Gödel's Lost Letter

Terence Tao wrote a great obituary for Conway last year: https://terrytao.wordpress.com/2020/04/12/john-conway/ "Gödel’s Lost Letter and P=NP" also does obituaries of mathematicians quite often, and goes more into detail on their work. https://rjlipton.wordpress.com/2020/04/14/john-horton-conway-1937-2020…

This is a sad story, mostly about everyone deciding what the most kind way is to treat broken work by someone whose best days are well behind them. Interesting discussion at: https://meta.mathoverflow.net/questions/3894/is-there-a-way-to-discuss… And mathematicians taking him seriously: https://rjlipton.wordpress.com/2018/09/26/reading-into-atiyahs-proof/

Dick Lipton and Ken Regan, for example: https://rjlipton.wordpress.com/ Scott Aaronson: https://www.scottaaronson.com/blog/ Jeremy Kun: https://jeremykun.com/ Also, although he's a mathematician rather than a computer scientist, there is Terry Tao: https://terrytao.wordpress.com/

> I would love to believe this is true, unfortunately I do not think it is. This is the blog post that introduced me to the idea that human + computer might be better a computer alone: https://rjlipton.wordpress.com/2015/07/28/playing-chess-with-the-devil… It's called Freestyle Chess or Advanced Chess. Humans have beaten the best computers this way, but I'm not sure it is clear that human + computer outperforms a computer alone consistently. Incidentally, that's a great blog to browse if you like chess and math/CS theory.

NB. There is an amazing discussion in the comments following one of the linked blogposts. https://rjlipton.wordpress.com/2009/12/26/mathematical-embarrassments/

Not that the author claimed it to be safe, but this probably isn't secure: https://rjlipton.wordpress.com/2012/03/01/do-gaps-between-primes-affec… The technique relies on starting a search for one of the prime factors of the modulus at a particular (large) place, although if I understand the code right, not a place related to the other factor, so maybe it's not devastating, or who knows. That article is still very worth reading. :)

With a follow-up article: https://rjlipton.wordpress.com/2015/02/07/still-a-brilliant-idea/

But at some point Advanced Chess will be 'dead' in the sense that the best human players no longer help the best software win: the human will pick better moves less than poorer moves, and blow games. It's hard to tell when, but given how large the margin is, and how finely balanced the best chess engines are now (Lipton has some interesting posts on computer chess, most recently https://rjlipton.wordpress.com/2014/12/28/the-new-chess-world-champion… ), I wonder if that moment has already passed?

Shor's algorithm is exponentially faster than the best existing classical algorithm for factoring. But that doesn't mean there's not an exponentially faster classical algorithm which would be comparable in speed to Shor's algorithm. Some noted experts believe this is possible, and even likely. Richard Lipton believes it, and claims that Peter Sarnak does too: http://rjlipton.wordpress.com/2013/04/27/sex-lies-and-quantum-computer… Yes, 't Hooft and others, including Leonid Levin and Oded Goldreich, believe that quantum computers won't function because of quantum mechanics breaking down. See, e.g.: http://www.wisdom.weizmann.ac.il/~oded/on-qc.html http://www.cs.bu.edu/fac/lnd/expo/qc.htm

Dear Mr. von Neumann: With the greatest sorrow I have learned of your illness. The news came to me as quite unexpected. Morgenstern already last summer told me of a bout of weakness you once had I am pretty sure that "already last summer" is poor English grammar. Tisk - this Godel character. Probably not worth a face to face. http://rjlipton.wordpress.com/the-gdel-letter/

For those interested in the details, the (excellent, as always) blog of Richard Lipton sheds some light http://rjlipton.wordpress.com/2011/11/29/a-breakthrough-on-matrix-prod…

See https://rjlipton.wordpress.com/2011/04/13/even-great-mathematicians-gu…

I wouldn't say Aaronson is the premier expert on computational complexity. That would probably be Lance Fortnow who literally writes the blog on computational complexity (http://blog.computationalcomplexity.org/). RJ Lipton is pretty notable, as well http://rjlipton.wordpress.com/.

You've got some great ideas, but some of them are stymied by academic ego. For example, in peer review, reviews are generally unnamed. A discussion thread based system can suffer from what peer review did - people refuse to comment negatively with an attached name. A great example of moving in this direction is the recent P=NP discussion that happened largely on Dick Lipton's blog (http://rjlipton.wordpress.com/2010/08/08/a-proof-that-p-is-not-equal-t…

Dick Lipton's blog ( http://rjlipton.wordpress.com/ )does this as well, and I think this is an essential part of it being as successful as it is. People like being mentioned and like reading about other people, and humanizing ideas goes a long way towards making them interesting.

Here is a good link. http://rjlipton.wordpress.com/2010/08/08/a-proof-that-p-is-not-equal-t… The suspence is killing me. As an observer I'm worried that there is some small error in the core of his argument that will cause the whole thing to unravel, as is often the case for proofs of this level. But I am hopeful that at least this may be a breakthrough that points to approaches to finally nail this great beast.

Also see here for a follow-up article: http://rjlipton.wordpress.com/2009/11/12/more-on-mathematical-diseases…

See: http://rjlipton.wordpress.com http://blog.computationalcomplexity.org Also check out the STOC, FOCS, and SODA conferences.

Two I have in my reader: Gödel’s Lost Letter and P=NP - http://rjlipton.wordpress.com/ Igvata - http://www.igvita.com/ (Ruby based, sometimes too terse)

Dick Lipton's blog has a huge collection of really amazing kiddie pool tastes and pointers within awesome nuggets of theoretical computer science, one of his posts that I want to walk through carefully when i have the time is the one on the quantum method, for proving various nifty algorithmic complexity results etc. http://rjlipton.wordpress.com/2009/03/28/erds-and-the-quantum-method/ sexy stuff

This is something that I've struggled with myself so I can only give you advice as I've learned it from my perspective. Math is a big field so you have to understand what parts you're really interested in and how you want information to be presented so that you'll learn it. For example, I'm interested in computation aspects of mathematics (information theory, computation group theory, abstract algebra, analysis, etc.) and I always prefer a "computer programmers" intuition in how to learn these subjects. That is, understanding how to 'program it', whatever that means for the different subjects I'm interested in. Here is a small list of intuition I've learned about finding good resources: * Books, and sometimes textbooks, are still a valuable resource. It's still the case that having a book on a subject that has curated content is better than the random Wikipedia articles or blog posts on the subject. Use Wikipedia, obviously, and look at blog posts, but I search for books in the subject area, especially if it's a field I'm not familiar with. * When looking for books, prefer books that have "elementary" in the title, as in "elementary introduction". The more "advanced" books are talking about the bounds of research in the area, often fussing over esoteric issues whereas the "elementary" books give the foundation of knowledge in that area. * Ideally for me, books would have "fundamental algorithms" somewhere in the title, as these books usually are exactly what I need to…

> This is like measuring the computational complexity of a mathematical proof - it has no complexity, it's delcarative. There are few things that are absolute in this universe, but computational complexity is one of them. There is plenty of complexity involved. Have your heard of proof complexity? How do you know that a proof is a proof and how is a proof created? It's pretty much the core of P vs. NP question (and first discussed by Stephen Cook, the same guy who formally framed P vs. NP): https://en.wikipedia.org/wiki/Proof_complexity Actually, the first one who asked the question of P vs. NP was Kurt Gödel (https://rjlipton.wordpress.com/the-gdel-letter/), and as a mathematician, he was very much interested in the complexity (although he didn't use that word) of proofs. He said that if P = NP (not in those words): > If there really were a machine with φ(n) ∼ k ⋅ n (or even ∼ k ⋅ n2), this would have consequences of the greatest importance. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem, the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine.

>Quantum Computers have been proved to be more powerful than classical. wrong. I take issue with that claim. Take for example, the Deutsch-Jozsa problem. Given some function f, on n bits to one bit, such that f is either {zero on all the possible inputs, or one on all inputs}, or f is zero on half the inputs and one on the other half. To tell which of those cases it is, it requires 2^(n-1) + 1 tests of f. You have to test it on half the inputs, plus one. With the added power of a quantum computer, we can solve it with only one call to f. Boom, there is exponential speedup with quantum computing. This is similar to what lies behind Shor's algorithm for factoring. Check it out, wikipedia has diagrams that I can't put in here. http://en.wikipedia.org/wiki/Deutsch%E2%80%93Jozsa_algorithm EDIT: It seems he addresses this exact problem here http://rjlipton.wordpress.com/2011/10/26/quantum-chocolate-boxes/ I apologize, I was a bit hasty to call him out, he does know what he is talking about.

This interesting article submitted here on HN is one I have often recommended to other readers, so I'm glad to see it on HN's main page. It's particularly interesting to read the comments here, mostly largely agreeing with Tao, as I am currently at Epsilon Camp, http://epsiloncamp.org/ the most advanced mathematics summer program for YOUNG learners in North America, and the parents of the campers here are all pondering the issue of their children's mathematical development. Plainly, at any given age, some young people are more advanced in their mathematical development than many of their age mates, but it is still to be seen how steadily and consistently the mathematical development of the most advanced young learners can be developed if optimal practices are applied to their education. Based on this published writing by Tao and various writings by other mathematicians, the Epsilon Camp program provides FAQ pages for parents, http://epsiloncamp.org/FAQ.php who in many cases are not themselves mathematicians, to serve as food for thought as precocious young mathematics learners are growing up. One of the issues for many of the parents from various parts of the United States is simply finding a flexible local school. Another issue, which the meetings of the parents at the camp has helped to handle, is sustaining friendship relationships among those most advanced young mathematics-learners as they disperse around the country at the end of the summer program. On the whole,…

this is great article for someone to undertand how cryptography and AES works - but as all great explanations - it fails to explain some key concepts that take it from 0 - to "got it!" for me (hacker with absolutely no idea of how cryptography works other than simple caesar ciphers and stuff from childhood books) - this actually reminds me of the article in rjlipton's blog that also made it up HN a couple of weeks ago (http://rjlipton.wordpress.com/2011/08/05/give-me-a-lever/) about the lever in any mathematical explanation. I'm going to use that a little leniently here and go so far as to say - the algorithm makes sense - but for someone who doesn't get cryptography - its the "key" that's the lever. I get the diffusion and confusion and most of the rest - that math is not the hardest bit. But how the key actually comes into being - and how the Standard ensures that only the 2 people in the conversation have the key - is what would really crack the entire thing for me.

The paper[1] claims that MIP* = RE. At first I wasn't sure whether that was true, because the paper is too long and complicated for me to verify. But then I found a series of questions and answers written by this quantum guy Scott Aaronson[2] and by this other quantum guy Kenneth Regan[3], who both know a bunch of quantum stuff. Each one of their blog posts makes me more certain about MIP* = RE, to the point that now I'm 99.9% sure it's true, even though I could never verify it myself. [1] https://arxiv.org/abs/2001.04383 [2] https://www.scottaaronson.com/blog/?p=4512 [3] https://rjlipton.wordpress.com/2020/01/15/halting-is-poly-time-quantum…

Some related links: An introduction to the area from before this new breakthrough -- https://www.ams.org/journals/notices/201910/rnoti-p1618.pdf Gil Kalai on the mathematical implications -- https://gilkalai.wordpress.com/2020/01/17/amazing-zhengfeng-ji-anand-n… Lance Fortnow -- https://blog.computationalcomplexity.org/2020/01/quantum-provers-to-in… KW Regen -- https://rjlipton.wordpress.com/2020/01/15/halting-is-poly-time-quantum… Boaz Barak -- https://windowsontheory.org/2020/01/14/mipre-connes-embedding-conjectu… Scott Aaronson -- https://www.scottaaronson.com/blog/?p=4512

The first part is actually quoted in the original post. The full letter is here: https://rjlipton.wordpress.com/the-gdel-letter/

I just found this week's blog posting by RJ Lipton: https://rjlipton.wordpress.com/2019/10/21/a-polemical-overreach/ The Mackenzie book chapter discusses both the Fetzer and DeMillo Lipton Perlis controversies. It's really interesting that the debate got into the questions like, can one formally verify a bridge?