Yet more errors in papers

Following up on my posts PostBQP Postscripts and More Wrong Things I Said In Papers, it felt like time for another post in which I publicly flog myself for mistakes in my research papers.  [Warning: The rest of this post is kinda, sorta technical.  Read at your own risk.]

(1) In my 2006 paper “Oracles are subtle but not malicious,” I claimed to show that if PP is contained in BQP/qpoly, then the counting hierarchy collapses to QMA (Theorem 5).  But on further reflection, I only know how to show a collapse of the counting hierarchy under the stronger assumption that PP is in BQP/poly.  If PP is in BQP/qpoly, then certainly P#P=PP=QMA, but I don’t know how to collapse any further levels of the counting hierarchy.  The issue is this: in QMA, we can indeed nondeterministically guess an (amplified) quantum advice state for a BQP/qpoly algorithm.  We can then verify that the advice state works to solve PP problems, by using (for example) the interactive protocol for the permanent, or some other #P-complete problem.  But having done that, how do we then unravel the higher levels of the counting hierarchy?  For example, how do we simulate PPPP in PPBQP=PP?  We don’t have any mechanism to pass the quantum advice up to the oracle PP machine, since queries to a PP oracle are by definition classical strings.  We could try to use tools from my later paper with Andy Drucker, passing a classical description of the quantum advice up to the oracle and then using the description to reconstruct the advice for ourselves.  But doing so just doesn’t seem to give us a complexity class that’s low for PP, which is what would be needed to unravel the counting hierarchy.  I still think this result might be recoverable, but a new idea is needed.

(2) In my 2008 algebrization paper with Avi Wigderson, one of the most surprising things we showed was a general connection between communication complexity lower bounds and algebraic query complexity lower bounds.  Specifically, given a Boolean oracle A:{0,1}n→{0,1}, let ~A be a low-degree extension of A over a finite field F (that is, ~A(x)=A(x) whenever x∈{0,1}n).  Then suppose we have an algorithm that’s able to learn some property of A, by making k black-box queries to ~A.  We observed that, in such a case, if Alice is given the top half of the truth table of A, and Bob is given the bottom half of the truth table, then there’s necessarily a communication protocol by which Alice and Bob can learn the same property of A, by exchanging at most O(kn log|F|) bits.  This connection is extremely model-independent: a randomized query algorithm gives rise to a randomized communication protocol, a quantum query algorithm gives rise to a quantum communication protocol, etc. etc.  The upshot is that, if you want to lower-bound the number of queries that an algorithm needs to make to the algebraic extension oracle ~A, in order to learn something about A, then it suffices to prove a suitable communication complexity lower bound.  And the latter, unlike algebraic query complexity, is a well-studied subject with countless results that one can take off the shelf.  We illustrated how one could use this connection to prove, for example, that there exists an oracle A such that NPA ⊄ BQP~A, for any low-degree extension ~A of A—a separation that we didn’t and don’t know how to prove any other way. Likewise, there exists an oracle B such that BQPB ⊄ BPP~B for any low-degree extension ~B of B.

The trouble is, our “proof sketches” for these separations (in Theorem 5.11) are inadequate, even for “sketches.”  They can often be fixed, but only by appealing to special properties of the communication complexity separations in question, properties that don’t necessarily hold for an arbitrary communication separation between two arbitrary models.

The issue is this: while it’s true, as we claimed, that a communication complexity lower bound implies an algebraic query complexity lower bound, it’s not true in general that a communication complexity upper bound implies an algebraic query complexity upper bound!  So, from a communication separation between models C and D, we certainly obtain a query complexity problem that’s not in D~A, but then the problem might not be in CA.  What tripped us up was that, in the cases we had in mind (e.g. Disjointness), it’s obvious that the query problem is in CA.  In other cases, however, such as Raz’s separation between quantum and randomized communication complexity, it probably isn’t even true.  In the latter case, to recover the desired conclusion about algebraic query complexity (namely, the existence of an oracle B such that BQPB ⊄ BPP~B), what seems to be needed is to start from a later quantum vs. classical communication complexity separation due to Klartag and Regev, and then convert their communication problem into a query problem using a recent approach by myself and Shalev Ben-David (see Section 4).  Unfortunately, my and Shalev’s approach only tells us nonconstructively that there exists a query problem with the desired separation, with no upper bound on the gate complexity of the quantum algorithm.  So strictly speaking, I still don’t know how to get a separation between the relativized complexity classes BQPB and BPP~B defined in terms of Turing machines.

In any case, I of course should have realized this issue with the algebrization paper the moment Shalev and I encountered the same issue when writing our later paper.  Let me acknowledge Shalev, as well as Robin Kothari, for helping to spur my realization of the issue.

In case it wasn’t clear, the mistakes I’ve detailed here have no effect on the main results of the papers in question (e.g., the existence of an oracle relative to which PP has linear-sized circuits; the existence and pervasiveness of the algebrization barrier).  The effect is entirely on various “bonus” results—results that, because they’re bonus, were gone over much less carefully by authors and reviewers alike.

Nevertheless, I’ve always felt like in science, the louder you are about your own mistakes, the better.  Hence this post.

76 Responses to “Yet more errors in papers”

  1. TrumpSupporter Says:


    Totally off-topic, but I have recently became aware of your Trump hatred and frankly, I am disappointed. In all fairness, at least you gave some reasoned explanations for your opposition to Trump, unlike Terry Tao who, in my opinion, tarnished what was until then an impeccable reputation by coming out against Trump so non-nonsensically.

    Over the years I have read your blog on and off. I think that both you and Terry Tao are vastly overrated. If your respective scholar outputs don’t change much in substance from where they are now, nobody will remember your names 100 years from now, unlike say Andrew Wiles’, Grigori Perelman’s or Donald Knuth’s -to speak of somebody who is a computer scientist.

    I find the new generation of leftist academics very corrosive for both academia and society. Leftists have been a majority in the faculty ranks for many decades now. This is particularly true at top schools. Only, the previous generation was respectful with people who disagreed with them. It was perfectly possible to be a student of one of those left wing academics without being oneself left wing. I don’t know about your current doctoral students’ political views, but I doubt any of them disagrees with your views. Even if they did, they would not say it openly. This is the ultimate self-selection mechanism that has made academia so detached from the real world that few people take it seriously anymore outside the ivory tower.

    I am lucky to have gone to graduate school in a different time. Today, I would strongly recommend anyone who isn’t a left wing zealot against pursuing graduate studies beyond a masters degree. It won’t be a pleasant experience with people like you or Terry Tao calling the shots.

    There are 63 million of us who voted for Trump. Each of us did it for his own reasons. Mine were clear -a once in a generation opportunity to give chemotherapy to the establishment that has ruined the United States since the end of the cold war. The events of the last few weeks have served as confirmation that I did the right thing. We are living under such a nasty deep state that only somebody with the strength of character Trump has is willing to put a fight against it. These weeks have served to confirm that the “military industrial complex” Eisenhower warned against is real. They have also shown that each president since had engaged in a Faustian bargain of shorts with it.

    My only fear is that Trump might get tired of fighting the swamp and will not run for re-election. That would be a tragedy.

    Anyhow. Totally off-topic but I wanted to share my views with you now. Better late than never.

  2. Scott Says:

    TrumpSupporter: The fact that you would lump me with Terry Tao, turns your attempts at insults into high compliments.

    I hope you’re happy with how the national chemotherapy is going—my hair has certainly fallen out, and the suspicion is reaffirmed daily that the oncologist you chose has all the expertise of Dr. Nick from The Simpsons.

  3. Matthew A Broome Says:

    Ahhh, Scott! Thanks for this post, but it should certainly not be labelled under ‘Embarrassing’! The great thing about being a scientist–as you well know–is that it is perfectly OK to make mistakes and get things wrong. This is the way we learn, and it’s a thrill.

  4. Joshua Zelinsky Says:

    @ TrumpSupporter #1

    “Over the years I have read your blog on and off. I think that both you and Terry Tao are vastly overrated. If your respective scholar outputs don’t change much in substance from where they are now, nobody will remember your names 100 years from now, unlike say Andrew Wiles’, Grigori Perelman’s or Donald Knuth’s -to speak of somebody who is a computer scientist. ”

    I’m not a computer scientist by profession so I can’t speak about Scott’s work with that much confidence (although the sheer fact that other mathematicians find Scott’s work interesting does undermine your claim a bit). I am a number theorist and can speak to the claim about Terry Tao. If all Terry Tao had done was his joint papers with Ben Green (including the Green-Tao theorem where he and Ben Green proved that there are arbitrarily long arithmetic progressions of primes) and his name would be remembered. The fact that you’ve made the above claim suggests that you either have no understanding of the mathematics that Tao has done or are so blinded by your political dislike of his non-mathematical opinions that you’ve decided to completely ignore how impressive his math is.

  5. John Sidles Says:

    Lol … the Donald Knuth who Marched for Science? The Terry Tao … who wrote the essay “It ought to be common knowledge that Donald Trump is not fit for the presidency of the United States of America“? Thanks for the reminders, TrumpSupporter!
    The study of errors, slips-of-the-tongue, delusions, and dreams has been crucial to the development of every psychological science.

    Hmmm … mathematics is a psychological science …isn’t it?

    In this regard, Terry Tao concluded his 2009 AAAS induction acceptance speech by asking a great question:

    Academia has not experienced massive change — on the scale of the industrial revolution — since the invention of the printing press. With the advent of the internet — the modern day analogue of the printing press, among other things — could it be revolutionised once again?

    Em-dashes as in the original 🙂 During the eight years since Tao’s 2009 speech, it has become apparent that mathematics is being revolutionized, both by the internet and by advances ins neurocognitive science.

    There exists so much recent scientific literature upon mathematical neurocognition, analyzing this topic from such various perspectives — these perspectives all being entirely legitimate (as it seems to me) — that it is no small task to pull together even the most basic reading list … perhaps in the next few days I will post a draft of one.

    Meanwhile, for the dilection of error-abhorring mathematical formalists, Freek Wiedijk’s (wonderful) web site “Formalizing 100 Theorems” lists precisely one computational-complexity theorem as a good candidate for formal checking, namely the proof that IP=PSPACE.

    Formalizing the proof of IP=PSPACE would make (as it seems to me) a terrifically challenging and interesting student research project, for multiple reasons — of which the (exceedingly remote) likelihood of finding an error is (for me anyway) among the least interesting and important.

    What are some of the most interesting and important reasons to introduce formal methods more widely into mathematics in general, and complexity theory in particular? Answering that question, with an appropriate appreciation of the recent advances in neurocognitive science, is what requires a pretty lengthy reading list! 🙂

  6. TrumpSupporter Says:

    Scott #2:

    Thank you for publishing my comment. I think that it says a lot about you.

    On the chemo session, take it easy. Besides, now that you live in a red state -I know Austin is a blue island in a red ocean- perhaps you should do an effort to venture outside the left wing echo chamber that academia is to understand why so many people in Texas have been voting red for such a long time. I know for a fact, although it is not your case, that Texas has become a refuge of sorts for non-liberals who don’t feel at home anymore in states like California, Washington or Massachusetts. On the insults thing, well not really, you know I am stating a fact that neither your nor Tao’s work at this stage is of the caliber of the aforementioned.

    Joshua Zelinsky #4:

    One of the tragedies of the left wing echo chamber, that Scott is fully aware of after his public confession on the challenges of growing up as a heterosexual shy geek -Scott, many of us identified with it so you know-, is that I have to maintain my identity hidden because I can assure you I don’t want to bring any SJW attention to my generally quiet life.

    So you know I know what I am talking about. First, as you say, the work was join work with Ben Green, not Tao’s sole work. Second, this result is of a lower impact than say proving the twin prime conjecture -a problem that remains open. Yitang Zhang’s work got as closer to the latter result than Tao’s and Tao knows it.

    Both Terry Tao and Scott are token professors of their respective fields (every age has had those). As revered and respected people like these tend to be during their lifetimes, nobody remembers them once they pass away unless they have made significant contributions to their fields. What is that we know today of high significance in number theory or quantum computing that we would not have known if neither Terry Tao or Scott had never been born? Not much really. On the other hand, one can make the claim that if Andrew Wiles had not been born, Fermat’s Last Theorem would still be a conjecture. Ditto of the Poincare conjecture and Perelman. That’s what we are talking about here.

  7. Scott Says:

    TrumpSupporter: I think you’re flat-out wrong about Tao, as Joshua Z. explained. If mathematicians don’t remember him a century from now, I suspect the likeliest reason will be that there are no more mathematicians.

    But as for me, if it even needed to be said explicitly that no one would remember my work a hundred years from now, I’d consider that an amazing compliment, surely a higher one than I deserved! It would be as if a rec letter said: “this candidate is not quite in the same league as Einstein or Darwin.” If that’s criticism, who needs praise? 😉

    I don’t see evidence for your claim that the math and physics communities, at least, have become more intolerantly leftist than they were a few generations ago. For godsakes, a few generations ago a large fraction of these communities would’ve been actual Marxists! And Einstein, Bohr, Grothendieck, Smale … (to pick a few examples) were MUCH more associated with lefty political causes than the vast majority of scientists I know, certainly more so than me or Terry Tao (!).

    (By the standards of the Union half of the United States, my politics are extremely moderate, even to the right on some issues. It’s only when you include the Confederacy in the comparison group that I become a “leftist” by default.)

    Anyway, what’s your evidence for my and Tao’s intolerance: that we engage and argue with people who disagree with us on our personal blogs?

  8. John Sidles Says:

    Lol … the discourse here on Shtetl Optimized strikingly affirms Elif Batuman’s SJ-positive assessment that “Everyone thinks they’re Dumbo“. As Batuman tells it:

    I found myself remembering the day in kindergarten when the teachers showed us Dumbo, and I realized for the first time that all the kids in the class, even the bullies, rooted for Dumbo, against Dumbo’s tormentors.

    Invariably they laughed and cheered, both when Dumbo succeeded and when bad things happened to his enemies.

    But they’re you, I thought to myself. How did they not know? They didn’t know.

    It was astounding, an astounding truth. Everyone thought they were Dumbo.

       — Elif Batuman, The Idiot (2017)

    Keeping Batuman’s Dumbo-principle in mind, an enjoyable, math-centric inherently SJ-positive exercise for Shtetl Optimized readers is this one:

    • List your favorite three 20th century mathematical theorems and/or postulates.

    • Consulting biographies as necessary, assess the SJ-positivity of that mathematician on a scale of 0-10.

    To set the scale, let us specify that the dedicated Nazi mathematician Oswald Teichmüller receives a scale of “0”, whereas the dedicated pacifist Emma Noether receives a “10”.

    Needless to say, SJ-positivity is built-in to the meta-structure of this question. Because as soon as a person — especially a young person — thinks widely about mathematical theorems, and thinks deeply and sympathetically about the lives of the people who proved those theorems, that person already is “swimming left”, Batuman-style … aren’t they?

    As a SJ-positive bonus, it’s pretty clear (to me anyway), that an overwhelming majority of the highest-rank mathematicians of the 20th century have been SJ-positive state-defying gadflies.

    Perhaps that’s because my three spur-of-the-moment choices were Noether’s theorem, KAM theory, and elliptic curve cryptography. Hey, I’m Dumbo too! 🙂

  9. TrumpSupporter Says:

    Scott #7,

    Again, thank you for willing to engage me on the merits.

    Several points:

    – You seem to implicitly concede that your current output doesn’t justify anybody remembering your name 100 years from now so I will make my case against Terry Tao given his current output -as an aside, I truly wish that that I am wrong in my assessment that nobody will remember your names 100 years from now because that would mean that in the next 20-30 years either of you you will have accomplished something worth remembering. The case against Tao is as old as the argument “quantity vs quality”. I do not question that Terry Tao has produced a lot of output that meets the standards used to make decisions about who gets tenure at America’s top universities. He has produced more than the average mathematician produces. A different question is what gets one into the pantheon of mathematics. Paul Erdos produced even more of this kind of output but he is no Gauss or Euler. Speaking of Euler, here is an example of mathematician who produced both quality and quantity. We don’t have anybody these days who comes remotely close to him in that regard. So my contention is that when undergraduates study mathematics 100 years from now, based on the his current output, professors will say “Terry who???” because frankly he hasn’t produced any revolutionary result unlike Wiles or Perelman. Another way you can test my hypothesis is by asking your learned friends outside the world of academic mathematics or academic computer science (say engineers or software developers who are current with academia). Most will know that Andrew Wiles proved Fermat’s Last Theorem. Most will also know that there is a crazy Russian genius who solved a very difficult problem who lives with his mother in Moscow (although maybe they won’t be able to name Perelman or the problem he solved). Fame is ephemeral, that applies to both pop culture as well as academia. Fame and results that stand the test of time are not necessarily correlated. Some times it happens, but in most cases that isn’t the case. Take for example the field of classical music. Bach wasn’t a particularly popular composer in his time (known in some circles but not wildly known; he was better known as a virtuoso organist during his lifetime). Now he is considered the best composer of classical music who ever lived. As some say, Mozart is what you hear when the angels play music with each other, Bach is what you hear when the angels play music to God. You get the idea. I stand by my challenge: what is that Terry Tao has produced of significance that improves our knowledge of mathematics in the way the proof of Fermat’s Last theorem did?

    – With respect to the evidence of intolerance. Let me see. I have seen my share of grad students who live their right wing leanings inside because when your advisor makes it well known who are the good guys and who are evil, then you know it is not wise to talk politics in the office. Let me offer you some anecdotal evidence at the risk of disclosing too much. I was in graduate school at a top notch university (top 10 in the Shanghai ranking) during the G W Bush years. I remember vividly to this day the lead to the 2004 election. One of the professors I was taking a class from back then came to class every day with a Kerry/Edwards sticker from the start of the academic year until election day. The day after the election, we had our regularly scheduled class with him and he opened by saying -I am paraphrasing- “with respect to what happened yesterday, I guess that for democracy to work, people need to be intelligent”. Now, you tell me upon hearing that, which student in his right mind would open up to this professor as being “right wing”.

    – As to what constitutes right wing, the example of the confederacy was cheap, among other reasons because it was the Democratic Party that started the whole American Civil war and that produced every single incarnation of the KKK ever since. Woodrow Wilson -you cannot get more learned than that- was very cozy with them, as you probably know.
    So this qualification aside, let me tell you why I would never vote for the Democratic Party, in its current form, even though I am not 100% on board with everything the Republican Party represents: its radical pro-abortion platform that was in display at the third presidential debate when Hillary Clinton refused to back down on partial birth abortion. The way I see it, the idea behind abortion, that after a human being has been conceived he/she doesn’t have intrinsic value only the value his/her mother assigns to him/her, is pure evil. While I do have religious beliefs, I have reached this conclusion on purely rational grounds, as did the pro-life atheists shown here . So abortion for me is a gateway issue: I cannot actively vote for a party that says that unborn children have no rights whatsoever. We already know the type of society that results from the eugenics ideal: lots of learned people, but also Aktion T4 -more info here . When I get to choose which party I vote for, that issue takes preference. When I do not like the Republican guy, as it happened in 2012 for example, I stay home. Trump’s unapologetic commitment to the prolife cause is another reason I back him -and put differently, his past pro-choice pronouncements were my most important obstacle from getting into the Trump train originally. I am a results oriented person. He has gotten Gorsuch in the Supreme Court and he is giving Planned Parenthood a run for their money. No other more traditionally conservative president would have been able to do this (certainly neither Ronald Reagan nor either of the Bushes did).

    Now tell me, say that I am one of your advisees. Can you honestly say that I would be free to say this to you without having a mark on me in the aftermath? Honestly, given your public statements, I don’t think so. Terry Tao is clear that he doesn’t think anyone in his right mind would vote for Trump.

    Your turn Scott.

  10. Scott Says:

    TrumpSupporter: Most of the public has heard of neither Tao, nor Erdös, nor Wiles, nor Perelman. But I’ll bet that the fraction who have heard of each is pretty similar — can anyone come up with a quick empirical test? Of course, by itself this means nothing: far more people have heard of Kim Kardashian. But there’s more than one way to be a first-rate mathematician. Some invest their whole careers in one problem; others (like Grothendieck) build abstract theory; and still others (like Erdös) pioneer a new and enduring style of mathematics across many thousands of results.

    As it happens, I work closely with a student who’s pro-life—as in, extremely active in the pro-life movement. I co-taught my first course with a far-right Republican. I got along and get along with them fine. Does that surprise you?

    Of course there are limits. If someone supports, say, extreme abortion restrictions (as opposed to what I’d consider the more moderate abortion restrictions enshrined in Roe v Wade), that’s a difference of opinion that can be reasonably discussed (or not discussed, if math is more interesting…).

    But what if a student were, say, a neo-Nazi? Or thought liberals should be roughed up at rallies, and reporters who ask questions about CBO scores should be body-slammed? At a sufficiently extreme ideological distance, you no longer even have reasonable assurance that the person you’re discussing math with won’t pull a gun on you, or try to have you imprisoned or deported from the country. And sadly, the basic norms of civilized society are precisely what Trump has eroded over and over and over, and what many of his supporters are thrilled that he erodes.

    The fact that Democrats used to be segregationists, while Republicans used to be abolitionists, is as relevant for present purposes as the fact that the color red used to represent Communists but now represents Republicans. You and I both know perfectly well that the parts of the US that wanted slavery then (white, rural, southern, low openness) have massive overlap with the parts that want deportations and Muslim bans and imprisonment of protesters now, and that the underlying impulses are substantively the same. Or do you imagine that Jeff Sessions, had he been alive then, would’ve been a warrior for abolition?

  11. John Sidles Says:

    An alt.Trumpian trait that is distressing to academics — who retain long memories of on-campus mass-murders — is the accelerating deployment of out-of-the-box Trump campaign rhetoric in service of justifying homicide.

    Here’s a horrifying recent example from a prominent alt.Trumpian weblog:

    Breivik: saint or monster?

    [Mass-murderer] Anders Breivik did not target innocents. He didn’t attack teenagers at a pop concert or families enjoying a night out on a public promenade. He struck a highly effective blow against the political machine that is still actively engaged in attacking his people and attempting to eradicate them.

    If you don’t believe violence is a legitimate way of resisting invasion, if you don’t think that making war on those making war on you is permissible, that’s your prerogative, but your opinion is both ahistorical and irrelevant.

    The fact is that Anders Breivik not only gave up his freedom to strike back at the quislings who are actively seeking to destroy your nation and your people, but he did so alone, and in the full knowledge that he would be hated for it by many of the very people he sought to save.

    Even more horrifying than today’s increasing prevalence of alt.murderous manifestos — for SJ-positive academicians anyway — are the hundreds of commenters who publicly and enthusiastically support Breivik’s canonization, as a patron-saint of Trumpish alt.Boeotian ideologies, makes horrifying reading.

    These abhorrently pro-Breivik comments call to mind Saunders Mac Lane’s personal account in “Mathematics at Göttingen under the Nazis” (Notices of the AMS, 1995)

    There was a book-burning in Göttingen on May 10, 1933. At about that time the copies of the Literary Digest which my mother sent me were no longer allowed to come. …

    My landlady regularly provided me with evening tea and talk; I discovered that two weeks of propaganda had converted her from mild conservative views to ardent Nazi discipleship.

    Mac Lane’s account vividly shows how small anti-SH acts — such as the the nationalistically alt.Trumpish act of suppressing the SJ-positive Literary Digest — can play a crucial catalytic role in potentiating later, far greater, calamities.

    Is it realistic, to imagine that the academic community can be willfully indifferent to its own, sobering, horrifying, all-too-alt.Trumpian, history?

  12. fred Says:

    Hi Scott,

    from a practical point of view on how to correct information that’s out there…

    You’re doing those corrections in this blog and in the versions of the papers posted on your website.
    Is this enough or do you think there’s a lack of tools to keep papers up to date? Can this be improved?
    For code, there is version control and github, etc.
    For papers, I’m only familiar with arxiv, which has versioning.

  13. TrumpSupporter Says:

    Scott #10

    Somehow related to your comment, I believe that any justification of abortion beyond the life of the mother being in physical danger -since somebody has to die, I am fine to letting the mother make the choice- is evil. With respect to cases of rape, I am all for having easy access to the day after pill but I am not OK with cases of abortion a human beyond that for having been conceived by a rape. And I say this as somebody who is, because of my personal experiences, very sympathetic to women who have suffered the horror of being raped. That’s why I am in favor of the usage of the day after pill in these cases (something that puts me at odds with some prolifers). With this said, I consider Roe v Wade to be the poster example of evil government. Not only because it created a non existing right to abortion, but because it usurped in a very vicious way the role of the US congress as the body that makes law. The US constitution gave the US Supreme Court the role of interpreting the law not of creating a trimester-based abortion law. If I were to point to the very moment in which the US Supreme Court started to become a super-legislature of 9, and therefore become to be seen as a fundamentally political body, that’s Roe v Wade.

    On the other stuff you comment, glad you are open to working with prolife students -as an another aside, I am also very actively involved in legal and peaceful prolife activism- but honestly, how many professors do you know that are this accommodating? Also, I am curious if your very public experience with the SJW movement made you reconsider your willingness to work with students who don’t share your political views.

    With respect to which mathematicians are better known and who deserves to be called a great mathematician, I am definitely in the quality trumps quantity camp. If you take the long view of history, which is what we are talking about here, most historians also do. When you consider the totality of mathematical knowledge, we celebrate Newton and Leibniz for having invented calculus, not Brook Taylor for having invented integration by parts (Taylor is celebrated for his work on the series expansion theorem that bears his name but not as the inventor of a technique to do integration). There are several reasons why the most valuable history is made when the events are distant. The most common reason given out is that there is a better perspective to analyze the impact of something. The reason given less often not no less true, particularly when it comes to analyzing historical figures, is that the people who could lobby for or against said figures are dead, therefore an objective analysis is more feasible.

    I do have my limits when it comes to whom I am willing to engage in a technical discussion, and obviously I wouldn’t entertain a conversation with neo-Nazis, but here is the problem. The moment the left wing echo chamber labels all Trump supporters with being neo-Nazis, that’s the moment something very valuable has been lost (that’s essentially what Terry Tao post in his infamous “proposition” post). If you take a look at the Republican primary, Trump got more votes than any other candidate in the history of the Republican primary. He got a total percentage of the vote similar to Romney and McCain. He over-performed the polls in those states -like New York or Connecticut- where the Republican primary electorate is made of mostly so called “RINO” or “Rockefeller” Republicans. And yet, people like you were not willing to entertain the notion that reasonable people – “reasonable” in your terms- would vote for Trump. And then you are surprised that the only high profile endorsement from the brainy class Trump got was Peter Thiel’s? I can assure you that if I were a billionaire, I wouldn’t mind to be open about my support for Trump. Since the likelihood of me becoming a billionaire equals the probability that the Earth is flat, we’ll have to have these conversations with you being openly anti-Trump and me being secretively pro-Trump.

    Finally, I do not agree with your analysis of the character of the Democrat and Republican parties. Political parties, like other organizations that have survived for a long time -I am thinking about say colleges and corporations- have a unique DNA that is set by their founders. They might change tactics at any given time to adapt to market conditions but their DNA remains intact. The Republican Party is the party of individual freedom and free enterprise that puts individuals ahead of government. The Democratic Party is the plantation party: its aspiration is to use the coercive force of government to tell people how to live their lives, be it as slaves (as in the case of the slavery era), segregated citizens (the Plessy era) or via identity politics (the current era). I am too much of freethinker to follow the dictates of those who think they know better than me, sorry.

  14. fred Says:


    who cares what one’s contributions will be seen 100 years from now?! We’re all gonna be dead by then anyway (unless you worry about your ancestors getting some kind of royalty? :P)

    And it’s not like we get to chose who we are or can see the future – the way science works, you could be the smartest guy in the world but focused on a topic that’s not gonna be relevant 100 years from now anyway.

    Also, every single piece of an engine matters to make the whole thing work.

    So, relax…

  15. TrumpSupporter Says:

    John Sidles #11:

    Seriously, what part of A->B doesn’t mean B->A don’t you understand? This notion that all Trump supporters are Anders Breivik’s fans is the kind of dialogue that makes for a non starter.

    I am curious by the way if you agree with the Hillary Clinton position that a 7, 8 , 9 month old unborn child doesn’t have any constitutional rights (or as Barbara Boxer would say, a child only has human rights the moment he gets out of the hospital). Because honestly, as horrifying as is to kill 70 people in a single day, I am also very appalled by the 50+ million unborn children that have been sacrificed at the altar of choice.

    The following proposition is probably true:

    “The relative and absolute numbers of Anders Breivik’s fans among Trump supporters is very small compared with the relative and absolute numbers of people who think that a 9 month old unborn children has no constitutional rights among Hillary Clinton voters”.

    And yet, we, Trump supporters, are the ones who have to put up with the nonsense.

  16. svat Says:

    Scott, on the topic of the post, thanks for posting this. You’re setting a good example to everyone on admitting mistakes (even though in your case they’re rather minor); the world needs more people to behave this way, when it seems everyone’s going in the opposite direction.

    On the political discussion into which the comment thread has immediately derailed, thanks for your long record of expressing your views, debating with those disagree, and so on. Who cares about who is remembered in 100 years, or about “ratings”? It’s part of being a good person to speak one’s mind and have a clear conscience; it’s great to see you doing that consistently for so many years now.

    If someone were to complain of leftist intolerance in academia then you’re probably among the worst examples one could pick. However, intelligent people, like Jonathan Haidt and the Scott Alexander whose novel you mentioned in the previous post, have painted a picture of intolerance existing in academia and in the elite generally: (just to pick a recent post)

    What are your views on theirs?

  17. TrumpSupporter Says:

    svat #16

    I am familiar with Jonathan Haidt’s work -one of the few high profile academics who openly recognizes that academia has a left wing bias problem-, but it is the first time I see Scott Alexander writing about this topic.

    I generally agree that conservatives are creating their own parallel institutions. I also disagree with this notion that conservatives are running away from “facts” unless your vision of “facts” is,

    – Religious belief vs atheism. None of the two positions can be shown to be true via reason or the scientific method. Yet the only version deemed “polite” in a public discussion in academia is to proclaim one’s atheism.

    – Prolife vs prochoice. Same thing. A very compelling case can be made that human life begins at conception. Yet the “keepers of neutrality” label prolifers as “anti woman bigots”.

    – Gay marriage. If you favor traditional marriage over the recent and invented “gay marriage” you get the Brendan Eich treatment even in the company you yourself started!

    – Global warming. A pointless debate since 15 years of falsifiable predictions of doomsday predictions have failed to materialize the doomsday scenarios. 2016 is over and Manhattan still stands not eaten by the Atlantic ocean. On a more serious note, the models that predict temperature ranges based on inputs like CO2 concentrations have been shown to be inaccurate. In an area like particle physics, the consistent mismatch of measurements with predictions would have made people abandon a theory. In the world of the “neutral gatekeepers” those who point out this mismatch are labelled as “science deniers”.

    I could go on. I agree that overtime this doesn’t bode well for either side. Institutional academia will increasingly become more left wing, but its reports will increasingly be ignored by the public at large. You have to look no further than the Nobel Peace Prize to see what is likely to happen. Up until 1994, this prize had had its share of controversy, but for the most part there was widespread agreement that winners of the prize deserved it. Then Arafat won it, and the prize became a joke. Sure, leftists still hail winning a Nobel Piece Prize as something worth celebrating regardless of who wins it. In the conservative world, the prize is not given automatically any credence. Thus, if Al Gore or Barack Obama win it, it is further proof that the prize is a joke. When Malala Yousafzai wins it, it is hailed as a reminder of what the prize used to be.

    One thing is for certain. History is full of examples that engaging in witch hunting is never a good idea. Currently it is the left that engages in witch hunting. The creation of parallel conservative institutions is nothing but a defensive move by those who have been excluded from the institutions that used to command widespread respect in America.

  18. John Sidles Says:

    TrumpSupporter asserts [bizarrely]  “The only version [of religion] deemed “polite” in a public discussion in academia is to proclaim one’s atheism.”

    Lol … TrumpSupporter, you’re posting these peculiar claims to the wrong weblog … considering that committed Christian Donald Knuth occupies the very highest level of Scott’s computer science pantheon (and mine too).

    Scot has described Knuth as

    Priest of Programming, Titan of Typesetting, Monarch of MMIX, intellectual heir to Turing and von Neumann, greatest living computer scientist by almost-universal assent … alright, you get the idea.

    Wherever you’ve been getting your alt.disinformation about academic norms, TrumpSupporter, perhaps you simply shouldn’t trust that alt.source anymore? Ditto for ceasing to trust the alt.source(s) of your comment’s alt.disinformation regarding climate-science! 🙂

  19. TrumpSupporter Says:

    John Sidles #18

    Donald Knuth is 79 and belongs to a different era. If you had the time to read the above link pointed to by svat, , you’ll see that during the 1990s a process began that transformed academia from being left of center to ultra left.

    The same link points out that “the 12% in the red line for 2014 is mostly made up of professors in schools of engineering and other professional schools; the percent conservative for the major humanities and social science departments is closer to 5%”.

    I wouldn’t consider Donald Knuth’s case as being representative of today’s environment. He got his tenure in the late 1960s. By the late 1970s he was already a computer science giant with little to fear from haters, . Still, he wasn’t very assertive about his faith until the late 1990s .

    So if anything, that you use as support to your position that a computer science giant -one whose name will probably be remembered 100 years from now- was only able to speak openly about his faith in the late 1990s speaks volumes of the situation a regular academic faces these days.

    I stand by what I said earlier: I would not advise anyone who is not a crazy leftist to pursue a graduate education beyond the masters degree at any top university unless that somebody’s passion for learning trumps all the hatred that person is likely to encounter at the PhD level and beyond. In addition, in computer science and mathematics, it is not even necessary these days to attend these institutions to learn high level mathematics for knowledge’s sake. The internet has transformed learning in these fields. A motivated learner is able to learn from the best online since all you need is there. Granted, you don’t have the one on one dialogue, but that didn’t prevent Yitang Zhang from producing high caliber mathematics.

    The barriers of entry for the motivated learned have never been lower. Of course I understand that there is the issue of making a living. Say that you are given a choice between making a big salary at say Microsoft through a software engineering job that leaves you plenty of time to learn new things on your own vs having to go through the pain of having a bunch of leftist judge whether you belong to their tribe during the tenure process. It is not obvious to me many people who are not leftists will pick going through the tenure track, specially since one of the good things American universities have is that if you ever produce anything like what Yitang Zhang produced but at your software engineering job, you won’t have lack of options if you latter want to move to academia. The day to day of the run of the mill professor could not be more different from the day to day of the highly compensated software engineer at a good company, where results are measures by profit not by “the opinion of your peers”.

  20. John Sidles Says:

    GASARCH wept.

    The above is a strictly New Testament \(\otimes\) Computer Science joke. Heck, the distressing prevalence of willful alt.Trumpish ignorance — in respect to pretty much all dimensions of academic life — makes pretty much everyone weep!

  21. Retired Nerd Says:

    Long time lurker, first time poster here. It occurs to me that TrumpSupporter has way, way too much time on his hands.

    And how is the unanswerable question of who and how many people will remember Terrence Tao a hundred years from now at all related to the question of whether Trump is qualified to be president?

    Back to the Quantum Computing?

  22. TrumpSupporter Says:

    John Sidles #20

    Not sure what you are trying to get to. If Donald Knuth, whom I met in person once, felt that he could only be very open about his Christian faith 6 years after retiring from his professorship at Stanford, you have a pretty strong case that being a Christian in academia back then was tough. It is probably tougher these days given the sharp shift to the left experienced in the 1990s. In fact, Knuth kind of admitted himself that much in this Google talk 8 years ago . I don’t remember the exact words but he said something like that when people learned he was a Christian they were kind of shocked because in academia people understood that one could be a good scientist and Jewish -no offense Scott, that’s what he said- but not Christian.

    By the way, I am curious about your opinion of the constitutional rights of 9 month old unborn children. Because if we are going to judge people for having radical, out of the mainstream beliefs, I cannot think of anything more radical than saying that children only get constitutional rights when they get out of the hospital. Please do not hide from this one. We are trying to explore each other’s views.

    Retired Nerd:

    I see your comment waiting moderation- as an aside, I have seen several of these, that show the email address of whoever is submitting the comment. I am not impressed by the accusation that I have “too much time in my hands”. It’s the typical trick of he who has no arguments whatsoever or who is embarrassed that his arguments have been debunked. I might or might not have “too much time in my hands”, that’s none of your business. For all you know, I could be one of those Silicon Valley millionaires who was able to retire from working for others at the age of 35. Please stick to the arguments.

  23. John Sidles Says:

    Trumpsupporter posts “I’m curious about your opinion of the constitutional rights of 9 month old unborn children. Please do not hide from this one.”

    Why don’t we both show Scott the courtesy, of waiting until he posts a lede relating to fetal morality?

    Until then, long may the wonderful, community-building, Shtetl Optimized traditions of ignorance-bashing, error-confessing, and quantum-pondering continue to prosper — traditions that many people (including me) appreciate and deeply cherish. 🙂

  24. TrumpSupporter Says:

    John Sidles #23

    Let’s apply some logic here. You avoiding answering the question altogether means very likely that you don’t think a 9 month old unborn baby has constitutional rights. Why?

    Polls consistently show that the public opposes in large majorities abortions in the third trimester. Here is an example of an article at Slate discussing this . Gallup polls show something similar. So, you’d be in the mainstream if you had said that you think 9 month old unborn babies have constitutional rights. Thus, the only reason I can think of for you to avoid answering the question is that you believe the opposite but you don’t want to admit it publicly.

    Now, why is this relevant? You are the one who brought to this discussion the connection Breivik with some extreme Trump supporters. It is only fair for me to point out that some anti-Trumpers are even more extreme because there have been around 50 million abortions since 1973. According to this around 1% of abortions in the US are late term abortions. That amounts to 500000 (five hundred thousand) unborn children whose lives were terminated when they were 6 months old and older. Again, Breivik killing 70 people is bad, but boy, what about those 500000 unborn children. You don’t feel for them?

    You tell me who has expressed here radical, out of the mainstream beliefs.

    I unequivocally and unambiguously condemn Breivik and the thinking he represents. Can I hear from you that you equally condemn the killing of 6 month (and older) unborn children? Probably not, but then again, in left land, reality is always reversed: the innocent is condemned and the guilty is spared of punishment.

  25. Scott Says:

    svat #16: Thanks so much for the kind words, which mean a lot to me.

    I think that leftist intolerance in academia is a real problem, even a problem that’s affected my life in serious ways. I’d also venture that, in the grand scheme of things, leftist intolerance in academia is less of a problem than the current control of all three branches of the US federal government by illiberal, anti-science authoritarian thugs and their enablers. Indeed, I believe one of the worst effects that leftist intolerance in academia has had, has been to provide unlimited free rhetorical ammunition to the forces that brought Trump to power. And I’m constantly amazed that more people don’t see this.

    Of course, I also think that leftist intolerance is an orders-of-magnitude worse problem in humanities and social sciences than it is in the hard sciences—in the hard sciences, we mostly just deal with the spillover effects. (Most people in hard sciences are rightly horrified about Trump, but half the country is, not to mention the rest of the planet; that hardly makes us “intolerant.”)

    And I think that holding up me and Terry Tao (!!!) as the exemplars of leftist intolerance—me, who’s spent much of his time FIGHTING leftist intolerance, and Terry, who’s spent his time advancing mathematics as much as any human alive—is particularly ironic stupidity.

  26. Scott Says:

    Retired Nerd #21:

      And how is the unanswerable question of who and how many people will remember Terrence Tao a hundred years from now at all related to the question of whether Trump is qualified to be president?

    Allow me to explain. In TrumpSupporter’s strange worldview, Terry Tao, and his totally unmemorable work on compressed sensing, the Navier-Stokes equation, arithmetic combinatorics, pseudorandomness in the primes, etc. etc., exemplify the decadence and corruption of modern academia (!!)—which, in turn, is just one aspect of the decadence and corruption of the entire modern world, which finally is why Trump needed to become president: namely, to take a sledgehammer to everything.

  27. TrumpSupporter Says:

    Scott #25

    I think there is going to be some agreement after all.

    I do agree that leftist intolerance is a much worse problem -as you say orders of magnitude higher- in the humanities and social sciences than in the hard science and engineering. Not only the aforementioned registration studies bear this out, but I can give personal testimony that in the hard science department where I did my PhD good ideas trumped bad ideas regardless of politics. What sets hard science and engineering apart is that there are both more objective ways to determine whether a problem is hard -see the Millennium problem list- and whether the problem has been solved whereas in humanities and social science is all about the opinion of your peers. It is no accident that engineering and professional schools are less affected by the problem of liberal bias. You shun away a brilliant scholar because of political bias in the hard sciences at your own risk.

    That’s not what I am talking about though. I am talking about the run-of-the mill scholars. Not every professor is you or Terry Tao. When rules are bent to accommodate somebody who is an OK scholar but not a superstar -and you know as well as I do that this happens at every level of the ladder everywhere: PhD, post-doc, faculty hiring decisions, even tenure decisions- then political bias -and other bias like the anti-Christian bias mentioned by Knuth-matters. Therefore in the long rung people who don’t have a liberal bent shun academia altogether unless they have a deep desire for becoming a professor. The problem here is that sooner or later people do what people do: get married, have families of their own, etc. At that point, putting up with the bias at the expense of one’s family well-being is heroic and not everybody has the stamina to do it.

    I re-iterate that I support you in the ordeal you went through with your SJW encounter. What you said is what every shy, male, heterosexual geek I know has experienced, including the thought that one would be better off being gay than going through that.

    On the disagreement part. Reasonable people can disagree -something that you concede but that Terry Tao doesn’t. We’ll have to agree to disagree on Trump. I have never felt more empowered personally since I became politically aware -until my late 20s I thought politics was for losers. Trump doesn’t prevent me from applying myself to my work and personal interests -that include computer science and mathematics. What Trump does, that Obama didn’t do and that Hillary Clinton wouldn’t have done either, is to remind people that the United States is the greatest country on Earth and that there is no other country I rather be a citizen of. I don’t believe in the internationalist idea that all countries are created equal. They are not. Just look at the UN committees on human rights to see what a joke looks like. Really, Iran lecturing the world on human rights? Give me a break.

    It has been a pleasure to discuss these matters with you and I wish you the very best going forward both on the personal as well as professional levels.

  28. TrumpSupporter Says:

    Scott #26

    I thought my other comment was going to be the last one in the thread but I have something to add,

    “Terry Tao, and his totally unmemorable work on compressed sensing, the Navier-Stokes equation, arithmetic combinatorics, pseudorandomness in the primes, etc. etc., exemplify the decadence and corruption of modern academia”

    They do actually. Compressed sensing for example was over-hyped among other reasons because Terry Tao co-wrote one of the seminal papers in the field, particularly after Terry Tao won the Fields Medal. A decade later, compressed sensing remains a curiosity that hasn’t found widespread usage because it is not a universal technique and it is very hard to implement in those applications where it is appropriate. Most practical sampling these days is done still via the Shannon theorem. If nothing dramatically changes in the long term, 100 years from now, compressed sensing will be a a footnote in the history of sampling.

    His work in Navier-Stokes, same thing. As shown with the work of Grigori Perelman solving the Poincare conjecture, history remembers him, not Richard Hamilton’s work on the Ricci flow that was instrumental for Perelman.

    I could go on, but you get the idea. I stand by what I said. I hope I am wrong, because that would mean that you guys have done something remarkable, but nobody will remember Terry Tao 100 years from now if his output doesn’t change significantly from its current form.

  29. Scott Says:

    TrumpSupporter, a few last responses:

    1. So by your own testimony, you completed a PhD in a hard science subject, and leftist intolerance presented no obstacle to doing so. If so, doesn’t that somewhat undercut your central claim?

    2. Nevertheless, you claim that your right-wing opinions mean that you can only comment here under cloak of anonymity. Of course, this is also extremely convenient for you; it allows you to sit in judgment over Paul Erdös and Terry Tao (“no one will remember their work a hundred years from now”), while leaving the reader to wonder whether you’ve done anything that anyone will remember next week.

    For better or worse, I decided a long time ago that, insofar as it’s humanly possible for me, I will not hold any opinions in private if I’m unwilling to defend them in public under my real name. I’ll let others judge how well I’ve lived up to that commitment.

    3. No, it wasn’t my own brush with SJWs that taught me that I should freely discuss science even with people of differing political opinions. Basic human decency, together with passion for science itself, led me to that conclusion a long time ago.

    4. Your idea that the Republicans are the “individual freedom party,” while Democrats are the “Plantation Party,” is a neat way to resolve the cognitive dissonance you must feel in supporting the clearly-identifiable ideological faction in the US that’s been on the wrong side of history over and over and over, since the Civil War and earlier. But I have one question: wouldn’t your theory, if true, predict that today it would be liberal Democrats rioting and lighting torches in support of Confederate flags and monuments to Robert E. Lee, while Republicans would cheer when the symbols glorifying slavery came down? So then why do we observe the exact opposite?

  30. Raoul Says:


    You sound like a pretty knowledgeable guy. That must make it pretty lonely in Trump supporter circles.

    I propose a $1 bet: I bet there is not a SINGLE Trump supporter in the world who can solve a cubic algebraic equation by hand, without looking it up.

  31. Raoul Ohio Says:

    EDITOR NOTE: 1. Please make previous comment: Raoul Ohio.
    2. I still see someone’s email in my submit comment window.

  32. TrumpSupporter Says:

    Scott #29

    I expected not to be back, but since you have questions, here come my answers:

    1- Not really. The fact is I left academia after my PhD to get an engineering job. I would say that the bias was only one reason among others I considered to leave academia for good. This type of decisions are never motivated by a single factor. I will say this: I do not regret my decision of doing the PhD since those years were the peak of my intellectual life in hard science but I do not regret either the decision to move on from academia. During those years I also did some scouting in the humanities and social sciences -attending talks, talking to professors- and that’s how I realized that the left wing bias problem in academia was way worse in the humanities. Also, even in my current job I cannot be openly pro-Trump. However, because my job security is linked to me producing profitable products vs some committee’s opinion, I feel more secure in the sense that I have more control over my own destiny than in academia. The free market is a great thing.

    2- Oh, I have no problem admitting that my own PhD research work is already forgotten. It was probably forgotten by the time I got my diploma in my hands. Same thing with the stuff I am currently doing. We get paid for doing it, the company makes a buck, other people make a buck (sales, marketing, support, etc) and that’s about it. My reason for saying things anonymously has a lot to do with my understanding of human nature. It is not that I am ashamed of my views -in fact my close friends are well aware of them- or that I wouldn’t be able to handle left wing arguments openly. I have these discussions with leftist friends of mine all the time. The problem is that I know a thing or too about mobs, like you learned during your own SJW ordeal. It would be malpractice to expose myself to that kind of thing. I enjoy my quiet life. I respect your decision of expressing yourself openly but different people can have different opinions about this. Anonymous political commentary is as old as the American republic itself.

    3- Good for you. As I said, that makes you better than other people I know in academia.

    4- No cognitive dissonance. I am stating facts that are obvious to anyone who has had a genuine interest in understanding the motivations and behavior of both parties throughout the respective histories. The only political party to have ever endorsed slavery and racism in its platform was the Democratic Party. The Civil Rights Act of 1964, that ended legalized racism in the US for good, was passed because a majority of Republicans were able to break a Democratic driven filibuster . As to what my theory would predict, I don’t agree with your statement. My theory would predict that on any given issue where an individual freedom clashes with the desires of statists to put their own interests above that freedom, Republicans will side with individual freedom while Democrats with statists. That’s why when slavery and racism were popular in the South, the Democrats were pro-slavery and pro-racism. Or why former KKK member Robert Byrd was never repudiated by the Democratic Party since he was always on board with statist positions. As to your observation; again, liberal democrats have reached the conclusion that today is political suicide to defend racism so, as a way to over-compensate for the past failings on this regard, they are loud voices on a non-issue. It’s a non-issue in mainstream America because mainstream America is not racist today. On the policy arena, racism was a hot topic in the 1960s, not today. Back then, the Democrats were on the pro racist side. It always amazes me that liberals are extraordinary at making noise in the present about past battles while they remain extraordinarily silent on the most pressing current moral issues. I still haven’t heard from you or John Sidles about the issue of aborting second and third trimester unborn babies. As Christopher Hitchens said, anybody who has ever watched a sonogram of a few weeks old unborn child knows that what is growing inside the mother womb is a new human life. Further, advances in the care of premature babies makes it possible for around one third of babies born at 22 weeks and older to survive -there was a study published in the New England Journal of Medicine about this two years ago Abortion terminates that new human life in the most gruesome way possible. To add insult to injury, companies then sell the body parts that practice generates to the highest bidder. That to me is a moral question of the first order in this day and age. Yet you want me to spend my time with what the liberal democrats who were racist in the 1960s think today about racism. I just don’t get it. I also remind you that social science is not like mathematics or physics. Theories in social science cannot be falsified in the way say the prediction that Manhattan would be eaten by the Atlantic ocean by 2016 can. Guess which of these two theories was falsified on December 31st 2016, mine or the doomsday prediction about Manhattan.

  33. John Sidles Says:

    TrumpSupporter (circa #28) asserts [entirely incorrectly] “Compressed sensing remains a curiosity that hasn’t found widespread usage because it is not a universal technique and it is very hard to implement in those applications where it is appropriate.”

    TrumpSupporter, your assessment might be improved by the references and worked examples of our own survey “Practical recipes for the model order reduction, dynamical simulation, and compressive sampling of large-scale open quantum systems” (2009, PDF on-line here); see in particular Section 4.6 “Quantum state reconstruction from sparse random projections”.

    Rather slowly, we’re rewriting this entire body of work in the “Yellow Book” language of algebraic geometry and geometric quantum mechanics. Indeed, for us apostates from “The Church of the Larger Hilbert Space”, there are good reasons to foresee that, in coming decades, the entire undergraduate quantum mechanics curriculum will convert to once-heretical, but now ultraorthodox “Congregation of Manifold Varieties”.

    A pedagogically pragmatic reason for this conversion, is to assist undergraduate students in achieving, as early and painlessly as feasible, sufficient mathematical maturity to participate in the accelerating varietal STEAM-revolution that is generating both “God-like” go-playing capacities, and efficient large-dimension quantum simulations.

    Nowadays, more-and-more researchers regard it as trivially obvious that projective varieties (aka tensor networks) inherit a natural Kählerian triple of symplectic, metric, and complex structures — which are precisely the structures one needs to unravel Hamiltonian quantum dynamics (via symplectic structures), with Lindbladian quantum informatics (via metric structures), by algebraically efficient algorithms (via complex structures).

    Even here on Shtetl Optimized, once-common Kähler-hostile jokes are heard chiefly from that ever-shrinking cohort of researchers who are not reading Yellow Books — a cohort that nowadays includes pretty much nobody … to the great benefit and dilection of contemporary quantum-physics students! 🙂

    After all, what is a wave function, but a machine for answering questions about operator expectation-values? And what is AlphaGo, but a machine for answering questions about go move-values? Why should these machines be computationally different? Why shouldn’t we joyfully exploit their natural Kählerian endowments?

    Through the lenses of computer languages like Google’s TensorFlow, and quantum-state descriptors like tensor-network states (aka projective varieties), we perceive that these the two superficially very different question-answering machines — quantum wave-functions and AI go-players  — in actuality differ mathematically only in inessential details.

  34. Joshua Zelinsky Says:

    Raoul/Raoul Ohio #31

    I’m not sure that’s a really great test. I suspect that many mathematicians would have trouble doing that without having a very large amount of time.

    Also, as a pure anecdote matter, I know at least two mathematicians who openly supported Trump (although in that context that undermines other aspects of TrumpSupporter’s argument.)

  35. TrumpSupporter Says:

    Raoul Ohio #30

    I don’t know. I only know how to solve algebraic quadratic equations. But that is besides the point. I still have to meet a single Hillary Clinton supporter, irrespective of their education attainment, that understands that any entity that spends more money than it takes in over a long period of time has only one way to go: bankrupt. That’s even more basic than your algebraic cubic equation challenge since it is basic arithmetic with integers.

    John Sidles #33

    At the risk of giving out more than I want, I understand where you are coming from, but I don’t think you see your own comment is self-refuting. The paper you refer to is from 2009 (8 years ago) when the compressed sensing fever was at its peak. I don’t see things have changed much since.

    If you take a look at the theory that is taught in those departments that teach practical sampling to build real life analog to digital converters, you have this , ie, still Shannon theory.

    The reason I can make a bet that compressed sensing will be seen as a mere historical footnote is because the theories and techniques that stand the test of time (such as Shannon’s information theory or Shannon/Nyquist sampling theorem) tend to have in common both simplicity and universality in the domain of application (some argue that both are related but to me they are two different dimensions). Compressed sensing has neither. It requires you to do a-priory modelling of the signal to sample as “sparse in some known basis”, then it requires you to do random projections (without which the L-1 optimization used to recover the signal won’t succeed). None of this is simple nor general. In addition you have the whole L-1 recovery process that has its own problems because L-1 efficient optimization itself relies in tricks more than general algorithms.

    Compare that with enforcing that a signal belongs to the Hilbert space of band limited signals. Whether you are talking about the time or spacial domains, anti-aliasing filters are well known creatures. Then reconstruction of signals from their samples is similarly done via reconstruction analog filters that are also well known.

    For compressed sensing/sparse modeling to replace Shannon sampling/Fourier analysis it would need to be able to do everything we can do today with the latter, at least as good and at least as easily. And even then, it would be tough. Typically paradigm shifts (say from centralized computing to client/server to distributed computing) require new problems for which the old techniques do not work. We are not there.

    Similarly, I believe that the whole AI/Machine Learning thing is over-hyped. Neural networks are a 1950s technology. Granted, we have more data and more powerful computers, but still old ways of thinking that result in the same old problems . I am of the Roger Penrose school of thinking on this one.

    You probably also get one of the reasons I left academia: the pressure to publish over hyped ideas that most reasonable people consider baloney or else! In physics this has led to the discussion, in the context of string theory, of whether empirical verification of physics theories is even necessary. Take that!

    Anyhow John: still waiting to hear your take on the constitutional rights of 9 month old unborn children. I am more interested in your perspective on this matter.

  36. John Sidles Says:

    TrumpSupporter (circa #35), you are perhaps correct that the compressed sampling literature can seem uninspiring, when it is assessed as the last word upon a narrowly circumscribed mathematical topic.

    Conversely, however, if we appreciate the compressed sensing literature as one of the first mathematical words upon an exceedingly open-ended topic — that topic being varietal dynamics broadly conceived — then compressed sample has been, and continues to be, a marvelously exciting topic.

    Much depends on how one regards this subject … to me, Terry Tao’s “first word” perspective seems like much more fun.

  37. TrumpSupporter Says:

    John Sidles #36

    Let me offer you a third option: a clear case of over-fitting, ie, of seeing a pattern where there is none.

    The older -and hopefully wiser- I become the more I have reached the conclusion that most problems in life are caused by the general public’s lack of deep understanding of mathematics. Seriously. By “deep” I mean the ability to interiorize its meaning and implications. Richard Hamming defined mathematics as the language of clear thinking. I guess another way of saying the same thing is that most people don’t think clearly all the time -I myself am guilty of this more than I would like to admit. This probably explains why I haven’t met a single Hillary Clinton supporter who applies basic arithmetic to the problem of the explosion of the national debt.

    Among the worst offenses I see people commit on a daily basis is over-fitting. Human beings are constantly looking for meaning in their lives and situations. There are very few true, convinced nihilists. Those that I know lead very unhappy lives (like Nietzsche or Jean Paul Sartre did). So it is not so much the search for meaning what I am talking about rather, the finding of too much meaning on random noise. That’s how I see what you describe here.

    Finally, still, I would love to hear your perspective on the constitutional rights of 9 month old unborn children.

  38. Daniel Seita Says:

    Trump Supporter #35:

    But that is besides the point. I still have to meet a single Hillary Clinton supporter, irrespective of their education attainment, that understands that any entity that spends more money than it takes in over a long period of time has only one way to go: bankrupt. That’s even more basic than your algebraic cubic equation challenge since it is basic arithmetic with integers.

    Perhaps I can serve as a counterexample to your point. I supported Clinton and I’m concerned about our government’s finances and debt.

  39. TrumpSupporter Says:

    Daniel Seita #38

    Every rule has its exception. However, it is not my experience that the average Democratic voter understands this. Every attempt I have had at having a reasonable discussion with these about the perils of doubling the national debt every 8 years (as it happened with G W Bush and Obama) or that social security and medicaid are in a path of insolvency end up with me being accused of not caring for the poor, not caring for the elderly, etc. Worse yet, I still hear from time to time that if only we were to reduce our military spending the money we spend in defense could be spent in the social programs and that by doing so there wouldn’t be any fiscal woos. Few people bother to see that 65% of the federal expenditures are in the form of entitlement spending ($2.45 trillion out of $3.8 trillion in 2015 and that the military is not the biggest budget item.

    One of the reasons many Republican voters hated G W Bush was his “increasing the debt as if there is no tomorrow” mindset. In all fairness, the last president to balance the budget was Bill Clinton, but there was no indication that Hillary Clinton intended to slow down the increase of the debt since she promised to increase entitlement spending.

  40. Patrick Says:


    I am not sure this is a good use of my time, but for some reason I feel compelled to respond to a number of your comments which strike me as particularly unreasonable. My response became fairly long, so I will split it into several comments, each dealing with a different topic.

    You have made a number of statements about the relative worth of mathematicians that I think you have not adequately defended. What criterion are you using to judge the work of mathematicians? The main criterion you have mentioned is “being famous 100 years from now.” Since you have apparently not clarified what this means, I have to ask: famous among what group of people? I would not be surprised if Terry Tao is not famous among laymen 100 years from now but I would be surprised if he is not famous among researchers in analytic number theory, additive combinatorics, PDE, etc.

    I don’t think it is very worthwhile to judge mathematicians by their fame among the general public because that sort of fame tends to reward weird personalities, solving problems that are easy-to-state and so on. In fact, two of the examples of great mathematicians that you provide are good examples of this. Does anyone doubt that Perelman would be less famous if he hadn’t turned down the Fields medal and Millenium prize? Or that Wiles would be less famous if Fermat’s Last Theorem was harder to state or had a less amusing history? And I am no expert in number theory, but my understanding is that Wiles is most renowned among number theorists for proving the Modularity Theorem, which by work of Ribet implies Fermat’s Last Theorem. In fact, if history had happened a little differently and Ribet’s work had come after Wiles, would you now be lauding Ribet as a great mathematician for proving Fermat’s Last Theorem?

    You also mention that Perelman will be better remembered than Richard Hamilton. Again, I have to ask: better remembered by who? Among the general public I am sure that Perelman will be better remembered because the story of a reclusive Russian genius who solved a famous problem and then turned down a million dollars is more entertaining than the story of a great differential geometer who invented some essential techniques in the field. And though I am certainly not an expert in differential geometry, I think many geometers would tell you that Bill Thurston is more important than both Hamilton and Perelman.

    You also asked for examples of things that Terry Tao did that “improves our knowledge of mathematics in the way the proof of Fermat’s Last theorem did.” Once again, what criterion are you using to judge “improves our knowledge of mathematics”? I think many people believe that Tao’s work in analytic number theory and PDEs has significantly improved our knowledge of mathematics. Maybe some of his theorems are harder to state to a non-expert than Fermat’s Last Theorem and in his PDE work there isn’t one theorem to single out as the most important like with Wiles and the Modularity theorem, but most mathematicians would agree that these things are not a good measure of their importance to mathematics. And let me ask you: do you think Wiles is more important than Tao because you are knowledgeable in both algebraic number theory and PDEs and decided that the Modularity theorem is more important than Tao’s work? And if not, how did you come to that conclusion strongly enough to disagree with experts who claim that Tao’s work is significant?

    I also want to ask why you brought up this issue in the first place. In your first comment you said “Over the years I have read your blog on and off. I think that both you and Terry Tao are vastly overrated.” I also have to ask here: who are the people overrating Scott? I have never heard anyone say that Scott will be remembered in 100 years or that he is as important in computer science as Donald Knuth or anything like that. But the truth is that almost nobody will be remembered in 100 years and almost nobody is as important as Donald Knuth. People do claim that Scott is a good researcher doing good and useful work. Do you disagree with that? If so, why?

    Also, mathematics is not a contest. There is enough room for both Wiles and Knuth and Tao and Aaronson. Whether or not Tao and Aaronson are overrated should not have anything to do with their freedom to express their political views on their personal blogs. After all, neither of them made any claims about their status as a mathematician when stating their political views so your remarks sound a lot like an ad hominem attack against those whose views you disagree with.

  41. Patrick Says:


    At one point you claimed that Scott and Terry Tao might discriminate against students based on their political views. In particular you said “It won’t be a pleasant experience with people like you or Terry Tao calling the shots.” Later Scott said that one of his students is strongly pro-life. Did that cause you to reevaluate your opinions at all?

    More broadly, do you have good evidence for your claim that it has gotten harder to be a conservative student? For instance, do you know of anyone being punished in some way because of their political views? By the way, I don’t mean just examples of people who felt uncomfortable to speak about their views. I think people being uncomfortable to express their opinions is bad, but it does not always imply that their is a huge risk to you because you hold those views.

    You also claimed that “Leftists have been a majority in the faculty ranks for many decades now. This is particularly true at top schools.” I wouldn’t be surprised if this is true, and I am certain it is true in the humanities. But do you actually have clear evidence that this is true in, say, engineering? Also, you seem to be implicitly assuming that it is bad if the majority of people in academia agree on a certain issue. I think that most faculty members (though not all), particularly at top schools, believe in evolution. However, 40% of Americans believe that God created humans in the last 10,000 years. Do you think that this is a problem and that it would be better if 40% of academics believed this as well? If not, why is this situation different from things like global warming, religion vs. atheism, etc?

  42. Timothy Gowers Says:

    At the risk of writing something relevant to your post, let me say that I hope that you will soon (if you haven’t already) post updated versions of the papers in question to arXiv, if for no other reason than to make a mockery of the idea that there is something super-important about having a “version of record” published in a traditional journal.

  43. Ian Says:

    Feeling sad, a simple comment of mine making a Dr. Nick reference seems to be stuck in moderation? Hope it wasn’t too out of line, but I was picking up on your cancer / Dr. Nick post.

  44. Raoul Ohio Says:

    Joshua Zelinsky #31:

    Good point. I came up with that off the top of my head as something slightly beyond what one would get in classes.

    1. After learning how completing the square leads to the quadratic formula (about 1970 or so), I tried to figure out the cubic formula. I gave up and looked for hints in a math handbook. They gave two answers. One involving a trig identity, and one “easy once you know it”. Try it yourself: Given

    Aw^3 + Bw^2 + Cw + D = 0,

    1.1 Use a division, a translation, and a scaling to eliminate three parameters and put it in a normal form. I suggest

    x^3 + 3Fx – 2 = 0.

    1.2. Substitute x = y + z, separate into two equations, and finish it up.

    2. As pointed out in Cornelius Lanczos’ “Applied Analysis” (still in print and must reading after 60 years), it is easy to solve the quartic equation if you know the cubic. Try that next, very easy. BTW, Lanczos was one of Einstein’s few pals, and wrote the famous letter to Roosevelt that Al signed.

    3. If you really have a lot of coffee, try to solve the quintic equation in terms of elliptic functions. That is hard. I think Mathematica can do it.

  45. Scott Says:

    Timothy #42: Thanks for the on-topic comment! 🙂

    Alas, for these particular decade-old papers, I’m not even 100% sure that I still have the TeX files, or if I do, then all the auxiliary files needed to recompile them…

  46. TrumpSupporter Says:

    I’ll make two replies too.

    Patrick #40

    You ask very good questions and believe they are fair questions. Let me see if I can clarify a bit where I am coming from.

    First, let me say that every time we talk about “better/worse” we are making subjective value judgements that are by definition, biased. I am not taking the post-modern position that objective truth doesn’t exist, rather, that by its very nature, speaking of “better and worse” is subjective. Let me illustrate this with an example from physics. Which one was a better physicist, Newton or Einstein? I find this question very hard answer because while it is obvious that Einstein’s modelling of the physical world as a space time continuum governed by special relativity has been able to produce better predictions -latest, and counting, is the gravitational waves measured by the LIGO experiment- it is no less true that Newton took the world of physics in an entirely new direction by mathematizing the physical world in such a way, that to this day, aeronautical and mechanical engineers still use Newton’s equations for their work when they design planes and cars. Still, I find the question about who is a better physicist Newton/Einstein or Stephen Hawking very easy to answer: the first two are better than the latter by a yuuuge margin. That’s why I make a reference to “100 years from now” or if you will “200 years from now”. I am using time as a measure of which results impact a discipline in such a way that the echos of said results are still felt after a long period of time when the humans that could argue one way or another due to their proximity to the people who came up with the results are no longer here to judge. The judge is the development of the discipline over time.

    This is perfectly compatible with saying, that under the standards used to measure excellence in research in 2016 in computer science and mathematics, both Scott and Terry Tao are stellar figures in their respective fields but none of them has produced the type of results that, in my own subjective judgement, will stand the test of time. Nobody remembers the names of Putnam fellows, even though becoming one is an extraordinary achievement in the context of undergraduate mathematics. We remember Richard Feynman not because he was a Putnam fellow or because he was a Nobel Prize winner, but mostly because of his work on Quantum Electrodynamics -he also had a colorful personality, but not every colorful personality in science is remembered.

    I guess that what I am trying to get at is this: correlation doesn’t imply causation. In the long view of history, we remember those scientists that have made such contributions to the field that their impact lasts generations. Along the way these people usually -but not always- win contemporary awards and prizes. All I am saying is that somebody having won awards and prizes is not the same as having made a dramatic contribution to the field and that the latter is what I use to make my value judgement as to who is the better scientist.

    Along these lines, I can affirm that the work of both Andrew Wiles and Perelman is of higher caliber than either Scott or Terry Tao in the latter two’s current form. I find it interesting that you try to minimize what Andrew Wiles actual contribution was since he made it very clear, in the Nova documentary “The Proof” that he was obsessed with Fermat’s Last Theorem since he was a child.

    Let me give you another example. Another of my mathematical heroes of all time is Kurt Godel. He didn’t win any major mathematical award -though he was eligible for winning the Fields Medal in 1936 in the first edition- but boy did his work have impact.

    Again, this is my totally subjective way of evaluating things. Different people can have different opinions about this, but I think it is very hard to argue against the long view as the better way to evaluate high caliber mathematical work.

  47. TrumpSupporter Says:

    Patrick #41

    “Later Scott said that one of his students is strongly pro-life. Did that cause you to reevaluate your opinions at all?”

    Not really. It says a lot positive things about Scott, but if I were a right wing student at UT Texas, Scott’s public expression of his left wing views would be a factor in me attempting to join his team or collaborate with him. I have been in situations where the boss was outspoken when it came to politics all while claiming to be “neutral” and in my experience the neutrality ends when you need your boss to stand up for you. I don’t know Scott personally and I don’t know for example how Scott would handle a situation in which his pro-life student were to get into trouble -through no fault of his own- in a demonstration against Planned Parenthood for example. The typical reaction in these cases that I have seen is to throw the student under the bus. This is why I found Terry Tao’s Trump post so disturbing. Up until that point I had no clue where his political sympathies lied. Given the stats in academia, his were likely to be liberal, but nobody could tell from his blog. Then the first time he becomes open about it, he eviscerates a political leader seen by many people, like yours truly, as the best thing that ever happened to the US since Reagan. I won’t end up well. Will it cause Terry Tao loss of fame? Probably no. Will it cause people with right wing sympathies to think twice before saying certain things to hum. Probably yes.

    “More broadly, do you have good evidence for your claim that it has gotten harder to be a conservative student? ”

    “You also claimed that “Leftists have been a majority in the faculty ranks for many decades now. This is particularly true at top schools.” I wouldn’t be surprised if this is true, and I am certain it is true in the humanities. But do you actually have clear evidence that this is true in, say, engineering?”

    The aforementioned contains stats for engineering and professional schools too.

    “Also, you seem to be implicitly assuming that it is bad if the majority of people in academia agree on a certain issue. ”

    Not necessarily, except if it is used as a non-nonsensical litmus test of sorts.

    Take for your example of evolution. I am still amazed to this day, that it has become to represent a litmus test as to who is pro-science and who isn’t. The theory of evolution in its current form is not falsifiable in the way say Maxwell’s equations or General Relativity are. I don’t know if it is true or not, but I don’t lose my sleep over it. On the other hand, say that the Newtonian approximation of general relativity were to be false. I would be scared of getting into a plane or taking my car every day knowing that at any given moment the acceleration due to gravity could deviate from the 9.8 m/s^2 value that every engineer assumes when they design planes and cars. I am not saying that evolution is false, I am saying that it is too much ado about nothing. Take a related issue -which is independent of it- the age of the universe. I do believe that it is around 14 billion years old, but I also think that the evidence supporting this contention is of a different order of magnitude than the argument for or against Darwinian evolution. And yet, for some reason, I have never heard one of these recalcitrant atheists in academia, particularly in the humanities, to start the conversation about his is anti-science: do you believe in the second law of thermodynamics is true?

    The obsession with evolution is a manifestation of another phenomenon I have observed throughout my life: when it comes to making the most important decisions about something, it is usually the least competent people who make them. The witch hunt on who is anti-science is led by humanities and social science academics using evolution as the litmus test, not by physicists using Newton’s laws, even though the latter have much more impact in the daily life of people than the former. Go figure!

  48. Scott Says:

    Ian #43: Sorry about that! I was totally confused by your comment, which just said “Hi, everybody!” I missed the Dr. Nick reference, and assumed you were testing the commenting system (given the recent problems we’ve had). I’d approve your comment now, except that it would completely mess up the comment numbering.

    The lesson that voice intonation is missing from blog discussions, and that therefore a joke that’s obvious to you might be lost on your readers, is one that I’ve often had to learn the hard way.

  49. Douglas Knight Says:

    Scott, the arxiv saves the tex source. Click on “other formats.”

  50. John Sidles Says:

    The tensor-product of Scott’s essay (on errors in research) and Tim Gowers’ comment (circa #42) affords an opportunity to praise an innocuously heretical remark that appeared, in-passing, on Tim Gower’s own recent weblog essay “Intransitive dice II

    I’ll start by collecting a few thoughts that have already been made in comments. And I’ll start that with some definitions. First of all, I’m going to change the definition of a die [emphasis added by me] …

    Here we see error-correcting cognition of a form that is central and crucial to progress in every STEAM enterprise — evolving the questions that are asked.

    Rather than post a long comment on definitional error-correction, I will recommend to Shtetl Optimized readers two recent STEAM-works that exercise definitional error-correction to wonderful effect (as it seems to me).

    The first work is a thoughtful essay from Google’s DeepMind research group — which is the research group that designed AlphaGo (2016) and AlphaGo Magister (2017) — titled “AlphaGo’s next move“.

    The second work — which was highly recommended by Lance Fortnow and Bill GASARCH’s weblog — is the hilariously expressed, wonderfully wise, YouTube video Vivekanand Vimal, PhD’17, speaks at Brandeis Commencement (note in particular that newly-graduated neuroscientist Vimal reaches out respectfully and generously to TrumpSupporter’s cohort).

    In aggregate, Scott’s original post, Tim Gowers’s Polymath essay, the Google/DeepMind/AlphaGo essay, and Vivekanand Vimal’s neuroscientific commencement address, each illuminate — in mutually supporting ways (as it seems to me) — the universal STEAM-maxim “make mistakes as fast as you can.”

    So, let’s all get busy! 🙂

  51. Jr Says:

    TrumpSupporter #27,

    I live in a European country where most of the population is convinced it is the greatest country in the world (you would not like it. Partial birth abortions are illegal but taxes are skyhigh) but where most don’t feel the need the demonstrate it in the American way. If you are really confident about how great your country is there is no need to keep repeating it. From that perspective Obama and Clinton seem over the top patriotic so it is amusing that you think they are unpatriotic.

    Also, to be correct, what Trump promised was to “Make America Great Again”. I have two points to make regarding that. For one thing it rather implies that America is not great at the moment, or at least it wasn’t until Trump was elected. Some people seem to portray any left-wing criticism of America as unpatriotic but think it is totally fine for right-wingers to come with harsh criticism. (If you really hate abortion so much, should you not view Ireland, where abortion is almost totally illegal, as a greater country than the US?)

    The second point I wish to make is that “Make America Great Again” suggests America used to be great. No, I don’t know if Trump has said at exactly which date America was great but do you, TrumpSupporter, accept that a black person or gay person might wonder if the time Trump is nostalgically looking back to might not be perceived as less great by them? Do you accept that America has never really been great for African-Americans and that for all we know Trump is pining for the 1950 or 60ies, which were great in some ways, but when black people were also actively oppressed?

  52. TrumpSupporter Says:

    Jr #51

    It would help to name the country you are talking about. It seems to me the only country in Europe that bans abortion entirely is Malta which at the same is regularly accused of being a tax haven.

    I know my share of Europeans, some of my closest friends are. Their very human tendency is to try to understand the politics of the United States under the prism of European politics. It is never a good idea to try to understand one complex society using the parameters that work to understand another.

    So you understand the difference. The United States was founded by people who were running away from political and religious prosecution in Europe. The US constitution was designed to be the “anti-Europe” of the XVIII’th century: Europe was totalitarian then, and each kingdom/principality had its own official religion imposed on its subjects. Thus, the US was founded of the idea that “all men are created equal” and the Constitution protected a set of individual freedoms early on -chief among them are freedom of religion and freedom of speech. Most Europeans I have talked to about these issues are amazed that the United States is such a hot bed for religious cults and by the vitriol they perceive in the political discourse. Both are as old as the American republic. Throughout the years, there have been several waves of immigration to the United States. In each case, the people who came via these waves, and particularly their descendants, have made theirs the American ideal. so in continues to live on.

    On the other hand, the people who live in Europe today have the DNA of the people that thought that putting up with a totalitarian ruler was a better proposition than risking leaving it behind to find who knows what in America. Thus, the political cultures of the United States and the different European countries could not be more different. In general Americans do not have the faith in rulers (ie, government) that the average European has. The average American doesn’t look up to government to solve his problems, rather, expects government to get out of the way and focus on the things it was setup up for: the military, administration of justice and protection of basic individual rights such as private property.

    This is not to say that the United States was born perfect. Clearly it wasn’t because of the issue of slavery first, and legalized racism later. The United States is a dynamic nation that has been adapting throughout the centuries with basically the same legal framework whereas take a country like France. Their revolution took place around the same time as the American Revolution. They had a series of kings and different political systems in the next 200 years. Now they have the V-th Republic and who knows what will happen in 2022 given that I have little hope that Macron will be able to change the old ways.

    All this to say that when Trump says he wants to “Make America Great Again” it is not understood in the terms you mention. I do not know any Trump supporter who would like to restore racist laws or to criminalize homosexuality. The issue of gay marriage is a totally different topic. I believe the European Court of Human Rights struck the right balance where the US Supreme Court didn’t: it recognized that criminalizing homosexuality was incompatible with individual freedom but it also said that gay marriage was a new thing and that it was up to each member state to figure out how to handle it .

    To me “Making America Great Again” meant:

    – Stop being the laughing stock in the world, as we were with Bush -because he didn’t have the guts to question the judgement of the deep state when he was asked to invade Iraq- and Obama -whom Europeans love because they see in him what Americans saw in Gorbachev: the best fifth columnist America has ever produced. You take a look at the video of Trump lecturing the ineffectual NATO leaders and you get an idea what being “great again” means.

    – Stop piling up debt. Seriously. Both Bush and Obama doubled the national debt in their respective presidencies. That is unsustainable.

    – Stop using the coercive force of government to impose secular ideology -such as abortion on demand or gay marriage- on individual states whose voters said explicitly in the their own elections that they don’t want. I am not going to defend slavery, but the way slavery was ended was via a constitutional amendment. Same thing with legalized racism: it was ended with a constitutional amendment -the XIV-th amendment- but the statist Democrats back then nullified it for almost 100 years with Plessy v Ferguson until Republicans helped overturn it with the Civil Rights Act of 1964. In the case of gay marriage, the US congress passed in 1996 with veto proof majorities the Defense of Marriage Act in 1996. By a 5-4 majority, liberal ideologues overturned the decision made by the representatives of the people.

    – Stop over-regulating the economy to the point that it grows at 2% due to lack of new business creation. Most Americans rather have plenty of opportunity than government paychecks (that’s something that divides the American and the European mindsets). Basically we the need to reverse this trend .

    I hope you get the idea.

  53. Jr Says:

    TS #47,

    I wonder whether you recognize the similarity in what you are saying about being uncomfortable having a boss with political views that differ from your own, and what the SJW activists are saying? Do you accept that a woman graduate student might be uncomfortable discussing the situation of women in science with a male supervisor or colleague, especially someone who has had outspoken non-feminist views?

    TS #15,

    I don’t believe something without a functional nervous system has human rights and that includes a foetus in early stages of development. I also believe a pregnant woman has human rights and her right to life takes precedence over the foetus and her right to her body takes precedence in the early stages of pregnancy. But in the latter stages restrictions in non-life threatening situations are another thing, as current American law recognizes.

    I do find it interesting that you would speak of “constitutional” rights. The American constitution makes no mention of foetuses having rights, and I don’t see it as implied in an obvious way from the text either. And since the constitution only regulates state action, with one exception, it would not stop abortions anyway even if an absolute right to life was read into the document. Your focus on seeing your preferred policies in the constitution surely mirrors the people who support Roe v. Wade.

  54. TrumpSupporter Says:

    Jr #52:

    How on Earth you make the connection between left-wing bias and misogyny is beyond me. Do you realize that the liberals that overwhelmingly populate academia also believe in identity politics and therefore should be welcoming to opening their ranks to women? Perhaps somebody who is currently a male in academia should be able to address your question better than me.

    On the issue of abortion. You say

    “I don’t believe something without a functional nervous system has human rights and that includes a foetus in early stages of development”.

    Would that also apply to people who have neurodegenerative diseases? I am trying to understand your position better. So for example, do you believe that people in advanced stages of Alzheimer do not have human rights either and that therefore can be disposed of? If you believe that having had first a functioning nervous system that degenerates later into a non functional nervous system is different, then I ask you to explain.

    My position is the following: once sperm and egg meet, a new human life is created via a continuous process that only ends with the death of said new human life. Terminating that life at any stage of the process is morally wrong. I am on record saying that I agree with exceptions such as when the life of the mother is in danger or the usage of the day after pill in the case of rape. However, how we think about this is of utmost important: a new human life with rights vs a “thing” that can be disposed when it is an inconvenience.

    Note that the delineation of having a “functional nervous system” is totally arbitrary -as an aside, here I believe we have an example of somebody that doesn’t understand the concept of with the y coordinate being the status of development of a human life and the x coordinate being time. There are no bright lines after conception happens. I can, for example, imagine the day when we have artificial wombs so that the entirety of a pregnancy can be carried out outside a human body. You sound no different to me from Peter Singer, who believes that parents should be free to kill their newly born children because they lack the autonomy of a full fledged human being. Both “having a functional nervous system” and “autonomy” seem totally arbitrary criteria to me. What these two criteria have in common is that they have been conceived from a position of strength (a non aborted person) to punish an innocent human life when said human life is most vulnerable.

  55. Jr Says:

    Well, they way we define death is that the central nervous system has stopped working.

    The lines we draw will be somewhat arbitrary, but I am comfortable that someone without feeling or sensation is not to be treated as a person with rights.

  56. TrumpSupporter Says:

    Jr #55

    So to be clear, when you say “I am comfortable that someone without feeling or sensation is not to be treated as a person with rights” you are fine with disposing of people with neurodegenerative diseases or in a coma, correct?

    If the answer is “Yes” then that’s a position I can respect on intellectual grounds, just as I respect Peter Singer’s. With this said, I find your and Singer’s position morally reprehensible. It is very ironic that Trump supporters’ get accused regularly of being neo-nazis when in fact it is people with a point of view similar to yours that started and executed .

    As a personal anecdote I would like to share that a good friend of mine who is a card carrying liberal, pro-Obama, pro-Clinton, certainly anti-Trump and who in general loves to work with very smart people confessed to me once that he is a secret admirer of William Shockley’s dysgenics agenda. I didn’t judge him when he said that because indeed, the logical conclusion when one thinks that not all human lives are intrinsically valuable regardless of any other consideration is to try to rank these lives according to some arbitrary criteria -in his case intelligence- and to try to get rid of those deemed less valuable.

  57. Jr Says:

    Also, I don’t think my point came across clearly (or at least it was not understood) about the female grad student. What I mean is that when you say that the presence of outspoken liberals make science an uncomfortable environment for conservatives, it sounds very similar to when left-wing snowflakes think the presence of conservatives creates a hostile environment.

  58. TrumpSupporter Says:

    Jr #55

    So to be clear, when you say “I am comfortable that someone without feeling or sensation is not to be treated as a person with rights” you are fine with disposing of people with neurodegenerative diseases or in a coma, correct?

    If the answer is “Yes” then that’s a position I can respect on intellectual grounds, just as I respect Peter Singer’s. With this said, I find your and Singer’s position morally reprehensible. It is very ironic that Trump supporters’ get accused regularly of being neo-nazis when in fact it is people with a point of view similar to yours that started and executed .

    As a personal anecdote I would like to share that a good friend of mine who is a card carrying liberal, pro-Obama, pro-Clinton, certainly anti-Trump and who in general loves to work with very smart people confessed to me once that he is a secret admirer of William Shockley’s dysgenics agenda. I didn’t judge him when he said that because indeed, the logical conclusion when one thinks that not all human lives are intrinsically valuable regardless of any other consideration is to try to rank these lives according to some arbitrary criteria -in his case intelligence- and to try to get rid of those deemed less valuable.

  59. Scott Says:

    Jr #53:

      I wonder whether you recognize the similarity in what you are saying about being uncomfortable having a boss with political views that differ from your own, and what the SJW activists are saying? Do you accept that a woman graduate student might be uncomfortable…

    That’s an excellent point, and one I’d been thinking about as well. I now have the “honor” of having been accused in this same comment section, in both cases by people who never met me, of

    (1) being unfit to mentor conservative students because of my pro-choice (and more generally progressive) views, but ALSO

    (2) being unfit to mentor female students because of having voiced some disagreements with one specific version of feminism.

    In both cases, I think people who know me would say that my actual experiences, mentoring both conservative and female students over a decade, put the lie to both charges.

    Thinking it over, I wonder if the people who level these sorts of accusations genuinely can’t imagine that two people would get so engrossed working on a research problem together that differences in their political views, or for that matter their sexes, would seem trivial by comparison. If so, then I daresay that lack of imagination says more about the accusers than about the accused.

  60. gentzen Says:

    TrumpSupporter #28, #35:

    It is nearly impossible to predict which mathematics will turn out to be popular or practically applicable. I wondered since a long time, why the aliasing formula for DFT (which is closely related to the Shannon/Nyquist sampling theorem) is rarely stated explicit, and almost never proved. Motivated by your strong statements, I condensed my perplexity now into an explicit question: Is the aliasing formula for DFT a folklore result? If anybody knows the answer, I really would like to hear. One partial answer might be that the phenomenon can be explained by looking at each separate frequency individually one at a time, and that this local explanation is preferred (by nearly everybody) over the global formula as more intuitive.

    And I can’t resist to quote Hardy’s famous misprediction

    No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.

  61. TrumpSupporter Says:

    gentzen #60

    I agree that it is hard to predict which mathematics will turn out to be popular or result in revolutionary applications. As I said earlier, I am making a value judgement based on my own experience.

    The answer to your question is that real life signals are analog and the DFT does its magic in the DFT-frequency domain so you need to approximate these samples of the DFT from what you can do in real life. Typically doing the DFT magic would require designing some sort of high precision analog filter bank, something that is very hard to build.

    On the other hand, there are decades of experience doing low pass analog filters. So the formula “low pass filter + sampling at the Nyquist rate” is better understood, implementation wise, than alternatives, including the idea using analog filter banks sub-sampled to then reconstruct the signal in the digital domain as if that been sampled using the Nyquist rate.

    When it comes to sampling, what engineers know how to build matters. It’s a similar problem that happens in computer science with algorithms. It is one thing what the theory says you can do, quite another what is practical in terms of implementation. Take Google’s MapReduce for example. For all the hype it generated both inside Google as well as in the open source world, it turned out to be a restrictive way of doing batch processing and Google ended up abandoning it .

  62. Scott Says:

    TrumpSupporter #46: You don’t need to keep reiterating that the discoveries of Euler, Gödel, Feynman, Knuth, Wiles, Perelman, and I-forget-who-else were all of a higher caliber than mine. I PLEAD GUILTY. As, of course, must virtually all of humanity.

    Even imagining that I had the ability to make discoveries at that level, you can see how I instead spend my time: arguing with anonymous commenters like yourself! 😉

    I can say only that I’m grateful for every second I’ve been able to spend doing research, both alone and in collaboration with students and colleagues; that outside friends and family that’s been the highlight of my life; that even if a problem isn’t the center of the world, it becomes that for you while you’re working on it; that my enjoyment would remain even if there were no external recognition whatsoever (as there basically wasn’t in the “first phase” of my research career, ages 15-20…); and finally, that even if only 20 people care enough to read and understand my proof of a theorem, that probably brings me greater satisfaction than if 20,000 people read my blog post.

    And to return to the point, I don’t see how any of this has any bearing on your arguments about leftism in academia! You yourself brought up Knuth as a true great. Well, did you know that Knuth uses his homepage to voice political views that could be characterized as noticeably more leftist than mine? In general, while as we both agree, scientists as a whole are markedly more leftist than the general population, within science I’ve noticed no correlation whatsoever between “greatness” of achievements and intensity/outspokenness of leftism, anywhere on the greatness spectrum from Einstein down. (Does anyone know of studies directly addressing that?)

  63. TrumpSupporter Says:

    Scott #62

    I am sorry I offended you with my repetitions. You have been very kind with letting me commenting anonymously here.

    But here is what I will say too. Do you think is fair to have leftist scientists who, in most cases, get their work funded by the American tax payer -either via government grants or via federal loans for the students that pay tuition at their schools- constantly call the 63 million people who voted for Donald Trump all sorts of names. In this thread John Sidles wasted no time to insinuate that we are a bunch of neo-Nazis and Raoul from Ohio wasted no time asking about solving cubic algebraic equations. You yourself said “I think people support Trump for the same reason why second-graders support the class clown who calls the teacher a fart-brain to her face.” That’s certainly an aspect of it, but I also enumerated a set of reasons of what “making America great again” means for me beyond that.

    Reasonable people can disagree about many things, including politics and religion. However, there is a difference between disagreeing in a respectful way and assigning epithets (racist, ignorant, having the mindset of a second year old) to those who don’t agree with our politics. The latter is not constructive.

    You speak of Donald Knuth. As I said, I met him once in person. He is a true gentleman. Even his website lists his political opinions in the form of questions without imputing motives to those who would answer the questions differently than he does. I have met other great scientists of that era who are also leftists. Same thing. It is not the issue of being a leftist, rather, the patronizing effect of creating an environment in which only left wing ideas are welcome in public. That, I believe, is the real problem. And that is what people like you or Terry Tao did in the lead to the 2016 election: to remind everyone that if you want to be “cool” in 2016 academic science, your duty is to denounce Donald Trump and to call his supporters all sorts of names, including, that these supporters lack the judgement to decide for themselves the leader they want for their country.

    One of the lines that touched many hearts in Trump’s speech at the 2016 RNC was,

    “Remember: all of the people telling you that you can’t have the country you want, are the same people telling you that I wouldn’t be standing here tonight. No longer can we rely on those elites in media, and politics, who will say anything to keep a rigged system in place. ”

    That’s what we are talking about here.

  64. wolfgang Says:

    >> pro-life

    I assume TrumpSupporter is also against the death penalty and in favor of universal healthcare?

    If not why not?

    Btw this is an honest question about one aspect of US politics I never understood …

  65. Anonymous Says:

    I’m also from Europe, so I neither would call me a Trump opponent nor a Trump supporter but rather an outside observer with interest and some level of anxiety. The anxiety is not new, though. Obama being a lame duck maybe was a bigger topic in Europe than in the U.S. (judging from watching the news in the U.S. while being there on conferences and talking to my American colleagues). So I understand that one of the main concerns of TrumpSupporter is the future of a country close to bankruptcy.

    But I also get his point about the way a professor expresses his (political) views publicly. I wouldn’t mind at all if my professor openly promotos leftism while I prefer the right side or vice versa. As Scott explained so well, when working together and getting lost in math and cs problems, things like that just don’t matter. The same applies for being pro or against SJW and so on. BUT there is a huge difference between expressing own (political) views and insulting all those who don’t agree. If I would support Trump I would never apply for a position under Scott’s supervision because he expressed repeatedly that one has to be dumb to support Trump. And while race, political opinion, and whether you are a Linux or Windows person might not get in the way of a fruitful collaboration, thinking the other person is dumb certainly is an obstacle. Especially for a brillant person like Scott. I was actually shocked when Scott blogged that he saw a student on campus wearing a Trump shirt and he wanted to go there to congratulate him for his bravery. If I would be a student on that campus I would feel mocked by such words. I get all of Scott’s arguments against Trump, most of them are totally reasonable. But I don’t understand how someone with that level of intelligence totally neglects that there might be also reasonable arguments on the other side and just judges everyone based on one single political view. This is especially disturbing in my eyes as this is basically a binary decision in the U.S. – and both candidates seemed to be not ideal choices. So yes, please feel free to express your views and opinions. I think they are always inspiring and raise attention to important problems. But although this is your blog and obviously your rules, ranting in such a way might affect people and decisions in a way you might not intend.

  66. TrumpSupporter Says:

    I see a comment from Anonymous awaiting moderation explaining the situation. I have another from myself that basically says the same thing. Hard to work for somebody who thinks you are dumb for supporting Trump. I had to experience a similar situation recently at work and thanks God it was fixed with my boss leaving the company for his own reasons. While his leaving had nothing to do with me being pro-Trump and he being anti-Trump, my general well-being at work increased dramatically.

  67. TrumpSupporter Says:

    wolfgang #64

    Among pro-lifers (understood as anti abortion on demand) there are very different views about these other issues, at least in the circles I do my prolife work. What unites us is a deep desire for the recognition that unborn children have constitutional rights and that we see as a great injustice to terminate a new human life who have committed no crimes for reasons of convenience to somebody else (be it the mother, the father who pressures the mother to have an abortion, social pressure, etc).

    My personal opinion on the other two issues is that I am in favor of the death penalty for extreme cases. For example, had Bin Laden been arrested in the US, I would see the death penalty as an appropriate punishment for having ordered 9/11. Which crimes are so outrageous as to deserve the death penalty as a punishment is a matter of ongoing debate, but on the principle I am in favor of the death penalty for serious crimes. Note that I don’t see any connection whatsoever with being prolife on the abortion debate. For a criminal to be convicted of the death penalty, said criminal has had to commit a horrendous crime. An aborted unborn children has committed no crime whatsoever other than being an inconvenience to somebody. It is hard for me to understand how “being an inconvenience” to a third party justifies terminating one’s life. Yet, Peter Singer thinks that way and Jr above too. I do not buy the “inconvenience” rationale but I am perfectly comfortable with “having been convicted of a heinous crime” as a reason to terminate another human being’s life.

    On the issue of universal healthcare. I think this is more related to the issue of being prolife/prochoice. A common criticism I hear from the proabortion crowd, one that I consider it a legitimate one, is that many prolifers are really pro-birth. Meaning, that both during the pregnancy and after the pregnancy they don’t want to spend a dime on expecting mothers and their kids. Don’t count me among them. Although I do not believe that socialized medicine systems like those they have in Europe in Canada are good -mostly because they free-load the research necessary to develop new treatments on the US-, I do believe in using government funds in one way or another to help those that through not fault of their own are going through a rough time. My thinking is one of “safety net” vs “entitlement”. This is another reason I liked Donald Trump vs the other Republicans: he said several times in the debates that he wouldn’t let people die in the streets for lack of health insurance as a code word for this thinking.

    There is no unanimity about these issues among prolifers. For example, Roman Catholics are very active in prolife circles and organize a lot of the big prolife events. Devote Roman Catholics follow the Vatican doctrine against the death penalty and the Roman Catholic Church’s social teaching that kind of favors government funded universal healthcare as an entitlement. I am not a Roman Catholic, so I don’t feel bound by these positions. I have reached my own through my own thinking.

  68. Anonymous Says:

    Just to illustrate my point for you, Scott: You complain right here in that comment section that people judge you just for what you blog here without knowing you personally; assuming things about you that – as you say – are just plainly wrong. That’s an excellent point. But you judge millions of people just by their decision in the last election. And Raoul Ohio did nothing else with his comment on cubic equations. So how is that justified? And yes, of course, people form an opinion about you by reading your blog. I guess that’s not surprising news. And students might make decisions based on that – for example not to attend your courses because they are afraid to be labeled as unfit or dumb. You talked about the importance of getting the best students (also from foreign countries) not that long ago. So why not promote open discussions on controversial topics like that instead of prejudging people with different opinions? I guess both sides could benefit from that.

  69. Scott Says:

    TrumpSupporter and Anonymous: Ironically, my actual views are extremely mainstream and moderate compared to the views of many other academics. So then why, in your view, should a student be afraid to work with me but not afraid to work with them?

    The only answer I’m getting is: because I discuss and defend my views openly on my blog.

    Not once did I call Trump supporters stupid. Some of them, like Peter Thiel, surely have high IQs. But it’s impossible to convey the substance of my position without saying I think they made one of the most grievous mistakes in the country’s history, eroding civil democratic norms and paving the way for ethnic-nationalist authoritarianism. That doesn’t mean they should be punished in the workplace, as I also said clearly in my post about Thiel (and took flak from the far left for saying—the usual, that I’m an entitled white male douchebro, etc etc. I can’t win, can I?)

    Look, the student/adviser relationship can be personal and idiosyncratic. There might be students who don’t want to work with me because they dislike the shape of my nose, or my habit of rocking back and forth in my chair. So I suppose I can only be grateful that, through a decade of expressing sometimes controversial views on this blog, I’ve continued to attract students from a wide variety of backgrounds, who often disagree with me about important things, but who share a passion for tackling CS theory problems. May I continue to do so.

  70. wolfgang Says:

    >> I don’t see any connection [between death penalty] whatsoever with being prolife on the abortion debate.

    In many cases there is a non-zero probability that an innocent human being gets put to death – this is the connection imho.

  71. TrumpSupporter Says:

    Scott #69

    I don’t know you personally, and it is not possible for me to judge what kind of adviser you are one-on-one. During my own career as graduate student, there were professors with stellar research output who were known to be a-holes as research advisers but people were willing to put up with them for the prestige of having been a student of professor X. My own advisor is a great person and it was the honor of my life to work with him during my PhD. One of the reasons I do not regret having done the PhD and for which I would do it again is that it was the privilege of a lifetime to work with him. To this day I do not know what his politics are. It wasn’t possible for me or other students who had worked for or with him to discern them from his teaching or his way of talking to students. From some comments you might think he is a liberal from others that he is a conservative, but no official clue where he lies. If I had to make a bet, using maximum a posteriori estimation, given the prior distribution of party registration among academics, my money is that he is a liberal, but I do not know for a fact.

    At the same time, I know what is like to be among the few non liberals in an environment where everybody who is loud about his politics is a liberal, anti-Trump and thinks that voting for Trump is for dumb people -that would be my current work environment. After my former anti-Trump boss left, my new boss is totally shut about his politics. Again, using maximum a posteriori estimation and given the prior distribution that can be obtained from the political donations database in my field of work, I would assume he is a liberal, but I have no clue.

    I personally don’t care about people’s politics. My significant other is an anti-Trump liberal you see, so I know what a respectful relationship between two people who disagree on politics look like. My own experience in academic and professional environments is that whenever you have an outspoken majority on politics, the minority is shut out of the most important things. That doesn’t mean you are one of those who does these things, all I am saying is that based on my experience and your public pronouncements, I would try to avoid being your PhD student (or Terry Tao’s) if I were in the position of having to choose an advisor at UT Austin of UCLA.

    And with this said, I am saying good bye for good. It has been great to talk to you and to have this discussion. I thank you for publishing my comments and engaging me.

  72. Anonymous Says:

    Please, Scott, I made it totally clear that I really appreciate and respect you for ‘discussing and defending your views openly on your blog’. And I’m happy for you that you think this never affected your work experience with anyone in a negative way. But you also cannot guarantee that there are indeed no students who feel threatened by what you say. And this is very much different from disliking your appearance or some charming habits. Maybe you did not literally called Trump voters dumb but you can hardly claim that you don’t prejudge them. And you let Julia write on your blog that
    “People are fucking morons whose brains are not built to see through bullshit.” in that very context. So yes, me as a student would be anxious to let you know about my political views because I would think that you would not consider me intelligent or fit enough only based on those. But not because of your own views (with which I actually mostly agree) but for all your insults of people not in line with you. I think you are intelligent enough (by far) to explain and debate your views and arguments on a more civilized level. And I certainly share your passion for CS theory problems, and I advise PhD students of my own. I just don’t share your passion for blind judgment. And regarding your ‘ I can’t win, can I?’: A dialogue normally is not about winning. It’s about understanding the other party better and maybe finding consensus in unexpected corners or inspiration to rethink the own position. But that requires some respect and a certain level of openness from both sides. It’s not about belittling the others or twist their words until they give up. It’s about learning from each other.

  73. OnOracles Says:

    If $X\subseteq Y$ are complexity classes then when does it hold $X^O\subseteq Y^O$? Is it true for almost all oracles $O$ or is there important non-trivial cases?

  74. Scott Says:

    OnOracles #73: You’re asking about the entire subject of relativization in complexity theory. For most inclusions X⊆Y that we know, the proof also shows that XO⊆YO for an arbitrary oracle O. However, there are famous exceptions, such as PSPACE⊆IP, which fail relative to some O’s (and even relative to a random O with probability 1). For more, see my P vs. NP survey or any graduate-level complexity theory textbook.

  75. OnOracles Says:

    thankyou (your math tex did not parse)

  76. Shtetl-Optimized » Blog Archive » Yet more mistakes in papers Says:

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