William Gasarch

You'll definitely enjoy this: https://www.cs.umd.edu/~gasarch/papers/poll.pdf Edit: sorry, misread your comment, thought you were wondering what was Knuth's opinion. Still, I'll leave the link here, since it's a great read nonetheless

Alan Perlas, Dick Lipton, and Richard De Millo addressed this question in their POPL3 paper, Social Processes and Proof of Theorems and Programs, https://www.cs.umd.edu/~gasarch/BLOGPAPERS/social.pdf. It is still a good read.

Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really clearly explaining the philosophy of how you prove something independent of the axioms. I found it very illuminating. Second, the article has some interesting notes on how mathematicians go about what axioms to select and which not to. Penelope Maddy's "Believing the Axioms" [2] is the classic on this topic (it has two parts). It is focused on set theory, so it has a nice description of Martin's axiom and the arguments for and against. It was nice to read because it is a deeply technical argument (set theory is hard!) but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. The arguments are sometimes aesthetic ("rules" versus "surprises"), sometimes pragmatic (what they can and cannot prove), and sometimes involve deep values of what the universe should be like (should higher cardinalities be like lower ones? weirder? simpler?). It might all seem abstract, but if your day job is programming, imagine an argument over how to architect a large and complex system. Perhaps both…

There is a section that uses the same type of thinking that you can use to prove the R(3, 3) = 6 case (often known as the Theorem of Friends and Strangers [0]), towards the end of the paper [1] (though in a pretty different way). Proving the general case of things is often very removed from the proofs of a specific case, so its interesting to see similar thinking in the general case. Also, I really like the simplicity of the proof of the Theorem of Friends and Strangers. You can show someone the proof with two colored pens in 10 minutes. But at the same time, it uses some interesting proof tools: contradiction, the pidgeonhole principle, creating a boundary by counterexample (ie that R(3, 3) must be greater than 5). So I like the Friends and Strangers proof for its elegance and how fun it is to show people, and I like Ramsey's proof because it captures some of the fun of the R(3, 3) case while proving something much, much more difficult. [0] https://en.wikipedia.org/wiki/Theorem_on_friends_and_strangers [1] https://www.cs.umd.edu/~gasarch/BLOGPAPERS/ramseyorig.pdf

This is a great question that permeates all verification work. In the lingo, we call the set of assumptions a proof makes the "trusted computing base" (see https://en.wikipedia.org/wiki/Trusted_computing_base ). I listed some of the assumptions we made in another comment https://news.ycombinator.com/item?id=10018985 . The question you ask was believed for a long time to be the death knell of formal verification. It was persuasively argued in "Social Processes and Proofs of Theorems and Programs" https://www.cs.umd.edu/~gasarch/BLOGPAPERS/social.pdf that any proof of a complex system is necessarily at least as complex. Thus there would be no reason to trust the proof any more than the original code. The breakthrough in verification came when we started using machine-checked proofs. In this approach, you write a short, simple, and trusted proof checker. You can then verify complex systems using complex proofs that are checked by the simple checker. Then the only possible errors in the proof can come from the proof checker being wrong. Because the checker is simple and general (ie, it can check all kinds of proofs, not just proofs about a particular program), it is more trustworthy. Just to reiterate (including some content from my other comment): we can be wrong only if we have written the wrong definition of linearizability, misstated the correctness theorem, or if there is a bug in the proof checker or in Coq's logic itself.

A lot of those discoveries were actually made elsewhere ( not the majority, but an embarrassingly significant amount) https://www.manchester.ac.uk/about/news/indians-predated-newton-discov… https://sd2.org/bibha-chowdhuri-a-woman-of-firsts-with-no-recognition/ > After the war ended, Cecil Powell, a British physicist, continued the research in England using similar methods with more sensitive plates, detecting a new particle and winning him the Nobel Prize in 1950. Chowdhury and Bose’s work was acknowledged in his book, but their recognition quickly faded. https://www.theguardian.com/world/article/2024/sep/01/hidden-story-anc… https://www.cs.umd.edu/~gasarch/BLOGPAPERS/fibfibs.pdf

This comment from a previous thread https://news.ycombinator.com/item?id=27846084 links to a two-part paper called "Believing the axioms", part 1: https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf I have only taken a short look and am not qualified to summarize, but it seems this would have an answer if there is one.

Before someone jumps at you for daring to ask this question... yes, there are many many math riddles, and indeed not all are equally important, and we may not always know in advance which ones are. Some turn out to be more "productive" in the sense of leading to development of techniques, connections to other fields, etc. Ramsey theory (the riddle discussed in the article) is one of these, here is just a short list of nontrivial applications to CS (admittedly, mostly to theory of CS): https://www.cs.umd.edu/~gasarch/TOPICS/ramsey/ramsey.html

Better link... https://www.cs.umd.edu/~gasarch/BLOGPAPERS/social.pdf

>> We don't prove things to someone. Proofs just are. I think you need to do a bit of reading, to see what others have said on this subject, because you seem convinced you have the right attitude and the right frame of mind- but you only manage to trip yourself into making absurd statements like the one above. Proofs just are? Where? I suggest you start here: https://www.cs.umd.edu/~gasarch/BLOGPAPERS/social.pdf And here, for a bit more context: https://m-cacm.acm.org/magazines/2021/7/253448-program-verification/fu…

a hard or impossible problem? a while ago when prepping for my APs, I came across the paper "Reflections on Trusting 'Trustlessness' in the Era of Crypto/Blockchains" (Georgiadis,E.) available here: https://www.cs.umd.edu/~gasarch/BLOGPAPERS/cbit-4-2.pdf and found some parts very illuminating. It makes reference to a paper by Kuhn called Backdoor Engineering and advances that point to a concept the author calls conceptual corruption. Under this light, zero trust architectures are impossible. My fellow classmate a CS nerd made an additional point about the paper reflections on trust. it was beyond me.

There's the classic "Believing the Axioms" by Penelope Maddy [0] [1], the excellent study guide "Teach Yourself Logic" [2], plenty of great articles on the Stanford Encyclopedia of Philosophy (e.g. [3] or [4]), Dana Scott's "Lambda Calculus: Some Models, Some Philosophy" [5], and too many other interesting rabbit holes to count (i.e. resolution, unification, cut elimination, nonstandard models of arithmetic...) [0] https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf [1] https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms2.pdf [2] https://www.logicmatters.net/tyl/ [3] https://plato.stanford.edu/entries/settheory-alternative/ [4] https://plato.stanford.edu/entries/platonism-mathematics/ [5] https://lawrencecpaulson.github.io/papers/Scott-Models.pdf

Trusting trustlessness is the paper you want to consult with https://www.cs.umd.edu/~gasarch/BLOGPAPERS/cbit-4-2.pdf

So according to this poll http://www.cs.umd.edu/~gasarch/papers/poll.pdf only about two thirds think P!=NP (unless you also count the independent group as also believing we cant prove it). So seems like the majority opinion but not universal.

Linked on the page is an opinion poll [0] with responses from some prominent mathematicians (including Shor, Conway and Knuth). I thought it was really interesting to hear their thoughts on the problem. [0] http://www.cs.umd.edu/~gasarch/papers/poll.pdf

I just did a search for "real-world applications of Ramsey theory" and this was the closest I got. Sorry. http://www.cs.umd.edu/~gasarch/BLOGPAPERS/ramseykings.pdf

A relevant paper to discuss alongside this one: https://www.cs.umd.edu/~gasarch/BLOGPAPERS/social.pdf Social Processes and Proofs of Theorems and Programs Even rigorous mathematics requires social processes to become 'reliable', ie truthful.

Yes! And 4 out of the 79 theorists from this survey[1] think that it is. [1] https://www.cs.umd.edu/~gasarch/papers/poll.pdf#page=3

FYI, that link gave me a 404. I found what appears to be the same paper here: https://www.cs.umd.edu/~gasarch/BLOGPAPERS/social.pdf

I think they do have some ideas. You might look at Believing The Axioms (http://cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf, but it's two parts) which studies set theorists who are investigating new axioms for set theory, especially large cardinal axioms, iirc. Honestly, that work is a ways beyond what I can understand, but it's clear that it's an active area of research. You might also read about Hugh Woodin has a very developed program arguing for axioms which would imply the negation of the Continuum Hypothesis. (This doesn't necessarily apply to this specific question, but in general, research into axiomatic foundations of math is ongoing).

http://www.cs.umd.edu/~gasarch/BLOGPAPERS/17.txt 2,2,1,3,3,4,2,1,4,1,3,2,2,3,4,4,3 2,4,2,1,1,2,3,3,4,4,3,4,1,1,3,2,2 3,1,3,4,4,4,1,1,1,4,3,2,4,1,2,3,2 4,1,2,3,1,3,2,3,4,1,2,1,4,4,1,3,4 3,1,1,1,2,3,2,4,2,3,4,4,4,3,2,2,1 1,3,3,2,4,3,1,4,4,1,2,2,1,2,3,2,3 3,1,4,2,4,2,2,3,3,2,1,4,1,4,4,1,3 4,3,1,2,1,1,1,3,2,4,4,3,2,3,4,1,2 4,4,3,3,4,2,4,2,3,1,2,3,2,1,2,1,1 1,2,2,2,1,3,4,4,1,4,3,3,3,4,2,4,1 4,2,3,4,3,2,1,3,2,3,1,4,3,2,1,4,1 4,2,4,1,2,4,1,4,3,2,2,1,3,3,3,3,2 1,3,2,1,2,4,4,1,3,3,3,4,2,2,1,1,4 1,4,1,4,3,3,3,4,3,2,1,1,2,1,4,2,4 2,4,1,2,2,4,3,2,1,3,4,3,1,4,1,3,3 3,3,2,3,4,1,3,2,2,2,4,2,3,1,1,4,4 2,3,4,4,3,1,4,1,4,2,4,1,1,2,2,3,1

interestingly, the solution given is not ordered along rows or columns. http://www.cs.umd.edu/~gasarch/BLOGPAPERS/17.txt any idea why not?

There's a funny poll on P=?NP that William Gasarch conducted among computer scientists: http://www.cs.umd.edu/~gasarch/papers/poll.pdf Some argue that the question is not as important as it is widely believed to be (see Sariel Har-Peled's opinion in the poll) - many NP hard problems have practical approximations, many P problems are prohibitively costly to solve.

I'm not sure if it is or isn't weird that he hasn't published on complexity theory. I recall Gasarch publishing a poll in SIGACT where a handful of the researchers polled claimed that it'll be resolved by someone outside of the field. Regardless, from reading the synopsis of the proof, I'm skeptical to the point of disinterest. (I apologize for not providing a link -- but the only URL I was able to find from Google: http://www.cs.umd.edu/~gasarch/papers/poll.pdf is broken.)

According to http://www.cs.umd.edu/~gasarch/BLOGPAPERS/17x17chart.pdf c would be either 17 or 18.

See: https://www.cs.umd.edu/~gasarch/BLOGPAPERS/social.pdf

Tang is smart, but the author of this piece is unfairly characterizing Aaronson's beliefs and the nature of the P=NP problem. I don't know a single mathematician or computer scientist researcher that thinks that P=NP is likely. I know they exist, based on this poll[0], but in practice P equaling NP would be incredibly surprising. And even if it does somehow, I highly doubt the equality would practically hold. The reason I think it is fair for Aaronson to operate as if P!=NP is that even if it is unproven, it doesn't mean we can't make judgements about its relative likelihood. I haven't proven that all beetles are slower than the fastest cheetah, but based on what I understand about cheetahs and beetles I can operate as if I do and change my mind if I'm proven wrong. I think it's less likely that P=NP than it is that a beetle can outrace a cheetah. [0] http://www.cs.umd.edu/~gasarch/papers/poll2012.pdf

of duck backs, water, and bridges bygone > deterministically achievable in sub polynomial time it's already been stated that 'sub polynomial' was a typo, the original sentence discussed subexp state of the art but i changed it to discuss my research instead and failed to remove the 'sub' 'deterministic' can also be appropriately called out as redundant when used with polynomial time because it is implied in the latter but i wanted to be explicit > my intended inference inference is defined as 'a conclusion reached on the basis of evidence and reasoning' this is a word structure i use in place of faith based assumptions: i think, i believe, etc; to establish a respect for evidence based conclusions part of my research's aim is interested in a polynomial time algo for integer factorisation, but before i am to conclude that there is such a thing i require evidence, hence it being the intended inference of my research stead a direct inference > 'it's hard for me it must be impossible for you' this is what you will hear very often from many in the know if you tell them you have interest in a poly factorisation algo, even in spite of the problem being open except in the case of one of the people that fanned all this interest Richard Karp: (Berkeley, unsure, P=/=NP) My intuitive belief is that P is unequal to NP, but the only supporting arguments I can offer are the failure of all efforts to place specific NP-complete problems in P by constructing…

I've placed 74 using an IP solver 1 1 . . . . . 1 1 . . . . . 1 . . . 1 1 1 1 . . . . . . . . . . . . 1 . 1 . . 1 . . . . . 1 . . . . 1 1 . . 1 . . 1 . . . 1 . . . . 1 . 1 . . . 1 . . . . 1 . . . 1 . . . . . . 1 . 1 . 1 . . . . . 1 . . . . 1 . . . 1 1 . . 1 . . . . . . . . . 1 . . . 1 1 . . . . 1 . . . . . . . . 1 1 . . 1 . . . 1 . . 1 . . . 1 . . . . . 1 1 1 . . . . . . . . . . 1 . . 1 . . 1 1 . . . . . . . . . . . 1 . 1 . . 1 . 1 . . . . . . 1 . . . . . 1 . 1 1 . 1 . . . 1 . . . . . . . . 1 . 1 1 . . 1 . . 1 . . . . . . . . . . 1 1 1 . . . . . 1 . 1 . . . . . . . 1 . 1 . . . . . . . 1 . 1 . . . . . 1 1 The solver chokes on anything larger than a 6x6 grid, but I've had good luck adding one row at a time to a smaller solution. http://pastebin.com/m4c4696af edit: He mentions that he already found a size 74 set in one of the linked pdfs. http://www.cs.umd.edu/~gasarch/BLOGPAPERS/17x17.pdf

The epiphany for me this week was a long list of rabbit holes from this long post: https://news.ycombinator.com/item?id=47324054 https://medium.com/@acidflask/this-guys-arrogance-takes-your-breath-aw… https://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EWD692.html https://www.cs.utexas.edu/~EWD/transcriptions/EWD06xx/EWD638.html https://www.cs.umd.edu/~gasarch/BLOGPAPERS/social.pdf https://blog.computationalcomplexity.org/2021/06/i-went-to-debate-abou… https://6826.csail.mit.edu/2020/papers/noproof.pdf https://news.ycombinator.com/item?id=9948767 Now I have no illusions about who looked stupid and who were stupid. It really doesn’t matter. The jury is still out.