Post AvpUUioTEcgiNXSvuy by tao@mathstodon.xyz
(DIR) More posts by tao@mathstodon.xyz
(DIR) Post #AvpUUioTEcgiNXSvuy by tao@mathstodon.xyz
2025-07-05T15:57:53Z
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"In the end, the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?" - George Orwell, "Nineteen Eighty-Four". (1/6)
(DIR) Post #AvpUUkK3cfY53nFfn6 by tao@mathstodon.xyz
2025-07-05T15:58:03Z
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In modern mathematics we take it for granted that there is a consensus standard of truth that virtually all mathematicians subscribe to, and which permit even the bitterest mathematical disputes to be resolvable by breaking down the mathematical arguments into atomic steps and comparing them against the (nearly universally accepted) axioms of mathematics. I myself experienced this first-hand back in 2015, when Edward Nelson, a renowned logician, announced a proof that Peano Arithmetic was in fact logically inconsistent; this was a shocking claim, and the argument initially looked quite serious; but Daniel Tausk and I found an error in the argument, which Nelson graciously acknowledged, withdrawing the claim. (The discussion where this took place can be found at https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html .) Despite Nelson's quite non-mainstream views on the consistency of arithmetic, he still firmly believed in the consensus standard of mathematical truth, even going so far as building his own proof assistant in which he had already partially formalized his argument (although, crucially, the most problematic portions of his argument had not actually been formalized). (2/6)
(DIR) Post #AvpUUlWr8ZNenmFPNY by tao@mathstodon.xyz
2025-07-05T15:58:18Z
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This ability to arrive at consensus by recognizing an objective standard of truth that all personal or political opinions are subordinate to is increasingly rare outside of mathematics, to the point where many non-mathematicians may not even be aware that such a epistemological framework is even possible: in so many spheres of discourse nowadays, it has become normalized to view objective standards primarily as ideological tools, cited to support one's preferred arguments or to attack one's opponents if they align with one's views, but ignored or discredited if they do not. The idea of a leading figure actually changing their views because the objective data was consistently pointing to a different conclusion has become almost quaint. (3/6)
(DIR) Post #AvpUUmk0d9UoYrPQWG by tao@mathstodon.xyz
2025-07-05T15:58:47Z
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And even in mathematics, our consensus standard (which was painfully acquired over many centuries of foundational debate, particularly during the "crisis in foundations" in the early twentieth century) is not completely universal and invulnerable. Consider for instance the affair of Mochizuki's claimed proof of the abc conjecture from 2012, which has not been withdrawn; the usual consensus-building steps of having good-faith technical discussions about key definitions and claims have failed to take place despite some valiant efforts, and what discussion remains has become increasingly personal and unproductive. (In particular, there is no viable path currently to formulating the arguments in a way that could conceivably be verifiable by a formal proof assistant.) (4/6)
(DIR) Post #AvpUUnyE3mSiNF4IJk by tao@mathstodon.xyz
2025-07-05T15:58:58Z
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More recently, we face the real and disturbing possibility that certain directions of mathematical inquiry - for instance, in developing reliable statistical tests for electoral integrity - may not only be defunded by public science agencies, but have their mathematical conclusions actually overruled by political ideology. Even if the supremacy of the objective mathematical standard of truth is technically acknowledged, it can still become weaponized: mathematical results which go against the prevailing ideology could be relentlessly critized for even the slightest typo or technical flaw in the presentation, whereas results that support this ideology could be uncritically embraced even they contain substantial gaps or ambiguities in interpretation. (5/6)
(DIR) Post #AvpUUoqSo7qb5SRbrE by tao@mathstodon.xyz
2025-07-05T15:59:10Z
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One potential bulwark against such politicization of mathematical truth is the broader adoption of formal proof verification, though even here there are some (fortunately still quite theoretical at present) potential "exploits", for instance through subtly altering the definitions of key concepts in Lean's core "Mathlib" library. (See this recent talk https://www.newton.ac.uk/seminar/46706/ "Can Mathematics Be Hacked? Infrastructure, Artificial Intelligence, and the Cybersecurity of Mathematical Knowledge" by Fenner Tanswell.) Still, I view an increased acceptance and deployment of formal methods as a net positive in this regard, even if it is not a "silver bullet".More generally, I think it is important to acknowledge just how precious the consensus objective standard of mathematical truth is, and how important it is to defend it. (This is not to say that such foundational matters should be completely immune from criticism or debate; but such discussion should be in good faith and grounded by genuine philosophical concerns, rather than driven by some external political agenda.) (6/6)
(DIR) Post #AvpUUpqr49kqCxdR6e by shironeko@fedi.tesaguri.club
2025-07-05T17:04:21.913646Z
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@tao one of the craziest thing I learned when I got into college and start reading papers is that the source code used to produce the results is almost always missing.
(DIR) Post #AvpnKlWMD7Il0gxXSi by bks@mastodon.social
2025-07-05T19:37:08Z
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@shironeko @tao Sometimes you're better off NOT seeing the code.
(DIR) Post #AvpnKmlzYTOytTHXTE by shironeko@fedi.tesaguri.club
2025-07-05T20:35:32.675162Z
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@bks @tao for them I guess, lol