system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5:Is it possible that something (nothing in particular) we consider "true" in math could not be true

For example, what if we found out 6^0 (6 to the 0th power) does not actually equal 1? How would that affect what we know about mathematics? Is this question even feasible?
gpt
There's a difference here between math and science.

In science, we make observations about the world and try to find equations that help us explain and predict what will happen. If we discovered tomorrow that gravity is twice as strong if you stand inside a giant magnet, that would totally cause scientists to revisit everything they know about gravity and magnetism. That's not very likely to happen, but surprising discoveries on a smaller scale do happen all the time in science, and the theories have to change to accommodate those new discoveries.

In math, though, mathematicians clearly define what they mean by a number, what they mean by raising a power, and so on. Everything we know in math we know for certain because it's either been defined that way, or proven to be that way. Math doesn't change based on observations of the world, math stays what it is, and we change the mathematical equations we use to explain the world.

So nobody could ever discover that 6^0 is not 1, because it's been defined that any number to the power of zero is 1. It didn't have to be defined that way, but it makes more sense than any other possible value.

Occasionally there have been cases where mathematicians believed something to be true but hadn't proven it, and it later turns out to be false. Occasionally they even believe something has been proven, but this is far less likely because there are a lot of smart mathematicians out there and an incorrect proof doesn't usually last long.