I'm sure many of y'all knew about this [cough] controversy. I'm completely
ignorant of the entire corpus and domain. But because I never stay in my lane,
I still find it interesting - and perhaps a decent foil for Brouwerianism.
(I've attached an attempt to define that, assisted by Claude, Perplexity, and
Asta.)
Anyway, here's Claude's summary of IUT and the controversy, including a few
threads provided by Asta to pull on, were one interested:
<Claude>
Based on the bibliography, here's a concise summary:
## What is IUT?
Inter-universal Teichmüller Theory (IUT) is Shinichi Mochizuki's framework developed to prove the abc conjecture, one
of number theory's most important open problems. The theory involves creating multiple "copies" or
"versions" of mathematical universes and studying deformations between them, particularly through
constructions called "Hodge theaters" and analysis of what Mochizuki calls the "log-theta-lattice."
## Main Critical Issues Identified
**The Scholze-Stix Gap**: The most prominent criticism comes from Peter Scholze
and Jakob Stix, who identified what they view as a fundamental flaw in
Mochizuki's proof—specifically in Corollary 3.12, which is crucial for deriving
the abc conjecture. They argue there's a logical gap that cannot be bridged.
**Verification Crisis**: The mathematical community has been unable to verify the proof despite
over a decade of effort. The bibliography references this as approaching "a replication crisis
in mathematics" (Bordg, 2021) and characterizes it as a case of "deep disagreement"
(Aberdein, 2022).
**Communication Breakdown**: The theory uses entirely novel terminology and framework
that most mathematicians cannot follow. References to "intellectual humility"
(Rittberg, 2018) suggest issues with how the theory has been presented and defended.
**Incompleteness**: Even Joshi's 2025 report, while attempting to defend aspects of IUT
against Scholze-Stix, acknowledges "incompleteness in Mochizuki's original
proof."
The controversy fundamentally centers on whether a valid logical path exists
from IUT's premises to the abc conjecture, with critics arguing it does not.
</Claude>
@article{Bordg2021ARCR,
title={A Replication Crisis in Mathematics?},
author={Anthony Bordg},
journal={The Mathematical Intelligencer},
year={2021},
volume={43},
pages={48 - 52},
url={https://api.semanticscholar.org/CorpusId:232126755}
}
@article{Aberdein2022DeepDIA,
title={Deep Disagreement in Mathematics},
author={Andrew Aberdein},
journal={Global Philosophy},
year={2022},
volume={33},
pages={1-27},
url={https://api.semanticscholar.org/CorpusId:253237235}
}
@article{Rittberg2018IntellectualHIP,
title={Intellectual humility in mathematics},
author={C. Rittberg},
journal={Synthese},
year={2018},
volume={199},
pages={5571 - 5601},
url={https://api.semanticscholar.org/CorpusId:189003361}
}
--
¡sıɹƎ ןıɐH ⊥ ɐןןǝdoɹ ǝ uǝןƃ
ὅτε oi μὲν ἄλλοι κύνες τοὺς ἐχϑροὺς δάκνουσιν, ἐγὰ δὲ τοὺς φίλους, ἵνα σώσω.
Hallmarks of Brouwerianism:
## 1. Constructivism
Constructivism: Mathematical objects and truths exist only as mental constructions of an ideal “creating subject,” grounded in the intuition of time and “twoity,” rather than in a mind‑independent realm or in mere formal systems. (2)(3)(4)(1)
- First Act ⇒ constructivism: By separating mathematics from language and logic and founding it on the inner intuition of twoity and the construction of the natural numbers, the First Act directly identifies existence with constructibility in intuition. (5)(3)(1)(2)
- Second Act ⇒ constructivism: By admitting only species and (lawlike or free) choice sequences over already constructed objects, the Second Act extends the domain of what exists strictly through further acts of construction, never by purely declarative axioms. (6)(7)(1)(2)
## 2. Communicability
Communicability: Legitimate mathematics consists of constructions that can, in principle, be expressed and transmitted so that other mathematicians can reproduce the same mental acts, even though mathematics itself is not identical with language. (3)(8)(2)
- First Act ⇒ communicability: By insisting that mathematics is fundamentally languageless yet allowing language only as an auxiliary tool, the First Act makes communication a matter of reliably encoding and decoding underlying constructions, not manipulating uninterpreted formulas. (8)(2)(3)
- Second Act ⇒ communicability: Species and choice sequences are introduced with clear rules or methods (even when they involve free choice), so that any creating subject who follows the same description can generate “the same” object up to the intended notion of equality, securing shared access. (7)(1)(2)(6)
## 3. Effectivity
Effectivity: Proofs and existence claims must supply a method or procedure (finite or rule‑governed step‑by‑step construction) whose execution constitutes the truth of the statement, ruling out purely non‑constructive uses of principles like unrestricted excluded middle. (9)(10)(1)(2)
- First Act ⇒ effectivity: By construing arithmetic as built from iterated temporal constructions of twoity, the First Act ties basic mathematics to explicitly iterable acts, which already limits acceptable proofs to those that correspond to such effective constructions. (4)(1)(2)(5)
- Second Act ⇒ effectivity: The Second Act’s admissible objects—choice sequences and species—are given only via rules, procedures, or explicitly described modes of free choice, so the associated proofs must track a way of carrying out or extending such sequences, not merely assert their existence. (1)(2)(6)(7)
## 4. Processual character
Processual: Mathematics is an extendable partial order constructed/understood by the creating subject(s); truths are not absolutely fixed but are “won” when a construction is carried out or an impossibility is realized, especially in the open-ended context of infinite choice sequences. (11)(2)(4)(1)
- First Act ⇒ processual: The First Act grounds mathematics in the unrolling intuition of behind/ahead (twoity), so that even finite mathematics is understood as a sequence of mental events clicking into place, not as inspection of a static structure. (2)(3)(4)(1)
- Second Act ⇒ processual: By introducing freely proceeding choice sequences and species over them, the Second Act treats many objects - especially those tied to the continuum - as inherently growable constructions whose possible extension stages may change what can be proved about them, making the extension process constitutive of their mathematical properties. (12)(6)(7)(1)(2)
[1. https://plato.stanford.edu/entries/intuitionism/](https://plato.stanford.edu/entries/intuitionism/)
[2. https://iep.utm.edu/intuitionism-math/](https://iep.utm.edu/intuitionism-math/)
[3. https://www.marxists.org/reference/subject/philosophy/works/ne/brouwer.htm. https://www.marxists.org/reference/subject/philosophy/works/ne/brouwer.htm](https://www.marxists.org/reference/subject/philosophy/works/ne/brouwer.htm)
[4. https://www.ebsco.com/research-starters/history/brouwer-develops-intuitionist-foundations-mathematics. https://www.ebsco.com/research-starters/history/brouwer-develops-intuitionist-foundations-mathematics. https://www.ebsco.com/research-starters/history/brouwer-develops-intuitionist-foundations-mathematics. https://www.ebsco.com/research-starters/history/brouwer-develops-intuitionist-foundations-mathematics](https://www.ebsco.com/research-starters/history/brouwer-develops-intuitionist-foundations-mathematics)
[5. https://www.math.ucla.edu/~joan/logicofbrouwerandheytingfin.ps](https://www.math.ucla.edu/~joan/logicofbrouwerandheytingfin.ps)
[6. https://eprints.illc.uva.nl/id/document/165](https://eprints.illc.uva.nl/id/document/165)
[7. https://www.math.ucla.edu/~joan/Oslonewzoom2022handout.pdf](https://www.math.ucla.edu/~joan/Oslonewzoom2022handout.pdf)
[8. https://www.jstage.jst.go.jp/article/jafpos1956/11/1/11_1_35/_pdf](https://www.jstage.jst.go.jp/article/jafpos1956/11/1/11_1_35/_pdf)
[9. https://web.math.princeton.edu/~nelson/papers/int.pdf](https://web.math.princeton.edu/~nelson/papers/int.pdf)
[10. https://en.wikipedia.org/wiki/Intuitionism](https://en.wikipedia.org/wiki/Intuitionism)
[11. https://arxiv.org/html/2509.22528](https://arxiv.org/html/2509.22528)
[12. https://www.sciencedirect.com/science/article/pii/S0019357718303276](https://www.sciencedirect.com/science/article/pii/S0019357718303276)
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