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From a mathematical point of view, one must be very careful, and we have abundant evidence of that.

However, we are fully justified in saying that if anybody came up with a mathematical system in which 2 + 2 != 4, we can dismiss it without having to do some sort of deep analysis of it. 2 + 2 = 4 is obvious. We can literally do it with 4 little objects right in front of us. If we can not accept that as obvious, we are hopelessly ignorant and have no reason to trust our fancy proofs, either. (Italicized to emphasize my main point.) If you can't trust that, you certainly can't trust the significantly more complicated number theory axioms do anything useful.

Note that 2 + 2 = 4 carries some implicit context when we say it without qualification, and subtly sliding in a context change is not a disproof. 2 + 2 = 1 modulo 3, but that's not what anybody means without qualification. Clearly we're operating on "the numbers I can hold in my hand" here, or some superset thereto. Note how I'm not even willing to say "the natural numbers" necessarily; it isn't obvious to me what some billion digit number added to some other billion digit number is. It's actually crucial to my point here that I'm not extending "obvious" out that far; I can only run an algorithm on that and trust the algorithm. But I'm just being disingenuous if I claim ignorance of 2 + 2. And being disingenuous like that tends to turn people off, and doesn't encourage them to try to learn more.




Let's say I have a new mathematical system (that I call the "brontosaurus"). It is made up of some fundamental elements ("a little end", "a big part in the middle", and "another little end") and some axioms ("and that's how dinosaurs are shaped"). Now, I claim that this is a complete formalization of all mathematics, such that if you prove some statement S involving quaternions, tesseracts, and turgid verbals, you can be assured that no falsehoods have crept in and therefore that S is true.

But in my system, it's not immediately apparent whether 2+2=4, if only because none of '2', '4', and '+' are part of the fundamental elements. So, how are you going to determine whether my system claims 2 + 2 = 4 or 2 + 2 != 4? You'll need to prove it, one way or the other (or both, in which case my system is screwed).

The proof that 2+2=4 has absolutely nothing to do with "claiming ignorance" and everything to do with whether or not you can "trust the algorithm".




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