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Let me try to explain with an example outside of mathematics:

        All swans are white.
For centuries (possibly millennia, as Juvenal thought it, too), that was obvious (in western Europe) to anyone studying nature. Then, Willem de Vlamingh returns from a journey to Australia with some dead black swans.

Now, there are various options. Some of them are:

        - You can drop your claim that all swans are white.
        - You state these aren’t swans because they aren’t white.
Both have merit, but even if you choose the latter, it is hard to keep claiming that it is obvious that all swans are white.

This is somewhat similar. Smart men have studied numbers for centuries, and have thought hard about what it means to be a number. They came up with a set of properties and laws that they thought was necessary and sufficient for a set of objects to behave like numbers. The fundamental theorem of arithmetic was not in that list, as it was thought one could derive it from simpler laws, or, possibly, nobody even thought of doing that, as they thought it to be obvious (I do not know enough of the history of mathematics to know which is true)

Suddenly, somebody comes up with a set of objects that abides by those laws, but for that set, the fundamental theorem of arithmetic does _not_ hold.

Now, you can claim that set of objects isn’t a set of numbers, but you can no longer claim that the fundamental theorem of arithmetic is obvious.




Gowers argues that even without considering generalizations of the reals, it is not "obvious". He argues that if the Theorem were "obvious", we should quickly be able to say that 23 x 1759 != 53 x 769, without multiplying them out.


Perhaps I'm dim but that seems like a ludicrous argument. The inequality is obvious, if you know that all the numbers are prime, and it's not if you don't. Ergo, what's non-obvious is that the big numbers are prime, not that fundamental theorem is true.

Shouldn't the standard of proof should be something like whether it's obvious that:

    23 * 7 * 11  =/=  17 * 7 * 13
? In this case the inequality is obvious, to the extent that you can convince yourself of it without multiplying anything. Doesn't this imply that therefore the theorem is obvious?


It is only obvious if you take the theorem as a given (as mathematicians typically do). However, I bet if you are given a slightly different system, it would be difficult to say whether this is true: For some natural number N, do there exist n primes p1,..,pn such that p1..pn = N and no other m primes q1,..,qm where q1..qm = N ? As an example, Gowers gives the complex numbers, where it may intuitively appear to be true ... but it is not.

Gowers mentions that, to him, an obvious argument would be: 23 x 22 != 21 x 25, since 2 divides 23x22, but 2 does not divide 21x25.


> Gowers mentions that, to him, an obvious argument would be: 23 x 22 != 21 x 25, since 2 divides 23x22, but 2 does not divide 21x25.

I don't follow the distinction - isn't the same true of my example, just swapping "2" for "11" or "23"?

I mean, given Gowers' standing I'm prepared to assume his premise is true, and I didn't follow the last argument (by comparison to other kinds of rings) at all so I'm guessing that it's the "real" reason why the theorem isn't obvious. But the preceding arguments made no sense to me at all.


The difference is that divisibility by 2 can be tested almost instantly. (Only the least significant digits of each factor matter.) However, testing each factor for divisibility is often as tedious as the original multiplication!

I think we're all at a loss for a good definition of "obvious".


But that's a weak argument. By that token, checking whether two numbers add up to a third or whether two numbers are equal or even whether two numbers have the same number of digits aren't obvious, if you make the numbers large enough.

"Can be answered quickly" is not a good indicator of "is obvious".


Well, yeah. Having the same sum, or having the same number of digits aren't really obvious properties of numbers.

Similarly, both being prime and having the same product aren't obvious properties either. Making the fundamental theorem of arithmetic non-obvious (or at least individual cases of it, but if the general case were obvious then the individual cases should also be obvious).


Oh dear. So you're taking the much stronger stance that it's not even just non-obvious to prove, it's non-obvious to intuit!

If you are being honest, the set of numbers the fundamental theorem of arithmetic applies to is much more different from sets of objects it does not apply to than black swans are from white swans.

Obviousness is probabilistic, so proving one seemingly obvious thing non-obvious doesn't mean we have to stop calling everything that seems obvious non-obvious, which is the implication of your analogy.

Sets of objects that don't follow the fundamental theorem of arithmetic are already super non-obvious themselves, so it seems strange to use their features as an excuse to invalidate an obvious feature of the set of integers.


Not only that, but there are other even more obvious properties of the set of integers that don't inhere to other sets of objects since discovered. So your claim is essentially that there are no obvious properties of integers, which leaves one feeling a little flat.


Many people would consider the set of numbers the fundamental theorem of arithmetic applies to be more different from other sets of objects that don't abide by the fundamental theorem of arithmetic than white swans differ from black swans. I'm afraid this isn't the last word on the debate.


Or explain why swans are black in one environment, white in another environment. Here is where knowledge increases. Otherwise, we would be counting all generalized claims, stereotypes as scientific hypotheses.




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