Math 201

In particular, as I alluded to yesterday, when I was twelve I had my first real encounter with the infinite – courtesy of Euclid.

Please understand that for me this was a very big deal. I had been wrestling for several years by then with people throwing around words like “infinity” and “God”, and I was getting quite uncomfortable with my growing understanding of the culturo-centrism around me. I found it seriously disturbing that we were being taught about all of the Greek gods to be found in Edith Hamilton’s “Mythology” in terms that made it clear that they were unreal – the product of an immature civilization, whereas (so we were told) our Judeo-Christian God was the real thing.

And yet it was clear even to my pre-adolescent mind that the complexity and subtle humanity to be found in Zeus, Apollo, Athena, Aphrodite and the other supposedly mythological inhabitants of an earlier Pantheon were much more in tune with the human condition that I saw around me every day than was the highly abstracted and invisible God that our society had more recently settled on.

So I guess you could say I was ready for some more tangible encounter with the infinite. And at twelve years old I found it, thanks to old Euclid of Alexandria.

Specifically, I learned that back around 300BC Euclid proved that there were an infinite number of prime numbers, and that he did so in a way that did not in any way resort to belief or doctrine – merely straightforward statements of fact and steps of logic.

I’m sure many of you know the proof – it is simple and elegant, and it goes like this: Suppose there are only a finite number of prime numbers. Well, that means you can write them all down in a list. It might be a very very long list, but it’s a finite list, so eventually – if you write long enough – you can write them all down, every one.

Euclid’s trick was to ask what happens when you multiply together all of the prime numbers on your list. Of course you get some humongously huge number, which you can call N. Then he looked at N+1, the number that is one greater than N.

Here’s where things get interesting. Clearly N+1 is not divisible by any of the prime numbers in your list – if you divide it by any of the numbers in the list, you will get a remainder of 1.

So either N+1 is a prime number, or it must have factors in it that are not in your list. Either way, there must be at least one prime number that you didn’t put in your list.

But that’s a contradiction – your list was supposed to contain all the prime numbers. In other words, it’s impossible to make a list of all the prime numbers. Every time you try, another one will pop up that wasn’t on your list.

And so the prime numbers must go on forever, infinitely.

When I was twelve and saw this, and understood what I was seeing, it was a huge relief. I realized you can talk about things that are not only vastly larger than us but are in fact infinite, bigger than the stars and galaxies, bigger than the universe itself, simply by stating some simple facts and applying a little reason.

Which to me made a lot more sense than trying to decide how the universe works by counting how many people are voting for Vishnu, Zeus, Yahweh, Jesus or the Buddha in any given country or century. Mathematics is beautiful because it is true, in the most powerful possible sense of that word.

Math is, quite specifically, the science that seeks verifiable answers to the question “What is true?”

Even about infinite things.

6 thoughts on “Math 201”

  1. That. Was. Awesome.

    (I have not thought of prime factorization in almost ten years. I also have never encountered this proof!)

  2. By the many arms of Vishnu, I swear, this thinking will never get you to Valhalla! At least not in this life…

  3. ROFL – I have been in the Walhalla temple.
    Maybe I can bring a small Ken bust with me next time and set him up next to Gauss (if that is ok for you Ken 🙂 )

  4. But wait a minute, as far as I remember Ken would have to be dead for at least twenty years to be eligible.
    And I believe he should better have been born in Europe too.

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