In a world like this one, take every ally you can get

Update (Sep. 17): Several people, here and elsewhere, wrote to tell me that while they completely agreed with my strategic and moral stance in this post, they think that it’s the ads of Republican Voters Against Trump, rather than the Lincoln Project, that have been most effective in changing Trump supporters’ minds. So please consider donating to RVAT instead or in addition! In fact, what the hell, I’ll match donations to RVAT up to $1000.

For the past few months, I’ve alternated between periods of debilitating depression and (thankfully) longer stretches when I’m more-or-less able to work. Triggers for my depressive episodes include reading social media, watching my 7-year daughter struggle with prolonged isolation, and (especially) contemplating the ongoing apocalypse in the American West, the hundreds of thousands of pointless covid deaths, and an election in 48 days that if I didn’t know such things were impossible in America would seem likely to produce a terrifying standoff as a despot and millions of his armed loyalists refuse to cede control. Meanwhile, catalysts for my relatively functional periods have included teaching my undergrad quantum information class, Zoom calls with my students, life on Venus?!? (my guess is no, but almost entirely due to priors), learning new math (fulfilling a decades-old goal, I’m finally working my way through Paul Cohen’s celebrated proof of the independence of the Continuum Hypothesis—more about that later!).

Of course, when you feel crushed by the weight of the world’s horribleness, it improves your mood to be able even just to prick the horribleness with a pin. So I was gratified that, in response to a previous post, Shtetl-Optimized readers contributed at least $3,000, the first $2,000 of which I matched, mostly to the Biden-Harris campaign but a little to the Lincoln Project.

Alas, a commenter was unhappy with the latter:

Lincoln Project? Really? … Pushing the Overton window rightward during a worldwide fascist dawn isn’t good. I have trouble understanding why even extremely smart people have trouble with this sort of thing.

Since this is actually important, I’d like to spend the rest of this post responding to it.

For me it’s simple.

What’s the goal right now? To defeat Trump. In the US right now, that’s the prerequisite to every other sane political goal.

What will it take to achieve that goal? Turnout, energizing the base, defending the election process … but also, if possible, persuading a sliver of Trump supporters in swing states to switch sides, or at least vote third party or abstain.

Who is actually effective at that goal? Well, no one knows for sure. But while I thought the Biden campaign had some semi-decent ads, the Lincoln Project’s best stuff seems better to me, just savagely good.

Why are they effective? The answer seems obvious: for the same reason why a jilted ex is a more dangerous foe than a stranger. If anyone understood how to deprogram a Republican from the Trump cult, who would it be: Alexandria Ocasio-Cortez, or a fellow Republican who successfully broke from the cult?

Do I agree with the Lincoln Republicans about most of the “normal” issues that Americans once argued about? Not at all. Do I hold them, in part, morally responsible for creating the preconditions to the current nightmare? Certainly.

And should any of that cause me to boycott them? Not in my moral universe. If Churchill and FDR could team up with Stalin, then surely we in the Resistance can temporarily ally ourselves with the rare Republicans who chose their stated principles over power when tested—their very rarity attesting to the nontriviality of their choice.

To my mind, turning one’s back on would-be allies, in a conflict whose stakes obviously overshadow what’s bad about those allies, is simultaneously one of the dumbest and the ugliest things that human beings can do. It abandons reason for moral purity and ends up achieving neither.

187 Responses to “In a world like this one, take every ally you can get”

  1. Bertie Says:


  2. Peter Norvig Says:

    I think a majority of the Lincoln Project ads have an audience of one. Certainly the latest two, “Broke” and “Don the Con”. That can be effective — they got Brad Parscale kicked off the campaign, which is good because he was pretty effective. But those ads aren’t getting any votes.

    There are some other Lincoln Project ads that I agree are good and look to me like they would be convincing to wavering or undecided voters. But I’m not the target audience.

  3. domotorp Says:

    Churchill probably shouldn’t stand as a positive example though, see

  4. anonymous Says:

    May I suggest, as a way of coping with the possibility of a Trump win, that you would try looking at the positives sides of Trump (and they do exist, even tho you feel overwhelmed by the negatives).

    Yes, he’s a total psychopath. But at least he knows how psychopaths feel and work and he understands dictators better than naive sane leaders like Obama. Trump’s foreign policy and his way of dealing with China are much better than any leader before him. I honestly view China as a much bigger threat to human rights and freedoms in the world, and as the biggest threat to the future of humanity and the US.

    He might seem like the most mad president and war mongering fool to ever be a president, but apparently his unpredictability and perceived madness is such a big deterrence that the U.S. enemies finally genuinely fear provoking him.

    He might have fucked the coronavirus response hard, but a vaccine might solve this problem before a new president comes anyway, and it’s not very likely a new president could improve the existing situation by far (or do you think wrecking the economy again is a good choice?)

  5. 4gravitons Says:

    Since you mentioned third parties…are you going to be involved in a vote trading site this election? If not, do you have any recommendations for good ones?

  6. RKeelan Says:

    I’m Canadian and not permitted to donate to candidates in the US election, but I’m persuaded that the most effective place to donate is down ballot Democrats in marginal districts whose only chance at winning lies in flipping Trump voters or activating Democrat-sympathetic non-voters.

    The rationale (which is not my idea, more on that below) is that national campaigns (Biden & Harris) can win by focusing on the traditional liberal base, and get very little value from the marginal dollar, because they already have so many. Meanwhile, a democrat running for state representative in Iowa, for example, has exactly one winning strategy: get a bunch of Latino meatpacking plant workers to vote for the first time. These are the kinds of voters that big national campaigns will never even try to reach. As an added bonus, down ballot campaigns have tiny budgets, so the marginal value of donations is much higher.

    If that sounds persuasive (or if it sounds dumb, but maybe like there’s a sensible idea in there somewhere), let the people whose idea this is explain it properly:

    Or you can check out @Pinboard on Twitter. Not my account, but I’m a huge fan.

  7. John Newton Says:

    Entirely endorse 100%. Electing Joe Biden and, if possible, winning the Senate, is the single most important goal in order to prevent “a worldwide fascist dawn”.

  8. Hank Rearden Says:

    I’m honestly shocked to read you write “if I didn’t know such things were impossible in America would seem likely to produce a terrifying standoff as a despot and millions of his armed loyalists refuse to cede control.” There is no more charitable way to put this other than that this is a pure, unadulterated falsehood. The biggest threat in American cities over the past 90 days has been getting murdered by a marauding band of rioters and looters the vast majority of whom lean left.

    Regarding your donation to the Lincoln Project, it seems pretty clear that the messaging, while savage, is meant to endear itself to the Jen Rubins and Bill Kristols of the world who didn’t vote for Trump in the first place. In other words, the Lincoln Project isn’t really about flipping Trump voters but rather perpetuating the smug sense of superiority “Never Trumpers” claim to have over actual Trump voters.

  9. Mo Nastri Says:

    I’m excited to see what you have to say about Cohen’s proof of the independence of the continuum!

  10. Anonymous Says:

    If Biden wins the White House, but the Democrats don’t take Congress, the country breaks. A GOP majority in the Senate would have zero incentive to allow even a single important bill to pass. They learned from the Obama era that if they sabotage a Democratic White House by blocking everything, then the GOP suffers zero blowback and simply makes the President look bad. The GOP learned from the Obama era that their quickest route back into power is to prevent a Democratic President from governing.

    And even if the Democrats win the Presidency and get a razor-thin majority in the Senate, they would need every single Democratic Senator (or Democratic-caucusing Senator) to agree to eliminate the filibuster. That would mean getting Angus King and Joe Manchin, among others, on board. Good luck with that.

    I don’t think people have fully internalized how close we are to representative democracy, at the federal level at least, suddenly ending after the election. There is a reason why Presidential systems are so unstable everywhere else in the world, and why every other representative democracy uses a Parliamentary system.

  11. Ethan Says:

    Peter Norvig #2

    Totally unrelated to the topic at hand, but since I quoted you here could you elaborate, to the extent you can without giving out trade secrets, how Google currently selects software engineers and to what extent it has moved away from Cambridge Tripos like methods, , to focus more on methods that predict more reliably on the job performance?

  12. Uncle Brad Says:

    I don’t suppose there are problems and solutions to go with those lecture notes. I have two children in college right now (one at UT) so I’m too broke to take the class myself.

  13. Elizabeth Says:

    It’s always good for successful/productive people to talk about the malaise or depression they’ve felt during these prolonged restrictions.   You may have seen the report in June that 25% of people surveyed age 18-24 had “seriously considered” suicide during this period,

    At first the WHO and the news media underplayed covid, but then they swung too far in the other direction and scared part of the population into a hysteria that continues to this day.   The final act of selfishness for the baby boomer generation is dismantling civilization for young people so that boomers can live a few more years (average age of death of covid ~= average life expectancy).  We should have returned to normal in May after we “flattened the curve” and overwhelming the hospitals turned out to be a non-issue.  
    But the covid hysteria continues, and based on youtube videos with millions of views, it’s obvious to many people that the democrats and mainstream media committed months ago to using this virus as their best chance to defeat Trump.  The WHO gets a pass, the CCP gets a pass, Cuomo gets a pass, the fake news gets a pass, rioters get a pass, instead we’re supposed to feel that all of this is somehow Trump related.  That 200k deaths deserve this much attention and sacrifice from all of us, when back in 2016 we had 250k – 500k deaths from medical mistakes, and 150k deaths from respiratory illness:

    If Hillary had won in 2016, we would have barely heard about covid this year at all.  The news would assure us that the death rate is very low and it mostly affects the elderly and infirm.  If tapes were released of this alternate world President Hillary saying in February that she didn’t want people to panic, the news media would defend that as justified, Snopes would fact check it as gloriously wise, etc. 
    I care relatively little about the outcome of the presidential election, but I’m extremely concerned about the rise of tyranny in the covid era.  I wouldn’t mind following leaders who are competent, but none of ours are.  Most of the state executive branches need to be punished and brought into check by the courts for their unconstitutional mandates.  The Democrats have been pretty clear about their plans for increasing tyranny, but the Republicans oversaw a huge increase in tyranny under W as well (patriot act, NSA spying, etc).  The issues I really care about are those that remove power from the government and restore it to individuals, and currently those issues have no champions.  

  14. Anonymous Says:

    Elizabeth #13:

    I have to push back on some of your comments here. The last I checked, Trump was not the leader of France, or Germany, or South Korea, or Taiwan, or India, and those countries have more than “barely heard about covid this year at all.” Saying that we wouldn’t have heard much about COVID if Hillary were president seems like enormous speculation.

    People are upset about Trump not wanting to “panic” people about COVID because he went on and held huge indoor rallies after he said that. Obviously no serious leaders wanted to panic their populations, but this meant that they also didn’t subject them to mass negligent homicide like Trump did.

    The deaths aren’t going to stop at 200k. We have no idea what the death toll is going to be at the end of this.

    As for this being the baby boomers sacrificing the young, it’s true that the death rate for young people is way lower than for boomers+, but I personally know people who are young and have survived, but who have suffered potentially permanent lung and/or heart damage from COVID. The rate of such complications for young people seems to be high. Who knows whether this will reduce their life expectancies down the road? This isn’t just an “old people disease.”

    As for arguing that “The Democrats have been pretty clear about their plans for increasing tyranny,” I don’t know that even the GOP has been “clear” about wanting to increase tyranny. What I do know is that the people who shout the loudest about tyranny are usually the first ones to welcome it in the door when it arrives – that’s the story of the GOP and Trump for the past few years.

  15. Scott Says:

    Uncle Brad #12:

      I don’t suppose there are problems and solutions to go with those lecture notes.

    You can find 11 problem sets right here (we haven’t posted solutions, for reasons I hope you understand). Enjoy!

  16. anon Says:

    Try considering hypothesis that are not what MSNBC is telling you to believe. Consider that
    neither side is a cult. You are not in a cult and the other side is not in a cult. But you only listen to people on “your side” and you think the other side are deluded or liars or believe liars.

    You do not credit the other side with rationality. Why can’t you deprogram anyone on the other side? Why can’t you just repeatedly call them stupid, racists, following Hitler and part of a cult and deprogram them?

    When they say. Well MSNBC and CNN have gotten it wrong every step of the way yet you still believe them.

    They said Trump would never win and then he did. Well that was because of Russians spending a few million dollars and somehow that was more effective than billions in campaign adds. It had nothing to do with Hillary being unlikeable and one of the worst campaigners in history who did not visit the flyover states who were supposed to vote for her.

    Now Biden is going to win because the polls that were wrong in 2016 are going to get it right. the people who we call racists and evil and stupid are going to answer honestly to pollsters. Biden who again is barely going to any states to campaign.

    Millions of Trump supporters will be violent? When it is leftist Antifa and black lives matter who are burning cities. Black lives matter who are 90% white calling black cops N* word. I have seen them twice harassing hispanic farmers at farmers markets. On their loud speakers the BLM did not say their demonstrations were infiltrated by right wing nazis. No they were haranging more priviliged white people to come out as human shields for the one teen black kid. Why can’t racists come out and deprogram and help overwhelm the police while the racist Target shopping mall is attacked and looted.

    You are right in that when mostly white losers living at home with their rich parents or on university campuses come out in large numbers and the police are forced from doing anything then eventually this will be met with the people who own 80% of the guns and who are 70% of the military.

    Meanwhile people in places suffering from major demonstration areas are going two to one against democrats. Democrats who were bailing out the leftist rioters. Leftist rioters who are now getting charged legally under federal law and arrested by federally deputized local police. A legal system that is slowly getting a handle on the leftist demonstrations.

  17. Scott Says:

    Elizabeth #13: Without the haphazard, partial, incompetent-compared-to-other-countries shutdown we’ve had, and based on an IFR of 0.65% or whatever, it wouldn’t have been 200,000 dead so far, but well over a million dead, more than the number dead from heart disease and cancer in the same period, and likely millions more (including younger people) with permanent damage. Another useful way to think about what’s happening is as a 9/11 every few days, on top of the background death rate, which with no shutdown, would’ve been multiple 9/11’s per day.

  18. Elizabeth Says:

    anonymous #14: I didn’t mean to imply that the virus is a non-issue around the world, just that it has become an overblown hysteria and the precautions should have ended back in May.   Recall the discussion about “flatten the curve”: health officials back then were open about the fact that the virus could no longer be stopped from becoming endemic, which is why we only aimed to flatten the curve to avoid overwhelming hospitals, instead of trying to change the area under the curve (total eventual deaths now that the virus is endemic around the world).    Everyone supported the initial lockdown.  When the goalpost shifted to turning society upside-down indefinitely for dubious benefit, many people rejected that as tyrannical overreach.  

    Regarding the message that people shouldn’t panic over covid, here’s Biden at a crowded rally on Feb 28 telling supporters not to panic:

    I apologize for using Breitbart as a source, but you can watch the video yourself, and the mainstream media doesn’t cover these facts for obvious reasons.  I remembered this incident from back when it happened, because I was a Bernie supporter and Biden was banking on the South Carolina primary so he didn’t want it to be altered or postponed.  Google and youtube censor this information, so I had to use DuckDuckGo just to search out the record that matched my memory.
    Does anyone care what Biden says or does?  Not really, he’s a non-entity.  Like a particle with no mass or energy, weakly interacting with the world.  Pointing out the hypocrisy is so unchallenging, so obvious to anyone who looks around, there’s not much satisfaction in it.  But it is surprising when other people don’t remember what really happened.  

  19. JimV Says:

    A smart and accomplished guy I knew at work mused one lunch hour on “Why doesn’t the USA send teams to explain our way of government to other countries so they can adopt it?” I didn’t quite have the heart to tell him the UN has such teams and they recommend parliamentary systems over our meshuggeneh Electoral College and two senators each for Wyoming, Rhode Island, New York and California; plus Australia has a better voting system. I understand the historical basis of our system, but at some point to survive as a civilized country we need a massive overhaul.

    If humans were perfect or even 50% better than they are any system would work, because those in power would insist on fairness even to non-supporters and disdain propaganda, but we aren’t. Only the best systems will work and we don’t have one.

    I should be trying to cheer you up and forestall depression, and wish I could, but depression seems like the realistic stance to me. As one of Walt Kelly’s (“Pogo” comic strip) characters said, “We have met the enemy and he is us.”

  20. anon Says:

    Elisabeth “The final act of selfishness for the baby boomer generation is dismantling civilization for young people so that boomers can live a few more years”

    I belong to a generation in between these boomers and these young people. When the time comes, I’d love to see my parents watch my children raise their own children. Please help me see that, by taking any reasonable precaution that can help mitigating this pandemia. I understand this can be boring for you to wear a mask or something. Please do it anyway. Thanks for your help.

  21. Elizabeth Says:

    Scott #17: I think even a worst-case scenario of a million deaths without any lockdown measures would be preferable to the way that the mental health of young people has been ravaged by the course of economic destruction and social isolation that we’ve followed.  I don’t necessarily dispute the 1 million deaths prediction, but I think Sweden’s approach which emphasized personal responsibility over government mandates was far more sensible, and seemed to bring them to herd immunity with 6000 deaths in a country of 10 million people.    Almost all covid deaths are accompanied by comorbidities, and the most common ones are obesity, hypertension, and diabetes.  This means that we all have the power to lower our covid death risk to nearly zero just by exercise and healthy eating.  But instead almost all that energy and intention goes into blaming and restricting others.

  22. tired Says:

    It is hard not to get depressed during these times. I’m horrified by the idea of voting for Biden, being one more example of a mediocre politician with vacuous politics. Obviously, voting for Trump is not happening (not to toot my own horn, but I actually have a human brain). So when do you say “enough is enough” and refuse to participate in a system (not just the current administration, I mean federal US politics) which has been nearly unable to solve almost any problem that the US has experienced over the past 2 or 3 decades (and probably more)? Gun control, climate change, health care, police violence, COVID, massive income inequality, blatant corruption, detainment and mutilation of immigrants, multiple recessions, homelessness, opioid crisis, student debt, etc., etc., etc. — what has actually changed here? I think the cliche is: “one definition of insanity is doing the same thing over and over again and expecting a different outcome”.

  23. arch1 Says:

    The mention of occasional debilitating depression made me think you might find the following inspiring. It is from Bela Bollobas’s preface to “Littlewood’s Miscellany,” by J. E. Littlewood (that said, I’m admittedly clueless as to how unusual such a result is)-

    “…Littlewood was adamant in his assertion that for about thirty years he was functioning at half his capacity, often spending hours in cinemas just to while away the tedious hours. … The year 1960 [his 75th] was a memorable one for him. … a brilliant neurologist, Dr. Beresford Davies … after experimenting with various drugs which were then comparatively new, hit upon the right combination for Littlewood’s case, and cured him. His freedom from depression led him to accept invitations to the United States, which he would never have contemplated during his illness. He made eight visits…”

  24. ira Says:

    1 Only people who haven’t been in real wars (whether violent or not) don’t understand the concept of ‘a tactical ally.’

    2 As you said, if a tactical non-aggression pact could be made with the literal (not figurative) Hitler, than, indeed, who is beyond the pale ?

    3 Swimming somewhat under the radar, Rick Wilson (apparently the founder of the Lincoln Project) said, ‘trickle-down economics is a complete failure. If Biden tries to implement it, we we will fight him non-stop.’ Their commitment doesn’t seem to be simply to defeat Trump.

    I wish the left had multiple Lincoln Projects. I’ll risk saying that the fact that they don’t is one of the reasons that the far-right has been hegemonic for the last fifty years. (It’s hard to appreciate just how far-right the GOP has become. I don’t think there’s any parallel in any of the other advanced industrial nations. As Chomsky said, ‘Eisenhower said ‘anyone who doesn’t accept the New Deal, has no place in American politics’. FFS, Bernie might not even measure up !’

  25. Anonymous Says:

    tired #22

    I certainly hear you on that. I am not an enthusiastic Biden supporter, to say the least.

    But look at your list of priorities: “Gun control, climate change, health care, police violence, COVID, massive income inequality, blatant corruption, detainment and mutilation of immigrants, multiple recessions, homelessness, opioid crisis, student debt, etc., etc.” Those priorities are overwhelmingly shared by the current platform (at least since the 2000s) of one of our two major political parties.

    But that political party – obviously, I’m talking about the Democratic party here – has never had the ability to enact that agenda, because even when they had nominal majorities in Congress and the White House in 2009–2010, they were stymied by the filibuster. It’s not a surprise that the Affordable Care Act got passed during the tiny window in which they had 60 votes in the Senate, just after Arlen Specter switched parties but before Ted Kennedy died.

    This is ultimately why the filibuster is so horrible. It makes people cynical, because it makes it look like even when they win majorities, they still can’t pass any laws. And it gives the minority party the ability to block everything and generate even more cynicism.

    The Democratic House majority passed serious climate legislation, for example, among many other bills that died because of the Senate filibuster. I would encourage you to look at all the bills the House passed during 2009–2010 that died because of the Senate filibuster.

    So the right attitude is, let’s get the Democrats back in charge again, and, if that’s successful, demand that they end the filibuster once and for all.

  26. ira Says:

    Only people who haven’t been in real wars (whether violent or not) don’t understand the concept of ‘a tactical ally.’

    2 As you said, if a tactical non-aggression pact could be made with the literal (not figurative) Hitler, than, indeed, who is beyond the pale ?

    3 Swimming somewhat under the radar, Rick Wilson (apparently the founder of the Lincoln Project) said, ‘trickle-down economics is a complete failure. If Biden tries to implement it, we we will fight him non-stop.’ Their commitment doesn’t seem to be simply to defeat Trump.

    I wish the left had multiple Lincoln Projects. I’ll risk saying that the fact that they don’t is one of the reasons that the far-right has been hegemonic for the last fifty years. (It’s hard to appreciate just how far-right the GOP has become. I don’t think there’s any parallel in any of the other advanced industrial nations. As Chomsky said, ‘Eisenhower said ‘anyone who doesn’t accept the New Deal, has no place in American politics’. FFS, Bernie might not even measure up !’

  27. Doug Says:

    Elizabeth #18: “I apologize for using Breitbart as a source…”

    You can’t apologize in the same post where you are doing thing. If your intention was to apologize, you could just have used the backspace key and found some reputable source. Surely one must exist?

  28. Scott Says:

    ira #26:

      As you said, if a tactical non-aggression pact could be made with the literal (not figurative) Hitler, than, indeed, who is beyond the pale ?

    I mean, Hitler kind of was beyond the pale. 🙂 Stalin’s non-aggression pact with him was one of the great strategic blunders of all time, and it’s very hard to imagine who the worse enemy would be, such that one could justify a tactical alliance with Hitler against that enemy. Even to ally ourselves with Stalin, against Hitler, was and should’ve been an agonizing choice (though on balance the right one). My point was just that, if our grandparents could make that choice with Stalin, then surely we can make it with Rick Wilson!

  29. Douglas Knight Says:

    Could you switch your youtube link to this, so that we don’t have to watch an ad before the ad?

  30. Scott Says:

    Douglas Knight #29: Thanks, done!

  31. Michael Says:

    Well said! Also, I’m really excited for you post about independence of the continuum hypothesis 🙂

  32. Nick Says:

    anonymous #4

    > Trump’s foreign policy and his way of dealing with China are much better than any leader before him. I honestly view China as a much bigger threat to human rights and freedoms in the world, and as the biggest threat to the future of humanity and the US.

    Maybe China is the big threat, and maybe Trump’s approach has been more effective so far. But do you think that a civil war is going to put America in a better position to deal with China? If China is such an existential threat, wouldn’t the smart move be to find someone capable of uniting and guiding America, rather than someone with a pathological desire to sow fear and division?

  33. James Cross Says:

    I feel the same about the Lincoln Project.

    I don’t understand why Democrats cannot seem to run ads like theirs. There is a candidate here in Georgia Jon Ossoff who I am going to vote for, but he is running against David Perdue, a Trump apologist with all the trimmings, yet Ossoff’s ads make Ossoff himself seem like he is the generic liberal do-gooder fighting corruption. Others are running some more hard-hitting ads but Ossoff ought to be relentlessly dying Perdue to Trump instead he is leaving it to the voters to infer that there is some connection between Perdue and Trump.

  34. PeterM Says:

    Wow! I have been hoping since I found your blog is that on some day you write about Cohen’s forcing. Thanks a lot!

  35. David Speyer Says:

    I can’t find the source I am remembering, but someone did a study of which anti-Trump ads appealed most to people who describe themselves as Republican. The Lincoln project did not rank very high; the winner was Republican Voters Against Trump .

  36. Derek R Says:

    “I beseech you, in the bowels of Christ, think it possible that you may be mistaken.”

    — Oliver Cromwell

  37. Gnarles Says:

    Ira: “It’s hard to appreciate just how far-right the GOP has become. I don’t think there’s any parallel in any of the other advanced industrial nations”

    On what issue is this true? I feel like the only issues where this is true are probably gun control and healthcare, where they’re rather extreme libertarians.

  38. Jonathan Paulson Says:

    I think your values are right but your facts are wrong. We should take whatever allies we can get. But the Lincoln Project ads are targeted at highly-educated politically-engaged voters (particularly the president himself), so they look good to those people, but they are not actually effective at targeting the marginal voters in the election (this according to the political people I follow on Twitter).

  39. Yoni Says:

    Hi Scott

    Been going through some depression / suicidal feelings myself over the past several months (for mostly very different reasons). Feels crap. Hang in there man. Wishing you all the best and a shana tova.

    תכלה שנה וקללותיה, תחל שנה וברכותיה

  40. Nick Nolan Says:

    Direct quote from Eitan Hersh[2].

    If you feel unfulfilled, melancholy, paralyzed by the sadness of the news and depth of our political problems, there is an alternative: actually doing politics. Citizens who want to empower their political values would be better off if they spent less time consuming politics as at-home amateurs and instead fell in line to help strengthen organizations and leaders. Rather than kibitzing with their social media friends, they could adopt some of the spirit of the party regulars, counting votes and building interpersonal relationships in their neighborhoods.

    The ratio of time spent on political consumption versus action is off. Maybe spend 4-6 hours per month in actual political work withing political organizations and take that time from following political news.

    The book that really changed my politics consumption: Politics Is for Power: How to Move Beyond Political Hobbyism, Take Action, and Make Real Change by Eitan Hersh. “A condemnation of political hobbyism—treating politics like entertainment—and a call to arms for well-meaning, well-informed citizens who consume political news, but do not take political action.”

    A third of Americans say they spend two plus hours a day on politics. Of these people, four out of five say that not one minute is spent on any kind of real political work.

    Hobbyism is a serious threat to democracy because it is taking well-meaning citizens away from pursuing power. And the power vacuum will be filled.

    Concern is the informed citizen who is already spending significant time and energy on politics, but without serious purpose. More likely than not, if you are reading this, this is about you.

    Typical explanations for what’s ailing American democracy let ordinary, engaged citizens off the hook. Our own behavior demands at least as much reform as any political institution. [2]

    Hobbyism also cultivates skills and attitudes that are counterproductive to building power. Rather than practicing patience and empathy … hobbyists cultivate outrage and seek instant gratification.[3]

    racial minorities have long operated in tension with the well-educated, cosmopolitan wing of the party. It’s a tension between those who have concrete demands from politics and seek empowerment and those who have enough power that politics is more about self-gratification than fighting for anything. Only if you don’t need more power than you already have could you possibly consider politics a form of consumption from the couch rather than a domain of goals and strategies.[3]


    1. on The Ezra Klein Show.

    2. Boston Review article

    3. The Atlantic College-Educated Voters Are Ruining American Politics

  41. Mitchell Porter Says:

    I share the concerns of Elizabeth #13 #18 #21 that lockdowns have been used in excess, but also note the observation of Anonymous #14 that explaining this solely in terms of American politics, neglects the near-global embrace of lockdowns.

    There’s also the question, if things should have been done differently, how should they have been done; and whether this better way of doing things, relies upon knowledge of Covid epidemiology that didn’t exist, back when decisions first had to be made.

    My tentative model of how coronavirus political response has varied, is as follows:

    In East Asia, lockdowns were used to suppress the initial outbreak, then further outbreaks were fought with a combination of digital tracing and social adherence to public health measures.

    But in the west, the greater liberty and diversity meant that these follow-up measures couldn’t be used as effectively. So lockdowns went from being a short-term policy to prevent hospital overload, to being the main tool for preventing spread of the virus, until vaccines arrive.

    However, the populist right in the west has resisted this policy of recurring open-ended lockdowns. To some extent I think this is a matter of social class. The progressive middle and upper classes can do their white-collar digital work from home, but everyone else still has to step outside, to go to work (if they have work) and to buy essentials.

    Meanwhile we have also learned that the virus kills those who are already weakened by old age, or by certain forms of ill health. The populist critique of lockdown policy therefore asserts that society-wide lockdowns are inappropriate; and attacks the media for focusing on total number of coronavirus cases, rather than on the much smaller number of people who are seriously ill, as well as for neglecting details like the age distribution.

    However, is there a clear *alternative policy*? People point to Sweden. One hears that in Sweden, owing to a political idiosyncrasy, anti-Covid public health measures could only be recommendations, but not enforceable regulations. So possibly that is the essence of the populist/conservative alternative policy – you don’t build an anti-Covid police state, any more than you have an anti-flu police state during flu season. You recommend masks and social distancing, you take special steps to protect the especially vulnerable, and that’s it, until vaccines are available.

    I presume that somewhere, there is an actual informed debate going on, about the pros and cons of these different policies, that talks about herd immunity and post-Covid syndrome and so forth, and cites what we have learned so far about the disease. But I don’t see that debate in our mainstream western media yet, just cheering or booing policy according to political leanings.

    In this comparison of eastern and western responses, I should also mention how it is in the poorer countries, that can afford lockdowns even less. I have only been following India, where it seems that of necessity, public policy must now be about protecting the vulnerable and treating the ill, rather than preventing the spread. But surely many smaller countries will have to make the same choice. Policy in poorer countries is also probably influenced by WHO recommendations, but I confess I haven’t been following those.

  42. James Cross Says:


    Sweden’s death rate is bad and far worse than other Nordic countries. They have also changed how they are doing reporting so it isn’t even clear that they are anywhere near herd immunity. To top it all, their economy shrank enormously. This is hardly a model for other countries.

    The problem in the US has been erratic implementations of lock downs, different times and degrees for different states, and inadequate ability to test and track. The initial lock down, if it had been rigorous adhered to and followed with testing and tracking, would have been followed with COVID rates like Canada or Germany.

  43. Sniffnoy Says:

    Gnarles #37:

    Well, I don’t know what you consider “far-right”. The two examples you give are actually not what I would consider examples, because as you say, those are examples of them being libertarian! What I would consider notable and scary is not them being libertarian, but rather them being illiberal and authoritarian. (Of course, I’m not Ira who you responded to, but…)

    But, OK, that wasn’t my main point. My main point is that object-level, standardly-debated issues are not the main thing you should look at. You should be looking primarily at the meta-level!

    I’ve seen this a bunch in defenses of Trump and it always confuses me — “Oh, but look, Trump is correct on this point and this point!” And, sure, Trump has gotten one or two things right, basically by coincidence, but that just like… doesn’t matter. Getting particular things right by coincidence in the present is another way of saying, reliably getting most things wrong in the future. What we need is not getting a few things right now, but getting things reliably right in the long term. And to do that, you need a good process.

    And yeah, even the best process will occasionally turn up wrong results. But to trash a good process that gets one particular thing wrong, in favor of a bad process that gets that thing right, would be a terrible mistake. You get right the one thing you wanted, but you will get most things wrong in the future.

    So, when evaluating the Republican party right now, the appropriate question is not, what do they think of gun control, or what do they think of healthcare. The appropriate question is, what do they think of the process of liberal democracy that exists in the United States, including the constitution that does much to specify it? What do they think of the republic they’re named after? And the answer, it seems, is that they are actively opposed to it. They are throwing their support behind Trump, who is acting to undermine its legitimacy and dismantle it, and they’re assisting him in his doing so.

    (Occasionally one sees defenses of Trump on the basis that Trump can’t be opposed to democracy because he was voted in; thus he represents the will of the people, so anything he does is democratic. While this could be considered “democratic” in the narrow sense, it’s not what the United States is supposed to be and it’s not the sort of democracy worth defending. Such a defense is implicitly advocating for illiberal democracy, like you got in Egypt after the Arab Spring. Or to use an extreme example, an elected monarchy would be “democratic” in this weak sense, but would not even slightly meet the liberal-democratic ideals of the United States, and would certainly not be in accordance with the current constitution.)

    So frankly, I don’t particularly care what the Republicans think about gun control or healthcare! I care whether they believe in rule of law or rule of man, and right now they seem to believe in the latter. That’s the thing to look at.

    (Of course, this manifests in somewhat more “object-level” issues as well; in some cases the line is a bit blurry. E.g., an issue that’s come into the spotlight recently is police reform. This could be considered an object-level issue because it doesn’t directly deal with free speech or the democratic process; but it could also be considered a meta-level issue as it’s directly related to rule of law vs. rule of man. Naturally, the Republican party has come down on the illiberal, rule-of-man side. That’s the sort of thing I would give as a more object-level example of them being authoritarian or “far-right”.)

  44. anonymous Says:

    Some days I think that if Trump had acted in the complete opposite way, and went full panic mode before it hit anyone and went into lockdown, the democrats would call him a dictator that is limiting the freedom of people, that he was doing it for political motives and whatnot. I think the democrats would have opposed him for the sake of opposing him, and they would cry out about the economic ruin and they would have taken the current Trump stance, if Trump took their current stance.

    I know that from Israel, where we were initially lucky enough to have the corrupt leader pick the correct choice (in my opinion) of doing a lockdown and closing the sky, the media and the left fought his decisions so hard, that everyone ended up so brainwashed against the lockdown that today we’re entering a second lockdown because hospitals are about to reach their capacity.

    We made such an incredible u-turn that I can’t help but blame the media and the left for gaslighting and brainwashing everyone just for the sake of achieving a political goal of taking down a leader they don’t like. The magnitude of this gaslighting and brainwashing was so strong that it had managed to reach Bibi and the health minister themselves, causing them to reopen too soo and act recklessly, joining the dumb crowd in thinking the virus is gone and it was nothing and the lockdown was a mistake.

  45. Jelmer Renema Says:

    @ Nick Nolan #40:

    100% agreed. Engaging in actual politics is good for the soul, especially local politics. It gives direction to all the pent-up frustrations about how badly things are going, you meet people you have something in common with despite different backgrounds, and if you’re lucky you get to bring about some concrete, positive change in the world.

  46. asdf Says:

    Scott, are you trying to read Cohen’s actual paper about CH’s independence, or just trying to understand the proof from a more contemporary perspective? This talk looks pretty accessible to someone with exposure to other parts of logic:

    Re Biden, we’re in something like a hostage situation (Biden being the hostage taker). Maybe the best thing to do is give in to the demands / pay the ransom, but we shouldn’t pretend to like it. And flipping Republicans only helps if the move rightward doesn’t dissuade a greater number of potential Democratic voters, which it very well might. Finally, Biden is ahead in the polls now only because of Trump’s Covid-19 failures tanking him. At the time the DNC gave Biden the nomination, the pandemic wasn’t yet foreseen, and Biden had no chance of beating Trump. So if Biden is elected, he has the virus to thank for it. From a perspective that Trump is terrible for reasons separately from his mishandling of Covid-19, we have to see Biden’s installation as a monstrous failure of the Democrats, who were then saved by a near deus ex machina of the pandemic. I’ll look at the other comments later.

  47. Scott Says:

    anonymous #44: My Israeli wife has been following Israeli covid news every day since this started. She and I are extremely curious to visit the alternative universe you inhabit, the one where covid spun out of control there because the far left (the true power in Israeli politics right now) pressured the hapless Bibi into reopening too soon. Is there a portal to your world somewhere on earth, and can we access it in a socially distanced way?

  48. Scott Says:

    asdf #46:

      Scott, are you trying to read Cohen’s actual paper about CH’s independence, or just trying to understand the proof from a more contemporary perspective? This talk looks pretty accessible to someone with exposure to other parts of logic…

    Thanks! I had read many articles that cover things at a similar level of detail to that talk, never feeling like I came away with a good enough understanding. So I’m now working through the following:

    (1) Cohen’s own Set Theory and the Continuum Hypothesis (which I read 20 years ago as a student, but without nearly enough understanding—this time is going much better!)
    (2) Ken Kunen’s Set Theory: An Introduction to Independence Proofs
    (3) Dana Scott’s 1967 paper giving an alternative presentation of the independence of CH

    As I do this, I’ve been availing myself of ridiculously generous and patient help-via-email from Timothy Chow, the author of A Beginner’s Guide to Forcing. When there were points that Tim and I couldn’t resolve, we’ve consulted with Bob Solovay and Andreas Blass.

    I believe I now fully understand Gödel’s proof of the consistency of CH and AC (the constructible universe), and I’ve “mechanically verified” Cohen’s original proof of the consistency of V≠L, but without yet feeling like I could’ve come up with it myself. The latter, the “Feynman test,” is one obvious test for whether you understand something. Another is whether you can explain, clearly and in your own words, why all the obvious simpler approaches to the problem wouldn’t have worked, why each weird definition that’s introduced is actually needed.

    All else equal, of course I’d prefer to learn a more modern approach. The problem, I’ve found, is that the “modern” treatments consistently bury all the real action under a mountain of abstract machinery of Boolean-valued models and posets and Baire category, which I can nod along with and pretend to understand but actually don’t. Whereas everything Cohen writes is elementary and self-contained enough that, with sufficient effort, I can at least mechanically verify that it’s true, even if I still don’t deeply understand it. But I’m only halfway through Kunen’s book (not yet at the forcing part), so let’s see if that one helps me!

  49. fred Says:

    “If Churchill and FDR could team up with Stalin”

    The immediate result of which was the staggering loss of lives on the Russian side.

  50. Ethan Says:

    Scott #48,

    Any chance this new focus on the work of Paul Cohen and Godel would move you more towards Roger Penrose’s argument that consciousness is not algorithmic that builds, in large part, on the implication of Godel’s incompleteness theorems that “computation” (that Turing machines do and what formal mathematics is) and “understanding” (coming up with the Godel sentence “this statement cannot be proved” which is what Godel’s mind did) are different things?

  51. fred Says:

    Hey Scott,

    is CH related to the following (a while back I started looking into rendering fractal volumes when I ran into this)?

  52. Sniffnoy Says:

    fred #51:

    No, Banach-Tarski does not in any way relate to CH. Banach-Tarski is simply true in standard mathematics; CH is undecidable. Banach-Tarski does famously rely on choice, though.

  53. Scott Says:

    Ethan #50: No. That argument I do understand, and I think it’s wrong, and I’ve written in detail about why!

    The only way I can think of that CH might conceivably be relevant to that discussion, would be if human intuition had some reliable way to “just see” the truth or falsehood of CH, even while formal reasoning within ZF left the question open. But I don’t see that human intuition does have such a power. Sure, we can make guesses and express our opinions about the question, or try to introduce new axioms to resolve it, but then so could a suitably programmed machine—or at least, nothing that we know says otherwise.

    Note also that the independence of CH is not an example of Gödel’s incompleteness theorem (even though Gödel was involved with it as well): it’s a different independence phenomenon.

  54. mjgeddes Says:

    To umderstand logic, here is how I connect math to the physical world:

    Algebra = QM [Matter-Energy]
    Geometry= Relativity [Space]
    Logic = Statistical Mechanics [Arrow of time]

    I think that ;

    Algebra refers to a notion of symmetry (matter-energy),
    Geometry refers to a notion of locality (space)
    Logic refers to a notion of ordering or compositionality (time)

    We know that alegbra is the math of quantum mechancs, whereas geometry is the math of relativity theory. But how could logic be the math of statistical mechanics? It sounds very weird, yet if my conjecture is correct, it must be true, indicating that there’s something major we don’t understand about both the arrow of time and mathematical logic.

  55. Scott Says:

    fred #51: Banach-Tarski is an application of the Axiom of Choice, not of CH. But sure, it’s the same intellectual neighborhood (indeed, Cohen’s work also established the first models of ZF where AC is false).

  56. Ethan Says:

    Scott #52

    We will have to continue to agree to disagree on the topic :-). The more, for lack of a better word, “mature” I become in my personal and professional lives, the more Roger Penrose’s argument resonates with me.

    One minor clarification,

    “Note also that the independence of CH is not an example of Gödel’s incompleteness theorem (even though Gödel was involved with it as well): it’s a different phenomenon.”

    I understand that. I never got to read even sketches of Godel and Paul Cohen’s proofs on the CH independence with respect to set theorem axioms because I lacked the necessary background and I never got the time to build it. I did read a sketch of the proof of Godel’s first incompleteness theorem during my time in graduate school in my spare time and I found it fascinating.

    The context of my question, which you already answered with your reference to whether one day human intuition will have the ability to “just see” the truth of the CH or its negation, is whether learning how Godel’s prodigious mind worked by understanding his proof that the existence of a cardinality between that of the integers and that of the reals could not be constructed from set theory, would make you closer to the Penrose argument.

    The fascinating part of this conversation is that Godel and Paul Cohen represent the two predominant worldviews in mathematics: Godel was the ultimate Platonist whereas Paul Cohen -as far as I know- was the ultimate formalist.

  57. Scott Says:

    Ethan #56: While Gödel was indeed about as Platonist as it’s possible to be, I wouldn’t call Cohen “the ultimate formalist.” Cohen pointed out that a serious difficulty with seeing set theory as just a formal game, is that it lets you prove statements even in number theory that seem clearly true and that you couldn’t prove otherwise. And he repeatedly expressed the view that 2ℵ_0 might eventually come to be seen as “clearly” greater than ℵω for any of the usual ordinals ω.

    As for Gödel’s mind: yes, he was clearly one of the greatest mathematicians who ever lived. But thousands of people today understand all his main results as deeply as he did, and if he hadn’t lived, someone else would’ve proved the incompleteness theorem (indeed, Tarski, Post, Skolem all got pretty close to it) and the consistency of AC and CH not long afterward.

  58. fred Says:

    What’s interesting is what it all means in terms of the limits of mathematics to describe reality, or whether reality can be mathematical.

  59. mjgeddes Says:

    Fred #58,

    The way to ‘concretize’ math is to embrace radical probabilism, giving up the notion that we can be certain of anything. We tend to think of math as certain, whereas physics is empirical, but if we give up the notion of certainty, then the boundary between math and physics dissolves…. math becomes physical; physics is math, math is physics, and we see that reality is indeed mathematical.

    Matter, Information and Math are all physical. You are simply ‘looking’ along a conceptual dimension marking different levels of abstraction. So we go from physics, to computer science to pure math, but the subject matter of all of these is physical. Computer science is simply a generalization of statistical mechanics, and then there’s a further generalization to a multi-verse of ‘possible worlds’, which we interpret as ‘math’.

    The connecting ontological links:

    Stat Mechanics —> Probability Theory — > Computational Complexity —> Computational Logic —> Mathematical Logic

    You can see a smooth transition between all of these domains, with no clear divisions.

    I’m pretty sure that to a post-human super-intelligence, all of this would be obvious to it after about 10 seconds of contemplation 😀

  60. Ethan Says:

    Scott #57

    Thank you for the clarification. I was less familiar with the nuances of Paul Cohen’s philosophical stance towards mathematics. I had just heard him described as a formalist several times.

    “he was clearly one of the greatest mathematicians who ever lived”

    100 % agreed.

    “But thousands of people today understand all his main results as deeply as he did, and if he hadn’t lived, someone else would’ve proved the incompleteness theorem (indeed, Tarski, Post, Skolem all got pretty close to it)”

    Speaking from total ignorance about the works of Tarski, Post, Skolem you refer to and thus unable to judge how close they became to getting to Godel’s incompleteness theorems, one thing that is true of people like Godel, Einstein, Newton, Gauss or any of the hand picked people to which the word “genius” truly applies -the word “genius” gets overused these days in my opinion-, is that the result of their insights and intuitions is always obvious in hindsight to the rest of us. Last year, I read William Byers’ book “How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics” and I see it more representative of how these people explain the process of their thinking (particularly Godel and Einstein, both of whom spoke very openly about the role intuition played in their ideas) than a mechanical computer running software. Or put it differently, a la Penrose, I don’t think that what people like them (and in fact most run of the mill humans) do in their thinking can be accurately captured by a Turing machine running a program.

    fred #58

    My understanding is that Galileo was the first big name physicist to notice that nature spoke the language of mathematics. That idea, namely, that the laws of physics whatever they are in the grand scheme of things since currently there is a lot of debate as to whether a so called theory of everything that explains both the macro and micro observations is even possible, can be put in the language of mathematics was revolutionary at the time. When Godel came with his incompleteness theorems my understanding is that there was a lot of debate about what it would mean for mathematics’ ability to fully describe the world of nature. People in more recent times have preferred to move on and pretend that Godel’s incompleteness results never happened or that they don’t apply to attempts to model reality.

    Something that is definitely true, is that the world of modeling reality in general has moved away from understanding in he sense of proposing and solving differential equations by hand or numerical methods that discretize these such as finite element to mindless machine learning . There seems to be a larger trend of “less human thinking and more mindless number crunching” going on. To me that’s unfortunate, but that’s the way things are these days. In the world of high tech there is a larger debate going on about whether people ran out of ideas in the 1970s and we are just recycling them -ie, that there hasn’t been any genuine innovation since. This 2012 debate between technology investor Peter Thiel and then Google’s Chairma Eric Schmidt sets very well, in my opinion, the terms of the debate . I tend to side with Peter Thiel.

  61. Douglas Knight Says:

    research suggests that it’s the ads of Republican Voters Against Trump, not the Lincoln Project, that have been most effective

    Did anyone actually cite this research? I can’t find it.

  62. Sniffnoy Says:

    Ethan #60:

    My understanding is that Galileo was the first big name physicist to notice that nature spoke the language of mathematics.

    This is very much not true. Thony Christie talks about this a fair bit here.

  63. Ethan Says:

    Douglas Knight #61

    I am not going to read all that. Can you point me to the exact place where you think my contention is refuted and on the basis of what evidence?

    Galileo’s wikipedia page is not as definitive as my statement but it says

    “Galileo was one of the first modern thinkers to clearly state that the laws of nature are mathematical. In The Assayer, he wrote “Philosophy is written in this grand book, the universe … It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures;….”[123] His mathematical analyses are a further development of a tradition employed by late scholastic natural philosophers, which Galileo learned when he studied philosophy.[124] His work marked another step towards the eventual separation of science from both philosophy and religion; a major development in human thought”

    Priority disputes are a fixture of science and mathematics. I give you that this kind of discoveries don’t happen in a vacuum and that the social environment matters a great deal (it’s no accident that the development of general relativity and quantum mechanics both happened in the first decades of the 20th century for example; the 19th century is the closest we had had until these days to the widely shared belief among elites in the supremacy of scientific knowledge over all other forms of knowledge). Same with the development of calculus; both Newton and Leibniz developed it around the same time independently. But to say that Galileo was not a pioneer (if not the most important pioneer) in the mathematization of nature is a stretch; one of those conspiracy theories cranks like to create to live in their own alternative universe. Again, I look forward to what evidence you can provide that contradicts the above.

  64. Ethan Says:

    Douglas Knight #61. My apologies. My previous comment on Galileo was directed at Sniffnoy #62.

  65. Filip Dimitrovski Says:

    I’m not from the USA, but has anyone in this thread even managed to change the minds of pro-Trump (grand)parents or relatives?

  66. Sniffnoy Says:

    Ethan #63:

    Sure, I can point you to the particular paragraphs that discuss it. On the basis of what evidence? Um, I don’t know, Thony doesn’t go listing all his sources right there, I just assume he knows what he’s talking about because he studies this stuff. Obviously that won’t always be reliable, some things are contentious, but this seems pretty easy to disprove. He certainly gives plenty of names of examples that one can use as a starting point for looking this up; that should be pretty easy to verify.

    Anyway, the relevant paragraphs start from the one beginning “The reference to mathematics here…” and end with the one ending “…many decades before Galileo put pen to paper”. Hope that’s helpful.

  67. Scott Says:

    Douglas Knight #61: Sorry, that claim came from David Speyer #35, but he’s forgotten the source, and I’d forgotten that he’d forgotten. So I’ve withdrawn the claim unless and until someone tracks it down.

  68. Ethan Says:

    Sniffnoy #66

    I see. Let’s agree on one thing: history is always written by the winners and the Roman Catholic Church during Galileo’s times had lots of influence onto what was recorded in the history books. Galileo was a devout Catholic so there might be some bias with the version about who deserves what credit on Galileo’s observation.

    There is also the fact that once some big “insight” has been found in science, it’s easy to do some revisionism and re-interpret the past in much more favorable light.


    “Aristotelians that astronomical mathematical models are mere calculating devices and not models of reality”

    Did those people from the Middle Ages truly believe that those mathematical models were models of reality? Doubtful, given medieval mainstream thinking. Take the term Renaissance which is the movement that preceded Galileo and the ensuing scientific revolution

    “The Italian artist and critic Giorgio Vasari (1511–1574) first used the term rinascita in his book The Lives of the Artists (published 1550). In the book Vasari attempted to define what he described as a break with the barbarities of Gothic art: the arts (he held) had fallen into decay with the collapse of the Roman Empire and only the Tuscan artists, beginning with Cimabue (1240–1301) and Giotto (1267–1337) began to reverse this decline in the arts. Vasari saw ancient art as central to the rebirth of Italian art.[125]

    However, only in the 19th century did the French word renaissance achieve popularity in describing the self-conscious cultural movement based on revival of Roman models that began in the late 13th century”

    Does Giorgio Vasari deserve credit for recognizing the Renaissance as a distinct period of history or do the people of the XIX-th century deserve the honor?

    The insight that “nature speaks the language of mathematics” is something way more profound than using mathematics to model reality. So much so, that people who defend string theory these days as a theory of everything even claim that the lack of testable predictions specific to string theory does not invalidate the theory because it is such a beautiful theory, from a math standpoint, that it must be a true representation of reality even though we cannot test it!

    You are on much better ground with this, but only partially,

    “Giambattista Benedetti (1530–1590), who produced all of the arguments and thought experiments on the subject for which Galileo is famous. There is much more, which I have already dealt with in an earlier post and won’t repeat here.Galileo knew of all of this work”

    The Wikipedia page says

    “It is thought that Galileo derived his initial theory of the speed of a freely falling body from his reading of Benedetti’s works”

    This happens doing cutting edge science all the time. Researchers always build on the work of others, standing on the “shoulders of giants”. Mathematics has a long history of priority controversies. Back when I was a graduate student, this controversy was front page news . The math community eventually came together to give Perelman due credit, but the whole thing left him such a bad taste that he abandoned mathematics altogether, refusing a Fields Medals and, what might be more important for some people, leaving an award worth $1 million on the table!

    So I think that people might have reasons of their own for attempting to discredit Galileo’s pioneering work, but this is the first time I encounter someone who tries to convince me that he didn’t deserve the credit for his pioneering ideas. I accept he didn’t get his ideas out of thin air and that the environment he lived in was in part responsible for his work, but that happens all the time in the history of science. The real question is whether had it not been for Galileo said clarity would have become mainstream or would have it taken longer to develop. Newton was born a year after Galileo’s death. Did Newton get inspiration from Galileo? I am sure he did. Would Newton have been able to produce his laws of motion without Galileo preceding him? It’s very hard to make these history fiction exercises!

  69. Scott Says:

    Ethan #68:

      The math community eventually came together to give Perelman due credit…

    That’s a very strange way to put it! The math community tried to give Perelman both a Fields Medal and the first-ever $1M Millennium Prize, but Perelman declined them! His reason, to the extent he articulated one, had to do with what he perceived as a failure to credit Hamilton enough, not with any failure to credit himself. Though I still don’t understand why, if Perelman felt that way, he didn’t simply accept the prize money and then give Hamilton half (or whatever fraction he deemed appropriate)…

  70. Ethan Says:

    Scott #69

    I have never been in a position that I was offered a Fields Medal and will never be unless the IMU changes its rules because I am too old, so I can’t tell :-).

    I agree that Perelman’s stated reason for refusing for refusing the Millennium Prize wasn’t very convincing. He was probably giving some sort of polite “no”. In addition to following the controversy at the time, I watched this documentary a few years ago (in Russian with English subtitles) that was filmed some time after the controversy regarding both prices was behind and my take is that probably the guy is someone with extremely high standards and refusing the prize was his way of saying “my integrity is not for sale; I don’t forget what you guys tried to do”. But again, this is just me thinking out loud because if it’s normally difficult to get into the head of someone, getting into the head of Perelman is probably just as hard as proving the CH!

  71. william e emba Says:

    Kunen 2nd edition is much better than Kunen 1st edition regarding the minutia of forcing. In addition, he switched from the awful definition of forcing membership using a complicated induction to the more natural and obvious simultaneous induction on membership and equality together (an exercise in the first edition).

    You may wish to check out Shoenfield “Unramified Forcing”. This was the original description of the poset method of forcing in full generality. His notation is unreadably ancient, but the proofs hit all the high points and should prove to be very helpful as you use other sources for the details. An equally clipped but modern write up is in Friedman’s Class Forcing book. (Don’t try to read the rest of the book, as you may be permanently discouraged.)

    I disrecommend any approach using Boolean-valued models. The idea is incredibly intuitive from a distance, but the details are a nightmare for beginners, and in practice there are very few natural Boolean algebras available. In the case of \(\neg\)CH there is actually a natural Boolean algebra involving the product measure algebra for \(2^{\omega_2}\). In addition to Scott’s paper, see Manin Introduction or Takeuti Two Applications for details. But if measure theory is not something you are comfortable with, this won’t help you.

  72. william e emba Says:

    I think a good way to understand forcing is to use Cohen’s original intuition as a guide. He conceived of forcing in “proof-theoretic” terms. Deductions from ZF or ZFC were to be entangled with extra information, and this extra information, if planned for properly, led to \(\neg\)AC or to \(\neg\)CH.

    This never went very far—Cohen himself switched to models very early—but it is still a good intuition to keep in mind. One does ordinary mathematical reasoning aka “proofs” on specific models. But in addition to the usual rules for what proves what inside \(M[G]\), one has extra rules relating \(M\), \(\bf P\), \(G\) and \(M[G]\).
    Did you ever learn semidirect products in group theory? It’s not found in most beginning textbooks. I suspect most first-timers are somewhat shell-shocked. The raw definition is quite unintuitive. Yet it’s actually a simple concept, what must be true given a group \(G=HN\)

  73. william e emba Says:

    (accidental premature posting continued)

    with \(N\) normal in \(G\). You have things on the \(H\) side are trivial, and on the \(N\) side they are all twisted up. But working through several examples makes it work.

    So too with forcing. You have ordinary deductions on one side, and then some oddball deductions on the other involving names and conditions. Give yourself a metaphor, say “scores” or “tolls” or “comments”, being applied to every deduction, and as with semidirect products for groups, one side will be completely ordinary, and the other side will be twisted.

    But the result is that proofs regarding \(M[G]\) are simply proofs in \(M\) with these twists added. Cohen’s original intuition in order to prove \(\neg\)CH was to introduce tags into the usual proofs. Quite simply, proofs follow a certain algebraic pattern, and Cohen respected that pattern but allowed for refinements.

    On a first go-round, perhaps skip \(\neg\)AC. It’s certainly way cool, but it’s not been as significant in the long run, and it’s an additional layer of complications.

  74. william e emba Says:

    Perelman declined the Fields medal on the grounds that the “community” (aka two Chinese mathematicians) tried to rip him off. He declined the Millennium Prize on the grounds that Hamilton deserved significant co-credit.

    Personally I think he’s as Aspie and just not into that kind of stuff, so looking for his “real” reasons are missing the point. He resigned his academic position at St Petersburg when he learned that he had been given a raise without his permission. He said he couldn’t trust people who would not keep their word regarding his contract. To him, the world must be perfectly logical and deductive, otherwise he will not participate.

  75. Tamás Varga Says:

    From the lecture notes:

    From the standpoint of decoherence theory we can say that an event has
    “definitely happened” only if there exist many records of the event spread
    through the environment, so that it’s no longer feasible to erase them all.

    This is analogous to what happens in a blockchain: a (payment) block is considered definitive whenever so many other blocks have been added after it in the chain that makes it infeasible for an attacker to go back and modify the block. (However, just like in decoherence theory, it’s never 100% impossible.)

    Maybe Satoshi got his inspiration from quantum theory? 🙂

  76. fred Says:

    There wouldn’t be mathematics without the human brain.
    Brains have evolved to be compressions of their environment, by finding correlations between patterns of sensory data, all used to come up with probability models (simulation of the environment) that increase likelihood of survival through reproduction.
    The richness in the brain’s basic neural structures allowing for that “compression” (aka models) is what allow for “mathematics” when the patterns of correlation are so clear and pure that the associated probability model reduces to total certainty.

    E.g. there had to be a direct isomorphism between Ramanujan’s brain neural structures and the structures of numbers, giving him incredibly deep and intuitive insights into number theory.

    So the limits of mathematics are really the limits of the connections inside our finite brains.
    Once our machines will hold inside them patterns that our limited human brains can’t hold, will it still be called mathematics?

  77. James Says:

    Scott, on the topic of BPP=BQP, have you read May and Schlieper’s preprint ( It seems you can tweak their argument slightly to put both Shor’s algorithms (most naturally the discrete logarithm one) and general quantum circuits in BPP.

    May and Schlieper’s argument is that Shor’s algorithm works even if the third register is universally hashed to a small number of qubits (they use just one). The problem they ran into is calculating the hash functions for a Mosca-Ekert circuit (which they use to reduce the qubits for the first and second registers to a constant. They needed a homomorphic universal hash function to put factorization in BPP.

    You can avoid the need for intermediate measurements as with the Mosca-Ekert by using circuits that extract just the period modulo a small prime number. The requirement in Theorem 6.1 that the hash function be universal can be dropped (it just increases the range of probabilities). Sticking to Theorem 6.1’s quantum circuit template, each multiplication can be hashed so that errors build up per layer in the circuit. Unitarity means that the growth in errors is at most additive and the number bins for the hash function can be chosen to be large enough that the errors in the final state are small enough to extract the period.

    And lastly, the specific template of the circuit can be dropped and qubits can be unwrapped out of and wrapped in to the hashed qubits. Because of unitarity, errors increase only additively and, using Gaussian Belief Propagation, you can show the time complexity is something like (g^2/epsilon) log g, where g is the number of gates and epsilon is the error in amplitude.

  78. venky Says:

    +1 for Scott’s #48 shoutout to Prof. Blass, whose two-year graduate sequence on logic, recursion theory, set theory, and infinitary combinatorics is a lifetime experience. As Scott says, understanding logic is fundamentally a psychological endeavor. The professor’s mastery of the subject and his kindness, which gave him an acute understanding of your psychological confusions, made the entire experience most beautiful and moving. He would wave his hands at you, and magically, the impossible would become trivial. He called it proof by hand waving. Scifi for real!

  79. Filip Dimitrovski Says:

    24 out of the 35 living Turing Award winners just endorsed Biden. And yes, the list includes Donald Knuth. The statement does not mention Trump at all.

  80. Scott Says:

    fred #76: It’s true that Ramanujan and Hardy used their amazing pattern-finding brains to discover that the number of partitions of n grows like \( e^{\pi \sqrt{2n/3} \). Once they did, though, we can see that it’s true, and we can see that extraterrestrials (or AIs) with completely different brain structures would’ve found the same formula if they asked the same question. So yes, I’ll still call what the machines do “mathematics” if it has that same property.

  81. Scott Says:

    James #77: Implement it, use it to factor 2048-bit RSA challenge numbers, and then you’ll have my full attention 😀

  82. Ethan Says:

    fred #76 and Scott #79

    Mostly so that those who are reading can have a different view -at this point my experience interacting with people who believe that all the capabilities of the brain can be captured by a Turing Machine is that they take this as a dogma- here comes the Penrose-Lucas argument against such proposition . The key point is that consciousness -the quality that humans have that everybody agrees is unique to humans- has not been shown to be algorithmic under the rigors we expect from mathematical proofs. Some people believe -I stress “believe”- that with a large enough computer that mimics the brain structure, consciousness will “emerge”. I don’t think that will be the case, but this is where reasonable people can reasonably disagree.

    Since this is a controversial topic, it’s a matter of taking sides, with both sides having solid people defending their point of view. I take Penrose’s side.

    There are real world implications with taking sides. Since I am with Penrose, I don’t think we will ever see a machine passing the Turing test in a meaningful way as long as computing is Turing computation (if I understand it correctly Turing computation captures what quantum computers do; Scott could correct me if I am wrong).

    Thus, not in the least bit worried about machines taking over humans although I do believe that we won’t see a third AI winter. There are a few applications for which machine learning has been shown to work extremely well by brute forcing with massive computing and massive amounts of data constructs that are decades old (I am thinking particularly about neural networks) but they won’t be fully replacing humans any time soon. The most viable applications will be in the realm of “assisting humans” which is what computers have been doing since the 1950s anyway first with business applications, then with solving complex modeling problems with super computers, etc.

    Unfortunately, the crowd that believes that brains are Turing machines have supporters who are very successful financially and are able to pass for “breakthroughs towards AGI” things that are merely well known architectures on steroids. I am talking about Elon Musk and all the hype that followed the release of GPT-3. The hype lasted about one week until people got to task to show that GPT-3 is the same mindless machine learning technology we know but with more data and compute for training. A software engineering marvel, but no genuine breakthrough towards AGI.

  83. Thomas Larsson Says:

    James Cross #42
    Sweden certainly made a number of mistakes when the pandemic surfaced in March, but during the summer the number of cases has been low, and on 9/11 the US surpassed Sweden in the number of Covid deaths per capita. On 9/15 the US also surpassed Italy, and it is schedule to overtake the UK in two weeks. So these days the US is a worse hellhole for Corona than Sweden or any other European nation.

  84. fred Says:

    Scott #80

    it’s true too that one quality of mathematics (and more generally modeling the world) is compactness and then there’s also the amazing plasticity of the brain.
    It takes one very special brain (or network of brains working in tight cooperation) to “grow” the necessary neural structures to come up with a clear proof (or theory) to some of mathematics deepest mysteries.
    But then the results can be “uploaded” on a much bigger set of “common” brains (e.g. thousands of grad students learning GR every year).
    One doesn’t need to go through every single insight that lead to the rise of Quantum Mechanics in early 20th century to use the results of QM.

    It’s the same with human language, which has probably evolved over thousands of years, but it takes just a few years for any human child brain to master it.

    When it comes to limits, it’s likely that humans that are already very good at abstract thinking will soon want to extent their brain hardware with machines (in terms of increasing the interface bandwidth, like with Neuralink, etc).

  85. fred Says:

    Ethan #82

    I don’t think that consciousness is a necessary ingredient of computation, or even has anything fundamental to do with computation.
    I’m saying that from a direct experiential point of view (available to anyone who’s ready to spend some minimum effort on introspection of how one’s brain works).

    My thoughts just arise in consciousness, they’re not driven by consciousness, i.e. my conscious experience has absolutely zero insights into how my thoughts come to be (it’s not always obvious because our regular state of consciousness is to be “lost in thoughts”, where “we” identify with them, but one can carefully deconstruct that process).
    That’s also why people talk about “instinct”, “gut feeling”, or “it just appeared” when trying to explain how they came up with a particularly powerful idea or a beautiful song, etc.
    And when we consider the subset of thoughts dealing with mathematics in particular, everything that’s true is relying on long logical chains of air-tight reasoning that leaves nothing to chance (or choice, whatever “choice” even means in a world that’s deterministic).

    A child doesn’t need consciousness to carry out a multiplication – we shouldn’t confuse the feeling of what we refer to as “conscious effort” with some actual cause for the motions of the atoms in our brains and bodies – this has more to do with the brain trying to assess at every moment in time whether the actual result of its actions is matching some expected result coming from its models of the world and itself (that “matching” is itself a thing that appears in consciousness, but it’s not driven by consciousness), and the fact that our unconscious brain is comprised of many computational modules all vying for resources based on some fluid rules of priorities (that competition itself appears in consciousness as a feeling of “attention”).

    And I don’t think that anyone would seriously claim that AlphaGo-Zero had to “grow” consciousness in order to beat the top human Go master, and similarly that world top Go master would probably have a very tough time explaining how his own moves unfold during a game (at best he will say that moves come from a mix of learned rules and ‘gut feeling’).

  86. fred Says:

    Scott #57

    “As for Gödel’s mind: yes, he was clearly one of the greatest mathematicians who ever lived. But thousands of people today understand all his main results as deeply as he did, and if he hadn’t lived, someone else would’ve proved the incompleteness theorem (indeed, Tarski, Post, Skolem all got pretty close to it) and the consistency of AC and CH not long afterward.”

    That’s ignoring one important ingredient:
    The right sort of brains like the ones of Godel, Tarski, Post, or Skolem can only exist if the total population of brains is large enough.
    With a current pool of 8 billions brains, the chance is quite high for any effective discovery to be done multiple times (we only know what we’ve discovered, we’ll never know what we’ve missed, haha), but in a world of just one million brains, it’s way harder.

    (that’s ignoring the fact that when the population is small, it’s much harder for a anyone who’s driven to actual do what they love for a living).

    Which begs the question – a 13 billion light year bubble of universe (not just ours, but any) is only big enough to evolve some “density” level of intelligent structures, which could be on average too low to discover “reliably” some really advanced ideas or technologies: AGI, intergalactic travel, unification of gravity and QM, or … simply just surviving longer than a few dozen thousand years.

  87. Ethan Says:

    fred #85

    We definitely have diverging views.

    I think that the non algorithmic nature of consciousness -which I take as a given all while acknowledging that there is controversy about this topic- is one reason why machines will never achieve anything remotely close to a human like intelligence, which is not to say, as I have said previously, that Artificial Intelligence / Machine Learning -AIML- techniques aren’t useful. They are, but all good uses of technology start with the premise of deeply understanding the limits of the technology at hand. Take using AIML for a task such as image recognition/classification. Not every problem is like the canonical “cat detection problem”. To the extent the problem is like a “cat detection problem”, then I think that AIML is justified provided that the use doesn’t involve violating anyone’s rights. Using AIML image recognition to decide whether someone has a tumor by inspecting an MRI image is more problematic. In these later cases, I don’t think we will ever see the human completely removed from the decision making process. AIML will help in simplifying the life of the radiologist but it won’t replace the radiologist.

    Something I strongly disagree with as well,

    “And when we consider the subset of thoughts dealing with mathematics in particular, everything that’s true is relying on long logical chains of air-tight reasoning that leaves nothing to chance (or choice, whatever “choice” even means in a world that’s deterministic).”

    I couldn’t disagree more. Anyone who has done math at the professional level (I have) will tell you otherwise, starting with the fact that even getting to the point of axioms that are widely agreed upon as true by professional mathematics is not a straight forward process. A big chunk of the conversation in this thread is about the so called Continuum Hypothesis which states (wikipedia definition) “There is no set whose cardinality is strictly between that of the integers and the real numbers.”. The joint work by Godel and Paul Cohen showed it to be independent from a set of axioms called ZFC. These axioms were proposed in the early XX-th century in an attempt to fix ongoing -at the time- problems with mathematics such as Russell’s paradox You can get a set of mathematical results by considering ZFC and the CH true, and you can get a different set of mathematical results by considering ZFC and the negation of the CH true. Since it is obvious that either the CH or its negation is true, one of the two sets will be false. Which one? Who knows! Note that mathematicians had been doing math for millennia before encountering those difficulties. The work of mathematicians involves working on questions like these. So far, only humans have been shown to produce original mathematical results. Then once humans produce them, you can program computers to execute programs that leverage them but I have yet to learn about a computer that has produced research mathematics of the kind professional mathematicians make a living on.

    “And I don’t think that anyone would seriously claim that AlphaGo-Zero had to “grow” consciousness in order to beat the top human Go master, and similarly that world top Go master would probably have a very tough time explaining how his own moves unfold during a game (at best he will say that moves come from a mix of learned rules and ‘gut feeling’)”

    Go, just as chess, are finite search problems -they don’t involve reasoning with infinite sets- that computers can solve. Some humans have intuitive ways to do things that computers such as Deep Blue and AlphaGo do on a more mindless way. Sure, but the converse hasn’t been shown to be true, namely, that computers can produce every single result that humans produce with intuition. Reasoning about infinite sets is one area where I can’t see computers -as long as they implement Turing computing- ever matching humans.

    Unless you have something more substantive to add, I will leave it here agreeing to disagreeing. This is a debate that has gone on for decades among the world’s top minds and we are not going to settle it here. I am very sure you are very convinced of your views just as I can assure you that I am convinced of mine. As I said, I operate in daily life under the assumption that the Penrose-Lucas argument is true and I don’t fear current computers implementing Turing computing reaching the so called “Singularity”, an event that some have described as the tech elite’s equivalent of “The Rapture”. Some in the tech elite believe in the Singularity with the same fervor some fundamentalist Christians believe in the Rapture. I believe in neither even though I am a Christian and a practitioner of technology.

  88. Douglas Knight Says:


    Galileo appears to me to be a revolution in empirical science, actually measuring numbers, such as the gravitational constant or the period of the pendulum.

    Of course, Archimedes said that (terrestrial) physics was written in mathematics and everyone agreed with him after his translation c1300. He had numbers and it’s tricky to say that Galileo had anything that Archimedes didn’t, he just got the ball rolling again. Maybe Simon Stevin had some more incremental extension of Archimedes, but I don’t know. Also, there was lots of new math in astronomy and geometric optics, such as perspective and Snell’s law. But these three topics seem to have been separate magisteria.

    Christie mentions the Oxford calculators, who studied the pure mathematics of the effect of constant acceleration. But they appeared to have done this because it was the simplest case, not because it was physically relevant. (Their idea that we should study velocity at all is a breakthrough in theoretical physics, but I don’t think they measured any examples.) Giambattista Benedetti had thought experiments showing that falling should depend only on density, not weight (on intensive properties, not the arbitrary ontology of extension). Various people did experiments showing this, and the further fact that it doesn’t depend on the density, either. But these were experiments towards a qualitative conclusion, without numbers to measure. In a further link, he mentions that Domingo de Soto, famous as an economist, not a scientist, asserted that objects fall with uniform acceleration. How much credit should we give this? Lots of people made assertions about how objects fall that could be ruled out by pure thought (such as exponential growth), but which they failed to rule out. I guess uniform acceleration is the simplest hypothesis that explains the ability to throw objects upward. It’s great that people did all this with mathematics, but I believe it was only Galileo who went on to numbers.

  89. mjgeddes Says:

    Part of the reason why mathematical logic seems so hard and airy-fairy perhaps, is because there’s no obvious physics connection for the really abstract stuff. So one might be led into thinking that it doesn’t have the same status as algebra and geometry, where there are clear physics applications. One might be tempted to think of logic as merely an invented language. That’s a reasonable position to take, as far as I can tell, Scott did take that position. Until recently, I would have thought the same thing. But I now think I was very very wrong.

    As I mention above, I now suspect that there’s a major connection between logic and physics that hasn’t been understood yet, something that when we do properly understand it will be a scientific revolution on the magnitude of the discovery of relativity or quantum mechanics! So I’m suggesting that there might be a *huge* explosive insight about a physics application for mathematical logic yet be discovered.

    We’ve seen this story before. Abstract mathematics which at first seemed to have nothing to do with the real world, later turned out to be crucial to physics at a fundamental level, examples being Riemannian geometry for relativity, and complex numbers for quantum mechanics. Might I suggest that sceptics about the platonic nature of logic failed to learn the lessons from that?

    Let’s go back to statistical mechanics and the arrow of time. Now stat mechanics as understood today uses geometry and analysis, but I’m suggesting a switch to mathematical logic 😉 At first it sounds crazy. What could set theory and category theory possibly have to do with statistical mechanics and the arrow of time? Something for you to think about…

  90. Martin Mertens Says:

    Hi Scott, I’m curious. When you want to understand a result like the independence of the CH do you read the original paper, a chapter in a textbook, or something else?

  91. Scott Says:

    Martin #90: As I said before, I’m reading Cohen’s original book, but also Kunen’s book, and also a bunch of articles that I find online, and I’m also emailing experts. But all of that is only preparatory to the real labor of understanding, which takes place with pen and blank paper.

  92. JimV Says:

    “Go, just as chess, are finite search problems -they don’t involve reasoning with infinite sets- that computers can solve. Some humans have intuitive ways to do things that computers such as Deep Blue and AlphaGo do on a more mindless way.”–Ethan @ #87

    According to my reading about AlphaGo, it used two neural networks, each of about 225,000 nodes. It used a Monte Carlo Tree search to look ahead at the results of possible moves, with the evaluation of the results (which projected position “looked best”) done by one of the neural networks and the choice of a move given a position by the other neural network. The networks gained their abilities by trial-and-error training–the more training, the better the neural networks worked. Since neural networks are based on biological neurons, it seems to me to suggest that we get our intuitions as to good and bad positions and best moves the same way. Why not?

    By the way, neurons and consciousness are not unique to humans. Flat worms have about 200 neurons, and are capable of memorizing mazes between them and food. Cats have about 250 million, and dogs up to about 500 million. Dogs can learn up to 500 command words (not all dogs). Humans have about 100 billion neurons. Cognitive ability in animals seems to vary directly with the number of neurons.

    I believe the basic algorithm by which intelligence works is trial and error, plus memory, which is how neural networks are trained. I could find examples from this site of mathematicians trying one approach, and then another, and then another before finding a proof. See Einstein’s Zurich Notebook for another example. It is also how biological evolution produced us and our nanotech brains.

  93. Jelmer Renema Says:

    @Douglas #88: I think you put too much emphasis on measuring numbers. Lots of early math and physics was done with ratios instead. I can think of a few reasons for this:

    First of all, it’s how the Greeks did it, because of their emphasis on geometry. If you read Eucid you notice that everything that isn’t geometry is formulated as geometry and lots of the geometry is formulated in terms of ‘construct a line segment that stands to a given line segment as such-and-such’. For lots of practical things that you’d want to use geometry for in the ancient and medieval world, like dividing a field into equal-area parts for an inheritance or setting out right angles for a street pattern, you don’t actually need units.

    Another good reason is that there wasn’t a shared system of units, so how would you even communicate the proportionality constant? If you can’t, the statement ‘x = 1/2 gt^2’ is just as good as ‘the proportion between falling time and distance is as 1 to 1, 2 to 4, 3 to 9’, which I think is where the oxford calculators got to. Further evidence for this is that values for pi were communicated, which being a dimensionless number you can do without reference to a system of units.

    Finally, just doing ratios allows you to do many more experiments, especially if you don’t have access to tables of physical or material constants or accurate measurement devices (don’t forget instrument making as a profession wasn’t a thing until Early Modern at the earliest). For example the way they tried to measure kinetic and gravitational energy was by dropping hollowed-out lead balls of different weight into wet sand from different heights and measuring the depth of the craters. There, the depth of the actual crater doesn’t matter if you don’t know all the details of the sand; all the physics is in the ratio instead.

  94. anonymous Says:

    Scott #47: You must have missed the news and the media in May. There was a powerful brainwashing that the coronavirus is over, that the lockdown was an overreaction, that everything health experts say is exaggerated and overreaction. You had news reporters grilling and criticizing health experts on prime time TV how their COVID models never happened and the hospitals never collapsed (well duh, because of the lockdown).

    You had 12 news giving prime time to Yoram Less, a complete madman that says coronavirus is just the flu, no lockdown is needed and that the government should ignore it completely, and instead of grilling him with facts and proving him wrong the news reporters gave him passive interviews saying “oh well if you say so”. Just like Fox News allow coronavirus deniers, the news were full of them in May, all the news, not just the left but the center and mainstream too.

    We were nearly done with it, nearly managed to contain it to know local transmissions, and at this point the pressure from the media won, and the government decided to reopen everything instantly without waiting, and you can look at the graph from May and see that we never recovered from this.

    Maybe it’s hard to get a sense of the Israeli mainstream consensus from there, but as someone living here it’s clear that the media is responsible for the rushed and reckless reopening. There was incredible pressure of false information.

  95. sc Says:

    mjgeddes #89: Still waiting for the breakthrough revealing that a quantum level we really live in a p-adic (non-archimedian) world. Looking forward to physicists trying to *measure* that p and find out it’s like, 31 or something. Would make an interesting hard sci-fi story imo

  96. Nancy Lebovitz Says:

    Thank you for supporting Republican Voters Against Trump. I’ve contributed $50.

    I wish RVAT got more attention.

    I’m sorry to say I don’t think they’re getting attention on the left because if my sample of left-wingers is typical, they don’t want to think of Republicans as human beings with normal motivations and a capacity for learning.

  97. Scott Says:

    anonymous #94: What you describe doesn’t sound to me like a left vs. right issue at all, but just a media stupidity issue.

  98. Ethan Says:

    fred #85

    One last clarification that in my mind is implicit to the following but that might not have been clear to other readers:

    “Go, just as chess, are finite search problems -they don’t involve reasoning with infinite sets- that computers can solve. Some humans have intuitive ways to do things that computers such as Deep Blue and AlphaGo do on a more mindless way. Sure, but the converse hasn’t been shown to be true, namely, that computers can produce every single result that humans produce with intuition. Reasoning about infinite sets is one area where I can’t see computers -as long as they implement Turing computing- ever matching humans.”

    In every case where some “big breakthrough” towards Artificial Intelligence has been claimed by the believers in “brains are just Turing machines” proposition such as the aforementioned (Big Blue, Alpha Go, to which you can add GPT-3), many very smart humans have been involved in designing and building the “magical gadget” that beat the lone human in his own game with a sophisticated mindless computer (playing chess, Go). How so many people fall for the fallacy “now that the magical gadget has been built by an army of super smart humans, suppose that the gadget exists in isolation; isn’t it amazing that AlphaGo was able to beat a Go master?”. Well, it isn’t. In addition to designing and building the magical gadget, we needed armies of super smart humans pushing Moore’s Law to its limits, coming with innovative computing architectures such as GPUs -that were used in the training of AlphaGo- and as I said, coming together to design -designing is a distinctive human endeavor- AlphaGo.

    So if you think about AlphaGo this way, what you see is a zillion humans who were required to beat the mental abilities of a single human, leveraging decades of advances in semiconductor technology and computing innovations to build a gadget that encoded all these advances plus the lessons of the best Go players in history. How is that some people believe that this is equivalent to “some gadget magically appeared out of thin air that beat a Go master” is beyond me, but I guess that’s why we are having the debate. At every step I see human ingenuity and creativity playing a significant role in building these so called “AI systems”. Some prefer to see only the final gadget these humans produced. It seems an instance of “when a wise man points at the moon, the less wise sees only the finger”.

    Take GPT-3. In the last 2 weeks, The Guardian both tried to scare readers with the alleged magical abilities of GPT-3 and then publish an op-ed that clarified that humans were heavily involved in the process of writing the op-ed we should be scared of, check .

    As I said, I don’t think I have anything else to add to the debate but I wanted to clarify the above in case it wasn’t obvious.

  99. Ethan Says:

    JimV #92

    Artificial neural networks are mathematical devices known as “non linear filters” whose coefficients are computed via training on data. They were first proposed in the 1940s. Their theoretical limits have been studied in detail by the experts during this time and are well known.

    The current explosion in AI was triggered by the empirical observation, as a result of this work , that their performance in the narrow tasks neural networks are good at -generalizing to new data that has the same statistical properties as the data used to train them- increased with the amount of training data and the number of parameters and that there didn’t seem to be a limit to how far you could go by adding more training data and more parameters (unlike other machine learning architectures whose performance tends to plateau after reaching a certain amount of training data irrespective of increasing the model’s size).

    I stand by contention that we won’t see anything remotely close to “consciousness” emerge from artificial neural networks irrespective of their size. Seeing artificial neural networks as equivalent to biological neural networks in every aspect -and making arguments of the type “if one day we build a computer that has as many neural networks as a human it will become like a human”-, is a belief, not something that can be concluded from the theoretical study of artificial neural networks. It’s a very popular belief among many in computer science -although if you listen to Donald Knuth here you will likely conclude he disagrees with such belief although the topic isn’t explicitly discussed- but it is just that, a belief, not a scientific fact.

  100. I Says:

    Scott #91:

    “… the real labor of understanding, which takes place with pen and blank paper.”
    Do you mean producing novel proofs? Also, do you conjecture related statements and prove them as you read or do you do that after you’ve read everything?

    JimV #92:
    Intelligence does vary with number of neurons, but its the number of cortical nerouns that are important. Packing density, conduction velocities and inter neuronal distance also matter. Here’s a good review on the modern perspective:

    mjgeddes #89:

    Do you mean something alike Boris Zilber’s program where you derive geometry and physics from logical principles e.g. starting with uncountably categorical theories? John Baez has an article on this:

  101. Douglas Knight Says:

    Jelmer Renema,
    Sure, ratios are numbers, which is why I mentioned Archimedes. But I don’t believe any such experiments were done before Galileo.

    ‘the proportion between falling time and distance is as 1 to 1, 2 to 4, 3 to 9’, which I think is where the oxford calculators got to.

    To repeat myself: No, they didn’t. The Oxford calculators said nothing about falling objects. They certainly didn’t measure it. Maybe de Soto asserted this, but he didn’t measure it.

    For example the way they tried to measure kinetic and gravitational energy was by dropping hollowed-out lead balls of different weight into wet sand from different heights and measuring the depth of the craters.


  102. Scott Says:

    I #100: Given how well-developed the theory of forcing now is, I have zero ambitions to prove anything that wouldn’t be obvious to the experts. But yes, understanding a proof doesn’t just mean mechanically verifying the steps; it means learning your way around the whole space of possible decisions to make—e.g., what would change if I repeat what Cohen is doing with random reals rather than generic reals? do you even need forcing to prove the Powerset axiom in the new model M[G], or do you only need it for the Replacement axioms?

    A major problem is that mathematicians have a deeply ingrained culture of covering their tracks—e.g., presenting a complicated definition without explaining the tree of decisions and false starts that led to that definition, which is the part that’s actually needed for understanding. (Cohen is better than most in this respect, but only somewhat.) So you often need to reconstruct the decision tree yourself, and you need to do it as you go along.

  103. matt Says:

    Scott: “A major problem is that mathematicians have a deeply ingrained culture of covering their tracks—e.g., presenting a complicated definition without explaining the tree of decisions and false starts that led to that definition, which is the part that’s actually needed for understanding. (Cohen is better than most in this respect, but only somewhat.) So you often need to reconstruct the decision tree yourself, and you need to do it as you go along.”

    Um, computer scientists can be equally guilty of this. They actually do it in a marginally different way: the bulk of the paper consists of trivialities with the key new insight buried deep in section IV.C.3.a.7 or something. It’s like they are writing a program in an overly high-level language: all the actual work occurs in some libraries or in some code inside one of the objects, but what you first see is just some very abstract calls. The physics approach to writing is sometimes decried as “stream of consciousness”, but it’s the best at evading this problem, I think.

  104. Scott Says:

    matt #103: While no field is immune from obscure writing, I’ve personally found the average CS theory paper far more readable than the average math paper (well, maybe that’s why I went into CS theory in the first place 🙂 ). Actually, one of the main difficulties I’ve had with presentations of forcing is exactly what you said: there’s page after page of verification of obvious-seeming properties of the forcing relation, and then suddenly a few confusing pages after which the independence of CH is apparently proved. Buried in this is how the heavy lifting got accomplished, which is similar to asking: which “obvious” properties are not true of the forcing relation, and what would’ve broken had the relation been defined slightly differently?

  105. Ethan Says:

    Scott #102, matt #103:

    Have you guys considered the possibility that there is no actual “decision tree” in the formalist sense in most cases and that most mathematicians -since Platonism is the majority view among them- come up with those ideas through a process that could better described as “revelation” and that after the are convinced of the truth of the revelation they work the result backwards from known results?

  106. mjgeddes Says:

    I #100,

    I’d be looking at something more like Baez’s ideas for using category theory to better understand classical physics:

    Mathematical logic looks me to be somewhat analogous to classical physics, the heart of which would be statistical mechanics.

    John Baez knows something. He’s just not telling us 😉

  107. Scott Says:

    Ethan #105: I also prove theorems for a living—not at the level of Paul Cohen, but enough to know that there’s always a decision tree. Even (say) Terry Tao and Michael Freedman, when I’ve discussed problems with them, were trying out ideas, rejecting ones that didn’t work, etc., “merely” better than me, not (so far as I could tell) receiving revelations from a burning bush. 😀 And I read Cohen’s retrospective essay about forcing, where he explained some of the branches of his own decision tree, and that helped me enormously—and if he’d explained 10 times as many branches, it would’ve helped me 10 times as much!

  108. Ethan Says:

    Scott #107

    “I also prove theorems for a living—not at the level of Paul Cohen, but enough to know that there’s always a decision tree”

    Lucky you : – ).

    I do not dispute that -ie, that there is always a decision tree of some sort. In fact Andrew Wiles said as much in the famous documentary “The Proof” that was aired by PBS. However, at the same time, as you can see in this excerpt of the same, he described the moment he was able to fix his proof of Fermat’s Last Theorem as a “revelation” . So could see the “revelation” as an efficient way to find the solution in a larger meta decision tree of all possible alternatives or something deeper, depending on your worldview :-).

  109. JimV Says:

    Last reply on the subject of general comments by Ethan:

    It was obvious from your previous comments that you have a fixed opinion which is never going to change. Ultimately you seem to believe that human intelligence cannot be understood and simulated by known physics. All I can say is that there is sufficient evidence consistent with the hypothesis that it can be to convince me. Among them:

    1) All my life in my own work and in others and in history, human inventions have progressed by trial and error. You have seen cars, computers and telephones evolve in your life time. Modern human bone structure dates to about 200,000 years ago; the first archeological evidence of the wheel and axle dates to 6000 years ago–but inventions snowball. With the wheel and axle you can have carts, gears, water mills, windmills, capstans, pulleys, … and PC hard drives. That’s what our progress consists of: trial and error passed on to the next generations by forms of memory, such as books. Biological evolution works the same way, with less effective memory but many more trials per minute and over a much longer time, to produce things such as nanotech neurons which we are not yet capable of. And yes, the functioning of neural networks is based on the functions of biological neurons and synapses. See Dr. Mitchell’s “Artificial Intelligence”.

    2) Like humans, computers with neural networks or without need an operating system which receives external inputs, passes them on to background routines, and communicates the results externally. That to me is the function of consciousness. The sensations by which we receive external inputs may be unique to our nervous systems and the physics of this universe, but it is the function which is important, and that can be and is implemented in computers.

    3) AlphaGo has the equivalent processing power of about a million neurons, the same as a mouse. Computers have a long way to go (a factor of about 100,000) to reach human brain equivalence, and may never reach that point. Still by devoting their processing to a single task, such as Go, they have demonstrated what to me is strong evidence of principle.

    During its match with the South Korean World Champion, AlphaGo made what Go experts called a “beautiful move”–unexpected, never seen before. The South Korean stated afterwards (I paraphrase from memory), “I never thought a computer could understand all the subtleties of Go well enough to beat me, but when I saw that move, I knew then that I would lose the series.”

    4) Applying this model to my own life, I reasoned that I could train my own neurons to do the complex motions of guitar finger-picking more quickly by practicing with my eyes closed. It worked.

    Any single piece of evidence can be discounted, as climate-change deniers do, but when it all fits into a logical structure, it tends to convince those who see the data.

    None of this of course is a proof that animal intelligence works by known physics and thus can be simulated by electronic devices. There are no proofs in science, just hypotheses which seem to work and fit all the known and constantly-generated evidence.

    The key point to me is to understand that everything our brains do can be explained in mechanical/physical/chemical/electro-magnetic terms. Most people who believe differently seem to reject such terms implicitly as if they could not possibly apply to the greatness of humans. What greatness? It took us over 100,000 years to invent the wheel; and a bunch of us elected Trump.

    With this perspective, statements I have seen such as, “Oh, AlphaGo is just processing an efficient search process” seem to me to be beside the point. Can they prove that is not what our brains do? It works, and well enough to impress Grand Masters!

    (That went on much longer than I thought it would, sorry. I’ll make another donation recommended by this site as a fine.)

  110. Cody Says:

    Sorry to hear you’re struggling with depression but at least good to hear it sounds like the productive moments are outweighing the bad for now. However, the comment about Trump and his “millions of armed supporters” refusing to cede control is laughable. The most recent Democratic nominee has urged Biden not to concede “under any circumstances.” Meanwhile, leftists are assaulting and even murdering Trump supporters in the streets. Not to mention some of the recent comments in response to the Supreme Court vacancy.

    Honestly, I’ve been struggling a bit with depression as well, and my primary trigger is the fact that, for the first time in my life, I no longer feel able to express my opinions openly in America. Not because I fear they will be proven wrong, but because I fear that to do so would put my career if not my physical safety at risk. So at the very least this concern about the result of the election not being respected seems like something that goes both ways. However, speaking as a former two-time Obama supporter, it seems like the left is much more likely to refuse to accept a Trump victory than the other way around.

  111. fred Says:

    Cody #110

    While I can’t say it’s been depressing me, it’s definitely been bothering the shit out of me. It’s a if all parts of society have been subjected to some sickening blackmail, something not seen since McCarthyism, except now we’re being judged based on how much we hate what traditional America stands for instead of how much we blindingly love what traditional America stands for:

    – At a personal level, in a work context, there’s a definite new feeling that we all have to be super careful what we say whenever we chat between coworkers in private channels, even if we’re of the same opinion, because you never know who’s listening. The simple fact that if someone is caught saying something like “All Lives Matter”, their spouse would lose their job…

    – When talking with family and friends from Europe, it’s clear that they’re being fed exclusively one interpretation of what’s going on here in America, namely the news channels over there lazily just cut and paste whatever CNN/NBC is putting out, not just in terms of content, but mainly in terms of what’s currently important and what’s not (sourcing your content from CNN and the NYT was an ok thing to do before Trump and before they realized they could make a ton of money by feeding into TDS).
    Like, as of two weeks ago, people in Europe seemed to think that covid19 bodies were literally piling up in the streets of America.

    – The suppression of public free speech has also been very concerning. Fewer and fewer people with a public platform and reasonable and rational minds dare say anything because whatever is currently acceptable is always a moving target – anything you put out there right now will be dug up sooner or later as the new proof of being a racist, a sexist, a Nazi, etc. By extension, no politician on the left willing to say “enough is enough”…

    – The Chinese Communist Party using what’s left of our free speech as a weapon against our democracy values, and then American companies and academia kowtowing to them.
    The NBA, Disney, Activision,… you name it, all turning their back on American values and spewing CCP propaganda for a few more dollars.
    And university professors never mentioning one negative thing about China because their revenue stream massively relies on the tuition fees of the sons and daughters of corrupt CCP members.

  112. fred Says:

    Scott #104

    “I’ve personally found the average CS theory paper far more readable than the average math paper

    And also because you can always say
    “Sorry, I’m not gonna read your CS paper, but why don’t you just go ahead and *actually* implement it and feed problem instances to it, and then let me know how that worked out?”.

  113. Scott Says:

    fred #112: Yeah, that’s not quite so easy with lower bounds papers… 🙂

  114. fred Says:

    Ethan #87

    The only thing I’ve said is that consciousness has probably nothing to do with “algorithms”.

    You seem to be stating some of the ideas in Hofsdadter’s GEB book – i.e. that the human brain does some “magic” that can’t be reproduced by algorithms.

    I’m not saying that’s not true, what I’m saying is that’s got nothing to do with consciousness.
    Bringing Godel and CH as some sort of proof that humans are conscious is suspicious because:
    – Being good at math doesn’t seem to make one more conscious, whatever that means: Buddha like, wiser, able to keep his shit together when it matters, keep on smiling when set on fire, …?
    – Most humans don’t do advanced math, babies don’t do advanced math, chimps don’t do advanced math, dogs don’t do advanced math,.. And they’re all not doing advanced math for one reason: the cognitive skills that matter 99.999% of the time in terms of survival (across almost all species) are about solving the equations of motions at an unconscious level as fast as possible.
    Unless you’re gonna tell me that brains (of birds, dogs, humans) are really about “magically” solving NP-hard problems efficiently. That ability would sure as hell make it easier to do math, but also allow for amazing feats that we simply don’t see in the living world… you’d think that such ability would make it impossible for a classical algorithm to crush a human Go master.

    And even if brains could achieve such feats, again: what does it have to do with consciousness? Just because “solving NP hard efficiently is a mystery” and “QM is mysterious” and “consciousness is a mystery”, doesn’t make them related.
    At best all we can say is that if I were to bash your brains in with a hammer, it’s true that 1) you probably will no longer be able to do advanced math (or whatever else brains do), and 2) you probably will no longer be conscious, heh.

  115. fred Says:

    Scott #113

    Haha, that was a call back to your comment #81

    “Implement it, use it to factor 2048-bit RSA challenge numbers, and then you’ll have my full attention”

  116. Ethan Says:

    JimV #109

    “It was obvious from your previous comments that you have a fixed opinion which is never going to change.”

    This personal attack was unwarranted. There is one peace of evidence that could make me change my mind which is if you could build a computer that could completely pass for a human in every aspect not only doing fast number crunching, namely, producing original work of the kind discussed here: proofs like Godel’s, Cohen’s or, if you have a more artistic side, original paintings like Vincent van Gogh’s or original music such as Bach’s, Beethoven’s or even more recent such as The Beatles’ or Guns and Roses’. GPT-3 is unable to produce an original op-ed without human editing despite having been trained with essentially all the text available in the public internet. Adding more parameters won’t change that because of artificial neural networks’ intrinsic theoretical limitations.

    In case English is not your first language or you are unfamiliar with what the expression “agreeing to disagreeing” means here is the Wikipedia page on the topic . I have said that a few comments ago and I reiterate it: I respectfully disagree with those who have different beliefs than mine. Personal attacks won’t make your arguments stronger.

    You have given reasons for your beliefs. I respect your beliefs even if you hadn’t given any reasons for them. Please understand that they are just that, “beliefs” not undisputed scientific facts. Science is a different thing altogether as Richard Feynman explains here . As I have said also before, the supporters of your point of view have more money than the supporters of my point of view but “popularity” and “my backers have more money” does not equate truth.

    My own belief in the can be falsified if you produce a Turning machine that is capable of understanding, something that would contradict the argument. Note that I stress “Turing machine”. At this time that’s the only theoretical model for computing we know how to implement. If you can come up with a different type of theoretical model for computation then the argument would no longer apply.

    And now I really stop. I won’t respond to any further personal attacks, only meaningful arguments that gets us past the point of regurgitating the propaganda points of the moneyed interests that want to make AlphaGo or GPT-3 pass for genuine intelligence.

  117. fred Says:

    Ethan #98

    “How is that some people believe that this is equivalent to “some gadget magically appeared out of thin air that beat a Go master” is beyond me […] At every step I see human ingenuity and creativity playing a significant role in building these so called “AI systems”.”.

    One one hand you’re saying that machines can’t match human genius, and then you’re also saying that, in case machines would someday match us, well, that would also prove human genius because we created them!

    Let’s take a step back then.
    Humans evolved from chimps, chimps evolved from small rodents, small rodents evolved from amphibious fish, fish evolved from smaller multi cellular organisms, … all the way back to the first DNA/RNA molecules. All reflecting the complexity and richness of the environment.
    At which point in that long chain has your “special sauce” been injected?
    You can ask the same thing during the development of a human, from a sperm cell, to a fetus, then a child, then a full adult.

    What I think is that consciousness is there for the ride all along, and is not related with intelligence or ability to solve problems and do math. The only thing that gets modulated by brain abilities is the content of consciousness, the richness of the experiences.

  118. william e emba Says:

    Zermelo did not invent his axiomatization in order to rescue set theory from paradoxes, but to respond to some of the charges against his first proof of the well-ordering theorem. His first proof is quite simple: given a set, just keep choosing new elements, and by transfinite induction over the ordinals this eventually fills out the set. Many people complained about the use of choice, but many also complained about the ordinals and transfinite induction. As ordinals were still new and poorly understood at the time, Zermelo for his second proof removed all traces of the ordinals, invented the mildly complicated “tower” proof, and laid out carefully just what made the proof work.

    At the end, he also answered the critics who thought the “paradoxes” were relevant, by pointing out that the known ones did not seem to occur in his system. That was a bonus, but it was not his motivation, as the harmlessness of the paradoxes was already known to Cantor and Russell. Indeed, when Russell discovered his own paradox, he informed Frege, not Cantor, for the simple reason it was a paradox for Frege, not Cantor.

    Both of Zermelo’s well-ordering theorem papers are available, in English translation even, in the From Frege to Gödel volume, so there’s no reason not to know this. But this meme seems to live forever.

  119. william e emba Says:

    You don’t plan to prove anything involving forcing? Raise your aim, Scott! Forcing is sometimes applied to computational complexity, and has been for several decades now. Look up “generic oracles”. There’s even a book by Krajíček on one aspect of this topic.

  120. Ethan Says:

    fred #114

    I don’t think I have much to add to the ongoing debate about humans being different than machines in terms of intellectual abilities. You’ve stated your views, I have stated mine and I think that at this point agreeing to disagreeing is the way to go because as I have said, this is a debate that goes all the way back to at least the time of Godel/Turing and the best minds have been unable to prove conclusively one way or another. I accept Penrose’s arguments as valid but not everybody agrees with them, obviously.

    On this point, which is totally unrelated to the matter at hand,

    “At which point in that long chain has your “special sauce” been injected?”

    As I have said previously I am a religious person of the Christian variety. Without giving too much info, I was raised Catholic but these days I consider myself an Evangelical Christian that attends Protestant services. Think of a “Donald Knuth” type of Christianity although I do not agree 100% with every one of this religious views. I had an around 10 year long stint as an atheist so nobody can accuse me of not having experimented with different worldviews. Out of all experiments, I settled with Protestantism as the spiritual approach that I believe better matches the human experience. And, as it couldn’t be otherwise, I totally respect people who have different religious worldviews including those who embrace atheism.

    I think God gave us body, mind and soul and that humans are special creatures in creation.

    How God injected the “special sauce” is part of the mystery of life. Evolution even if true -I do not take the view of some fundamentalist Christians that it must be false because it contradicts a literal interpretation of the Book of Genesis; my objections to taking evolution too seriously have to do with ; ie, I would like to have formulas like Newton’s or Einstein that help us make predictions as to how humans in 1 million years will be like- only deals with our bodies.

    Even if evolution is true as stated, it doesn’t have anything to say about our minds or souls.

    I think that some people’s obsession with evolution (both supporters and detractors) as the gold standard of science is a disservice to the scientific enterprise.

    I would be happy to use Einstein’s theory of general relativity -we know Newton’s law is a low order approximation- as the gold standard of science that I would defend against its deniers. In fact, I believe it so much to be true that I bet my life on it every time I drive my car or take a plane, both of which have been designed on the basis of Newton’s laws ability to model gravity.

    For some reason I can’t fully understand, some people want to use evolution as the gold standard to argue about the validity of the scientific method. Somehow I think it has to do with its low quality as a theory -it lacks a solid mathematical foundation. If you truly think about it, the biggest impact Darwin has had in society is outside the hard scientific realm and into the social realm: social Darwinism and eugenics. Note that I am not saying that the latter is a reason to reject evolution, rather, that evolution’s “physics envy” makes it of very little use outside intellectual circles who love to argue about evolution. In the applied realm, evolution hasn’t had the impact Newton’s theories of Einstein’s -with marvels such as LIGO- have had, thus I don’t worry about it too much.

  121. James Gallagher Says:

    It isn’t that hard to understand what consciousness is, we shut it off every night when we sleep and turn it on when we wake.

    It’s the brain remembering what happened in the past, each millisecond of our existence.

    When we dream we just don’t have such hard reality to guide what happens, so fun stuff occurs .

  122. Scott Says:

    william emba #119: I’m well aware of generic oracles in complexity theory—although for such a striking idea, the results obtained by their use have been a little disappointing so far. Perhaps I should’ve explained earlier: reading about generic oracles, and more generally, about the extensive analogies between forcing in set theory and oracles in complexity theory, was exactly what gave me the crazy idea in the first place that forcing, and Cohen’s proof of the independence of CH, were things that I could not only admire from afar but actually potentially understand.

  123. Scott Says:

    william emba #118: Thanks; that’s all extremely interesting historical background! Having said that, before “independence from ZF” could be construed by anyone as pointing toward “fundamental unanswerability,” the Z (later ZF) system didn’t merely have to be proposed in some paper; rather it had to be widely accepted. So when did that start happening—around the 1920s, after Frankel’s addition of Replacement and von Neumann’s addition of Regularity, and when Hilbert’s school and Ramsey and Post and others started seriously working on the Entscheidungsproblem?

    I recently read a quote from von Neumann about how the whole point of axiomatic set theory was to (1) allow the rigorous rederivation of Cantor’s achievements while also (2) avoiding the “antinomies” (Russell’s paradox, Burali-Forti, etc.), and that one comment instantly made a lot of what had confused me in my study of set theory snap into place. Suddenly I could explain in detail why each of the ZF axioms should be there, and why it should take the specific form it does, in terms of optimization for a goal that no one else had been so clear about.

    But I acknowledge, and am interested to learn, that the history that actually happened was somewhat more complicated than the history that would’ve made my understanding of the math the clearest. 🙂

  124. Scott Says:

    Ethan #116:

      There is one peace of evidence that could make me change my mind which is if you could build a computer that could completely pass for a human in every aspect not only doing fast number crunching, namely, producing original work of the kind discussed here: proofs like Godel’s, Cohen’s or, if you have a more artistic side, original paintings like Vincent van Gogh’s or original music such as Bach’s, Beethoven’s or even more recent such as The Beatles’ or Guns and Roses’.

    Alright then, suppose GPT-6 generates paintings that you can’t distinguish from van Gogh’s (or better yet, from those of the next van Gogh), or music that you can’t distinguish from the next Beatles’, in a fair test. How sincere are you in your implicit promise that that would be enough, that then you’d change your mind?

    And don’t just give me a cached answer, but actually consult your feelings here. Rather than admit you were wrong, would you more likely be ready with a custom-made explanation about how no, sorry, it still doesn’t count, because once you understand how the program works, you know that whatever paintings and symphonies it produced, impressive though they might seem to a non-expert, couldn’t have involved any “true” creativity?

    And would you go on to explain to me how the “van Gogh / Beatles / Gödel” criterion, the one you’d previously agreed to, was only meant to apply during periods when the President and the Senate majority are of different political parties? (Sorry, cheap shot! 😀 )

  125. Ethan Says:

    Scott #123

    “But I acknowledge, and am interested to learn, that the history that actually happened was somewhat more complicated than the history that would’ve made my understanding of the math the clearest.”

    One of my big “aha” moments was to realize that the actual history that led to many of the most important discoveries in science is always more complicated than what the version that is passed along in textbooks.

    For example, when it comes to how Einstein was able to convince people that general relativity could make testable predictions of its own, everybody has been told about the total solar eclipse of May 29, 1919 that captured the deflection of sunlight accurately as predicted by the theory. What I did not know until a few years ago is that this wasn’t his first attempt at running the experiment with one of the previous attemps having been interrupted by the first world war if I understood correctly because Russian soldiers confiscated Einstein’s instruments. In fact, apparently David Hilbert got general relativity’s equations a few days before Einstein’s after both of them had been discussing the topic following a conference Hilbert invited Einstein to but he accepted to give Einstein credit given that Hilbert’s contribution was coming up with the equations first but the ideas were Einstein’s.

    Another example is the Wright brothers pioneering work in aviation. The textbook version doesn’t mention the Europeans of the early 20th century, particularly the French, didn’t believe them or that they spent years in the court system defending their patented work.

    I could go on, but that’s the kind of story I have seen in every case where I have digged deeper into the actual events that led to a great scientific discovery or innovation. The textbook versions that survive and that I learned as a student typically presented a very embellished version of the actual -and messy- events that preceded said discoveries.

    This led me to conclude that research is really, really, like really hard, particularly interesting research.

  126. mjgeddes Says:

    Fred and Ethan,

    The explanation of consciousness is simple: it’s the properties of a complex adaptive system that enables the system to track it’s own time evolution. This is based on coordination, control and communication. A system consists of multiple sub-systems, and what consciousness does is that it enables all these sub-systems to cooperate. (1) Coordination – is the global workspace, (2) Control – is based on generative modeling (simulations of possible futures and pasts), and (3) Communication is higher order theory – (‘reflections’ or ‘thoughts about thoughts’ via translations between different levels of model abstraction).

    In a nutshell, consciousness is explained by a combination of global workspace, generative modeling, and higher-order theory (HOT). The poetry of science: Consciousness is coordination, control, communication in a complex system: the ‘cognitive arrow of time’. It’s a wrap! 😀

    And yes, of course, general intelligence could in principle be emulated on a Turing machine. I thought when I first read Penrose’s books that his theory was probably nuts, but but I now I’m sure of it 😉 His predictions haven’t aged well at all, whereas AI continues to exceed expectations ( Go mastery and good language models happened far sooner than sceptics predicted).

    It’s a curious sociological fact that some really smart mathematicians seem so ambivalent about AGI. In a way it’s understandable. Relative to virtually all other humans, mathematics do in a sense have ‘super-powers’. But once AGI comes along, then AGI will likely rocket far beyond even the best human mathematicians, and take over the research.

    So Scott, math is your super-power right? So if friendly AGI came along; in terms of quality of life, in *absolute* terms you’d probably be massively better off, but in *relative* terms (compared to other humans) you’d lose your super-powers, and you’d simply be in the same boat as the rest of us ordinary non-math mortals, mere spectators to super-smart entities at the research frontiers. That loss of status might perhaps make you somewhat ambivalent towards AGI.

    [I don’t think AGI would make human mathematicians completely redundant, because I think there would still always be a place for public education and teaching; making the results understandable to humans might always be best done by other humans, but of course science communication and teaching are *different* skills to pure research, which I think would indeed be fully taken over by the AGIs]

  127. Ethan Says:

    Scott #124

    Wow! That was a question : – ). I will pass on the political cheap shot, but I am eager to answer the rest!

    One of the problems that GPT-3 has -and I know that you know this from following you over the years but for those who lack the context- is the same problem that artificial neural networks have in general: they get trained for a very specific narrow task. In GPT-3’s case that narrow task is NLP (natural language processing) and training these neural networks is more an art -and it’s “human art”!- than a science where different edge cases are covered with additional data. The idea is that artificial neural networks generalize to new data that has the same statistical properties as the data they were trained on. So by training and training on additional edge cases you get a system that keeps imitating humans better but because humans, I believe, are infinite and “non predictable” in the statistical sense, I think that irrespective of the edge cases neural networks are trained on, humans will always surprise with cases not covered by the best trained neural networks.

    So to answer your question more specifically, I think that GPT-6 would have to be probably “infinite” to fully replace a human. A different question is whether you can get there asymptotically and with such power of approximation that GPT-6 and the human are epsilon indistinguishable but I doubt we will see a single system that achieves that. You’ll probably need several of them: one for NLP, another one for generating music, another one for generating math. In math’s specific case, I sincerely doubt we will ever see a GPT-6 system that is able to produce true original math research -precisely because of that “non predictability” in the statistical sense of humans- but I can see artificial neural networks replacing numerical solvers in compute intensive tasks such as weather modelling.

    So! To this,

    “And don’t just give me a cached answer, but actually consult your feelings here. Rather than admit you were wrong, would you more likely be ready with a custom-made explanation about how no, sorry, it still doesn’t count, because once you understand how the program works, you know that whatever paintings and symphonies it produced, impressive though they might seem to a non-expert, couldn’t have involved any “true” creativity?”

    The best I can say is that I have put my mouth where my money is. I have an ongoing bet with a former colleague I made around 2015 (a year when the current explosion of AI was in full force) about this very issue and I bet $1000 that we wouldn’t see anything remotely close to your GPT-6 by 2045. I am happy to have another bet with you. You know how to find me and I am a man of my word. So if by 2050 (to give the current wave hype another 30 years), something like the promised GPT-6 is produced, I will be happy to pay you $1000!

  128. Ethan Says:

    mjgeddes #126

    I won’t hammer the points I have already hammered, but just to be precise,

    “Go mastery”

    Can we at least agree that “Go” -just as chess- is a finite search problem that by definition sooner or later some computer -programmed by very smart humans- was destined to beat the best human player?

    “good language models”

    Are you referring to GPT-3? It’s not just The Guardian creating undue controversy, serious people took GPT-3 to task shortly after it was released and concluded

    “GPT-3 is quite impressive in some areas, and still clearly subhuman in others. My hope is that with a better understanding of its strengths and weaknesses, we software engineers will be better equipped to use modern language models in real products.

    As I write this, the GPT-3 API is still in a closed beta, so you have to join a waitlist to use it. I recommend that you sign up here and check it out when you get the chance.”

    So I don’t think that we are qualitatively closer to AGI than we were when Penrose wrote “The Empreror’s New Mind” which, for historical context, was written when what latter became known as the second AI winter was about to hit. AlphaGo and GPT-3 is what you get when you stretch neural networks to their limits with more data and more compute. What the current AI wave has that didn’t exist in 1980s is massive cheap computation and tons of data. Even the guys who got the 2018 Turing award for their contribution to neural networks got it for things they did mostly in the 1980s.

  129. John Baez Says:

    mjgeddes #106 wrote:

    > John Baez knows something. He’s just not telling us.

    I’m certainly trying to tell you.

  130. mjgeddes Says:

    Ethan #128,

    I have access to the OpenAI API, and I’ve experimented with GPT-3, I think it’s quite good. For summarization of known facts, it’s quite useful. It does often degenerate into gibberish if you ask it to try and prove something beyond what is known, or if you want it come up with something needing original thought. But even then, I still think it’s a useful brain-storming tool. The output doesn’t need any editing if you just want an essay on known facts about something – it can produce good essays.

    For language translation, I think it’s an improvement on Google Translate (although it’s limited to fewer languages than Google Translate). I’ve tested it on the toughest of translation tasks – including translating Japanese and Chinese poems into English; whereas Google Translate fails at this, GPT-3 is able to come up with reasonable translations, which capture the spirit of the poems. The quality of the translations would still be below that if the best human translators, but the quality is reasonable.

    Sadly, our access may be coming to an end, once the Beta ends, the cheapest pricing OpenAI are offering is $100 a month, putting it out of reach of hobbyists – OpenAI obviously want people to come up with profitable applications. I’m sceptical that many people will be successful, but there has already been at least one KillerApp (AIDUungeon) so who knows?

  131. John Stricker Says:

    Thank you Cody #110 and Fred #111!

  132. Nancy Lebovitz Says:

    It’s impressive that computers can win at games people have been playing for a long time.

    It will be more impressive if a computer can invent a game people want to play. I realize that “invent a game” is a fuzzy concept since games are related to previous games. Perhaps a strong human consensus that a game is something new would be enough. Or maybe just that people prefer the computer game to previous similar games.

  133. Ethan Says:

    mjgeddes #130

    Just to be clear, I have never said that machine learning or GPT-3 isn’t useful but as I said in a previous comment, we should be very clear about what it can do and what it won’t do. Over-hyping technology is one of Silicon Valley’s worst sins.

    Take translations. It’s great you have this level of accuracy on translations thanks to Machine Learning. At the same time, as Jaron Lanier, , rightly argues here , it took a lot of human translation to train the machine learning systems to get to that level of accuracy. And nobody has compensated those humans for their work!

    In today’s practical machine learning applications, the generation of what is known as “ground truth” (ie the data that captures the human intelligence that is then used to feed the training systems) is generally done in low cost countries -ie, developing countries. If you think about it, it’s another scheme by which the wealthy countries exploit the intelligence of the poor countries. You need massive amounts of this kind of data to build meaningful machine learning systems and still I believe -for the aforementioned reasons- that they will never fully replace a human.

    The mindset “here is some magical gadget that appeared out of nowhere” is at the root of many societal problems. That’s how we got Google and Facebook spying on Americans so effectively that all NSA had to do to spy on all of us was to piggyback on their efforts. And you want to argue that we, as a society, are better off as a result of Google and Facebook invasion of our lives, that’s a conversation I am happy to have because I don’t know anyone whose opinion I respect -irrespective of politics- who believes that in 2020.

  134. william e emba Says:

    Einstein did not personally attempt to participate in the 1914 Crimea eclipse expedition to test GR. He financed a German astronomer, whose equipment was confiscated as the Great War had broken out during his travels, and he was suddenly the Enemy.

    This may or may not have been lucky for Einstein, since his 1913 prediction was half the correct 1915 prediction. The warpage of spacetime is doubly relevant to photons, as opposed to the mere warpage of space by itself, which is usually sufficient for velocities well below \(c\).

    Hilbert may or may not have written down the field equations before Einstein. However, he published after Einstein. There is debate regarding Hilbert’s revisions of his paper. The debate is not helped by the fact that some of the pro-Hilbert people are openly anti-Semitic.

    In addition, Hilbert’s theory at the time was not covariant. This in itself is a no-go to many people.

    On the other hand, it seems every last formulaic aspect of SR was anticipated, and the names of the anticipators has usually been kept. But Einstein cut through the formulas and saw the physical essence and wrapped up the piecemeal dilly-dallying into a remarkably clear and simple theory.

    Mostl interesting is probably Paul Gerber’s 1898 gravitomagnetic theory and its prediction for the excess perihelion shift of Mercury: algebraically it is identical to Einstein’s, assuming speed of gravity = speed of light! The theory was completely forgotten until rediscovered by anti-Semitic antirelativists who made a big stink about it.

    Another fantastic pre-Einstein prediction was Max Abraham regarding gravitational radiation. He

  135. william e emba Says:

    (dagnabit hovering by accident over “Submit Comment” once again)

    Another fantastic pre-Einstein-GR prediction was Max Abraham regarding gravitational radiation, circa 1913. Many people had made comments in that general direction. Abraham was rather detailed. He even realized that conservation of angular momentum meant the radiation had to be quadrupolar!

  136. william e emba Says:

    The Wright brothers and their patent litigation with Curtiss and others did a lot of damage to the nascent American aircraft industry. It did not help that the various judges were incapable of understanding the technical issues (sort of like today’s judges on software and the Internet). When the US got into WWI, the government strong-armed the parties involved to share their patents in a patent pool.

  137. mjgeddes Says:

    John #129,

    I think you’re absolutely on target here:

    “I think we need a ‘compositional’ approach to classical mechanics. A classical system is typically built from parts, and we describe the whole system by describing its parts and then saying how they are put together. But this aspect of classical mechanics is typically left informal. You learn how it works in a physics class by doing lots of homework problems, but the rules are never completely spelled out, which is one reason physics is hard.”

    What if the most elegant formalism for classical physics was never actually found? Laypersons tend to think of classical physics as ‘solved’, but what if a ‘wrong turn’ was taken right at the start and the key insights to think about fields, forces and statistical mechanics in the right way were actually missed entirely?

    And that’s perhaps why science is currently unable to properly understand complex systems!

  138. william e emba Says:

    mjgeddes #137:

    You comments about Physics are interesting speculation, hard to answer. However, look a few departments over, in Economics. There you can find a loud rather vocal minority that believes that unregulated markets are perfect, government intervention is always wrong, and so on, and all this is mathematical theorems about some fantasy utopia they believe in. Never mind that the assumptions of the relevant theorems don’t actually apply in the Real World, and that there are tons of theorems which more accurately model the Real World and come to opposite conclusions. Or when the experiments they propose are actually done, and fail spectacularly, it only means the implementers didn’t go far enough.

    They are committed. They should be committed.

  139. Graham Says:

    You’ve confused the words “could” and “should” in your second to last paragraph. FDR and Churchill “could” team up with Stalin, but this does not imply that you “should” team up with the Never Trumpers.

  140. Ronak Says:

    Oh hey I ended up on the main blog what’s up

    Let me begin my clarifying two things:
    1. I didn’t accuse Scott Aaronson of supporting the right-wingers.
    2. I wasn’t making an argument from moral purity, I was making an argument from strategy.

    Anyway, the video I linked (that got edited out in the main post) makes the case that the Lincoln project is gaining a disproportionate amount of traction within the democratic party in exchange for a really small sliver of voters.
    Giving them money and promoting them on one of the biggest blogs in this part of the onlines is giving them more power.

    The overton window in this country goes between democrats and republicans. This is moving the Overton window rightward.

    The real questions are as follows
    1. (Large-scale, nebulous, intended for democrats at large) What do you get by defeating Trump while losing the possibility of redistributive policies? This is happening across the world; what will it take to turn the tide?
    2. (Small-scale, intended for Scott) Are there no other organisations that perform effective work for a single-digit percentage of [subgroup]? Maybe organisations that support the USPS, or help register black voters?

  141. Ronak Says:

    Also, on the subject of efficacy, here’s something I found from the first result for the google search for “lincoln project spending breakdown”

    > The Lincoln Project’s strategy of frequently producing viral attack ads against President Donald Trump but spending relatively little to place them on traditional airwaves in swing states has come under scrutiny.

    (Emphasis mine.)

  142. Sniffnoy Says:

    Ethan #68:

    Dude, you are really privileging the hypothesis here, starting with the assumption that Galileo must have been first and demanding I provide conclusive proof he didn’t. That is not a good way to argue, and frankly I’m not interested in undertaking such an argument. I think what I’ve presented is sufficient to convince someone taking a more neutral look at things, and I’m not going to argue it any further.

    I do, however, have a few other comments.

    First off, this comment of yours is question-begging:

    So I think that people might have reasons of their own for attempting to discredit Galileo’s pioneering work, but this is the first time I encounter someone who tries to convince me that he didn’t deserve the credit for his pioneering ideas.

    If you’re in the middle of an argument over whether the idea that the universe is mathematical is in fact, a pioneering idea of Galileo’s, please don’t make statements like this that rely on the claim that it is. For instance, you could rephrase that last clause:

    …this is the first time I encounter someone who tries to convince me that he didn’t deserve the credit for the idea that the universe is mathematical.

    See? Same information content but without the question-begging.

    Now obviously, this does not affect the substance of the argument, but doing this sort of thing is a good way to annoy people and to get them to conclude that you’re not worth arguing with.

    I’d also like to say that I’m surprised you find my claims so surprising. Popular versions of history, the history of science included, are so often distorted and shrouded in legend, and firsts so often misattributed, that I would in general consider “this person who you thought was first at this thing actually wasn’t” to be a fairly unremarkable claim for anything that’s several centuries old (and frankly, often things that are only a few decades old). It’s definitely surprising to me that the prior you put on Galileo first being specifically is that high!

    Douglas #88:

    Sure, Galileo did a bunch of important work on the laws of fall; getting actual numbers there is certainly pretty key. That wasn’t the question though. The question is, did he originate the idea that the universe is mathematical. And the answer there would seem to be no.

    Jelmer Renema #93:

    I mean fundamentally all measurements are measurements of dimensionless numbers, no?

  143. Sniffnoy Says:

    Ronak #140:

    If you think that Democrats consider redistributive policy to be the one important thing, such that beating Trump is hardly even worth it if it means a decrease in redistributive policy, then, um, I don’t think you have a very good picture of the Democratic party — or, really, US politics in general? The Democrats do not have some coherent philosophy based around redistribution, nor do the Republicans have some coherent philosophy based around opposition to such; rather, both parties are incoherent messes formed by the alliance of various factions.

    (I mean — you’re addressing Scott Aaronson, have you seen anything to indicate that he in particular is laser-focused on redistributive policy?)

    Anyway, as a Democrat myself, if you want to see a substantial part of my answer, see my comment #43 above. TLDR, the preservation of liberal democracy is more important than any particular individual policy that might come out of it. (I’m also probably not what you’d consider pro-redistribution; I’m in favor of particular redistributive policies but not others, based on various considerations, rather than being in favor of policies based on how redistributive they are. Or in short, I’m a liberal, not a leftist…)

  144. Ronak Says:

    Sniffnoy #143:

    I am in fact aware that democrats are centre-left at best on these issues.

    The party has two opposing forces, rightward and leftward.
    On the right is the republicans, on the left is nothing to speak of.
    The democrat party is the left end of the policy Overton window. If we’re letting that end be pulled to the right, what are we getting out of it? What are we losing from it?

    If your politics is “Trump bad”/”democracy, however badly implemented, good” and only that, we have nothing more to speak of I agree.

    If your politics includes a conditional block “if worldwide dawn of fascism bad then…” maybe we can talk about explanations and solutions.

  145. Sniffnoy Says:

    Ronak #144:

    You appear to be using a left-right political spectrum. I think this is a bad concept that obscures more than it illuminates. (Although, I’ve started saying “traditionalism-authoritarianism” rather than just “traditionalism” to be clearer about what I’m referring to. 😛 ) More specifically, you appear to be claiming that redistribution will prevent fascism or is somehow its opposite. This is… not a claim I think you will get many others to agree with. I’m trying to figure out how to read this. Possibilities:

    1. You really are claiming that redistribution will prevent fascism-as-commonly-understood. I’d call this an extraordinary claim requiring serious argument to back it up.
    2. You are using the word “fascism” in some way other than how it is commonly understood.
    3. You are taking one aspect of fascism-as-commonly-understood (lack of redistribution, which I wouldn’t even say is an aspect of fascism-as-commonly-understood, but let’s just spot you that) and considering that to be the important, bad, identifying part, with the other parts not particularly important or bad

    In cases (2) or (3) you are communicating unclearly and should alter or clarify your terminology (at which point, if one of these cases holds, I suspect it may become apparent that your priorities and those of Democrats in general are more different than you realize). In case (1) you have a lot more arguing to do.

    I mean, to be honest, what it looks like you’re doing is just taking a left-right political spectrum, identifying fascism with the right end and redistribution with the left end, thus making them “opposites”. :-/ But, I’m hoping that is just a false impression created by your omission of detail and you really do have more reasoning than that…

  146. Jelmer Renema Says:

    @ Snifnoy 142:

    I don’t understand what you mean with your statement that all measurements are measurements of dimensionless numbers. If you mean that ultimately we can only measure ‘how often does quantity X fit into quantity Y’ (e.g. ‘how often does my measurement stick fit into the length of this field’) then this just rephrases of the problem of not having a shared system of units.

    My point above was that the way of Greeks did geometry set up so that it didn’t need reference to a shared standard unit. Note that in ruler and compass geometry the ruler is unmarked.

  147. John Stricker Says:

    Jelmer Renema #146:

    Interesting point, which made me reread your comment #93.

  148. Sniffnoy Says:

    Jelmer Renema #146:

    Yeah, I was just pointing out the old point that all you can technically measure directly are dimensionless numbers, but you make a good point that practically speaking there can be quite a difference between measuring immediately-available ratios on the one hand, vs measurements which, though still technically ratios, require a shared system of units on the other hand.

  149. Douglas Knight Says:

    Of course, the Greeks did have units. Eratosthenes measured the circumference of the earth.

    If you really were interested only in the narrow question of who first said that nature ran on mathematics, you should have just named Archimedes. By citing a long essay, or even just a few paragraphs that named many people, you are implying that you are making a broader claim. I can’t really tell what that claim is. Which is pretty typical of people citations of Christie.

  150. James Says:

    Scott #81,

    The RSA algorithm I linked to would be too slow for a desktop computer past a few hundred digits. I would have just coded it, but I’m really that bad at coding anything. (That and you would need to understand Barvinok’s algorithm well enough to calculate the special case for discrete logs/RSA and get the complexity down to something a good GPU could handle. And I thought it might be a good idea to try to get a “this can’t work because of this” from someone before figuring out how to code around LattE’s count program.)

    Does that mean you don’t see any complexity theoretic barrier to approaching BPP=BQP via randomly approximating the effect of each quantum gate (so you don’t lose too much entanglement information)? It’s not clear to me how I could show the technique avoided the algebrization barrier.

  151. Ronak Says:

    Sniffnoy #145

    You are right that I didn’t elaborate on the connection between fascism and redistributive policies.

    I have strong opinions on this, based on detailed arguments by people smarter than I am; but I didn’t get into it.

    That’s because I first want answers to the following two questions that I posed, that were not answered:
    > This is happening across the world; what will it take to turn the tide?
    >> The democrat party is the left end of the policy Overton window. If we’re letting that end be pulled to the right, what are we getting out of it? What are we losing from it?

    I have not been entirely clear, and I apologise for that.

  152. Sniffnoy Says:

    Douglas Knight #149:

    Well, the question isn’t who said it first so much as who said it first last. 🙂 I.e., who first said it and got it to stick. If Archimedes said it, but then the idea died out, and Galileo reinvented or reintroduced it, he would still deserve credit for that. But if the idea was already circulating in the scholarly culture of the time, he wouldn’t. My understanding is that the case is more like the latter.

  153. Jelmer Renema Says:

    @ Sniffnoy 145:

    Regardless of whether Ronak is right or not, I don’t think his claim is extraordinary or unusual at all (assuming for the moment your interpretation 1 is correct). To make the point how commonplace it is: I was taught this in school as one of the key explanations for the rise of fascism the first time around. In fact, the idea that a strong welfare state acts as a safeguard against political extremism used to be so common that Europe built it’s postwar social order around it!

    As the Germans say ‘Erst kommt das Fressen, dann kommt die Moral’ (feeding comes first, then morality).

  154. Jelmer Renema Says:

    @ Douglas Knight 149:

    Yes, and to this day we’re still arguing over exactly which version of the stadion he used.

  155. Sniffnoy Says:

    Jelmer Renema #153:

    Oh, yeah, duh, wasn’t thinking of that somehow. Thanks. Although I do feel like it’s worth pointing out that with that argument, that, well, who says the revolution being prevented would have been a fascist one? :-/

    (Tangential, and I’ve said this elsewhere, but I want to make a note of how much I dislike the word “extremism” and how it conflates two different ideas. On the one hand you just have, like, object-level positions that are extreme, taking some set of principles to their conclusion as far as you can, which, like, there’s nothing wrong with that. On the other hand you have “meta-level extremism” — resorting to threats and violence and all that… which might better be called “illiberalism” (OK, a component of it). The word “extremism” conflates these and it bothers me. Admittedly these do often seem to go together in humans, but there’s no necessary logical connection.)

  156. Adam Says:

    Not sure if this is too late, but…

    I’ve donated $2,000 to Republicans Against Trump. I can send a receipt if you want. Love to get a match!

  157. Jelmer Renema Says:

    @ Sniffnoy 155:

    Yes, of course you’re right this argument was initially made against communism (Marshall plan and all that). But I think the point is more that the stability of any regime depends to a large extent on the ability to ‘deliver the goods’ to the man in the street, irrespective of what regime ends up toppling it. I think there are good historical arguments to give why the US is sliding towards fascism in particular, but that’s another story.

    As for ‘extremism’, 100% agreed. It’s one of those words like ‘totalitarian’ or ‘terrorist’ that serves to obscure meaningful differences rather than highlight them. I try to use it as little as possible.

    Incidentally, there *are* ethical system (like Aristotle’s) which are built around the idea that vice is virtue taken to excess, so for some people who use the word extremism it might actually be a meaningful reflection of their worldview.

  158. Scott Says:

    Adam #156: Thanks! On top of the $2000 I gave to the Biden-Harris campaign and the Lincoln Project, I just donated $1000 to Republican Voters Against Trump.

    Would anyone else like to match the rest of Adam’s $2000?

  159. Ethan Says:

    Somehow related to this conversation (in case anyone is reading),

    I forgot to mention this topic in earlier conversations. Because Machine Learning is essentially “mindless” and it’s as “smart” or “dumb” as the labeled data it is being fed with and the decisions made by the human model designers who privilege certain kinds of labeled data over others in their model performance optimizations, the issue of “bias in AI” as this phenomenon is being called is not a minor one. You won’t get rid of it with more training data and bigger models as long as what these AI systems do is mindless Turing computation.

  160. yme Says:

    What other kind of computation is there?

  161. Ethan Says:

    yme #160

    That’s a question for Scott. As far as I can understand, Turing computing captures, from the the theoretical side, all computers we know how to build, whether traditional John von Neumann computers or quantum computers.

    Scott, do you agree?

  162. Scott Says:

    yme #160 and Ethan #161: There are other kinds of computation besides Turing’s — as just one example, computation that manipulates real or complex numbers to infinite precision — but for all the other kinds, one can give reasons why they don’t capture the apparent power of the physical world as well as Turing computation does.

  163. Ethan Says:

    Scott #162

    Thanks for confirming that Turing computation is the gold standard for modeling the computers we can build.

    I had not heard about the other types before. That’s interesting! Is there any serious work on them or people just focus on Turing computation for the most interesting theoretical computer science work these days given what you just said?

  164. yme Says:

    Ethan #161: “You won’t get rid of it with more training data and bigger models as long as what these AI systems do is mindless Turing computation.”

    This gave me the impression that you thought there was some other kind of computation, which a future AI system might use, and which might enable it to overcome the limitations of current AI systems, which you believe are inherent in all Turing machines. But maybe I misunderstood you.

    I don’t know what computers will or won’t be able to do in the future, but I don’t understand Penrose’s argument that they couldn’t possibly ever equal people’s abilities. Sure, computers can’t solve the halting problem, but neither can people. People have enough trouble understanding programs that they designed and wrote themselves, which is why so many program have bugs. They certainly don’t have the ability to understand arbitrary programs given to them by an adversary.

  165. Ethan Says:

    yme #164

    Online forums are like email, prone to misunderstanding :-).

    What I meant by that is that my understanding is that the only physical computers we know to build are Turning computers, thus the theoretical results derived from the halting problem -in fact the Godel incompleteness theorems at large- apply to these computers not to other computers who might implement other forms of computation.

    When I have debated this issue in the past, some people that lacked deep knowledge in theoretical computer science -but who had drank the singularity kool-aid- wanted to make the argument that quantum computers don’t implement Turning computers thus that they are not limited by these theoretical results.

    Note that the Penrose argument, is just that that, an argument, not a rigorous proof but one I personally find convincing.

    At its core, what the Penrose argument says is that understanding -a distinctively human ability that our minds are able to do- cannot be captured by what Turing machines (therefore every physical computer we know how to build) do.

    The idea is that in the process of proving its first incompleteness theorem, Godel’s mind constructed a statement that says “this statement cannot be proved” that you must believe to be true based on the construction of the system (Peano arithmetic axioms + the induction principle + the appropriate encoding).

    The statement “this statement cannot be proved” (which is true) is equivalent to “this statement cannot be computed by a Turing machine” but humans can build such a statement out of “understanding the system”.

    You have a list of arguments and counter-arguments here

    For example, I don’t find this particularly compelling as a refutation of Penrose’s argument

    “Searle criticized Penrose’s appeal to Gödel as resting on the fallacy that all computational algorithms must be capable of mathematical description. As a counter-example, Searle cited the assignment of license plate numbers to specific vehicle identification numbers, as part of vehicle registration. According to Searle, no mathematical function can be used to connect a known VIN with its LPN, but the process of assignment is quite simple—namely, “first come, first served”—and can be performed entirely by a computer.”

    First, Searle is a human (not a computer) who after analyzing Penrose’s argument came up with the example, second the algorithm “first come, first served” comes up from a human after having thought about the problem (there you have what in AI is called “ground truth” which is the human encoding of “intelligence” in AI), it doesn’t “magically appear out of nowhere”.

    One pattern I have noticed among people who believe in the feasibility of AGI -I am not saying you or Scott fit this pattern because I don’t know any of you well enough to make a judgement, I am talking about people I have known well personally- is their belief in “things appearing magically out of nowhere” whereas to me the debate in mathematics surrounding the axiom of choice is indicative that even if you have a Platonist view that all possible algorithms already exist in one magical list (that onto itself is a strong belief), picking one such algorithm from the list is an act of faith and thus “picking the right right algorithm that solves a particular problem” is an act of faith plus an act of utmost human intelligence.

  166. Ethan Says:

    yme #164

    To follow up, it turns out that indeed, the cardinality of the set of algorithms, is countably infinite (ie, the cardinality of the natural number aka aleph_0) for reasons explained here , thus implementing any one of them assumes that the axiom of choice is true (an act of faith) because you can only implement an algorithm that you can “pick” from the set of all possible algorithms to then code it in the concrete computer language of your choice.

    And then picking the actual algorithm that solves a particular problem is a distinctively human intelligence trait -ie understanding. Machine learning only lets you make predictions -in the statistical sense- to new situations that are statistically similar to the situations contained in the training dataset, not new situations that are statistically different.

    So to summarize again: so long as the only computers we know how to build physically implement Turing computation, and given Penrose’s argument, I can’t see how AGI is feasible. I could be wrong, and as I said earlier, GPT-6 would have to do pretty extraordinary things for me to change my mind on this or a new form of implementable computation not subject to the limits of Turing computing would have to be proposed that overcomes said limits.

  167. yme Says:

    Gödel’s statement is not exactly, “this statement cannot be proved.” It’s more like, “this statement cannot be proved within this particular formalization of mathematics.” So we can conclude, I think, that whatever went on in his mind when he proved his incompleteness theorem was not equivalent to that particular formalization of mathematics. But I don’t see how we can rule out the possibility that his mental processes were equivalent to some other formalization of mathematics.

    I agree with you that Searle’s response isn’t convincing, but maybe not for the same reasons. I’m not sure what you mean by “things appearing magically out of nowhere” or “picking one such algorithm from the list is an act of faith.” Are you just saying that computers will not develop human-level abilities on their own, somehow, perhaps via a process analogous to biological evolution? Or are you saying that computers will never have human-level abilities even if people try their best to give them those abilities, via programming that is as clever as the programmers are capable of? Or do you believe that computers cannot, in principle, display human-level abilities, even assuming an omniscient programmer, the same way they can’t solve the halting problem?

    These are three separate possibilities.

    (When we say that the halting problem is unsolvable, we mean that there is no Turing machine that gives the correct answer in all cases. It’s not just a matter of us not knowing which Turing machine is the right one. There simply isn’t any right one.)

  168. Ethan Says:

    yme #167

    By “intelligent computers” we mean, by definition, hardware that can be built. If we stay in the non physical reality, we can imagine anything and everything as possible, including flying humans, spaghetti monsters and what have you. Ie, it becomes science fiction, not reality.

    Turing computation is the theoretical model that captures the computers we know how to build at this time. Someone once told me that the field of computer science is inherently psychotic and the more I interact with people who believe that AGI is feasible with computers we know how to build, the more I think this person was into something.

    To your questions (my answers are going to sound repetitive because I have already addressed them in previous comments). By way of prefacing my answers I will say that one way to think about “artificial intelligence” is as “cargo cult intelligence”. The use of the words “cargo cult” to refer to science was coined by Richard Feynman to refer to endeavors -particularly social sciences- that claimed to be scientific by mimicking in a shallow way the work of physicists -more here . With artificial intelligence I think we have something very similar going on among the believers in AGI.

    “Are you just saying that computers will not develop human-level abilities on their own, somehow, perhaps via a process analogous to biological evolution?”

    The notion that you can map 1-1 “humans” with “computers that humans build” is a belief, not a scientific fact. Therefore, any conclusions you might reach from studying humans do not necessarily carry over to computers because of the “cargo cult” argument.

    “Or are you saying that computers will never have human-level abilities even if people try their best to give them those abilities, via programming that is as clever as the programmers are capable of?”

    As I said, to me the Penrose argument means that the computers we know how to build will never be able to match human mental abilities. As I also said, this argument is controversial and I am happy to agree to disagree.

    “Or do you believe that computers cannot, in principle, display human-level abilities, even assuming an omniscient programmer, the same way they can’t solve the halting problem?”

    Already answered. It has to do with physical computers implementing Turing computing. It’s irrespective of the underlying technology (von Veumann vs Quantum), operating system, computer language or the super human abilities of the programmer. The physical hardware has inherent limitations by way of implementing Turing computing.

    I will make an analogy to the field of information theory. One of the early pioneers in the field of artificial intelligence, Claude Shannon, is better known for inventing the field of information theory, check . Information theory is the branch of applied math that deals with encoding and transmitting information. Shannon’s results made possible the digital world we live in even though most practicing programmers have never heard his name.

    One of the fundamental concepts introduced by information theory is the notion of a fundamental theoretical limit in matters such as how much you can compress a source of information or how fast you can transmit error free that information over a noisy channel (such as an internet connection) that cannot be overcome with technology (ie, the things we know how to build). This limit is irrespective of the encoding mechanisms used to put the source of information into bits or the error correction encoding used to transmit it.

    In my view, the Penrose argument decouples the abilities of physical computers from the abilities of human minds because of Godel’s incompleteness theorem. And this gap cannot be filled by more powerful computers as long as physical computation implements Turing computing. As I said, I am open to being proved wrong but I will need to see GPT-6 be good at things that are distinctively human, not just “imitation games”. I am not losing my sleep over it, to be honest with you.

  169. yme Says:

    Sorry if I’m annoying you. I don’t mean to.

    I agree that Penrose’s argument is intended to show that people can do things that are, in principle and not only in practice, impossible for computers to do. I just don’t see how it does show that.

    Gödel proved that any sufficiently powerful formal system cannot be both consistent and complete. Consistent means that, for any statement that the system can prove true, it can’t also prove it false. Complete means that every meaningful statement can either be proved true or be proved false by the system. But human mathematicians sometimes make mistakes, so they’re not always consistent. And even if some of them never make a mistake, there are certainly plenty of mathematical statements that they can neither prove nor disprove. So how does Gödel’s incompleteness theorem show that human mathematicians can do things that no formal system can do? Maybe one of the statements that they can neither prove nor disprove is the Gödel sentence for the formal system that’s equivalent to what goes on in their head when they do mathematics. (By “Gödel sentence,” I mean the one that says, “this statement cannot be proved within this formal system.”)

    I don’t think it’s obvious that we will be able to build and program computers to do everything that people can do. But I don’t see any reason why that would be fundamentally impossible either. So I don’t think it’s hopeless to try. Certainly not psychotic.

  170. Ethan Says:

    yme #167

    And to follow up furthermore. Except for the fact that Penrose’s argument poses a hard limit into what real life computers can do -irrespective of the smartness of their programmers- which is a controversial point, the rest of what I say is not really controversial among top AI experts.

    Watch this talk by Cassie Kozyrkov, Chief Decision Scientist at Google . The first point of her talk is “forget about science fiction” where she emphasizes that AI is human all the way through.

    Then read this page about John McCarthy by the ACM on his Turing Award. Most readers here I am sure know that the Turing Award is considered computer science’s equivalent to the Nobel Prize in terms of prestige “Given that McCarthy was primarily a mathematician and technologist who had little use for puffery, it is ironic that his most widely recognized contribution turned out to be in the field of marketing, specifically in choosing a brand name for the field. Having noticed that the title of the Automata Studies book didn’t stir up much excitement, when he subsequently moved to Dartmouth College he introduced the name artificial intelligence at a 1956 conference there and saw that it was embraced both by people working in the field and the general public.”

    The name “artificial intelligence” was a marketing device since the very beginning for a particular way of using computers. Another Turing Award winner, Donald Knuth, calls machine learning (a subset of AI which happens to be the form of AI making the news and the hype these days) constructing algorithms out of data in this other interview . As you can see if you watch the relevant part of the interview, he is not particularly fond of machine learning replacing natural language programming.

    So, one has to differentiate popular culture notions of AI from what the experts in the field think AI is. I recall LOL-ing when I first watched Monty Python’s movie Life of Brian . Some took it as a critique of the Christian religion. While it is impossible not to see elements of that in the movie, to me it is first and foremost a critique of the most extremist followers of the same, particularly those who blindly follow a distorted view of Christianity in many cases in opposition of Jesus’ own message as it is contained in the New Testament of the Christian Bible.

    In the case of AI I think we are in a similar situation. The field, however you want to call it, is understood by the people who are credited with pioneering it in a previous era and the top contemporary experts who work on it as being about very specific and narrowly defined things. Then you have moneyed people and propagandists who seem to lack understanding of these limits -or ignoring them if they understand them- hyping it purely for business reasons as something that it is not. The hype that followed the release of GPT-3 (a technology that now Microsoft has licensed exclusively) is an example of what I am talking about. The only developments that have made GPT-3 possible today vs 30 years ago is not major conceptual breakthroughs towards AGI but the massive availability of data (thanks to the success of the internet) and cheap compute of the kind (GPUs) that make the training of large artificial neural networks covering many edge cases possible.

  171. Ethan Says:

    yme #169

    I have another comment waiting moderation before you posted this. Just to be clear on this point,

    “But I don’t see any reason why that would be fundamentally impossible either. So I don’t think it’s hopeless to try. Certainly not psychotic.”

    People should do whatever the heck they want to do. I am not one who believes in telling people what to do. This is not a conversation about budget allocation. If Elon Musk (or Microsoft) want to spend tons of money in pushing machine learning to its limits that’s their right as far as I am concerned.

    What I am concerned about is things like these,

    Namely, that some people with power and influence believe that AI has capabilities it doesn’t have and we create another IQ/SAT/GRE fiasco on steroids. Something akin what is already happening in China.

    It has taken decades of debates, and lots of pain and individual people’s futures ruined, to discard the notion of humans having an intrinsic biological limit to their intellectual capacity that can be measured via timed multiple choice tests.

    Since I do believe that there won’t be a full blown 3rd AI winter, now is the time to have these conversations so that AI is used where it is proper, not in areas with the ability to destroy people’s futures.

  172. yme Says:

    Ethan #166:

    I don’t understand why you bring up the axiom of choice, or what it has to do with faith, but in any case, it deals with making infinitely many choices, not just one, even if the one choice is from an infinite set. The wikipedia article about it has a clear explanation.

  173. Ethan Says:

    yme #172

    I don’t want to keep beating a dead horse, but if the axiom of choice doesn’t apply, then it means that the number of algorithms that can be implemented is finite subset of all possible algorithms. I don’t think that’s a correct assumption. If the subset of implementable algorithms (among all possible algorithms) is infinite, it seems to me the axiom of choice is needed to ensure that all these algorithms can be brought from the Platonic world of algorithms to the real world of practical computers, not only the one needed to solve a particular problem.

  174. Revisiting the Continuum Hypothesis - Your Cheer Says:

    […] Scott Aaronson mentioned in a blog post (see here) that  he has read and understood the proof that CH is independent of set […]

  175. yme Says:

    I don’t think the axiom of choice has anything to do with practical computer programming. It’s just a mathematical axiom, which says that certain sets exist (in the abstract, mathematical, sense of “exist”) even though we can’t explicitly specify each of their elements. If we can explicitly specify a set’s elements, the axiom of choice isn’t needed in order to deduce the set’s existence, even if the set is infinite; other axioms of set theory suffice.

    When people program computers, they don’t just assert that a program exists; they explicitly write down the complete text of the program. So the program clearly exists, because it’s right there.

    I still don’t see what any of this has to do with the question of whether computers might one day be able to do everything that people can do, or whether that’s impossible for fundamental reasons.

  176. fred Says:

    On the effectiveness of the Lincoln Project ads:

  177. Ethan Says:

    yme #175

    I don’t want to keep beating a dead horse, but I don’t think you fully appreciate what you wrote here:

    “When people program computers, they don’t just assert that a program exists; they explicitly write down the complete text of the program. So the program clearly exists, because it’s right there.”

    The same algorithm can be represented in very different languages. It is “right there” only after it has been written in a particular language, otherwise, it is not “right there”, it’s in the Platonic world of algorithms. Implementations of algorithms don’t “magically appear out of nowhere”. Since the number of algorithms is countably infinite, to be “right there” you need to be able to pick anyone one of those algorithms and translate into an algorithm that can run on an actual physical computer. That exercise, in my view, requires the axiom of choice. And the actual matching of a particular algorithm to the right problem is an exercise of intelligence that, because both the number of problems and the number of algorithms are infinite, is intelligence outside the current methods of machine learning. This is the problem of gadgets like GPT-3, as impressive as they seem to the lay person. GPT-3 continues to be unimpressive to the experts in AI -besides being a technology marvel- despite having been trained with all the text available in the internet.

    If you talk to people who implement practical machine learning, they will tell you that building a model that identifies say “stop signals” on a set of few images is a relatively simple task. The problem -which is an obstacle to the developing of self driven cars- is to build a model that detects any possible “stop signal” a car can find anywhere in the US because such a task requires covering what in the world of practical software engineering is called “edge cases”. Covering edge cases is a very difficult task because any new model has to cover all previous edge cases plus new ones. Even if you could make the types of “stop signals” that can exist in the US a finite number, you have all sorts of things going on (weather conditions, wild life altering the signals, etc) that make the total possible of stop sign signals recognizable de facto “infinite”.

    This intrinsic limitation of all statistic learning techniques (whether it’s neural networks or others) is one reason why we seem to have hit for the n-th time another “AI wall”. Check . I don’t think this time we will see a full blown “AI winter” like it happened in the 1960s and 1980s (after all, large companies like Google, Facebook and Microsoft already leverage AI at the large scale) but it seems that we have come full circle with the wave started in 2012 with . It’s always a similar story: some prototype shows promise in a very narrow task, the propagandists of AI go around saying “if we can do this with such a small investment, imagine what we could do if you gave us all this much”, money pours in, promises fail to materialize, people move on to other things.

    That “imagine if” is why some people call computer science psychotic. In civil engineering, for example, there is no “imagine if we build the Empire State Building”. You build it or you don’t.

  178. yme Says:

    How did you arrive at your view of what the axiom of choice is all about? I think you’re ascribing to it a lot more, I don’t know, philosophical significance let’s say, than it really has.

    Why do you keep saying that the products of human intelligence don’t magically appear out of nowhere? Who thinks that they do? Obviously, they are produced by people. And if, one day, computers will be able to match human intelligence, the products of computer intelligence will also not magically appear out of nowhere. They’ll be produced by computers (which, everyone agrees, were produced by people).

    It seems to me that, on the contrary, your position is closer to “things can magically appear out of nowhere” than mine is. You think that there’s something special about people, and specifically their intelligence, that we will never understand—that, in principle, cannot be understood—well enough to implement on a computer. Well, how do we do it, then?

    I don’t know why I’m even asking. You’ve already mentioned earlier that it has something to do with souls and God and such, which seems pretty magical to me. But if I can’t even convince you that you’re wrong about the axiom of choice, which is just math, I certainly won’t convince you that you’re wrong about religion.

    Sorry. This is just frustrating, that’s all.

    Of course, I agree with you that hype is bad. People shouldn’t claim that their AI systems are better than they actually are. But what’s wrong with imagination? How will we ever make progress, if we never imagine that one day we’ll be able to do things that today we can’t?

    Anyway, regarding self-driving cars, people unfortunately do crash their cars sometimes. So a computer doesn’t have to be a perfect driver in order to match people’s abilities in that area.

  179. Ethan Says:

    yme #178

    For people who want to understand -meaning people like Nobel Prize winner Roger Penrose who himself is an atheist-, my position is clear and doesn’t merit further explanation. So I am happy to leaving it here, agreeing to disagreeing.

    I will just make one comment related to your general approach in this conversation,

    “I don’t know why I’m even asking. You’ve already mentioned earlier that it has something to do with souls and God and such, which seems pretty magical to me. But if I can’t even convince you that you’re wrong about the axiom of choice, which is just math, I certainly won’t convince you that you’re wrong about religion.

    Sorry. This is just frustrating, that’s all.”

    When I engage in debates of controversial topics, my goal is never to convince anyone of my views, rather explain them. Occasionally I learn something, but the experiences I learn the most when it comes to forming my own opinion about controversial topics is when I play the role of a spectator and I watch two experts defending the opposing views. I have watched Penrose make his case against other experts who hold different views and in my view, he always comes out of the debate with the upper hand given the clarity of his arguments whereas all I hear from the other side are empty promises of the kind that pushed John McCarthy, name the field “artificial intelligence” for purely marketing reasons: “what if”, “imagine this or that”. I can imagine many things, such as the Flying Spaghetti Monster. Imagining and being able to build with physical hardware are two distinct things. And the problem with the whole approach that those of your side of the argument use is that it inevitably disappoints because the limits of statistical learning theory are well understood on the theoretical side since the 1960s (the wikipedia article has a good summary here ) that are irrespective of the technology used to implement it. What made the deep learning revolution started in 2012 by AlexNet possible is the empirical observation that neural networks didn’t seem to have a plateauing in performance when more data was used for training and models could be make bigger which worked well for a while when there was additional data to train with massive -and cheap- computers. Fast forward to 2020 and there is no more data and a wall has been hit when it comes to the amount of available compute/parameters. GPT-3 is a good as NLP is going to get for a while given that the information available in the internet is not growing at the same rate it did 20 years ago and Moore’s Law.

    Whether I think they are right or wrong, everybody is entitled to their opinion for reasons I might or might not understand. Going around seeking to “convince” as opposed to “explain” is usually the wrong attitude. If you want to convince me that Penrose is wrong, go and build GPT-6 that falsifies his argument derived from Godel’s incompleteness theorems. To me that’s about as hard as making a plane that flies on the assumption that g, the acceleration due to gravity, is 1 m/s^2, instead of the empirically observed 9.8 m/s^2 but that again is why I have heard people whose opinion I greatly respect refer to computer science as a psychotic field.

  180. Scott Says:

    Ethan #177:

      Since the number of algorithms is countably infinite, to be “right there” you need to be able to pick anyone one of those algorithms and translate into an algorithm that can run on an actual physical computer. That exercise, in my view, requires the axiom of choice.

    Sorry, but you’re talkin’ nonsense here, and it’s not a matter of interpretation or opinion. The Axiom of Choice is specifically about the possibility of making infinitely many arbitrary choices. Whatever you think is involved in writing a finite computer program—even there was Penrosian creativity that could never be captured algorithmically—we still wouldn’t be talking about AC as a statement of transfinite set theory, but about something else. As the host of this discussion, I ask you to acknowledge this straightforward mathematical fact.

  181. Ethan Says:

    yme #178

    And in case my previous comment wasn’t clear enough:

    – We know how to build Turing computers and we know, because of Godel’s incompleteness theorems, that these computers have limits as to the kind of true statements they can produce.

    – We know that brains, particularly Godel’s as the first brain able to do this, are not constrained by the same limits that affect Turing computers because they are capable of “understanding” a distinctive, non algorithm (ie non Turing computable) feature of human brains.

    I can’t see how people keep hammering that building computers that can match human brains abilities completely with Turing computers is possible. It is one thing to stretch current computing paradigms with existing technology (that’s what GPT-3 does), quite another to argue that GPT-3 is a step towards AGI. It isn’t. GPT-3 is a proof of concept of how much you can stretch existing technology to implement the aforementioned theory of statistical learning. It’s a great achievement as it pertains to technology but not an intellectual breakthrough towards AGI.

  182. Ethan Says:

    Scott #179,

    Let me ask you a follow up question to this because it could be I am not understanding the axiom of choice properly

    “The Axiom of Choice is specifically about the possibility of making infinitely many arbitrary choices.”

    Say we have two sets:

    1- the set of all possible algorithms

    2- the set of all possible implementable algorithms on a particular technology (say a given computer architecture, and a given language), which is necessarily a subset of 1-

    Then if the first set exists in a Platonic plane (and most mathematicians are Platonists; I am not using this as an appeal to popularity argument, only to say that mathematics is being done by practitioners of mathematics under this assumption), I can’t see how you can go from the first set to the second set without the axiom of choice, namely, how do you bring any one of the algorithms in 1- to life without it? Being able to pick any one of them for implementation, means being able to pick all of them, namely being able to pick an infinite set of algorithms for their implementation out of the set of all possible algorithms.

    Thank you in advance for the clarification.

  183. Ethan Says:

    Scott #179,

    I think I can answer my own question. 2- is probably a set of limited size associated to the limited resources. So a world of limited resources, necessarily means 2- is a finite set, big, but finite. So yeah, the axiom of choice doesn’t apply to that situation. My bad!

    You still have the problem of human creativity/consciousness :).

  184. Scott Says:

    Ethan #182: The way I’d put it is that “the set of all algorithms” is simply ill-specified, until we explain better what we mean. In practice, if someone used that phrase, I’d assume they meant “the set of all Turing machines” or “the set of all LISP programs” or (most likely of all) “fix any of those sets; for my purposes it won’t matter which one.” Strictly speaking, though, we don’t have any mathematical handle on what “algorithms” are until we fix some particular method of encoding them. It’s just like how we don’t have a handle on “the set of all ideas,” but we do have some handle on “the set of all grammatical English sentences,” and we might hope that every idea can be encoded by some grammatical English sentence.

    So yes, this can be an interesting discussion to have, and one could even discuss the relation to AI—e.g., how should an AI encode “algorithms” or “ideas” abstractly? Using some internal linguistic-like representation, as some people argue that the brain does?

    My point was just that in any case, none of this has anything to do with the Axiom of Choice, which is not about arbitrary choices in general, but specifically about the possibility of making infinitely many of them.

  185. Ethan Says:

    Scott #184

    Thank you for the additional explanations. I am happy the misunderstanding on my side is cleared.

    With respect to the “set of all algorithms”, I read somewhere (which is where I got the idea of invoking the axiom of choice) that since at the end of the day all code that runs on a computer ends up becoming a series of 0s and 1s then you can make a mapping of strings to the integers whether it positive integers or -if two’s complement representation is used- to the positive and negative integers, thus you can make the case that the number of algorithms is infinite. But as you said this doesn’t mean much because you need a specific machine to run them, something that makes the possible set of algorithms runnable in that machine instantly finite, making the appeal to the axiom of choice pointless even if you could define “the set of all algorithms” properly because the number of choices is finite yo begin with.

  186. Alistair Stewart Says:

    Hi Scott,

    I’ve enjoyed the odd read of your blog for years, and Quantum Computing since Democritus is one of my favourite books of all time!

    Personally I do not see much threat from the right in the last few years, and especially not a “worldwide fascist dawn”. I suspect many votes for Trump are actually votes of protest against the identity politics of modern cultural and academic institutions, which seem to be increasing racism in the name of anti-racism. If you want to help gain votes against Trump, I don’t think the most effective approach is to try to change the programming of those on the right, it’s to bring some of the more extreme ideals of the left back towards the center. There’s a lot more I could say, but I think that gets to the core of our current problems.

    If it doesn’t sounds right to you, consider that the main cultural and media institutions are not filled with Nazis, they’re filled with left leaning people. I don’t see any popular far right movements or rhetoric going about. Only far right, and centrist (with a slight spectrum from center left to center right, but all generally quite willing to discuss problems and solutions rather than shout each other down)

  187. Scott Says:

    Alistair #186:

      If you want to help gain votes against Trump, I don’t think the most effective approach is to try to change the programming of those on the right, it’s to bring some of the more extreme ideals of the left back towards the center.

    Whaddya think I’ve spent a lot of my time doing on this very blog for the past 6 years, even as thousands of wokeists loudly condemned me for it? 🙂

    Anyway, I’ll probably close this thread soon, since I don’t like to maintain lots of open threads simultaneously.