Mathematician Claims Proof of Connection between Prime Numbers

[KichigaiNeko]

I have no real concept of what this is all about but thought it may be of interest for our mathematicians

A Japanese mathematician claims to have the proof for the ABC conjecture, a statement about the relationship between prime numbers that has been called the most important unsolved problem in number theory.

If Shinichi Mochizuki's 500-page proof stands up to scrutiny, mathematicians say it will represent one of the most astounding achievements of mathematics of the twenty-first century.

The proof will also have ramifications all over mathematics, and even in the real-world field of data encryption.

[Categories+Sheaves]

So much algebraic geometry... I've heard the term 'hodge theory' before but I have no idea what it means...

There is some mind-blowingly deep shit here (and I'm not just saying that because I just finished a beer). Here's the abstract of the first paper in his four-paper series:

---

Abstract. The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichm¨uller theory for number fields equipped with an elliptic curve — which we refer to as “inter-universal Teichm¨uller theory” — by applying the theory of semi-graphs of anabelioids, Frobenioids, the ´etale theta function, and log-shells developed in earlier papers by the author. We begin by fixing what we call “initial Θ-data”, which consists of an elliptic curve EF over a number field F, and a prime number l ≥ 5, as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via finite ´etale coverings to the once-punctured elliptic curve XF determined by EF . These finite ´etale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring Fl = Z/lZ acting on the l-torsion points of the elliptic curve.
We then construct “Θ±ell NF-Hodge theaters” associated to the given Θ-data. These Θ±ell NF-Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number field — which may be thought of as corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local field associated to the number field — are, in some sense, “dismantled” or “disentangled” from one another. All Θ±ell NF-Hodge theaters are isomorphic to one another, but may also be related to one another by means of a “Θ-link”, which relates certain Frobenioid-theoretic portions of one Θ±ell NF-Hodge theater to another is a fashion that is not compatible with the respective conventional ring/scheme theory structures. In particular, it is a highly nontrivial problem to relate the ring structures on either side of the Θ-link to one another. This will be achieved, up to certain “relatively mild indeterminacies”, in future papers in the series by applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an “alien ring structure” [associated, say, to the domain of the Θ-link] in terms of a given ring structure [associated, say, to the codomain of the Θ-link] will be applied in the final paper of the series to obtain results in diophantine geometry. Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tempered fundamental group of a p-adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest.

---

And this is where I feel bad about not knowing more
Time to dust off my Hartshorne...

[FallentoReason]

Is this one of the challenges that get rewarded with a million dollars?

[Angrboda]

It's been decades since I considered myself a mathematician, but I still love math, even though I am completely incapable in my dotage.

Thank you for bringing this result to my attention. I have no idea what it means, but I'm tingly all over.

---

Gödel wrote the following reply to Russell’s assertion in his autobiography, “Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal ‘not’ was laid up in heaven, where virtuous logicians might hope to meet it hereafter.”

Concerning my “unadulterated” Platonism, it is no more “unadulterated” than Russel’s own in 1921 when in the Introduction [to Mathematical Philosophy, 1919, p. 169] he said “[Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.]” At that time evidently Russell had met the “not’ even in this world, but later on under the influence of Wittgenstein he chose to overlook it.

---

"Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician."

Gottlob Frege

[Categories+Sheaves]

1. It's not a $1m problem, but since Hodge Theory is being thrown around, those sorts of problems (i.e. the Hodge Conjecture) aren't too far out of sight.
2. My favorite economist gives a good take on this news here. I thought the discussion about 'set-theoretic foundations' in his papers was something spurious, but it's actually a big deal (whatever it is...).
3. Mochizuki's illustration of what these inter-universal Teichmüller shenanigans are. Might be related to the set-theoretic business (he might need the existence of non-well-founded sets, for example)

[Categories+Sheaves]

Update: looks like I didn't read a link on that landsburg page well enough. Mochizuki is relying on non-well-founded sets. The Axiom of foundation going out the window? That's going to be controversial.

Again, neat stuff happening here. It's going to be a while before the mathematical community is able to fully digest this (esp. since this is a brand-new theory).

[Angrboda]

2. My favorite economist gives a good take on this news here. I thought the discussion about 'set-theoretic foundations' in his papers was something spurious, but it's actually a big deal (whatever it is...).

His notion that axiomatic systems are not fundamental, it's what's underneath them, echoes a question I've had (and use as a frequent example). In epistemology, there are various "theories of truth". What does it mean, what is it, what are its rules. One of the modern theories of truth and logic is that there are truth bearing entities (propositions, statements, sentences...) that are both true and false at the same time. These entities are called dialetheas, and the theories of truth based on them are called Dialetheism (most of which intersect at the liar's paradox and the strengthened liar's paradox; see also, paraconsistent logic). Australian philosopher Graham Priest is a major advocate of Dialetheism, and in one example he shows how, by reconfiguring the rules of classical logic (which result in a breakdown known as logical explosion under dialetheism), he can create a system in which dialetheas occur, but the sense of the inferences is preserved (no explosion). My question has always been, since it seems that the rules of logic in some sense define the nature of truth (in whole or in part), what is that 'something' that is being preserved when the rules of logic are redrawn?

[Categories+Sheaves]

It's not so much that the rules of logic are being redrawn--mathematicians LOVE stuff like proof-by-contradiction (we aren't going to have to give that up--at least not yet, anyway). If you're familiar with the classic paradoxes of naive set theory, weird stuff happens (e.g. Russell's, a relative of the liar) if you don't restrict the domain you're quantifying over. Non-well-founded sets have this eerie infinite regress quality to them, (even if they don't necessarily lead to contradictions) and I remember coming across some links to papers/sites trying to rehabilitate non-well-founded sets, so this stuff isn't completely out there.

And I guess the other point: mathematicians don't seem to use the phrase 'foundational' (at least in terms of set theory) in the same way philosophers do. The axioms don't have any particular epistemic priority or apodictic quality to them. We're adding some extra objects to our mathematical universe, so we have some new truths (or at least, some previously unprovable truths may now be provable). The more you assume (or the more you admit--in the sense of things being admissible or not) the more you can prove. Are you cool/familiar with paradigmatic descriptions of math/science?

[Angrboda]

Oh, just in passing, the term "foundational" has two different meanings in philosophy. I think one is akin to the mathematical, but that may be because I studied math more than I studied philosophy.

[CliveStaples]

So much algebraic geometry... I've heard the term 'hodge theory' before but I have no idea what it means...

There is some mind-blowingly deep shit here (and I'm not just saying that because I just finished a beer). Here's the abstract of the first paper in his four-paper series:

---

Abstract. The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichm¨uller theory for number fields equipped with an elliptic curve — which we refer to as “inter-universal Teichm¨uller theory” — by applying the theory of semi-graphs of anabelioids, Frobenioids, the ´etale theta function, and log-shells developed in earlier papers by the author. We begin by fixing what we call “initial Θ-data”, which consists of an elliptic curve EF over a number field F, and a prime number l ≥ 5, as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via finite ´etale coverings to the once-punctured elliptic curve XF determined by EF . These finite ´etale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring Fl = Z/lZ acting on the l-torsion points of the elliptic curve.
We then construct “Θ±ell NF-Hodge theaters” associated to the given Θ-data. These Θ±ell NF-Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number field — which may be thought of as corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local field associated to the number field — are, in some sense, “dismantled” or “disentangled” from one another. All Θ±ell NF-Hodge theaters are isomorphic to one another, but may also be related to one another by means of a “Θ-link”, which relates certain Frobenioid-theoretic portions of one Θ±ell NF-Hodge theater to another is a fashion that is not compatible with the respective conventional ring/scheme theory structures. In particular, it is a highly nontrivial problem to relate the ring structures on either side of the Θ-link to one another. This will be achieved, up to certain “relatively mild indeterminacies”, in future papers in the series by applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an “alien ring structure” [associated, say, to the domain of the Θ-link] in terms of a given ring structure [associated, say, to the codomain of the Θ-link] will be applied in the final paper of the series to obtain results in diophantine geometry. Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tempered fundamental group of a p-adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest.

---

And this is where I feel bad about not knowing more
Time to dust off my Hartshorne...

How much of this is legit? Algebraic geometry isn't my strong suit, but "alien ring structure" sounds like "vortex math" crackpottery.