are there rules that cant be rigorously explained?

Question

this is sort of a shower thought question that came to mind, and I would prefer if it wasn't taken violently seriously, but are their rules that cant be rigorously explained. like for example the limit of a constant times a function is equal to the constant times the limit of a function. Intuitively it makes sense and can be explained. but how about rigorously???

Answer 1

Pretty much any "rule" in maths is either a definition, an axiom, or can be rigorously proved.

Let's look at your example, $\lim_{x\to a}cf(x)=c\lim_{x\to a}f(x)$. To prove it rigorously, we may use the $\epsilon-\delta$ definition for limits:

If for any $\epsilon \gt 0$, there is a $\delta \gt 0$ such that $|f(x)-L|\lt \epsilon$ whenever $|x-a|\lt \delta$, then we say $\lim_{x\to a}f(x)=L$.

Proof:

Suppose $\lim_{x\to a}f(x)=L.$ In this case $ |cf(x)-cL|<|c|\epsilon$ whenever $|x-a|\lt \delta$. By the arbitrariness of $\epsilon$, the "rule" is now proven.

Answer 2

The field of analysis studies that sort of question carefully. Usually, in second-year pure maths, 'epsilon-delta' proofs introduce you to rigour. They came into maths in the nineteenth century, for example Minkowski.
Other times, for example with parallel lines in geometry, you can't prove it either way. So make it an axiom. If you assume there is one parallel line through a point, you get the geometry of the plane. If you assume there is none, you get the geometry of the surface of the earth. If you assume a line has many parallel lines through the same point, you get a different, hyperbolic surface. All three geometries are self-consistent, and apply in different situations. GPS satellites have to deal with hyperbolic space, ships sail on a sphere not a plane; and architects require flat surfaces.