And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.
I gave Peano as an example. I don't mind the assumptions, as long as they're reasonable and presented before the proof. Another comment pointed me to the fact that they were mentioned in an earlier paragraph, so my issue is resolved.
It is right there in the first section of the article.
"In this discussion, we assume that we already know some basic properties of arithmetic operations such as the distributive property of multiplication over subtraction, existence of the additive inverse of real numbers, etc."
Hi! I am the author of this blog post. The assumptions in the third paragraph were there at least since 16 Feb 2019. See https://github.com/susam/susam.in/commits/master/content/blo... for the change history of this post.
if you're going to prove such a fundamental thing, can you please provide the axioms that we start from? I.e. "we know" that a - a = 0, multiplication is distributive, and a x - b = - a x b. These seem arbitrary properties and "equally" fundamental to -a x -b = ab. Either start from peano and prove everything along the way, or tell the reader your assumptions. Don't just divine things along the way.
EDIT: Assumptions are in the third paragraph of the post. I highly doubt they were there when I wrote the comment. Either way, my concern has been resolved.