Which obviously leads to the epistemological problem that the article points out. You had extremely good mathematicians like Scholze look at it and thought he found a flaw, then one guy from Arizona disagreeing that it is a fatal flaw and claiming to have fixed it, which Scholze doesn't agree with.

So what do you really make of it if only a handful of mathematicians can engage with it, and they can't even agree with each other. Probably the biggest value of IUT is that it puts to the test what even counts as a proof.

[0]: https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%2...

You can express `a + b` or `a * b` in their regular algebraic notation or you can express them as a lambda expressions

ADD = λab.(a S)n

MUL = λxyz.x(yz)

Manipulating these expressions instead of algebra, you can suddenly compute things such as "+ * +" (Plus times plus). That will yield you another expression for sure, but we don't even know what that means.

So maybe an analogy would be, it's like you developed a field where, from that mess, you could derive important insights and even turn them back into proofs

And there's debate on whether all invariants truly are maintained throughout the entire process

My point is that even with a formal advanced education on an area (say -- physics), you are bound to weight the "consensual appraisal" of the authors of a claim to make up your mind.

With my PhD (and having left academia after it), I can easily determine that homeopathy is physical bullshit. It may very well work for different reasons (placebo effect), but when I read about people who try to explain that cosmic rays this or quantum energy that -- I know they are idiots.

And then came Jacques Benveniste who was published in Nature with his "water memory" hypothesis. Holy shit - physics has just changed! Why? Because it was in Nature.

Nature then realized this is a scam, retracted the paper, and physics is back on its (unfortunately) boring track (there is nothing that physicists hope more than a radical revolution in physics)

So we need to trust people who have the right titles, university affiliation, track record on TV and whatnot. In biology, despite having worked in some areas there, I have even less trust in myself.

Then comes the "-logy" soft science where it is not difficult to check that what they say makes sense or not just by looking at the math (which is simple, but can be very deceiving)

And then comes math itself. There can be a definitive answer, as you say, but the proof is completely beyond mere mortals. If you look at the Fermat's conjecture proof (a very simple conjecture, which also makes its beauty), it is completely unreadable for someone who is not in the subject. I did a lot of math and past the 10th line I am done.

For example the axiom of infinity (by finitists), the power set axiom and first-order theories in general (the downward Löwenheim-Skolem theorem implies that the infinity and power set axioms can't guarantee the existence on an uncountable power set), the fact that ZF doesn't allow a set of everything, and in particular no proper set complements, the fact that the axiom of regularity seems to be useless, etc.

Of course most ordinary mathematicians don't care about all that, because they don't care about ZF(C) or set theory or the foundation of mathematics in general. They rather care about problems in their specific field, like algebraic topology or whatnot.

Comparing the axioms of math to relativity in physics is just nonsensical. Math is independent of observation, if a proof is formally correct now, it will always be correct under that chosen axiomatic system. Sure, we can play with different axioms (as others commented, it's common to drop the axiom of choice), but that doesn't invalidate the previous work at all.

Just because Puerto Ricans and Spaniards speak dialects of the same language doesn't make our culture all that similar, as you surely know. I would even say there is little in common culturally.