and call that new number N:

N = m1 x m2 x m3 x … x M + 1

For example, if our largest prime happened to be 7, we’d do this:

N = 2 x 3 x 5 x 7 + 1 = 211

What can we say about N? Clearly it is larger than M. If it is a prime, we have found a prime larger than M. What if it is not prime? Well, then it has to be divisible by some primes. We know that if we divide it by any of the primes in our pool (m1, m2, etc), we will have a remainder of 1. Thus whatever it is divisible by is not in that pool. Again, we have found a prime larger than M. (In our example above, 211 itself is a prime and as you might note, it is larger than 7).

Either way, we have a prime larger than M. Therefore, our initial assumption is wrong, and there are indeed infinitely many prime numbers. That was Euclid’s elegant proof by contradiction.

As for self-reference, the interesting number paradox is actually a version of a more serious paradox that Bertrand Russell first discovered early last century. In essence, and shorn of mathematical syntax, Russell’s Paradox goes something like this.

I spoke of a “pool" of numbers above; the more accurate mathematical term for that is “set". Russell said, think of a set, call it R, that contains all sets that are not members of themselves. For example, take the set of all essays written about Russell, or the set of round dishes in our kitchen, or the set of harmonicas in this bag I keep my instruments in—none of these contains itself, so all three are members of R.

Question: is R a member of itself? If it is, then by definition it is not a member of itself, so fling it out. But if it isn’t, then by definition again it must be a member of itself, so we should fling it in there. Whereupon it must be flung out… and on we go like that, taking it out and immediately putting it back in and taking it out again.

We have a contradiction, a paradox. Now you might think this is an artificial construct that has no bearing on reality, but thinking along these lines has led some great mathematicians down some deep mathematical paths indeed.

Kurt Gödel was one; his Incompleteness Theorem is one of the most profound truths of mathematics.

I won’t get into Gödel’s work here. Instead, I’ll leave you with something to think about the next time you visit your hairdresser or barber. For one well-known way to help you wrap your hairy head around Russell’s Paradox involves barbers.

Let’s say Karan is the only barber in town. Naturally he gets plenty of business, even if in these corona times he cannot operate his salon. He has taken to advertising himself with what he thinks is a pretty nifty slogan: “I shave only the dudes who don’t shave themselves!"

Which is fine as far as slogans go. But Karan’s wife Rukhsana is an accomplished logician. One day, she asks him: “Sweetheart mine, who’s shaving you?" Karan is about to retort “Me!", but something sticks in his throat and he realizes what Rukhsana is asking.

For if he shaves himself, he must not shave himself. And if he doesn’t shave himself, he must shave himself. Karan’s pithy slogan, you see, binds him in a paradox.

So how would you answer Rukhsana’s question? Who’s shaving Karan?

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun

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