Sandifer: Right, but when we’re talking about a world where people are – where there are subtle evils and mocking misrepresentations, isn’t a degree of skepticism a good thing? Isn’t a resistance to anyone’s demand that you should obey them a good thing?
Day: No! Not if what is being told is true! How is being skeptical about the truth a good thing? All that means is that you’re opening yourself up to something that is false. Are you skeptical about 2+2?
Sandifer: I am not skeptical about 2+2, but the question becomes…
Day: Right, but would you consider it to be good for someone to be skeptical about 2+2?
Sandifer: But at least within this story, we’re not talking…
Day: Then you agree with me.
Sandifer sensibly ignores Day’s point because it is a distraction. 2+2=4 is an analytical truth whereas the issues that Sandifer was referring to is not even obviously something to which truth values can be applied (whether you should obey your superior in general) but if it was would be a synthetic truth. However I would also say that she is wrong to be not skeptical about 2+2=4. The right thing to say is either “Yes, I am skeptical about 2+2=4 although I have strong reasons to regard it as true universally”. I’ll need to explain that particularly as I’m obviously very much in the pro-mathematics-isn’t-it-great-camp.
We don’t know why mathematics is so useful and so powerful. There is no philosophical or professional consensus on the nature of mathematics and what is worse (or great) is there is not likely to be (see here for a tiny bit of the background). So there is no strong, secure and reliable foundation for our confidence in mathematical truth.
That doesn’t mean that 2+2=4 is an empirical or synthetic truth. It is still true because of how we set things up but we should still question why it is true and whether there are circumstances when it is not true. There are certainly circumstances when it is not applicable. Take 1 puddle and another puddle to it and what do you get? One bigger puddle – yeah but that is cheating and I was playing tricksy word games with “add”. Fair-cop but we also don’t know which circumstances of applying “1” or “add” to actual things MUST count as being correct applications of even basic arithmetic and which count as misleading word games. Nor are these trivial problems – substantial statistical problem occur because people don’t necessarily think about the kinds of numbers they add together e.g. an airline adding all the passengers they have had in a year mistakenly thinking that number gives them the number of customers they have had (it doesn’t because some passengers are the same person).
Worse yet simply taking 2+2=4 on faith is an intellectual dead end. This doesn’t mean mathematicians sat around unsure of what the answer to 2+2 is. Skepticism does not mean a kind of willful ignorance. Skepticism means that we can accept provisionally the truth of something and still dig into the “why?” of that truth. To question a truth does not mean you can’t live with that truth or even regard it as almost certainly true. One of the greatest pieces of methodological skepticism can be found in the work of Rene Descartes – Descartes believed in God and catholicism but still could adopt a position of doubt on a whole range of truths (including god) precisely so he could learn and discover.
As it happens 2+2=4 lacked solid logic for thousands of years. There is a stereotype of mathematics as being built, step=by-step from fundamental axioms by logical deduction. This picture of mathematics arose because of one book Euclid’s Elements. In the Element’s Euclid derives core geometrical truths from a series of self-evident axioms.
BUT – that was for geometry. The same axiomatic approach to arithmetic did not occur until THOUSANDS of years later. Indeed a proper treatment was not completed until 1889 with Peano’s Axioms. By which point proper skepticism applied to Euclid’s Elements had already taken modern thought to places that are frankly freaky – e.g. non-eculidean geometry – which took us to relativity and a world of ideas that were previously almost unthinkable.
So DO be skeptical about 2+2=4 – not because it isn’t true but because it is. Be sketpcial and to quote one last book “oh the places you will go”. (and what a wonderful suggestion of non-euclidean geometry that cover has)