>>12466349It's not, he just didn't articulate it properly. While you can construct a special snowflake algebraic structure for every taste you can potentially have, the main one that society has accepted to use is the one he's talking about, namely the one that forces associativity as the ultimate undeniable axiom along with a few other things. It is not as much of an issue of "is a construction with associativity valid" but rather "why is the standard that we've all decided to use demanding it", which is a very legitimate criticism. Wildberger's criticisms stem from the exact same problem, it's not that the reals should be outlawed and that you can't have your special cases where you implement them, but the standard of math that is used all over the world - the standard that we use in empirical applications - there is no point of having them in there because we will literally never use them. We will never use but an approximation of it. It is physically impossible within this universe to use it because the universe itself is finitist. It's an unbounded infinity of space but everything inside is finite, which is (basically) what OP meant by mentioning thermodynamics. So if we're going to accept one and only one set of axioms who're the standard then that standard should by all means be related to our physical reality, else it's pretty much useless. Here's the problem in all of the above - that the academia refuses to accept this because it'd take too much energy for the establishment to re-calibrate according to it and the path of least resistance is simply ignoring it. You can't be a one man army like Wildberger either or you'll get called a schizo. The only place that such a discussion can happen and attack this cemented establishment is on a decentralized forum where there's no repercussions to your image, which is the reason why this has not happened so far in our society.
