What is so special about IUT? They say the theory is "out of this world" but in what sense exactly? Did Mochizuki found a new interesting way to look at some ideas?
It's extremely complicated. The original document he wrote up was 500 pages of maths introducing effectively an entirely new theory. I studied number theory in uni, tried to read it, and understood barely anything of it.
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.
It kind of introduces a fun thought experiment, of a super high-level, complex equivalent of the Monty Hall Problem (which is so counterintuitive that even very intelligent and mathematically literate people will outright refuse to accept the established truth). How would we ever establish truth on something so monstrously complicated that only ~10-100 people in the world could possibly understand and at the same time so divisive that there cannot be a strong consensus?
I'm not a mathematician (but I've seen exact sequences and commutative diagrams) and to me the stuff out of his IUT papers[0] looks borderline LLM-generated. I can only imagine what the LaTeX source looks like.
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
Barrin92|8 months ago
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.
BobaFloutist|8 months ago
enricozb|8 months ago
[0]: https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%2...
IsTom|8 months ago
sebstefan|8 months ago
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
bmn__|8 months ago
Yes, we do. https://youtu.be/RcVA8Nj6HEo?t=1017