Six announcements

  1. I did a podcast interview with Julia Galef for her series “Rationally Speaking.”  See also here for the transcript (which I read rather than having to listen to myself stutter).  The interview is all about Aumann’s Theorem, and whether rational people can agree to disagree.  It covers a lot of the same ground as my recent post on the same topic, except with less technical detail about agreement theory and more … well, agreement.  At Julia’s suggestion, we’re planning to do a follow-up podcast about the particular intractability of online disagreements.  I feel confident that we’ll solve that problem once and for all.  (Update: Also check out this YouTube video, where Julia offers additional thoughts about what we discussed.)
  2. When Julia asked me to recommend a book at the end of the interview, I picked probably my favorite contemporary novel: The Mind-Body Problem by Rebecca Newberger Goldstein.  Embarrassingly, I hadn’t realized that Rebecca had already been on Julia’s show twice as a guest!  Anyway, one of the thrills of my life over the last year has been to get to know Rebecca a little, as well as her husband, who’s some guy named Steve Pinker.  Like, they both live right here in Boston!  You can talk to them!  I was especially pleased two weeks ago to learn that Rebecca won the National Humanities Medal—as I told Julia, Rebecca Goldstein getting a medal at the White House is the sort of thing I imagine happening in my ideal fantasy world, making it a pleasant surprise that it happened in this one.  Huge congratulations to Rebecca!
  3. The NSA has released probably its most explicit public statement so far about its plans to move to quantum-resistant cryptography.  For more see Bruce Schneier’s Crypto-Gram.  Hat tip for this item goes to reader Ole Aamot, one of the only people I’ve ever encountered whose name alphabetically precedes mine.
  4. Last Tuesday, I got to hear Ayaan Hirsi Ali speak at MIT about her new book, Heretic, and then spend almost an hour talking to students who had come to argue with her.  I found her clear, articulate, and courageous (as I guess one has to be in her line of work, even with armed cops on either side of the lecture hall).  After the shameful decision of Brandeis in caving in to pressure and cancelling Hirsi Ali’s commencement speech, I thought it spoke well of MIT that they let her speak at all.  The bar shouldn’t be that low, but it is.
  5. From far away on the political spectrum, I also heard Noam Chomsky talk last week (my first time hearing him live), about the current state of linguistics.  Much of the talk, it struck me, could have been given in the 1950s with essentially zero change (and I suspect Chomsky would agree), though a few parts of it were newer, such as the speculation that human languages have many of the features they do in order to minimize the amount of computation that the speaker needs to perform.  The talk was full of declarations that there had been no useful work whatsoever on various questions (e.g., about the evolutionary function of language), that they were total mysteries and would perhaps remain total mysteries forever.
  6. Many of you have surely heard by now that Terry Tao solved the Erdös Discrepancy Problem, by showing that for every infinite sequence of heads and tails and every positive integer C, there’s a positive integer k such that, if you look at the subsequence formed by every kth flip, there comes a point where the heads outnumber tails or vice versa by at least C.  This resolves a problem that’s been open for more than 80 years.  For more details, see this post by Timothy Gowers.  Notably, Tao’s proof builds, in part, on a recent Polymath collaborative online effort.  It was a big deal last year when Konev and Lisitsa used a SAT-solver to prove that there’s always a subsequence with discrepancy at least 3; Tao’s result now improves on that bound by ∞.

75 Responses to “Six announcements”

  1. Raoul Ohio Says:

    Re point 6: Reports on this will hopefully include the Tao (8 years old?) / Erdös (somewhat older) photograph. Anyone have a link to a good online version?

  2. Michael Says:

    I think there is a typo in the sixth part. It is too easy to misread
    for every infinite sequence of heads and tails, there’s a positive integer k such that, if you look at the subsequence formed by every kth flip, the heads outnumber tails or vice versa by an unboundedly large amount.
    as «for all sequences for exists k for all d exists n: x_k+…+x_{kn} is more than n/2+d», which is clearly false (with such quantifier ordering).

  3. Pete Says:

    Annoying (but hopefully interesting) nitpicking time!

    Ole Aamot’s name is alphabetized after nearly everyone: ‘Aa’ (usually spelt ‘Å’ when not in someone’s name) is the last letter of the Norwegian alphabet.

    It’s still ASCIIbetized before you, I suppose.

  4. Lewikee Says:

    Regarding point 6:

    Is it customary to write “we show that…” in an academic paper even when one is the sole author? Also, to make sure I understand the proof, is it correct to say that it was proved that even a carefully constructed ±1 sequence (perhaps drawn from a generator that minimizes auto-correlation?) would yield an unbounded sum of every kth element, given some k?

  5. jonas Says:

    Good timing. I heard that the Erdős discrepancy problem is solved just half an hour ago. Then I checked your blog because I thought you’d mention it, and even if not, you have links to lots of other mathy blogs that do.

  6. Scott Says:

    Michael #2: Thanks! Edited to clarify the statement.

  7. anon Says:

    This Tao is a mathematical monster..
    What surprises me is that he seems also a fairly normal person. The other mathematical “monsters” I encountered were all less good than Tao and completely weird persons.

  8. kodlu Says:

    #6: yes, when the sum is taken over all natural numbers. as of now there is no effective lower bound as a function of $n$ if one only considers sequences in $\{-1,+1\}^n$

  9. Alphamale Says:

    Regarding the NSA announcement, do you think it means Factorization and Discrete Log has an NSA algorithm which they themselves think could be found by public in near future?

  10. asdf Says:

    Raoul Ohio #1:

    via here:

  11. HR Says:

    Raoul #1: He was ten, I think. Here is the photo:

  12. James Cross Says:

    The evolutionary function of language is a mystery?

    Of course, I wasn’t at the talk so I don’t know what Chomsky said but the evolutionary function of language doesn’t seem like much of a mystery to me. Imagine where humans would be without language.

    Now its origin and evolutionary history is another matter.

    I am particularly drawn to the idea that its origin goes pretty far back and may have co-evolved with the making of tools. Of course, that language may have been more gestural with grunts and noises since the modern vocal chords hadn’t evolved yet. Modern languages I think probably developed in the last 100,000 years possibly originating in Southern Africa.

  13. Scott Says:

    James #12: I was there, and he said (indeed, repeated over and over) that the evolutionary function of language is a complete mystery. He also said it’s a myth that the purpose of language is communication—or at any rate, we have no idea what its purpose is. Under questioning, however, he seemed to lean toward the view that language evolved as an internal mechanism of thought, and then just happened to be useful for communication also, rather than that there was any selection pressure favoring the latter. And I believe this echoes what he’s said on numerous other occasions (anyone care to dig up references?).

    To my mind, Steven Pinker’s 1994 book The Language Instinct does a wonderful job of distilling everything in what Chomsky said in the 50s and 60s that was new and valuable and important, while jettisoning the weird, anti-Darwinian, quasi-mystical parts.

  14. Serge Says:

    It seems useless to me to separate the communication function of language from its thinking properties. Both functionalities are completely entangled, for if you have no communication protocol then there’s little data to process! A lonely thinking being is a myth… On the other hand, it’s more than obvious that language is vital to the preservation of our specie. I would like to hear Chomsky’s arguments against this.

  15. Jay Says:

    James #12,

    To Chomsky, the human langage is an exaptation of some information processing ability. When he questions the evolutionary function, what he questions is not the value of the exaptation, but the reason why the underlying information processing ability first evolved, possibly far before it serves communication purpose.

  16. Jay Says:

    Serge #14,

    Not sure this is what Chomsky would argue first, but the cortical regions associated with human langage understanding are also associated with all sort of information processing beside verbal langage.

  17. James Cross Says:

    I wasn’t doubting what you heard. I was just making sure what you wrote reflected what you heard. I am getting the Kindle version of the Pinker book and researching some of Chomsky’s views.

    It would seem to me that language would have evolved from signaling mechanisms similar to what we find in other animals. Some of these are gestural and some sound-based. I have a family of crows that hang out in my backyard and they emit a wide variety of signals – assembly calls, warning calls, and threatening calls (I think I even identified one used to threaten a small hawk recently). Any social organism would have an evolutionary advantage if they were able to communicate even in fairly primitive ways. So I would think there would be an evolutionary advantage related to communication built in from the beginning.

    One could argue that there might have been no additional advantage conferred from the development of the sophistication of language that we possess beyond the primitive signaling, particularly little additional advantage in its early stages of development. That is why I find the link to tool making interesting. If language is tied to tool making, then the sophistication of language may have been initially a by-product of tool making abilities acquisition.

    I would expect if this theory is even partially correct there would have been a transitional language corresponding to the primitive tools of Homo Erectus that would have had evolutionary advantages for coordinating hunting and gathering which also began to be important around this same time.

    So the idea that communication that language made possible would have no evolutionary advantage doesn’t make much sense to me.

    Now it seems sometimes Chomsky argues that language wasn’t designed for communication because it has features in it that are inefficient for communication. However, a lot of what evolution produces is inefficient and makeshift so I don’t think one necessarily follows from the other.

  18. James Cross Says:

    Jay #15

    I’ll check out the links but it seems to me that information processing ability is characteristic of the nervous system, not just the human nervous system but all nervous systems. This probably goes back at least as far as the first worms (if not further) and its function is control of the internal state of the organism, acquiring food, avoiding predators. Language and mind just builds off this.

  19. William Hird Says:

    Hey y’all, why doesn’t someone invite Chomsky into the blog and talk to him here instead of trying to guess what he means on these very subtle points ?

  20. Serge Says:

    Jay #16,
    I find it interesting that human language could be both a communication protocol and a processing language. This is exactly the purpose of what mathematicians like to call a good notation, one which allows you to convey your thoughts as efficiently as to perform mental calculations. But I doubt anybody – including Chomsky – could use such an argument to refute the evolutionary advantage of human language…


    a few parts of it were newer, such as the speculation that human languages have many of the features they do in order to minimize the amount of computation that the speaker needs to perform.

    This is also a characteristic property of a “good” mathematical notation. It’s surely one of the reasons why constructed languages (such as Esperanto) aren’t nearly as popular as natural languages (such as English). Whenever you restrict the number of built-in constructs – think of Turing machines, lambda-calculus, assembly language – you need to think longer to convey the most basic concepts… except when you’re among the experts of the language – but then it will have taken you longer to reach that level.

  21. James Cross Says:

    Jay #15

    I took a look at the links and there is much I would agree with in those papers.

    I also believe there was a significant change to the human species exactly in the 70-100,000 year ago range. I speculate about it myself and think the change was broader than just language.

    Where I might disagree somewhat is about whether there might have been some antecedent to the modern languages that appeared at that time. There may have been something intermediate in the late period of Homo erectus. But as I indicated before it could not have been the spoken because the species lacked the vocal chords. It would had to have been gestures and vocalizations.

  22. Raoul Ohio Says:

    James #17: I’m with you on that as a plausible theory. I would go so far as to say that it is the obvious guess.

    This is another example of a common phenomena in the social sciences: The theories proffered are all cases of thinking too hard, and ignore the obvious most likely explanation. There appears to be no awareness of Occam’s razor.

    Here is an interesting example froom a recent Scientific American article about the origin of dogs. Several theories are mentioned, some invoking religion for no particular reason.

    Unmentioned is what seems to me the obvious theory: Early humans and wolves are the only top predators that track and hunt in groups. They obviously were in competition, so interactions would be common. Each had advantages the other did not have. A mixed group of wolves and humans would be much more efficient than a non mixed group of either. Thus wolf/human cooperation is sort of like a global minimum energy state. The details of how you got to the minimum might not be known: did humans go out and steal wolf puppies for trackers, or did the wolfs start trailing along on hunts? Don’t know, probably both, and the global minimum is dogs.

  23. Alphamale Says:

    Scott would there be reason NSA is switching?

  24. pete Says:

    Extrapolating wildly from the relationship between neocortex size and the maximal stable social group size across several species ( ) a thoroughgoing sociobiologist might like the hypothesis that the evolutionary function of language is the maintenance of social order and stability, and that information processing, representation, abstraction, etc. are mere side effects – depressing, but plausible IMO.

  25. Scott Says:

    Alphamale #9, #23:

      Regarding the NSA announcement, do you think it means Factorization and Discrete Log has an NSA algorithm which they themselves think could be found by public in near future?

    No, I don’t.

      Scott would there be reason NSA is switching?

    Because a lot of people are expressing a worry that QCs will become practical within another couple decades, and there are government and private entities that want to keep messages secure for longer than that?

  26. Jay Says:


    (interesting blog 🙂 )

    > information processing ability is characteristic of the nervous system

    Of course, I should have added *cognitive* somewhere, e.g. we’re talking about information processing that operates on and modify the conscious content. Specifically, Chomsky et al. propose that it’s a “merge” process that is at the root of human langage.

    >there might have been some antecedent to the modern languages that appeared at that time

    Maybe, and Chomsky et al. present that as a standard view they wish to challenge. To tell the truth, I’m not convinced they present strong arguments against this particular idea.

    >I find the link to tool making interesting.

    Actually it’s one of the possible scenario fully compatible with Chomsky et al.’s view, especially as “associating gestures with objects” is one of the non verbal ability we can show impaired in aphasic persons (see link in #16). In this scenario the “merge” function would have evolved for tool or fire making, then a further exaptation of this ability would have lead to human langage. The recent discovery of Homo Nadeli, if he’s as old as one can hope, might be interpreted as a further support to this idea.


    >I doubt anybody – including Chomsky – could use such an argument to refute the evolutionary advantage of human language

    They don’t, see #15.

  27. Alphamale Says:

    “Because a lot of people are expressing a worry that QCs will become practical within another couple decades”

    But there is no evidence for this. People already knew how to do mechanical calculation thousands of years back (abacus which is not automated but works on rules). There really is no evidence for quantum computation factoring or discrete log beyond 5 bits of information.

  28. Scott Says:

    Alphamale #27: In the same sense, there was “no evidence” for a nuclear fission chain reaction in 1933, or for powered flight in 1890. Obviously, the question with future technologies is not what exists today; it’s what could exist: what’s consistent with the laws of physics, what there’s a plan for working towards, what would be built out of components that have been demonstrated individually. I’d hope an aggressive, go-getter alpha male like yourself would understand that! 🙂

    In particular, even if you only think there’s (say) a 10% chance of a scalable QC being developed in the next 20 years, depending on your application that might already be motivation enough to switch to quantum-resistant crypto.

  29. JollyJoker Says:

    Raoul #22: Wolves eat carrion, too. Assuming primitive humans left remains after butchering something, wolves would quickly learn that following human hunters is useful.

  30. Alphamale Says:

    Professor dont get me wrong, there was evidence of fission reactors

    and there was evidence of steam power or battery power or flight but with a different power source but no one has any evidence of quantum computation employing QFTs.

    No one knows if the alignments and cancellations needed in quantum computations could ever be scaled (infact you may have brought this up yourself in

  31. Jay Says:


    Nope, the evidences for natural fission reactor were found well after first power plants were constructed. Actually it’s explicitly stated in your link (sentence #3…).

  32. Scott Says:

    Alphamale #30: I don’t know what you mean by “employing QFTs” (quantum Fourier transforms or quantum field theories??). But in any case, there’s enormous evidence for cancellation of amplitudes at the level of up to 10-20 particles, and even for simple systems (e.g., the currents in superconducting Josephson junctions) with millions or billions of particles. If everything continues to work the way textbook QM says it should in more complicated situations—i.e., if there are no new surprises—then scalable QC is possible.

    Of course, maybe there is a surprise in store that would make QC impossible—as I’ve said many times, that would be the scientific thrill of my life! But if you had messages that needed to remain secret for the next 30 years, then it would be crazy to bet on such a surprise. If there were a quantum-resistant cryptosystem that you could switch to without too much cost (and it’s increasingly looking like there are, or will be), why not switch just to be safe? Indeed, the same argument would apply even to technologies that are much more speculative than QCs. When we talk about crypto, we have to be paranoid; it’s the people who think some particular attack can’t work who have the burden of proof.

    For more, I’ll just direct you to the archives of this blog—we’ve had variants of this same discussion like 5,000 times.

    PS. That MathOverflow post has nothing whatsoever to do with what we’re talking about here.

  33. James Cross Says:

    Jay #26 Raoul #22

    I take back one thing I said or implied. The ability of other species to sound alarms probably has little to do with development of language since this ability seems to come from a different part of the brain.

    The tool making theory, however, seems even more plausible to me. For tool making to persist across generations it would have to be taught. The way it would be taught would be by another watching someone else who was already skilled in making tools. This would require watching the hands and imitating their motions. It would be a very small step from that to using the hands to signal other things and over time we would develop a sign language capability. Sign language uses the same part of the brain as spoken language. If we add to that some rudimentary control of sounds – whistles, clicks, etc – that might combine with gestures and hand motions, particularly for drawing attention when another’s gaze was averted or out of sight, we might have a selection pressure for the anatomical modifications that allowed full speech.

    One other interesting point that I don’t often see noted in discussions is language is not exactly innate in humans like an instinct. While it is frequently pointed out that we could take children from any culture and place them in another at any early age and they would be able to learn to speak the language, children who are not exposed to any language at early ages never acquire a full language capability. So full language capability develops after birth and requires a nurturing environment of the human family.

  34. Jay Says:

    James #33,

    >children who are not exposed to any language at early ages never acquire a full language capability

    Amazingly, there are now good evidences that this is not true. Certainly feral children, who are deprived from social interactions with humans, never acquire language. But what do you guess would happen if a group of deaf children, to whom no one taught langage, can interact?

  35. Jr Says:

    So you don’t find Hirsi Ali to be Islamophobic?

    Also, I note a reference to the stock market in your podcast. I am sure that under certain assumptions you can prove that people won’t engage in stock trade but it seems to me that these assumptions must be stronger than mere rationality. One agent might sell because they need money, or because if they get a job for Apple, they don’t want to also own Apple stock for insurance reasons.

  36. Lou Scheffer Says:

    James #33 writes “For tool making to persist across generations it would have to be taught. The way it would be taught would be by another watching someone else who was already skilled in making tools.”

    For an example of complex behavior learned through actions, not words, watch what this human baby does after watching parents go through CPR class.
    This makes it pretty clear you could pass on quite complex behaviors with no language at all. Then sign language, then spoken language could evolve from these mechanisms.

  37. Kyle Cranmer Says:

    Regarding names that start With “Aa”, the ATLAS collaboration at CERN has first author “Aad”. Hard to beat that.

  38. Scott Says:

    Jr #35:

      So you don’t find Hirsi Ali to be Islamophobic?

    She’s used harsher language in talking about Islam than I would use. But maybe she could be cut some slack in that regard, given the horrific conditions that she escaped from and the Islamic justifications that she was given for them—justifications that, alas, are considered compelling in large parts of the world. In any case, I’d prefer to talk about ideas on their merits rather than using broad-brush labels.

    And I do try to be consistent about this. E.g., I’ve never felt any particular urge to join other Jews in pressuring people into apologizing for antisemitic remarks. I tend to think: if the remarks were factually false or morally evil, then condemn or ridicule them on those grounds! And if they were neither false nor evil, let them stand. Whether the remarks were insensitive, hurtful, etc. seem to me like side questions that are mostly beside the point—or rather, that would be mostly beside the point, in the kind of world I want to live in.

  39. Edward Brown Says:

    I think you are taking the argument a bit too far, or are taking the phrase “Agree to Disagree” to focus solely on the word disagree. What is your concept of that phrase? Or, more exactly, before you heard of Aumann’s Theorem, what did you imagine “Agree to Disagree” would look like for two rationalists discussing?

    Imagine two twins separated, one raised in England and one in the US. Before they meet, they each think the word color is spelled differently. What is the correct spelling of the word color or colour? After they meet and discuss, they will leave, each still believing their previous spelling was correct, and will likely continue to use their spelling, yet they will now agree there is another acceptable way of spelling it _in_a_different_context_. That is perfectly acceptable with Aumann’s Theorem, and that is exactly what I would expect two rationalists “agreeing to disagree” would look like. They did not converge on one correct way to spell the word color. Yet they fully agree. But will also likely maintain the same actions in their different contexts in the future, but now agreeing there is another equally valid way to look at it if they were in a different context.

    I can see a lot of “Aumann discussions” just ending with: sure, I now see that if I grew up in your shoes that is a rational position, and now you see that if you grew up in my shoes that is a rational position. We agree to disagree.

  40. Nick Read Says:

    Re ideas on impossibility of heavier-than-air powered flight in the 19th century and earlier: it’s amusing to consider that birds and other animals were doing it all along—but most humans just assumed it couldn’t be scaled up to the point where we could do it ourselves (after all, steam engines are too heavy relative to power output, so that’s that). da Vinci was probably the honorable exception. Then there were some with impossibility “proofs”, I suppose, . . . 😉

  41. Jr Says:

    Brown #39,

    In your example the twins don’t disagree after their discussion. They would simply agree that the British and the American spelling are different.

  42. Anonymous Says:

    Scott #38: “I tend to think: if the remarks were factually false or morally evil, then condemn or ridicule them on those grounds! And if they were neither false nor evil, let them stand.”

    I’m confused by your wording. Does this mean there are examples of remarks you’d condemn that are both factually true and morally evil?

  43. Scott Says:

    Jr #41: Yes, thank you! There’s no actual factual question at issue there.

  44. Scott Says:

    Anonymous #42: There are obvious examples where it can be morally evil to speak a truth (“why yes, Mr. Ax Murderer, the child is in that room!”). But I was thinking more about morally evil statements that don’t purport to be factual at all (example: “all left-handed people should be rounded up and killed”).

  45. John Sidles Says:

    Anonymous wonders “[Are there] examples of remarks you’d condemn that are both factually true and morally evil?”

    The poet replies

    Auguries of Innocence

    A truth that’s told with bad intent
      Beats all the Lies you can invent

        — William Blake (circa 1789)

    Initial question  Does Blake’s criterion encompass internet quibbling and cherry-picking?

    Note  Michael Harris’ recent Mathematics Without Apologies (2015) is worth searching for words like intention, yoga, and “foundations” — and even Steven Pinker’s cheesecake — whose meanings in Harris’ view (and mine too) are interlocking.

    For example, Harris quotes (approvingly?) the critic Roger Fry:

    If by some miracle beauty could be generated without effort, the whole world would be richer. […] [A]s things are at present, with our gaping admiration for professional skill, we are in less danger of finding a prophet whose utterance is spoiled by imperfect articulation than of being drowned beneath floods of uninspired rhetoric.”

    Further questions  Is truth irrelevant, when it is expressed without regard for intention? Or are we better off (individually and/or collectively) just asking for “more truthy-cheesecake, please?”

  46. Edward Brown Says:

    Scott #43:

    You said “it applies anytime your opinions are functions of the state of the world”. Spelling is very much a function of the state of our surrounding culture.

    The point is that they left in agreement, not by either abandoning their original spelling, but just by agreeing that there is another acceptable way in a different context. That is exactly what I would imagine “agreeing to disagree” would mean in a rational context: they don’t abandon their original belief, but merely append the other’s as an acceptable alternative in a different context. And they know that the other knows that they know that the other knows … etc. so their priors match now on this topic, even though they didn’t converge on one spelling.

    You skipped the question: Before you heard about Aumann’s Theorem, what did you imagine “Agree to Disagree” would look like for two rationalists discussing?

  47. Scott Says:

    Edward #46: No, you’re again playing semantic games, which sound good but don’t reflect the math.

    Within the context of Aumann’s Theorem, “the world” (the thing that has a state ω∈Ω) means the entire universe—the whole domain of discourse that the two participants in the conversation are both talking about.

    Thus, in your example, “‘color’ is the correct spelling everywhere” would be one statement about the world, which could be true for some ω’s and false for others.

    “‘colour’ is the correct spelling everywhere” would be a second such statement.

    “‘color’ is correct in some places while ‘colour’ is correct in others” would be a third statement.

    If Alice had only ever seen it spelled ‘color,’ then her credence would presumably be split between the first and third statements—and, of course, among countless other possibilities, e.g. that it can be spelled “color” or “kuller,” but never “colour” (but to keep our own meta-conversation tractable, we need to focus in on a few statements of interest to us). Likewise, if Bob had only ever seen it spelled ‘colour,’ then his credence would be split between the second and third statements.

    But this means that, as soon as Alice and Bob start talking, they can come to agreement (and common knowledge of agreement) that the third statement is correct, while the first and second are incorrect.

    You might object that, if Alice had never seen it spelled ‘colour,’ then the possibility that it might be spelled that way in other parts of the world would never even occur to her. She wouldn’t have a prior probability for its being true, because she wouldn’t even have considered it.

    Yet, while this is true of humans, it’s emphatically not true of the idealized Bayesian agents that Aumann’s Theorem talks about. Such an agent, by definition, has a prior for everything. And she’s only ever ‘surprised’ because something happens that she assigned a tiny probability to, never because something happens that she didn’t assign a probability at all.

    Again, no one is pretending for a microsecond that this captures real human behavior! As I said in both the interview and my blog post, the question is just whether this is a standard of rationality that we should aspire to.

    Lastly, regarding your question: alas, I don’t remember thinking deeply—or even shallowly—about what the phrase “agreeing to disagree” truly means before I’d encountered Aumann’s theorem. (I was a bad Bayesian, with no prior over the possible meanings. 🙂 ) But if you’d asked me, I’m sure I would have said that “color” vs. “colour” was a trivial and uninteresting example—exactly like two people preferring different flavors (flavours?) of ice cream. I.e., sure, woohoo, we’ve found a difference between people or cultures. But it’s not one that corresponds to two rational agents holding different beliefs about the actual reality of their shared world. So it doesn’t seem like something that even the most thoroughgoing rationalist could get uncomfortable about. Vive la différence!

  48. Scott Says:

    Addendum: What I could have said in the interview is that Aumann’s Theorem applies to moral questions if, and only if, you’re a moral realist—i.e., if you believe that complete knowledge of the objective facts of the world would also resolve all moral questions in a universal way. If you reject moral realism, then agreeing to disagree about moral questions becomes no more problematic from a Bayesian perspective than agreeing to disagree about which ice cream flavor is tastiest, or whether it’s flavor or flavour.

    The difference between morality and the other examples, is just that I suspect there are few if any spelling realists or tastiest-ice-cream-flavor realists.

  49. Edward Brown Says:

    Thank you. It makes more sense now.

    And I was indeed thinking ahead to moral questions, as I was feeling uncomfortably like Aumann’s theorem was claiming there needed to be a convergence. But your follow up helped resolve that as well. I guess I’m not a moral realist, for I feel the best they could do is say statements like “if you take such and such as a premise for morality, then this and that will follow”, and one could never reason out the premise itself.

    I chose spelling so it wouldn’t get mired in side discussions. I’m not sure I’d go so far as to say morality is as much a social convention as spelling, as the latter is more arbitrary in some sense I’m not sure I could quantify easily. But I always find it frustrating when people try to argue morality with logic. It’s like the people who leave little booklets about their faith to encourage convertion… they point out real situations in the world or that people may have, then they always quote the bible, book of mormon, etc. as if we already agree that is a place to start from, and arrive at a conclusion. That may seem silly, but when I hear vegan’s proselytizing it comes off the same way to me. And any arguments I make in return, probably come off similarly. (For similar reasons, even though I agree with many of your social stances, I cringe a bit when you word it as society progressing when there is a social shift in the premises we use to discuss morality. You even hope we can look ‘ahead’ to the next shift. This appears to presupposed there is a correct fitness function of morality that we are trying to optimize.)

    I usually come to your blog to hear about snippets from the edge of complexity theory. But I am glad you feel open enough to share thoughts on other subjects. I don’t find philosophy as interesting as Julia Galef does, but it is always neat to hear someone speak enthusiastically about a piece of a subject matter that excites them. I ended up watching several of her youtube videos.

    Thank you for sharing.

  50. Jair Says:

    Edward, I think it is pretty odd to demand that we attach a numerical variable to morality before we can admit any moral progress. I think it is pretty evident that abolition, for example, was a moral action since it improved the lives of many human beings. I don’t mean improvement in some numerical sense, although I’m sure you could try to quantify it somehow. I mean it in the same sense that Liszt was a better pianist than most six year olds. This is a true statement in the broadest sense of the word, but we don’t demand some kind of quantifiable measure in this case, and we do not necessarily need it for moral questions either. We can certainly use numerics to measure our progress in various ways, but sometimes simple human understanding can tell us true things about the world.

  51. Ole Aamot Says:

    On “Aa”-names: There’s also several people whose last name is Aaberge. One of them was in my mentor group at University of Oslo, Norway.

    The last name Aamot is composed of the Norwegian words “å” (river) and “mot” (meeting) and means where two rivers meet and form a big river.

    I saw Scott Aaronson during Justin Dove’s presentation at the QI/QC group meeting when I visited MIT in June 2014 and got his book Quantum Computing Since Democritus from MIT’s book store. IIRC Aaronson was first on the shelf.

  52. Scott Says:

    Edward #49: Thanks for your comment.

      For similar reasons, even though I agree with many of your social stances, I cringe a bit when you word it as society progressing when there is a social shift in the premises we use to discuss morality. You even hope we can look ‘ahead’ to the next shift. This appears to presupposed there is a correct fitness function of morality that we are trying to optimize.

    As Jair #50 alluded to, this sounds sophisticated, but if we’re never allowed to say that “social shifts in the premises we use to discuss morality” represent progress, then we’d have to agree that (e.g.) abolition and women’s suffrage weren’t progress, and that re-instituting slavery and a male-only franchise wouldn’t in any sense be a “bad” or “backwards” step, but just another damn thing that would happen … which seems more politically incendiary than any idea I’ve ever toyed with in the decade-long history of this blog!

  53. Raoul Ohio Says:

    They’re back:

  54. Raoul Ohio Says:

    Ole, #51. In that case, you might want to study Bessel functions, and other confluent hypergeometric functions.

  55. Job Says:

    The problem in #6 is interesting.

    Computationally, we could say that no Turing Machine can be constructed such that the ratio of odd/even numbers in its output is guaranteed to remain within a fixed bound, when externally sampled at an interval k. Right?

    In a sense, the TM can’t guarantee a condition derived from k because it does not know its value – even though it has full control over its output.

  56. Edward Brown Says:

    Scott #52,
    “…we’d have to agree that (e.g.) abolition and women’s suffrage weren’t progress”

    WE can agree that those are progress, because we appear to have a close agreement on the premise of what is moral. Are you a moral realist? I am not. What would be incendiary in present day would be to say that abolition or women’s suffrage was not a good thing. I am not making that incendiary comment, as I do believe women’s suffrage and abolition of slavery are good things. I am merely making what I feel should be a non-incendiary statement that one cannot prove with logic that those things are moral without inserting premises of what is moral. If someone is a moral realist, then they must believe they can indeed derive morale premises from complete knowledge of the objective facts of the world. That would be amazing! And if someone could actually do so, I hope they share it with the world.

    Hopefully that makes what I was trying to say more clear. Unless you are a moral realist, I have a feeling we are actually in agreement here, and I was just a bit unclear.

    Jair #50,

    I agree with you that we can discuss morality even if we cannot reduce it to purely objective numbers. But whether we discuss it qualitatively or quantitatively, to discuss it with logic requires invoking a premise of what is moral. And I don’t see how these premises themselves could possibly be derived from pure logic without somehow inserting at least a partial definition.

    “I think it is pretty evident that abolition, for example, was a moral action since it improved the lives of many human beings.”

    You are actually making my point there. Here you used the qualitative premise that one indication of a moral action is that it “improved the lives of many human beings”. We can agree on that qualitative premise. Society now agrees slavery is morally wrong, so it is easy for us to sit back and praise ourselves and our society for realizing this and maybe even delude ourselves into thinking that we’ve proven it logically.

    We should not be flippant about what we call logical, just because we happen to agree with the changes around us.

    What will happen if in the future people decide it is “pretty evident” that all life is equally valuable. No amount of logical arguments could dissuade people otherwise, because this is not a question of logic. They would say I am a selfish murdering materialist for eradicating all the termites in my house to save its structure just for my material comfort. You killed all those living things they scream! I respond, well of course, I do not deny that, but I don’t value those insect lives more than my non-living house structure. I reject their premise. I do not see how logic can resolve this. And I feel we should admit to ourselves that morality can’t be reduced to pure logic, before we finally find ourselves on the other side. As Scott has said before, it is all too easy to pat ourselves on the back for believing something we feel is progressive when it doesn’t take courage to do so. There is a scary other side to that as well. One day society might come for my milk and eggs and it will be too late to point out that this is not a debate of logic.

  57. John Sidles Says:

    Raoul Ohio opines (#53) “They’re back [D-Wave]”

    Lol … yes, but why are they back?

    The “small-m” mathematics of Michael Harris’s writings suggest that the yoga of Kalai-style “small-s” skepticism may be helping Google’s computer scientists to appreciate D-Wave’s noisy (yet performative) quantum-gate devices as avatars of noisy yet (yet performative) binary-gate algorithms.

    Scott is hopeful (# 32) “Maybe there is a surprise in store that would make QC impossible — as I’ve said many times, that would be the scientific thrill of my life!”

    Many people share that hope (including me) … and it is mathematically plausible that the Google / NASA / D-Wave partnership is working effectively to speed that thrilling day.

  58. Michael Gogins Says:

    This business about moral realism has been investigated extensively by philosophers for quite a while now (understatement alert). The upshot, as far as contemporary academic philosophy is concerned, is that the issue (like many metaphysical issues) has not been persuasively resolved. Therefore, to think that only science, or even the philosophical foundations of the scientific world-view, can create a plausible argument for (or against) moral realism is… simply to beg the question.

    That means moral realism is roughly as rational, or not, as its antithesis.

  59. Jair Says:


    “And I don’t see how these premises themselves could possibly be derived from pure logic without somehow inserting at least a partial definition.”

    Well, I agree. My partial definition is that, generally, causing human beings to suffer is immoral. Obviously this “definition” is incomplete – it does not address the issue of animal rights, or what exactly counts as human, or when it might be necessary to cause some suffering in order to alleviate the suffering of others. But it’s at least enough to allow me to feel confident when I say that, e.g., slavery is immoral, and I won’t lose sleep at night wondering if this is just a fluke of our culture. If you don’t agree with me that causing suffering is generally a bad thing, then I have to wonder if you’re talking about morality at all. No, I can’t prove mathematically to you that it’s immoral to strangle your elderly neighbors, but to quote Scott, you can’t even prove mathematically that bears crap in the woods.

  60. Daniel TTY Says:

    I wondered if the Erdos discrepancy theorem, EDT, (I use theorem here since it’s now proved to be true) have any consequences on the frequentist definition of probability?
    Since for an infinite sequence of numbers to have well-defined probabilities for each of the numbers, the frequencies for each numbers in any subsequence should converge to the value obtained in the infinite sequence. However, EDT shows that the difference in the occurrence of numbers can be as large as we want.

  61. John Sidles Says:

    Teresa Mendes, a Great Truth that is dual to your youtube video Physics needs a paradigm shift is the idea that “physics doesn’t need a paradigm shift.” E.g., in Harris’ Mathematics Without Apologies we read:

    This [modern algebraic geometry] is nothing as brutal as a paradigm shift; each generation’s new perspective is meant to be more encompassing, as if mathematicians were collectively climbing and simultaneously building a ladder that at each rung offers a broadening panorama and the growing conviction that the process will never end.

    For young research students especially, a major practical advantage of Harris-style research strategies is that researchers need not wait around for paradigm shifts to arrive, but rather can work individually, on a daily basis, to incrementally immerse-and-dissolve existing paradigms in a rising tide of understanding.

    Conclusion  Sales of Harris’ book might greatly increase if he retitled it (as it might have been titled originally) The Four Habits of Highly Effective Mathematicians: Intention, Yoga, Performativity, and Narrative (four habits that Harris’ book scrupulously indexes).

  62. Scott Says:

    Daniel #60: No, I don’t see any such implication. For one thing, for a discrepancy even to be “noticeable” statistically, the discrepancy would need to exceed √n, not just be unbounded.

  63. Arko Says:

    I am not expert in language, but language has a tendency to standardize communication. To that effect, its evolution might have been a natural tendency of the human brain to decrease the entropy in communication. Of course, I am being sloppy here.

  64. John Sidles Says:

    Arko observes: “Language has a tendency to standardize communication”

    Reflections  Arko’s fine comment encourages us to reflect that languages have purposes beyond communication (else why do people talk to themselves and sing in the shower, for example?), moreover languages themselves and human uses of those language coevolve, so it’s unsurprising that new uses of language are continually being conceived.

    Thus it is natural to ask: What elements of language are presently being standardized, with a view to evolving new, possibly noncommunicative, linguistic capacities?

    Readings in performative elaboration  The notion of performative elaboration — along with a description of the crucial role(s) that performative elaboration plays in standardizing mathematical language — is set forth in a crucial passage (as I read it) in the free-as-in-freedom textbook Homotopy Type Theory: Univalent Foundations of Mathematics (Univalent Foundations Collective, 2013, known as “the HoTT Book”)

    Section A.2.11  Inference of implicit arguments, typical ambiguity, ensuring that symbols are only defined once, etc., are collectively called elaboration. Elaboration must take place prior to checking a derivation, and is thus not usually presented as part of the core type theory. However, it is essentially impossible to use any implementation of type theory which does not perform elaboration; see [refs] for further discussion.

    Conclusion  The HoTT-style performative elaboration of Kalai-style (small-s) quantum skepticism presents significant career opportunities for young quantum researchers, in concrete sense that the Harris-style “Four Habits of Highly Effective Mathematicians” (of comment #61) compose a concrete faith and practice for performatively elaborating Kalai-style (small-s) quantum skepticism.

    Open question  What role(s) will accelerating computer capacities for performative elaboration play in the development of AI? Don’t ask me … yet surely any AI that we would call “human-level” would be expert at performative elaboration.

  65. John Sidles Says:

    In further regard to the computational complexity of performative elaboration (per #63-#64), a recent preprint by Leonardo de Moura, Jeremy Avigad, Soonho Kong and Cody Roux provides a student-friendly account of ongoing research in this area:

    Elaboration in Dependent Type Theory

    We describe the elaboration algorithm that is used in Lean, a new interactive theorem prover based on dependent type theory. To be practical, interactive theorem provers must provide mechanisms to resolve ambiguities and infer implicit information, thereby supporting convenient input of expressions and proofs. […]

    Even second-order unification is known to be generally undecidable [ref], but the elaborator merely needs to perform well on instances that come up in practice. […]

    A definition or proof may give rise to hundreds of constraints requiring a mixture of higher-order unification, disambiguation of overloaded symbols, insertion of coercions, type class inference, and computational reduction. The net effect is then a difficult constraint-solving problem with a combinatorial explosion of options.

    Lean’s elaborator manages to solve such problems, and it is fast: Lean’s entire standard library compiles in seconds.

    Conclusions  (1) Performative elaboration is a Knuthian “science” in the sense that we can (and do) teach it to computers, and (2) elaboration seemingly belongs to that vast class of problems that are formally hard yet practically easy, and (3) elaboration is among the most natural of hard-yet-easy problems, since it performs many of the tasks that are generally associated to the term “mathematical maturity.”

    So perhaps Tim Gowers and his colleagues are right to foresee the feasibility, this century, of creatively elaborative AI?

  66. John Sidles Says:

    Elaboration  As a third (and final) comment regarding elaboration, technically minded Shtetl Optimized readers can find student-friendly introductions to this topic at Microsoft Research’s interactive GitHub tutorial The Lean Theorem Prover, and philosophically minded Shtetl Optimized readers can find a student-friendly survey on Lean-collaborator Jeremy Avigad’s home page, as his preprint Mathematics and language.”

    First conclusion:  the Wittgenstein-Knuth connexion  Mathematical philosophy (in the sense of Wittgenstein) is becoming a performative science (in the sense of Knuth).

    The quantum connexion  The connexion with quantum information theory arises when we wonder: “These resources make mathematical elaboration computationally performative … what other STEM enterprises can similarly be elaborated computationally?” For our small group up in the Pacific Northwest, one answer is quantum transport theory … and our evolving seminar notes (that can always be read by clicking on my name above) describe some of the practical implications of Gil Kalai’s small-s quantum skepticism for implementing this computational program.

    The elaboration of history  Needless to say, we are not the first to make these connections. Back in the mid-1990s, DARPA/NRO’s Dennis Healy (now deceased) foresightedly supported we would today call “computational elaboration” in the photonic simulation package “Meep” (DARPA/NRO N00014-05-1-0700), and NRO’s research investment in computational elaboration continues today in the MURI program “Homotopy Type Theory: Unified Foundations of Mathematics and Computation” (which supports Jeremy Avigad’s research, among others).

    Performative imagination and ingenuity  As Dennis Healy put it in a 2007 lecture (shortly before his passing):

    Mathematical Toolkit
    for the 21st Century Quantum Mechanic

    Address for DARPATech 2007
    In the end, the quantum limit has little to do with physics. It has everything to do with human imagination and ingenuity.

    If we are going to go after fundamental limits, we must have tools that allow us to break through our classical conditioning and begin to think like quantum creatures.

    Second conclusion:  the Healy-Kalai connexion  Quantum small-s skepticism (in the sense of Gil Kalai) is evolving to become crucial tool of computationally performative quantum science (in the sense of Donald Knuth/Dennis Healy).

  67. scott signal Says:

    What’s the big deal about the recent experiment making 2 qubits on silicon? Overhyped or interesting? Thanks. -asdf

  68. John Sidles Says:

    Today’s chess-related essay by Ken Regan at Gödel’s Lost Letter and P=NP “Depth of Satisficing” can be read (by me anyway) as a data-rich analysis of computational elaboration in grandmaster chess, and it stimulated a comment to that effect (with student-friendly references), which is intended to be read as a continuation of remarks #65 and #66 above.

    Prediction  What naturality has been to 20th century mathematics, computational elaboration will be to the entirety of 21st century STEM enterprises; indeed the latter disciplines could not prosper until the former mathematical idea had been well-framed.

    Appreciation and thanks are extended to Gödel’s Lost Letter and to Shtetl Optimized for providing venues that so wonderfully engage STEM researchers (students especially) with these emerging ideas.

  69. Kenneth W. Regan Says:

    On “Aa” names: For years it was the height of poetry that baseball’s MLB home run king Henry Aaron came first in the MLB Encyclopedia—until he was pipped by David Aardsma, who’s pitched for various teams including Atlanta this season. If he were Norwegian we could try to banish Aardsma to the back of the A’s, but the name is Dutch.

  70. Kenneth W. Regan Says:

    More name fun: Julia Galef is not (?) related to my Princeton ’81 classmate David Galef, whose recent piece “Diction Slips” drew a comment (#2) from the Dana Scott, nothing to do with the Dana and Scott…congrats on appearing on her show.

  71. Bram Cohen Says:

    Scott, I’ve spent some time now puzzling over how people ‘agree to disagree’ and have come up with some probably highly unoriginal thoughts.

    The biggest problem seems to be dishonesty and authority. If we follow a protocol of repeatedly stating back and forth what our beliefs are after incorporating in the other person’s belief then a dishonest person can trivially game the protocol by always projecting 100% confidence. Other people who are only a little bit credulous will then get quickly hoodwinked by them, and people who are mostly rational and mostly believe that the slightly credulous people are rational but have a seemingly minor argument about what authorities are reliable will come to a radically different belief about the state of things. When it comes to prominent disagreement about facts, such as climate change and the existence of commonplace divine intervention, this is exactly what we see, with the dishonest authorities of course being the ones I disagree with.

    In those cases we also see that one side even refuses to pay homage to the concept of rationality, with the rebuttal to any such discussion often amounting to ‘you’re an annoying dweeb’. Perhaps many people view rhetoric and argumentation as part of the truth-finding process and simply don’t care about or don’t believe in an objective reality which exists outside of our opinion of it.

    Another possible source of disagreement is that due to ever-increasing entropy authorities are always in the process of losing their minds, so people, or at least rational people, are constantly in the process of questioning, and sometimes disagreeing, with them as part of a defensive posture in case authorities go off the rails.

    Also there’s the phenomenon that a person can sometimes come up with some new piece of information which they themselves are utterly convinced of for good reason but the claim unfortunately sounds completely outrageous. The problem with outrageous claims isn’t so much that they have low priors: People can simply be reassured that the priors have been met. The problem is that only a small number of kooks who make a large number of outrageous claims because they’re insane or dishonest rapidly make it so that the vast majority of outrageous claims come from kooks rather than well-informed honest rational agents, and hence those agents have a hard time convincing others. Of course, someone who has a history of proving to be right about outrageous claims gets given a lot more benefit of the doubt. I’ve found this to be the case with myself, in that previously when I made claims about, say, what BitTorrent would be able to do people just rolled their eyes and didn’t say anything, whereas now when I make claims about, say, what my new live streaming protocol can do people give me the benefit of the doubt and don’t say anything. It doesn’t work universally though. For example I came up with a radically new approach to steganography which crypto people can understand and think is great but unfortunately stego people are utterly mystified by and don’t seem to really believe works.

    Perhaps the most hopeful theory is that people who disagree with the consensus are sacrificing their own accuracy for the greater good of making the group consensus more accurate. If we vote on the state of something then the result is as accurate as it can possibly be if we assume that we didn’t know which person we were going to be before we were born. Any one individual is better off ignoring their own opinion and going with the opinion of the group as a whole, but if everybody did this then hardly anybody would be giving their opinion into the vote and the measure would be highly inaccurate. Everybody who disagrees is sacrificing a lot of their own accuracy now for a little bit of accuracy of everybody else later.

  72. Jay Says:

    …or arguments have more to do with tribe formation than truth seeking.

  73. jonas Says:

    Your blog main page has an image of the cover page of your book at a prominent place. This image has a broken link. (Probably your publisher broke it.) Perhaps you should fix it.

  74. John Sidles Says:

    Consonant with this column’s Shtetl Optimized keywords “Announcements, Nerd Interest, Quantum”, physicist John Martinis — a.k.a. The Man Who Will Build Google’s Elusive Quantum Computer and the subject of last year’s Shtetl Optimized column “Raise a martini glass for Google and Martinis!” (September 6th, 2014), and the director of Google/NASA/USRA joint Quantum Artificial Intelligence Lab — has just posted an exceptionally interesting arxiv preprint (as I read it) Qubit metrology for building a fault-tolerant quantum computer (1510.01406).

    There are plenty of different ways to read Martinis’ preprint. For quantum universalists (like me), it’s natural to read Martinis’ preprint as a roadmap for closure of a virtuous circle traversing both small-“o” optimism and small-“s” skepticism in regard to quantum computing.

  75. Jon Lennox Says:

    Probably a million people have pointed this out to you already, but today’s xkcd is of interest.