Disturbing the foundations of mathematics

Question

I was curious of knowing if it is possible that an event "x" could disturb so greatly mathematics that we could be casting doubts on all the achieved results from the very beginning. I'm not sure if my question is clear enought, if not, I'll reformulate.

Answer 1

Yes, if we can show that induction is inconsistent then pretty much everything we did will require a good thorough review.

On a slightly less catastrophic scale, if $\sf PA$ or $\sf PRA$ would be proved inconsistent, it might also require a thorough checking of a lot of mathematics done in the past few centuries.

It is unlikely, though, that any bridges would collapse, and computer software to contain new bugs, in case that it does happen.

Answer 2

Why not? It may have happened before.

I'm not enough of a historian to be able to say how well attested this is, but the "folklore" asserts that the discovery of the irrationality of $\sqrt2$ caused an immense upheaval in Ancient Greek mathematics. It appears that large parts of their geometry were founded upon the assumption that one can always find a "common measure" for any two line segments: that is, a line segment sufficiently small that each of the given segments is an integer multiple of it. When it was found that this is not always true - and, moreover, that it is not true in such an elementary geometric case as the side and diagonal of a square - Greek geometry had to be completely recast, leading eventually to Eudoxus' theory of proportion.

Answer 3

There are all sorts of "doomsday" scenarios that one can imagine. For example, what if someone finds two 100-digit numbers $a$ and $b$ such that $a \times b \ne b \times a$, as computed using the usual algorithm for multiplication? I imagine that something like this would cast doubt on quite a large portion of mathematics. I don't think it's very likely at all, but once we are willing to accept that weak systems like $\mathsf{PRA}$ might be inconsistent, then I don't see any systematic way to rule out such unlikely events.

So although the question is fun, I think it's a bit too broad. It's just like asking whether it is possible that an event 'x' could disturb science so greatly that we could be casting doubts on all the achieved results from the very beginning. Sure–what if things suddenly started falling upward instead of downward for no discernible reason, etc.