Why isn’t the fundamental theorem of arithmetic obvious? (2011)

For example, 24 is divisible by 6, though 24 has the prime factorization 2 * 2 * 2 * 3, with no 6 to be found.

"Right, but all the prime factors of 6 appear in the prime factorization of 24. If X is divisible by the prime p, then there is a unique prime factorization of X, which must include a factor of p", you may reply.

Well, this is the non-obvious fact. If you don't already know for certain that prime factorizations are unique (and this is the claim in contention), then it's not obvious why X couldn't have multiple prime factorizations, some including p directly and some not including p directly even though p ends up dividing the overall product.

The non-obvious fact is that a prime can't divide a product unless it divides one of the factors. Why should that be true?

(It IS true of the integers, but it's NOT true of other very similar structures. The reason it's true of the integers, the special property that makes the whole thing tick, is, ultimately, that Euclid's algorithm shows us how the values not divisible by prime p are in fact invertible modulo p (and vice versa), so that the values not divisible by p are closed under multiplication. But that's not at all immediately obvious!)

I especially disagree with his interpretation of the layman's experience. They're not assuming the proof or begging the question; they have a secondary unrealized assumption that is not at all their fault.

The fundamental theorem of arithmetic is "obvious" because we grow up learning a system where if p|ab then p|a or p|b. As mathematicians, we know that Euclid's lemma is a requirement for a unique factoring domain, and we understand that the choice of set can affect whether the lemma is true.

For a non-mathematician, this is an inherent part of their conception of numbers, factoring, and their definition of the word prime.

Put yourself in the position of talking to someone who thinks prime factorization is "obvious". Where will things go wrong in their explanation? In answer 2: they have a completely deterministic way to prime factor, and it relies on Euclid's lemma.

So you ask them: "Well we know 6 is divisible by 2. What if we break it into 2 numbers that aren't divisible by 2?"

"Well then this wouldn't work. But that's not possible."

"But what if it was?"

"But that's not how numbers work."

"But what if I took each of those numbers and replaced them with two numbers and that I'm going to call one big number. And when I multiply those together I'm going to multiply the first number of each pair together, then multiply the second number of each pair together. Then I'll multiply the result of that second multiplication with -5, and add that final answer with the product of the first pair I did earlier."

"uh"

"If I do that, then do you believe that I could get two factors of 6 where neither pair of numbers is completely divisible by 2? Try (1,1) and (1,-1) and you'll see that it works: 1 x 1 = 1, 1 x -1 = -1. That -1 x -5 = 5, plus the original 1 gives me 6. See! It's not obvious."

"Yeah, uh, you didn't tell me I could make a new type of number and multiplication up on the spot."

"No no no, it's all mathematically sound. You see, those second numbers are just the coefficients of the square root of negative 5."

"Okay, so is there any rule I've learned that I can trust?"

"So I bet you think it's obvious that when you add two numbers, you can always get an answer..."

We are essentially telling kids that (American) football is a game where two groups of people line up, block, run formations and routes, give the ball to someone, and try to get it into the end zone for points.

Then, when it's your turn to go on offense, you drop back, throw a pass to your undefended wide receiver, and act amazed that the other team didn't even consider that you might throw the ball simply because you didn't forbid it.

The fundamental theorem of arithmetic was proven around 1,800 years before sqrt(-5) was even conceived of. Over two millennia before Ring Theory. Those proofs were correct for the systems in which they were written. They might even be obvious within that system. That that they are not generally true does not change things.

If you want to argue about what #600000 and #FF0000 should look like, you're back to a rigorous theory of color.

I do say that. It's right there at the bottom of my comment.