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The Inconsistency of Arithmetic (2011)
- ivraatiems 5y agoNote that the actual message appears to be in the comments below the article: https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039590 https://golem.ph.utexas.edu/category/2011/09/the_inconsisten...
- irjustin 5y agoThanks I was confused by that.
- otterley 5y agoMore of this, please.
- Cogito 5y agoIf you're not familiar, John Baez is consistently a nexus for interesting conversations like this one. Read the archives and his twitter for more.
- civilized 5y agoWhat's really great about this is Terry Tao, a mathematician known primarily for analytics and combinatorics, can, in the space of two blog comments, completely filet a Princeton prof's flawed argument in the largely unrelated field of mathematical logic.
- mhh__ 5y agoYou're not wrong but as a professional mathematician it's not like Tao is exactly stumbling in blind. Logic is the basis of the bases
- catgary 5y agoYeah, my understanding is that analysts are often quite strong in that domain of mathematical logic (there’s a similar where algebraic topologists are often incredibly comfortable with categorical logic).
- lvs 5y agoFileting someone's argument is not a positive goal. It's an egotistical one. Finding truth is a good goal. That seems to be what is praiseworthy here.
- treyhuffine 5y agoAnd allowing one's ego to not get in the way of accepting the truth, even when it contradicts your original belief.
- civilized 5y agoEasy, there. It's just evocative language. I could have said "spot the core flaw" to the same effect.
- ameister14 5y agoNot trying to attack you, just an aside - I don't know that 'spot the core flaw' is the same as 'filet someone's flawed argument.' The latter implies 'ooh, he won;' the former only 'ah, there was a problem and he spotted it.'
- lvs 5y agoThen I apologize for responding to the language that you wrote and not other language you could have used.
- deleted 5y ago[deleted]
- derefr 5y ago"Finding truth" isn't quite what happened here, though. Finding a contradiction/other invalidation of someone else's proof, doesn't advance the knowledge frontier. It means that there was a flaw in process of proving the original proof, but it doesn't disprove the idea behind the original proof. You'd need your own separate proof to do that—probably a proof by contradiction, of one of the main lemmas of the proof. The useful skill, here, is one shared by logicians and compilers: the ability to quickly find flaws that invalidate a proof, without needing to fully understand what the proof is trying to prove. Said skill, if it could be more widely-learned, would be very helpful in iterating toward sound proofs (or impossibility proofs for such.) But when it's only a skill in the hands of another party external to the one writing the proof, that iterative aspect isn't really there.
- tomjakubowski 5y agoTao is respectful in his criticism. And Nelson returns the respect in humbly withdrawing his claim. That is the great thing about this exchange to me.
- civilized 5y agoThe tone of the exchange is great, although normal in the company of accomplished pure mathematicians.
- SquishyPanda23 5y agoTao also writes about how to do this sort of error spotting in a proof you suspect to be wrong: https://terrytao.wordpress.com/advice-on-writing-papers/on-local-and-global-errors-in-mathematical-papers-and-how-to-detect-them/ https://terrytao.wordpress.com/advice-on-writing-papers/on-l... In general he thinks a lot about tool kits and I think what we're seeing here is the math equivalent to being great at debugging.
- CTmystery 5y agoWhat I find most admirable is not the complete fileting, but the humble acceptance of error on something that proof author had obviously put much effort into. It sucks to hear that something you've worked hard on is fundamentally flawed, especially on such public display
- Causality1 5y agoIt sucks to hear that something you've worked hard on is fundamentally flawed, especially on such public display Exactly what I say at parent-teacher conferences.
- quercusa 5y agoAre you the teacher or the parent? I can't decide which I think would be funnier.
- deleted 5y ago[deleted]
- Rioghasarig 5y agoEven the smartest people can be wrong and it can take an outside perspective for them to realize this. This isn't an example of someone "fileting" another's argument. It's an example of two professionals have a conversation and arriving at an agreement. Nelson was making an extraordinary claim so the odds were definitely against him.
- singhrac 5y agoWorth noting that this is from 2011 (though still interesting and fun). Nelson passed away in 2014.
- _moof 5y agoI really hope I survive for longer than three years the next time I get completely destroyed.
- hypersoar 5y agoHuh, I remember when this happened, but I hadn't realized it happened in some blog commemts. For those who are unaware, the comment pointing the flaw in Nelson's proof is Terrance Tao, one of the most extraordinary mathematicians in the world. He's won a good fraction of the field's medals, including the Fields Medal. A nice, down-to-earth guy, too, by all accounts. I jokingly call this the time he literally saved mathematics.
- spoonjim 5y agoIt's sad that this will disappear from history with the blog.
- totetsu 5y agohttp://web.archive.org/web/*/https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html http://web.archive.org/web/*/https://golem.ph.utexas.edu/cat... Maybe a donation to the internet archive could ensure it stays around for posterity
- warent 5y agoIs it possible for me to understand as a layperson why this is so important?
- floatingatoll 5y agoClassy retractions are rare, or at least uncommon. We should all be more like them.
- Uehreka 5y agoYeah but GP said this was the time Terry Tao saved mathematics. I’d like to know what he meant by that.
- mabbo 5y agoI'll take a stab as a complete amateur. Peano arithmetic is the basis for a lot of math. There are very basic assumptions underlying it, but using just those assumptions you can make the natural numbers and perform math on them. This was a bold attempt to prove that they are not consistent. That is, you can use the base peano axioms to prove 1 = 0 and anything else you want. Terry Tao pointed out a mistake the author made, and thus 'saved' mathematics. Peano arithmetic remains consistent.
- newsbinator 5y agoThe very rare time on social media I come across a statement like this, I instantly respect and follow the person saying it.
- mixedmath 5y agoI was beginning my math PhD around this time, and I remember seeing so much math discussion on google+ (which is mentioned frequently in the linked page). I don't know why this was the case, but it really was true --- there was so much professional math activity on google+.
- JadeNB 5y agoI think John Baez and Terry Tao were, between them, a giant chunk of the reason for all that activity.
- CRConrad 5y agoI followed Baez, but never knew Tao was there too. :-(
- jonnycomputer 5y agogoogle+ had a lot of topic-oriented discussion. it was actually pretty cool.
- wolverine876 5y ago> there was so much professional math activity on google+. Is it all lost?
- andybak 5y agoNot sure what's in it but there's https://archive.org/details/archiveteam_googleplus https://archive.org/details/archiveteam_googleplus
- mixedmath 5y agoNow there are so many math fossils on google+. Activity is elsewhere. The math blogosphere remains pretty healthy. Thankfully many mathematicians maintain useful and interesting personal sites. If Wordpress suddenly died, there would be a bigger problem (many mathematicians don't self-host).
- nsonha 5y agois it just me or Google+ seems popular among academics?
- MayeulC 5y agoWas. It was shuttered by Google in 2019. It started to gain traction in some technical circles; I think Linus Torvalds used it to some extent. That was the only "social network" I was interested in back then, as it seemed more "serious".
- yablak 5y agoThis is why I hacker news.
- jonnycomputer 5y agoThe difference between a crank and someone like Nelson is the ability to admit that their crazy disruptive idea, their challenge to orthodoxy, is after-all wrong. We need people to challenge orthodoxy; its healthy. But we also need people to admit when they are wrong. Nelson has my respect.
- busterarm 5y agoPeople challenge orthodoxy and are willing to admit their crazy ideas are wrong all the time. The problem is that outside the field of mathematics, challenging the orthodoxy worked up an angry mob and they probably torched the person's house before it could be worked out.
- chithanh 5y agoWhat is true is that since we have computer-based theorem provers / proof checkers, "working it out" becomes a matter of getting your challenge past one of those, and the orthodoxy will have a hard time arguing against that. But OTOH you see many people in other disciplines challenging the orthodoxy (in Physics: string theorists, MOND proponents, etc.) and while their views aren't always taken fully serious, their houses aren't being torched either.
- busterarm 5y agoClearly some sciences are better than others. You can put air quotes around sciences if it helps make my point any clearer.
- tzs 5y agoAnother major difference is that Nelson's mistake was in a field in which he was an expert. Cranks are almost never experts in the fields they are making claims about.
- jonnycomputer 5y agoTrue. In another context, I recall long conversations between different physical anthropologists in a forum about one person's unorthodox belief that humans' closest common ancestor is shared with the orangutan, not the chimpanzee. While there was some occasional irritation, for the most part the conversations were civil, and informative, because the person who held this view was very knowledgeable about the field.
- SavantIdiot 5y ago"You are quite right, and my original response was wrong. I withdraw my claim" For homework, everyone on HN who posts should try saying this at least once this month, because you will be wrong at least once. I know I will.
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- ukj 5y agoNelson later withdrew his proof (after Terrence Tao corrected him). Instead, he took a philosophical position on the matter of whether Mathematics is discovered or invented: https://web.math.princeton.edu/~nelson/papers/rome.pdf https://web.math.princeton.edu/~nelson/papers/rome.pdf He sided with Computer Scientists.
- depressedpanda 5y agoThat was a great read, thanks for the link. > Plato was a dreadful fellow, the source of a persistent evil from which the world has not yet been liberated, [...] Does anyone more familiar with Plato know what Nelson could have meant by that? Edit: Alright, I think I found the reason: "platonism" is a form of mathematical realism, while Nelson strongly favored mathematical formalism.
- gameswithgo 5y agoI started reading this post, following along just fine, then the notation and terminology entered, and I started looking it up, ok, following along still but it is hard, then it quickly devolved where I didn't understand anything, and I felt really bad. Then saw the first comment by Terry Tao and felt a lot better.
- H8crilA 5y agoThis is debunked, but maybe a comment about how this relates to Godel's theorems: 1) the first Godel's theorem says that if a theory of arithmetics is complete then it must be inconsistent, 2) the second is that a theory of arithmetics cannot prove its own consistency, specifically if it proves its own consistency then it is actually inconsistent. This thesis (with an invalid proof) says "no consistent theory of arithmetics can exist", regardless of whether it's completely complete.
- vymague 5y agoThe renaming of the title is uncalled for. This response by Edward Nelson was the highlight: > You are quite right, and my original response was wrong. Thank you for spotting my error. > I withdraw my claim.
- woopwoop 5y agoOP here. I agree. I'm a mathematician, and like most mathematicians I find discussions of foundations actually kind of tedious (although obviously a proof that Peano arithmetic was inconsistent would be a big deal). Also only a handful of HN readers have anything at all to gain from the technical discussion here. What is of interest to the HN readership is the sociology of mathematics on display. Hence the original title.
- commandlinefan 5y agoI've always been bothered (maybe not the right word... intrigued?) by how it's taken as axiomatic that a negative multiplied by another negative must be positive. I can see why it's defined that way, and can't see any other way to define it, but it also leads to a lot of irritating arithmetical inconsistencies.
- lupire 5y agoHuh?
- commandlinefan 5y agoWell, classical Euclidean geometry relies on the parallel postulate, although it can't itself be proven. Abstract mathematicians have put together alternate non-euclidean geometries that are consistent but don't rely on the parallel postulate - kind of along the same lines as Nelson here "rejecting" induction just to see what would happen. I've always wondered what might happen if you "allowed", in the same sense, a square root to be a real rather than imaginary number. Maybe nothing, but I always wondered why nobody ever tried.
- depressedpanda 5y agoWhy do you assume nobody's ever tried?
- commandlinefan 5y agoWell, I guess I've never seen anybody try. Maybe they did!