[HN Gopher] Mathematical proof is a social compact ___________________________________________________________________ Mathematical proof is a social compact Author : digital55 Score : 101 points Date : 2023-08-31 16:58 UTC (6 hours ago) (HTM) web link (www.quantamagazine.org) (TXT) w3m dump (www.quantamagazine.org) | reso wrote: | By late-undergrad, it was intuitive to me that "proof" means "all | mathematicians who reads this agrees with it". Mathematics is | unique in that, mostly, the field can achieve consensus on | results, which we then call "proofs". | | But similarly, it makes sense that, even if a result is is "true" | in a universal, objective sense, if the mathematician cannot | communicate this in a convincing fashion to the rest of the | mathematics world, I don't think we can call that result | "proved". | [deleted] | SleekEagle wrote: | Not that you were, but I don't quite understand why people get | so caught up on this fact. There are objective facts about the | nature of reality, and we are all (or at least competent | practitioners in the field) are thoroughly convinced that we | have identified a subset of these facts. | | These presumed facts have helped us do things like go to the | moon and build skyscrapers, but then someone comes along with | the old "but how do you _actually_ know " argument of a college | freshman, and then we get into a conversation about the | potential social relativism of math. | | All the while, people will see a half-assed psychology study | with a questionable procedure, weak at best, erroneous at worst | statistics and therefore tenuous at best conclusions, and this | study is taken to be "true" and might legitimately impact | notable institutions. Yet when we're talking about extremely | complicated topics that exist on the edge of the horizon of | human intuition, no matter how obvious the impact some people | just refuse to accept things as objective simply because they | fail to intuitively understand them. | | Foundational fields like mathematics and physics are as | objective as we can get. If you don't accept that, your belief | about what is objectively true ends at cogito ergo sum and | that's that. This has always been such a pointless conversation | in my mind. | joelfried wrote: | We communicate with words, and people as a whole are used to | being lied to and gaslit regularly especially by those in | power. It's true that mathematics and the hard sciences have | mechanisms for understanding that are on a different scale | than, say, ethics and morality. However, it takes time for | people -- especially those currently engaged in questioning | the nature of their reality[1] -- to accept that in this | specific instance lying and gaslighting are a lot harder[2]. | | The people who eventually accept and internalize the | distinction around things that can be objectively shown to be | true are those who by in the large have done some of the work | to understand these things themselves. Godel's Incompleteness | Theorem is beautiful but it takes work to understand and if | it didn't, it wouldn't be much of a meaningful breakthrough. | Nobody is proving that 3+5=8 and then 4+5=9. | | So what the average person sees is a high level language they | can't speak with people being absolutely positive that this | thing is special and true and incontrovertible. That raises | red flags when you're dealing with folks talking about normal | everyday stuff, doesn't it? It's a lot harder to say "but I | don't understand" and a lot easier to say "but what if you're | wrong" socially. | | [1] As all college first years do, right? [2] Let's face it, | lying to people is never impossible, it's just harder to be | successful when you can be fact checked. | smif wrote: | > There are objective facts about the nature of reality | | This is a pretty bold claim and you would have to do a bit of | work to make it more convincing. Besides, it's not really how | science works. Different theories wax and wane over time all | the time. What we're really doing with science is coming up | with ever better models that give us greater predictive | power. | | You could argue that at the macro scale we're asymptotically | approaching some kind of objective truth with the whole | endeavor of science, but you can't simply tunnel across that | gap and make the leap to say that we know there is an | objective truth. | | The best that we can do is probably say something along the | lines of "these are the best methods of getting closer to the | truth that we have available - if anyone claims to have | better methods, they are very likely wrong", but you still | need to have the humility to accept that even the best models | that we have to date are not infallible. They do not give us | the objective truth, nor do they promise to, but they are the | most effective tools that we have available to us at the | current time. This is critically not the same as claiming | that they give us the objective truth. | | You could say that for all intents and purposes/everyday | usage, "sure, these are objective facts about reality" - but | I don't actually think that helps anyone and it serves to | foment mistrust towards science and scientists. I think the | best chance we have at preserving the status of science and | scientists in our society starts by being honest about what | it actually is giving us - which is quite a lot, but let's | not oversell it for the sake of convience or whatever. | alphanumeric0 wrote: | Is it a bold claim? | | On that account, do you lean more towards flat earth | theory? | reso wrote: | This fact is interesting because many people grew up with an | idea of mathematics as having discovered fundamental truth, | and some of us, when we get deep into the field, realize that | Math is made up of human ideas, exactly the same as a very | complex boardgame we play in our heads, and that the belief | that these boardgames represent something fundamental about | reality is it's own leap of faith. | BeetleB wrote: | > All the while, people will see a half-assed psychology | study with a questionable procedure, weak at best, erroneous | at worst statistics and therefore tenuous at best | conclusions, and this study is taken to be "true" | | ... | | > Yet when we're talking about extremely complicated topics | that exist on the edge of the horizon of human intuition, no | matter how obvious the impact some people just refuse to | accept things as objective simply because they fail to | intuitively understand them. | | I think the intersection of these two groups is the null set. | | > Foundational fields like mathematics and physics are as | objective as we can get. | | Objective? Yes. But objective does not equate to "true"[1]. | One requires data, and the other lives only in the mind. It | is not at all problematic to ponder over whether mathematics | is "true" - most mathematicians have an opinion one way or | another, and they are not unanimous in their opinions. | | [1] True w.r.t reality, not true in the logic sense. | hackandthink wrote: | >What AI can do that's new is to verify what we believe to be | true | | This is not AI. | | Combining Theorem Provers with AI is promising: | | https://leandojo.org/ | Tainnor wrote: | I find his scepticism about proof assistants like Lean a bit | weird. Of course, there is never absolute certainty, but there | are degrees. A proof in Lean is a quite strong guarantee, you'd | basically have to have a bug in Lean's core for it to be wrong, | which is possible but less likely than a flaw in a proof that | hasn't seen much scrutiny because it's rather unimportant (of | course, "big" results get so much scrutiny that it's also very | unlikely that they were wrong). | svat wrote: | I wouldn't characterize him as being sceptical about Lean; this | is what he says: | | > _Some terrific developments have happened with proof | verification. Like [the proof assistant] Lean, which has | allowed mathematicians to verify many proofs, while also | helping the authors better understand their own work, because | they have to break down some of their ideas into simpler steps | to feed into Lean for verification. [...] I'm not saying that I | believe something like Lean is going to make a lot of errors. | [...] They can be a very valuable tool for getting things right | -- particularly for verifying mathematics that rests heavily on | new definitions that are not easy to analyze at first sight. | There's no debate that it's helpful to have new perspectives, | new tools and new technology in our armory._ | | In between (the "[...]"s above), he says: | | > _But is this foolproof? Is a proof a proof just because Lean | agrees it's one? In some ways, it's as good as the people who | convert the proof into inputs for Lean. Which sounds very much | like how we do traditional mathematics. [...] I'm just not sure | it's any more secure than most things done by humans. [...] I'm | afraid I have a lot of skepticism about the role of computers. | [...] what I shy away from is the concept that we're now going | to have perfect logical machines that produce correct theorems. | You have to acknowledge that we can't be sure things are | correct with computers. Our future has to rely on the sense of | community that we have relied on throughout the history of | science: that we bounce things off one another. That we talk to | people who look at the same thing from a completely different | perspective. And so on._ | | The latter part is less about the correctness and more about | community: the main goal of mathematics is not merely a "true" | theorems, but (human) mathematical understanding. I wrote an | answer about this on math.SE in 2014 (mostly quoting Thurston | and Rota): https://math.stackexchange.com/questions/632705/why- | are-math... | [deleted] | epgui wrote: | I agree very much with you... but it seems to me like theorem | provers / computers can be a really helpful tool in building | trust and understanding between people or between groups of | people, in ways that printed paper can't quite match. | Tainnor wrote: | I agree that a big part of proving thing is understanding | intuitively "why" they are true. That's part of the reasons | why I don't think we'll just write proofs in Lean in 50 years | and leave it at that (the other reason is that, at least for | now, those systems are very "stupid" and need a lot of | handholding to prove facts that any maths undergraduate would | understand to be true). | | But when he says that he doesn't believe that a proof in Lean | is "more secure than most things done by humans" I | fundamentally disagree. | svat wrote: | My reading is that for an individual theorem or proof, | having it checked by computer may well be more "secure". | But a culture where all mathematics is done (only) by | computer would be worse, because you'd have outsourced the | thinking and quest for understanding which was the point in | the first place. | Tainnor wrote: | The way theorem provers work right now, it's still a | human that has to come up with the proof, the theorem | prover just verifies it. | | But yes, I wouldn't want computer proofs to "replace" the | proofs we have now. | jeremyjh wrote: | I read it more as him saying that at the end of the day a human | is inputting symbols into Lean, and a human is interpreting the | meaning of the proof that Lean has verified. Not that the | mechanics of the proof checking are in doubt, but that that | meaning of it all is still a social consensus. | epgui wrote: | > A proof in Lean is a quite strong guarantee, you'd basically | have to have a bug in Lean's core for it to be wrong | | That, or an error in the specification (ie.: not the | implementation). | | (I very much am with you that his attitude / way of framing it | is a bit weird) | Jweb_Guru wrote: | I will go further (since there are certainly bugs in the | kernels of proof assistants, more frequently than you might | imagine). _Even if there is a bug in Lean 's kernel_, it will | almost never invalidate a Lean proof, because bugs in the | Lean kernel (or the kernels of other proof assistants) are, | just like with normal programs, almost always bugs dealing | with edge cases testing the boundaries of mathematics or | syntax, not things that show up in everyday proofs. Coq has | fixed numerous proofs of False and, to my knowledge, these | fixes have never ended up invalidating a "real" proof that | wasn't specifically designed to test these kinds of edge | cases. Proofs are far more often broken by changes to the | surface syntax (e.g. inference algorithms or the tactic | language) than by changes to the kernel. | | Obviously, this is not foolproof against malicious | "adversarial" mathematicians, so knowing something was proved | in Lean still requires some degree of checking. But it does | make it pretty extraordinarily unlikely that a false proof in | a proof assistant was arrived at by accident, which is far | from the case with the "old" way of doing mathematics. | | (I'd also add that the concern about getting the | specification of a design wrong is something that rarely | lasts longer than attempts to actually use it to prove | anything nontrivial, since the proofs won't go through, but | there at least can be a lot of debate about how to interpret | specifications in some fields [e.g. computer science] or | whether certain specs are "actually" equivalent to the | statements someone wants to prove, especially for stuff like | category theory where manual proofs have a good deal of | handwaving... but these are minor technical issues compared | to the viewpoint of the article). | Legend2440 wrote: | It's the same "computers will never take MY job" attitude | that you see from programmers, artists, doctors, etc. | davidgrenier wrote: | I think his argument was restricted to a human-produced | mathematical result being ported to a Lean program where one | would be just as likely to commit a mistake. However I disagree | as well, I recall the difficulty of expressing what I wanted to | Coq being a barrier to expressing it incorrectly. | Tainnor wrote: | Yeah it's likely he hasn't worked much with Lean and may have | some misconceptions around it. | SkiFire13 wrote: | The thing about proof checkers is that they won't accept | invalid proofs (assuming their internal axioms are | consistents...), so if you make a mistake there it will | simply reject your proof. The only place where it won't catch | some mistakes is in proposition you want to prove, because | the only way for it to know what you want to prove if for you | to tell it, but that's much easier to humanly verify. | joe__f wrote: | What is the meaning of the word 'Compact' in the title here? | math_dandy wrote: | An agreement, basically. | | https://legal-dictionary.thefreedictionary.com/Compact | passion__desire wrote: | One metaphor that I have seen ubiquitous in Nature is Consensus | Reality. Fiat currency, quantum systems agreeing on each | other's value, social contract, etc. It's everywhere and now | social concept of proof. | | Here is Daniel Dennett talking it. https://youtu.be/32u12zjgJww | | I like the no miracles argument in favour of science. If our | scientific theories don't track reality, how are they | successful in prediction the future. | | Similarly the social aspect of mathematical proofs can be | replaced with requirements like being in congruence with | established facts (i.e explanatory power), predictive power, | and efficient representation (i.e. elegant compression) which | means faster programs / less computational steps required. | davidgrenier wrote: | Good teacher, his Number Theory book felt really good though I | have no comparable in Number Theory. I must say Number Theory and | Combinatorics are the most difficult topics I got acquainted with | in undergrad. | Tainnor wrote: | Number Theory is really weird because the definitions and | theorems are so often straightforward and easy to understand | (you could explain Fermat's Last Theorem to a bright school | kid), but the proofs can be devilishly complicated. | epgui wrote: | I mean... you can abuse language and math notation in ways that | you can't do for computer code. Math notation is actually | terribly and surprisingly informal compared to code. I'd just | argue that many[*] of these "social disagreements" would go away | within a computational framework. I think the future of | mathematics is in computer proofs. | | [*] Note that the claim is "many" (ie.: a subset), not "all". | catskul2 wrote: | My brain does not like the phrase "social compact" probably | because I've heard "social contract" so often, and rarely if ever | hear "compact" used in this way. On the other hand I hear "pact" | much more often. | Tainnor wrote: | It's especially weird because "compact" has a very specific | meaning in mathematics. | droptablemain wrote: | Construct seems more fitting. I also read the headline and | thought it sounded like a mistake. | [deleted] | SamBam wrote: | I read a "social compact" as more of a "social agreement," | which gets at the bottom of consensus that he's talking | about. | | A "social construct" is quite different, that's something | entirely constructed by society, like manners or nationalism. | epgui wrote: | I made the same comment as you in response to parent | commenter (now deleted)... But then I looked at some | definitions of "social construct" and it seems to have | broader semantics than I had thought. | qsort wrote: | > But is this foolproof? Is a proof a proof just because Lean | agrees it's one? In some ways, it's as good as the people who | convert the proof into inputs for Lean. | | Having studied CS in school I'm sort of triggered by this. Might | be the editorialization, but this is a statement I have a problem | with. | | I am not one of those who think that computers will save us all. | My point of view is that computers are meaningless machines that | do meaningless operations on meaningless symbols unless proven | otherwise. This is what gets drilled in your head in any semi- | decent CS program and it's a point of view I came to agree with | completely. | | But we have proven otherwise. | | Proof checkers like Lean, Coq, Matita, Isabelle and the like are | not like normal code, they are more similar to type-systems and | code expressed in their formalisms is directly connected to a | valid mathematical proof (see the Curry-Howard isomorphism). | | They are usually constructed as a tiny, hand-proven core, and | then built on their own formalism, ensuring that what is exposed | to the end user is perfectly grounded. | | If a program is accepted by the proof checker, then _by | construction_ it must be a valid mathematical proof. | | Of course, computers are physical machines that can potentially | make all sorts of mistakes. Hardware failures, cosmic rays, | whatever. But the probability of that happening on a large scale | is the same probability that seven billion people collectively | hallucinated elementary theory and it's in fact not true that | there are infinitely many prime numbers. | | edit: Just a clarification: it's only this particular statement I | have a problem with. The article is very much worth your time. Do | not get derailed by the title, it's not some sort of "math is | racist" nonsense. | cscheid wrote: | I'm with you almost everywhere, but it's worth pondering this: | | > If a program is accepted by the proof checker, then by | construction it must be a valid mathematical proof. | | I'd phrase it as "If a program is accepted by a _correct_ proof | checker, then ...". That makes it clear that we can't escape | having to consider how and why the proof checker is considered | correct. | | Edit: this is not entirely academic. from what I understand, | the unsoundness of Hindley-Milner in the presence of | polymorphic references was not immediately known (https://en.wi | kipedia.org/wiki/Value_restriction#A_Counter_Ex...). | markisus wrote: | This is a great point. In a computer proof system, you | produce judgements like "the term x has type is T", where the | type "T" encodes the statement of a theorem and "x" is the | proof. We must be certain that whenever the type checker | verifies that "x has type T", it means that the theorem "T" | is actually true. Such a meta-proof must be done on pen and | paper, before any code is written at all. Any bug in this | meta-proof would destroy our ability to rely on the computer | proof system. | cscheid wrote: | > Such a meta-proof must be done on pen and paper, before | any code is written at all. | | Not so fast; it could be done on a _different automated | system_ (say, a smaller one) than the one you're building, | in which case you're now relying on that system's | correctness. Or it could even not be proven at all. This is | not too different from what mathematicians do when they | (often) say "assuming P != NP", or "assuming GRH", or | "assuming FLT", etc etc. It's simply that it's worth being | careful and precise. | wk_end wrote: | To be ruthlessly, uselessly pedantic - after all, we're | mathematicians - there's reasonable definitions of "academic" | where logical unsoundness is still academic if it never | interfered with the reasoning behind any proofs of interest | ;) | | But: so long as we're accepting that unsoundness in your | checker or its underlying theory are intrinsically deal | breakers, there's definitely a long history of this, perhaps | somewhat more relevant than the HM example, since no proof | checkers of note, AFAIK, have incorporated mutation into | their type theory. | | For one thing, the implementation can very easily have bugs. | Coq itself certainly has had soundness bugs occasionally [0]. | I'm sure Agda, Lean, Idris, etc. have too, but I've followed | them less closely. | | But even the underlying mathematics have been tricky. | Girard's Paradox broke Martin-Lof's type theory, which is why | in these dependently typed proof assistants you have to deal | with the bizarre "Tower of Universes"; and Girard's Paradox | is an analogue of Russell's Paradox which broke more naive | set theories. And then Russell himself and his system of | universal mathematics was very famously struck down by Godel. | | But we've definitely gotten it right this time... | | [0] https://github.com/coq/coq/issues/4294 | tshaddox wrote: | > there's reasonable definitions of "academic" where | logical unsoundness is still academic if it never | interfered with the reasoning behind any proofs of interest | ;) | | I wouldn't really so. If you were relying on the type | system to tell you about any errors you made in your | program, the revelation that your type system was unsound | is not "only academic" simply because your program | coincidentally didn't have errors. Just like it wouldn't be | "only academic" if you discovered that your car's airbags | weren't functional for the last 10 years even though you | never had any accidents. | llm_thr wrote: | [dead] | Ericson2314 wrote: | Yeah any statement that goes back and forth a bit too fast | between Lean and ChatGPT raises my suspicions. | staunton wrote: | The thing that these systems offer is "having to check less | code". In some situations, that's huge. In others, it's not | worth the effort it takes to get there and maybe doesn't | actually reduce the complexity of checking that it all makes | sense. | | In other words, maybe you proved something beyond reasonable | doubt, but that's completely useless if you misunderstand _what | it was_ that you proved. | cvoss wrote: | > the same probability that seven billion people collectively | hallucinated elementary theory | | Just for fun, it's not unprecedented that the entire human race | is disposed to mislearn basic truths about the universe which | have to be unlearned when you get to the point where it | matters. See quantum mechanics. | | One hopes that conscientious mathematicians are careful to | select axiomatizations (such as ZF set theory or the type | system of Lean) which faithfully reflect reality. And, within | such a system, we can be extremely confident in what we're | doing. But the distinct possibility remains that the | axiomatization doesn't actually match our universe and, say, | admits a proof of false or implies something not physically | realizable, such as that a certain computation should | terminate, when no real life physical machine obeying the laws | of physics can ever do so. | mcguire wrote: | Be careful with things like "faithfully reflect reality". :-) | | Euclidean geometry is intuitively good and seems to work on | relatively small areas of the Earth's surface, but non- | Euclidean geometries are inconsistent with it and more | faithfully reflect a reality that lacks infinite flat planes. | mcphage wrote: | Or even Newtonian Physics, which faithfully reflects | reality--as long as you're not too large, or you're not | traveling too fast. And then it starts to break down. | User23 wrote: | It's unfortunate that semiotic has become a joke word. C.S. | Peirce did seminal work tying it to logic. | oulipo wrote: | I think his argument is exactly that, those formal systems can | suffer from errors (even though they've been checked), because | the compiler could have errors, the processor could have | errors, etc | | So in the end, it's always a question of "community" and | "belief that things seen by sufficiently many people are | right", eg that after 10 years of using a CPU, we've identified | and fixed most bugs, and so the program is even more likely to | be valid | mcguire wrote: | Once upon a time, the professor of a class on logic I was | taking pointed out that, if you proved some famous open | conjecture (I think Fermat's was the example) using some simple | proof that everyone had just missed so far, mathematicians | would pat you on the head, maybe give you an award or | something, and you (and the conjecture) would become another | forgotten footnote. | | Major mathematical advances come from difficult problems when a | new branch of mathematics is invented to solve them. Think of | calculus, non-Euclidean geometry, and Turing machines (and | their equivalents). | | The problem I see with Coq, the proof checker I've worked with | the most, and the others I've seen is that the proofs they | result in are neither good, understandable programs nor good, | understandable proofs. They are as complicated as complex | programs, but written in something that isn't a half-way clean | programming language, but they're also not written as a | traditional proof. That goes back up to the article's point | that the purpose of a proof is to convince and enlighten | others. | | Sure, I'd have to say a proposition proven in Coq is _true,_ | but how do you get from that to the ideas in the proof letting | you go on to other problems? | sebzim4500 wrote: | I guess the hope is that eventually there will be a language | which is roughly as easy to read/write as normal proofs in | English but can be machine checked. | | Lean comes closer than Coq, but is still far from this ideal. | It may be that AI will need to be involved (as in, there | could be a tactic that generates proof terms using an | transformer or something). | hgsgm wrote: | Like Cobol? | | Coqobol? | passion__desire wrote: | There is something similar Nima Arkani-Hamed talks about | Feynman equations during his amplituhedron lecture, pages and | pages of equations being reduced to simple representation. | | @56:26 in https://youtu.be/sv7Tvpbx3lc | madsbuch wrote: | I don't think the point is to either have good math or good | code. | | When you _develop software_ in Coq and the likes, it is | library code. It is high impact code that is guaranteed to be | correct. It is written once and not maintained - You would | never build the new Twitter in Coq. | | When you _develop math_ on Coq, you build abstractions. You | just need to know that theorems that you are working on. Then | you pull in all the lemmas in coq to help you solve that | theorem. You do not need to understand the lammas coq uses to | help you prove your theorem - coq does that for you. So your | math is extremely concise. This is in opposition to normal | math, where you need to be able to explain the hierarchy your | theorem works on top of. | hgsgm wrote: | Normal math also concisely cites pre-existing theorems | markisus wrote: | I think the quote is about the fallibility of the humans who | convert theorems into statements of type theory. You could end | up with a valid theorem, but not the theorem you meant to | prove. This would be a bug in the statement of the theorem, not | the proof. | | For example, you might want to prove that a certain sorting | algorithm is correct. You formalize the specification as "for | every two integers i, j, output[i] <= output[j]" and prove that | outputs of the algorithm satisfy this spec. However, this is | not a correct characterization of sorting, since the algorithm | might return the empty list. | Tainnor wrote: | 1. Sure, but verifying a spec is still _much_ easier than | verifying a whole proof (as we traditionally do). | | 2. In that example that you gave, you would have additional | evidence: if you use your system and every sort returns an | empty list, you'd probably notice it quite quickly. You can | also do manual or automated testing to see that it does the | right thing for example inputs. If you then consider it to be | unlikely that someone writes code that would exactly only | work for example inputs and only satisfy the specification in | the dumbest possible way, then you get some really good | evidence that your algorithm is actually correct. 100% | certainty is never possible, but there are still degrees. | [deleted] | markisus wrote: | Agreed. I think the interviewee is a little pessimistic | when he says "I'm just not sure it's any more secure than | most things done by humans." about computer assistance. If | the ABC conjecture were to be proven in Lean, I'm pretty | sure it would be accepted as true by an overwhelming | majority of the mathematical community. | hgsgm wrote: | The ABC conjecture "proof" is rejected because it uses | definitions and arguments no one understands. | | It would not necessarily be hard to program in those | incoherent definitions and get them to prove a result. | That wouldn't mean much. | | This is different from just programming in the conjecture | and letting Lean find a complete proof. | crvdgc wrote: | It's an important problem and people in formal verification | are aware of it. One example of tackling this is that the | Lean LTE team accompanies the proof with "several files | corresponding to the main players in the statement" and | "[they] should be (approximately) readable by mathematicians | who have minimal experience with Lean [...] to make it easy | for non-experts to look through the examples folder, then | look through the concise final statement in challenge.lean, | and be reasonably confident that the challenge was | accomplished". | | Details in this blog post: https://leanprover- | community.github.io/blog/posts/lte-exampl... | dcre wrote: | I will always remember when I first got into serious proof-based | math in the first semester of college, and I had to work hard to | develop my sense of what counts as a sufficient proof. I would | read the proofs in the textbook and hit the QED at the end and | not understand why it was enough. Eventually I came to understand | something like the framing here, which is that a proof is about | persuasion, persuasion is about judgment, and judgment (by | definition, maybe) can't be pinned down to clear rules. | | The University of Chicago has a special math class format called | Inquiry-Based Learning designed around this idea, where you work | _together_ in class to put proofs together and work out a shared | understanding of what is sufficient. I didn 't take it but I wish | I had. You can read some people's experiences with it here[0]. | | [0]: | https://www.reddit.com/r/uchicago/comments/i1id9e/what_was_y... | pjacotg wrote: | I recently read this book review [0] where a mathematical proof | was described as a dialogue between two semi-adversarial but | collaborative actors, the Prover and the Skeptic, who together | aim to aquire mathematical insight. | | I thought it was an interesting perspective. | | [0] https://jdh.hamkins.org/book-review-catarina-dutilh- | novaes-t... | hgsgm wrote: | This also what philosophical discussion/debate is. | sctb wrote: | > Moreover, one could play word games with the mathematical | language, creating problematic statements like "this statement is | false" (if it's true, then it's false; if it's false, then it's | true) that indicated there were problems with the axiomatic | system. | | I hope this isn't too far off topic, but can someone clarify | exactly how this problem indicts axioms? As an uninformed and | naive musing, it occurs to me that an issue with the statement | "this statement is false" is _this_. The whole of the statement, | that is, the thing having truth or falsehood, cannot be addressed | by one of its components. | Tainnor wrote: | You're precisely right that it's the self-referentiality that | gets us in trouble. Systems which don't have that don't suffer | from Godel-related problems (e.g. propositional logic, or | arithmetic without multiplication). | | Unfortunately, what Godel showed is that any logical system | that is strong enough to capture addition and multiplication | over natural numbers - and _nothing more_ - includes self- | referentiality. | | That includes the language of arithmetic - you can write down | formulas that refer to themselves - and Turing machines (via | the recursion theorem, any Turing Machine is equivalent to a | machine that knows its own source code). The constructions are | a bit technical, but it's possible. | finnh wrote: | Yeah I always thought that the construction of Godel numbers | was always the weakest part of the proof / the biggest leap | of faith / the part your prof would just hand-wave as being a | valid move. | | Of course once you get into Turing machines it all flows more | naturally, what with all of us being accustomed to "code is | just data". | Tainnor wrote: | I agree that Turing Machines feel more natural to | programmers than first-order logic (although "natural" | doesn't necessarily mean "rigorously proven"), but there | are no leaps of faith involved in Godel's construction. | | You can write down "P is provable" as a first-order | sentence of arithmetic (which involves some number | theoretic tricks), and you can also do the diagonalisation | trick that gives you self-referentiality. That's really all | you need. | rmzz wrote: | [dead] | kmod wrote: | IANA mathematician, but I read "axiomatic system" broadly as | including not just the axioms but also the logic in which they | are based. My understanding is that a common interpretation is | that ZFC axioms are a list of 10 strings of symbols, which only | have some sort of meaning when you pick a logic that gives | meaning to these symbols. But I think also that this particular | understanding of what axioms are ("formalism") is just one way | of understanding truth in mathematics, and there are others. | https://en.wikipedia.org/wiki/Philosophy_of_mathematics | | As for this particular issue I think this wikipedia page is | relevant: https://en.wikipedia.org/wiki/Impredicativity | thorel wrote: | This idea of giving "meaning" to a set of axioms is precisely | captured by the notion of "interpretation" in logic [1]. The | rough idea is to map the symbols of the formal language to | some pre-existing objects. As you say, this gives one way of | formalizing truth: a sentence (string of symbols that respect | the syntax of your language) is true if it holds for the | objects the sentence is referring to (via a chosen | interpretation). This notion of truth is sometimes referred | to as _semantic_ truth. | | An alternative approach is purely syntactic and sees a | logical system as collection of valid transformation rules | that can be applied to the axioms. In this view, a sentence | is true if it can be obtained from the axioms by applying a | sequence of valid transformation rules. This purely syntactic | notion of truth is known as "provability". | | Then the key question is to ask whether the two notions | coincide: one way to state Godel's first incompleteness | theorem is that it shows the two notions do not coincide. | | [1] https://en.wikipedia.org/wiki/Interpretation_(logic) | Tainnor wrote: | > Then the key question is to ask whether the two notions | coincide: one way to state Godel's first incompleteness | theorem is that it shows the two notions do not coincide | | It's even more subtle than that. They do coincide in a | sense, which is proven by Godel's _completeness_ theorem | (well, at least in First-Order Logic). That one just says | that a sentence is provable from some axioms exactly iff it | 's true in every interpretation that satisfy the axioms. | | So one thing that Godel's first incompleteness theorem | shows it's that it's impossible to uniquely characterise | even a simple structure such as the natural numbers by some | "reasonable"[0] axioms - precisely because there will | always be sentences that are correct in some | interpretations but not in others. | | Unless you use second-order logic - in which case the whole | enterprise breaks down for different reasons (because | completeness doesn't hold for second order logic). | | [0] reasonable basically means that it must be possible to | verify whether a sentence is an axiom or not, otherwise you | could just say that "every true sentence is an axiom" | thorel wrote: | Agreed, thanks for the clarifications. Another result | worth mentioning, which also shows that you cannot hope | to uniquely characterize a structure by "reasonable" | axioms is the Lowenheim-Skolem theorem which predates | Godel's incompleteness (although the history of these | results is somewhat convoluted). | | There, the obstacle is in some sense of a simplest | nature: if your set of axioms admits a countable model, | then it admits models of all infinite cardinalities. In | other words, it shows that there is something | fundamentally impossible in trying to capture an infinite | structure (like numbers) by finite means (e.g. | recursively axiomatizable). | Tainnor wrote: | Lowenheim-Skolem is such a mind-blowing result. It's a | shame it's not talked about more often. | ginnungagap wrote: | > In other words, it shows that there is something | fundamentally impossible in trying to capture an infinite | structure (like numbers) by finite means (e.g. | recursively axiomatizable). | | Let me just point out for other readers that Lowenheim- | Skolem applies to ANY first order theory (in a countable | language, or also in an uncountable language if stated in | the form that a theory with an infinite model with | cardinality at least that of the language has infinite | models in all cardinalities at least as big as that of | the language), it doesn't care about how complex the | axioms are from a computability point of view | Tainnor wrote: | You're correct, but I think the spirit of what GP wrote | is still true. | | You can't capture an infinite structure fully by | "finitary" methods, either you use FOL and then you run | into L-S, or you use higher-order logic (for which L-S | doesn't apply) but then you don't have a complete proof | system anymore. | | To tie it all together, L-S and incompleteness are about | different flavours of "not being able to capture | something". L-S is about models of different | cardinalities. These models do still all satisfy exactly | the same sentences. Incompleteness is about different | models actually satisfying different sentences. | qsort wrote: | This is probably the editorialization. | | He's alluding to the fact that unlike what one might naively | intuit, it's impossible to formulate a set of axioms that can | formally express all mathematics. | | Because of Godel's incompleteness theorems, any formal system | that's sufficiently powerful to formulate basic arithmetic (see | for instance Robinson arithmetic) and consistent (that is, it | cannot prove false statements) is both incomplete (meaning that | it's possible to formulate a wff within that system that the | system itself can neither prove nor disprove) and can't prove | it's own consistency. | | This fact is often portrayed in popular media as being a "bug" | or a "missing foundation" of mathematics. That is inaccurate -- | it's just a property of how logical systems work -- but it does | prove that the search for the holy grail of a grand unified | system of axioms for all of mathematics is destined to remain | fruitless. | | Modern mathematics is most often implicitly assumed to be | formulated in a formal system called ZFC, but there are | alternatives. | Tainnor wrote: | I think the second theorem is the "worse" result for | mathematics. It shows that it's completely impossible to use | a weaker system, which is hopefully self-evident, to prove | the consistency of a stronger, but weirder system like ZFC. | | Hilbert wanted to show that, even if you don't "believe" in | ZFC, we could at least try to convince you that it doesn't | lead to contradictions, but that fundamentally didn't work. | qsort wrote: | Oh this is interesting, I didn't know about this historical | detail. | | It makes sense, Godel's first could be mostly seen as a | version of the halting problem, while the second is "game | over" for a Hilbert-like foundation program. | | p.s. Just wanted to point out that it's funny how we wrote | _two_ mostly identical sets of comments. Great minds think | alike? :) | Tainnor wrote: | > Oh this is interesting, I didn't know about this | historical detail. | | That's the characterisation I got from Peter Smith's book | about Godel's theorems. I didn't verify original sources | or anything, but it sounds very plausible to me. | | And it also kind of answers the question about why we | should care: if a system can't prove its own consistency, | well that's not terribly interesting (even if it could, | we would have to believe the system a priori to trust its | own consistency proof). But if it also can't prove a | stronger system consistent, then that's much more | interesting. | | > Great minds think alike? | | That would be a bit too self-aggrandizing for me. :D I'm | very curious about logic (and maths in general), that's | why I know some stuff about it, but I'm no maths genius. | SleekEagle wrote: | Is the fact that these systems cannot prove their own | consistency actually a feature of the incompleteness theorem? | I thought it effectively boiled down to "you can keep either | consistency or completeness, not both". It's been a while | since my metamathematics course ... | qsort wrote: | It depends on exactly what you mean with those terms. | | When you say "you can keep consistency or completeness but | not both" you are essentially stating Godel's first. | | Morally speaking, Godel's first is "no system is complete", | but there are two exceptions: if the system isn't powerful | enough to formulate the Godel sentence then the proof | doesn't work, and any inconsistent system is trivially | complete because _ex falso quodlibet sequitur_ , i.e. any | wff is true. | | The part about a system not being able to prove its own | consistency is Godel's second theorem. | | But the theorem only does what it says on the tin: the | system not being able to prove its own consistency doesn't | mean that it being inconsistent! [1] This is the case for | ZFC, for example. We can't prove ZFC's consistency within | ZFC: that would violate Godel's second, and we know that | _either_ ZFC is inconsistent (that is, you can derive an | absurd from the axioms) _or_ ZFC is incomplete (that is, | there exists a well-formed formula of ZFC that cannot be | proved or disproved within ZFC). | | We don't know which one it is. | | [1] This implication and similar ones are often made in | pop-culture presentations of the foundation problem. I | generally vigorously object to them because they're not | only imprecise -- that's understandable -- but they leave a | layman with a completely wrong impression. | thorel wrote: | The article is a bit oversimplifying in summarizing the | axiomatic crisis as being problem with sentences like "this | statement is false". | | This being said, your intuition is absolutely correct, the crux | of the issue is with 'this'. What mathematicians realized is | that if you are not careful with your choice of axioms, the | resulting logical system becomes too "powerful" in the sense | that it becomes self-referential: you can construct sentences | that refer to themselves in a self-defeating manner. | | As others have mentioned, this is the idea underlying Godel's | incompleteness theorem but also, to some extent, Russel's | paradox that came before and is what the article is referring | to. In Russel's paradox, the contradiction comes from | constructing the set of all sets that contain themselves. | Tainnor wrote: | Maybe in analogy to Russell's paradox and how you can "fix" | it by distinguishing sets and classes, you can "fix" a Godel | sentence by adding it as an axiom, but then you'll just get a | new Godel sentence... and so on. | [deleted] | pelorat wrote: | Math is the description of how one number relates to another | number. | | I'm not a math person but this Shinichi Mochizuki sounds like a | hack job. | pphysch wrote: | Mochizuki is an extreme case, but the problem of "bullshitting" | is omnipresent in academia, and particularly (pure) mathematics | which is a) perceived as consequential and b) impossible for | laypeople to verify. | brazzy wrote: | The vast majority of academic math has little or nothing to do | with numbers. | simonh wrote: | Whether a task is meaningful or meaningless depends on how we | think about meaning. I tend to think about it in terms of | correspondences between structures. Such structures are usually | what we think of as information, so this comment has meaning when | interpreted using the corresponding knowledge of English in your | brain. A computer simulation of the weather has meaning to the | extent that it corresponds to actual weather, and its behaviour. | The environmental mapping data structures in a Roomba's memory | has meaning to the extent it corresponds to the environment it is | navigating. | | So meaning is about correspondences, but also about achieving | goals. We communicate for many reasons, maybe in this case to | help us understand problems better. The weather simulation helps | us plan activities. The Roomba's map helps it clean the room. So | there's also an element of intentionality. | | What is the intention behind these mathematical proofs? What are | their correspondences to other information and goals? | pphysch wrote: | > Initially, Newton and Leibniz came up with objects called | infinitesimals. It made their equations work, but by today's | standards it wasn't sensible or rigorous. | | It would be great if mathematics was widely presented and taught | in this messy, practical, human, truthful way rather than the | purist, mystical, mythical manner. | | Calculus wasn't "discovered", it was painstakingly _developed_ by | some blokes that were trying to solve physical problems. | svat wrote: | Nice interview / thoughts. There was a great article almost | exactly 10 years ago, revolving around the same example | (Mochizuki's claimed proof of the abc conjecture) that this | article starts with: http://projectwordsworth.com/the-paradox-of- | the-proof/ (It's still worth reading, even as background to the | posted article.) | [deleted] | mcguire wrote: | Catarina Dutilh Novaes has been arguing much the same thing. She | has a 2021 book out, _The Dialogical Roots of Deduction,_ which | is on my list but I haven 't gotten there yet. | | I also haven't watched this video, but I'm linking it because I | can't find an earlier one where she points out that the basis of | logic is in rhetoric. | | https://youtu.be/0IOhYneseiM?si=vEfJ-JFCPF05zzNO | zodzedzi wrote: | Off-topic question: I've been seeing this "si" URL parameter | pop up a lot in shared youtube links lately. Which app exactly | pulled the URL for you? | nuc1e0n wrote: | Seems that Lean could be considered as a programming language for | mathematical proofs. ___________________________________________________________________ (page generated 2023-08-31 23:00 UTC)