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One of the pleasant properties of the "game played with symbols" is that it doesn't matter why you're playing it, everyone still gets the same answers out. You can play it because you think it describes some "real" but abstract thing, or because you think the purpose of mathematics is to predict the universe and that by manipulating symbols you can do that, or you can play it because you like symbols. Nobody cares, they can use your results anyway. The same is not true of intuitive reasoning.

There's more than one way to provide a foundation for mathematics. The most widely-referenced at the moment is axiomatic set theory, but 2000 years of valuable results in mathematics were obtained without it, and with only the occasional mishap. The clever (and perhaps surprising) thing about axiomatic set theory is that it could be "slid in" underneath all of that reasoning, in a way that avoided fundamentally changing what mathematicians accept as a proof in most fields.

The metamath project seeks to compile proofs from ZF(C) of everything. It's interesting that even where such elementary proofs don't already exist, because mathematicians simply haven't written out the full detail of every proof in predicate calculus, nobody expects the project to fail to produce them. Mathematicians "can tell" that they're making arguments that formalise even without formalising them, obviously with some small scope for error.

As such, it doesn't matter that Euclid wasn't reasoning about a set that models a particular theory, because somebody who does that, or in general who reasons from axioms, can get the same results.

Sometimes people who care about axioms don't get the same results. In Euclid's scheme, Pasch's Axiom doesn't follow, which Euclid didn't notice. AFAIK that is not because he was stating true facts in a sensible order and it would not have been sensible to state this. He just overlooked it, it was so self-evident that he didn't even notice it evidencing itself. I think it's fairly clear that Pasch improved on Euclid's work by driving out such details. Euclid intended his list of axioms to include everything that he was taking for granted, so it's useful to reason just from the axioms you've identified instead of from anything that's self-evident, and thereby identify any mistakes Euclid may have made in declaring the list complete.

Or take the "axiom" that Euclid himself was troubled by, the parallel postulate. When considering non-Euclidean geometries in general, part of their value is that they have some things in common with Euclidean ones, and some things different. How is the difference characterised? By different axioms. Now, if Euclid felt that an axiom was something inherently true then that's fine as far as it goes, but if he held to his opinion that the parallel postulate is true then that would have rendered him incapable of considering a non-Euclidean geometry in light of his other axioms. That's a limitation of refusing to consider axioms to be negotiable. I never met Euclid, but I find it hard to believe that a great mind would be inherently limited in that way. He got a certain distance in the time available to him, but did not discover everything interesting about his procedure for reasoning. Discovering more interesting things caused modern mathematicians to start viewing axioms differently, and to view what mathematicians had been doing for 2000 years differently.

I also agree with axioms-as-definitions. You can by all means write down your axioms and rules of procedure, and use them on the basis that they're worthwhile in themselves, or that any foundation that provides a model for them will do, and you don't care to address the philosophical question of what that foundation might be. I don't think these parts of what the author says are controversial, the bit that's tricky is to reject the formal foundations entirely. I don't know what the author means by "beginning to the study of mathematics", but if he's talking about the training of a student then I doubt anybody would argue that children should be taught ZF before learning to count. As such it follows that ZF does not come first, if any formalism does it's PA.

I bet that fewer than 5% of mathematicians have ever employed even one of these "Axioms" explicitly in their published work

This sounds like a point that, if you want to make it seriously, you can investigate by statistical sampling. It's an interesting point, and let's assume it's true, but ultimately if you write $x

otin x$ you are not explicitly appealing to the axiom of regularity but you are appealing to a result that you have seen proved (with a very short proof) from ZF, and anyone likely to read your paper has seen this proof too. And so on upwards to results with much longer proofs. As metamath shows, there's no fixed boundary between results that can be formalised and results that can't.

The lack of explicit appeal doesn't prove whether or not the axioms are fundamental to the work. However, any given paper relies on some set of results, and if you replaced ZFC with something else that produces those same results then you wouldn't need to change the paper. That's what those playing with foundations are up to. It's perfectly reasonable to state discontent with foundations, but the difficult and enlightening task would be to provide an alternative. A naive notion of classes in place of things "too big to be sets" may or may not do the job. The author asserts that it does (by way of an example, the full list of tricks to form his foundation presumably is longer).

So, I think more than lip service is paid to foundations, but as against that results are accepted whose proof could in fact be more rigorous in the sense that they're not yet computer-verifiable in symbolic logic but could be made so in the opinion of both author and readers. Take from that what you will as to whether the formal work and/or the opinion that the formalization could be done, are "necessary". In the mean time, the author's main point is true that most mathematicians don't spend a lot of time worrying about foundations, and seem to do all right.