If only it were limited to facts learned from math education. For example, there’s the Jordan curve theorem (https://en.wikipedia.org/wiki/Jordan_curve_theorem), which I guess most four-year olds ‘know to be true’ from their experience with coloring books.
> It is easy to establish this result for polygons, but the problem came in generalizing it to all kinds of badly behaved curves, which include nowhere differentiable curves, such as the Koch snowflake and other fractal curves, or even a Jordan curve of positive area constructed by Osgood (1903).
So to some extent, the reason why such an "obvious" statement requires a complicated proof is because our everyday notions of what a "closed curve" is are much more restricted than what we consider in mathematics. This is kind of common in maths, especially in fields with a lot of visual intuition.
