Renormalization for mathematicians

Solution 1:

Maybe you should start reading a bit about the history of QFT as it would give the justification for the mathematics behind renormalization. A good starting point could be:

I - https://plato.stanford.edu/entries/quantum-field-theory/qft-history.html

You won't, however, be completely free from physics while trying to grasp the concepts. I found a paper that claims to explain QFT and RG (renormalization group) in just 11 pages.

II - https://arxiv.org/pdf/1311.4897.pdf

Not sure if it is enough, but you might give it a try. There is also the text from Weinberg about QFT development:

III - http://www.fafnir.phyast.pitt.edu/py3765/WeinbergQFThistory.pdf

That may help on organizing the subjects in your head.