Mathematics

Why is it necessary that 2+2=4? Because it is difficult to conceive how 2+2 could have been other than 4? But how do we know that this is not just due to our limitations? The fact that we, i.e. our brains, cannot imagine a different result does not per se mean that it is logically impossible for 2+2 not to be 4 (given the standard semantics of course).

Comments

We need to keep two questions straight here: (i) why is it necessary that 2+2=4; (ii) why should we believe it's necessary that 2+2=4.

The first question assumes that this is, in fact, a necessary truth, and asks what grounds the necessity. The second asks how we know.

On the first: why is it necessary that 2+2=4? Part of the problem is to decide what would count as an answer. I'll leave it to wiser heads than mine to offer a thought. But I think your worry actually lies with (ii).

Now of course, in some sense of "could," it could be that we're all utterly deluded, and in fact 2+2 isn't equal to 4 at all. All that this means is that I can't rule out beyond any possible doubt that we're utterly addled in ways we can't even imagine. But to quote my old colleague Dudley Shapere for the mty-nth time, the possibility of a doubt isn't a reason for doubt. Translation: the mere fact that we can dimly imagine that we're utterly and totally confused about even the most basic things doesn't mean we should quake in our epistemic lizard-skin boots.

Which is a good thing. Because once we put into doubt whether we know that 2+2=4, it's hard to see what would be left with to base any reasoning on. After all, if I'm wrong in thinking that 2+2=4, maybe I'm wrong when I think that from "If P then Q" and "P", it follows that "Q" is true. But in that case, the game is up and we have no reason to take any of our reasoning -- even about the frailty of our reasoning -- seriously.

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