All swans are white.
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.
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.
Shouldn't the standard of proof should be something like whether it's obvious that:
23 * 7 * 11 =/= 17 * 7 * 13
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.
I think we're all at a loss for a good definition of "obvious".
"Can be answered quickly" is not a good indicator of "is obvious".
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).
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.