TITLE: Renormalization for mathematicians
QUESTION [13 upvotes]: Can someone explain to me the processes of renormalization and regularization used in quantum field theory and similar fields in a way that a pure mathematician might make sense of it? Is there a mathematically rigorous way to go about doing them? Could you give an example of a basic renormalization and explain why it was necessary?

REPLY [6 votes]: 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.