Renormalization is a beautiful hack, but it is a hack.

Let's take infinity, subtract other infinitites add a bit of alchemy and here is our prediction. <- from mathematical perspective this is a very sketchy business.

But it works. And it works in other fields. For example in the theory of phase transitions (I have PhD on the topic :))

And it is pretty common in Physics to work with sketchy constructions for decades before mathematicians catch up.

I love the example of "Generalized Functions" (examples are Delta function and Step Function) that were widely in Physics decades before a consistent mathematical theory was developed.

I hope, at some point we will have a similar story with Renormalization theory.