Most philosophers won’t hesitate to nominate 2+2=4 as the paragon of a mathematical theorem : a complete misunderstanding.

We can either see 2+2=4 as a computation, i.e., an activity not involving any sense, any reasoning : analytic in the kantian acception. Or we can see 2+2=4 as the result of a reasoning — hence synthetic — however based on experience : everything can be checked, there is no room for doubt. This synthetic a posteriori is not typical of mathematics, which is naturally synthetic a priori. This means that mathematics cannot be justified — hence the failure of the foundational programs of a century ago.

If these programs didn’t succeed in alleviating our — however, unreasonable — doubts, they however individuated, inside mathematics, a synthetic a posteriori layer. This on a large scale, not limited to finite computations. How is it possible to deal with infinity and still be based on experience ? And what does this mean ?