gjm

Karma: 31,019

Hi. I’m Gareth McCaughan. I’ve been a consistent reader and occasional commenter since the Overcoming Bias days. My LW username is “gjm” (not “Gjm” despite the wiki software’s preference for that capitalization). Elsewehere I generally go by one of “g”, “gjm”, or “gjm11”. The URL listed here is for my website and blog, neither of which has been substantially updated for several years. I live near Cambridge (UK) and work for Hewlett-Packard (who acquired the company that acquired what remained of the small company I used to work for, after they were acquired by someone else). My business cards say “mathematician” but in practice my work is a mixture of simulation, data analysis, algorithm design, software development, problem-solving, and whatever random engineering no one else is doing. I am married and have a daughter born in mid-2006. The best way to contact me is by email: firstname dot lastname at pobox dot com. I am happy to be emailed out of the blue by interesting people. If you are an LW regular you are probably an interesting person in the relevant sense even if you think you aren’t.

If you’re wondering why some of my very old posts and comments are at surprisingly negative scores, it’s because for some time I was the favourite target of old-LW’s resident neoreactionary troll, sockpuppeteer and mass-downvoter.

gjm Jun 22, 2024, 1:25 AM
10 points
0
on: Claude 3.5 Sonnet
It’s pretty good. I tried it on a few mathematical questions.
First of all, a version of the standard AIW problem from the recent “Alice in Wonderland” paper. It got this right (not very surprisingly as other leading models also do, at least much of the time). Then a version of the “AIW+” problem which is much more confusing. Its answer was wrong, but its method (which it explained) was pretty much OK and I am not sure it was any wronger than I would be on average trying to answer that question in real time.
Then some more conceptual mathematical puzzles. I took them from recent videos on Michael Penn’s YouTube channel. (His videos are commonly about undergraduate or easyish-olympiad-style pure mathematics. They seem unlikely to be in Claude’s training data, though of course other things containing the same problems might be.)
One pretty straightforward one: how many distinct factorials can you find that all end in the same number of zeros? It wrote down the correct formula for the number of zeros, then started enumerating particular numbers and got some things wrong, tried to do pattern-spotting, and gave a hilariously wrong answer; when gently nudged, it corrected itself kinda-adequately and gave an almost-correct answer (which it corrected properly when nudged again) but I didn’t get much feeling of real understanding.
Another (an exercise from Knuth’s TAOCP; he rates its difficulty HM22, meaning it needs higher mathematics and should take you 25 minutes or so; it’s about the relationship between two functions whose Taylor series coefficients differ by a factor H(n), the n’th harmonic number) it solved straight off and quite neatly.
Another (find all functions with (f(x)-f(y))/(x-y) = f’((x+y)/2) for all distinct x,y) it initially “solved” with a solution with a completely invalid step. When I said I couldn’t follow that step, it gave a fairly neat solution that works if you assume f is real-analytic (has a Taylor series expansion everywhere). This is also the first thing that occurred to me when I thought about the problem. When asked for a solution that doesn’t make that assumption, it unfortunately gave another invalid solution, and when prodded about that it gave another invalid one. Further prompting, even giving it a pretty big hint in the direction of a nice neat solution (better than Penn’s :-)), didn’t manage to produce a genuinely correct solution.
I rate it “not terribly good undergraduate at a good university”, I think, but—as with all these models to date—with tragically little “self-awareness”, in the sense that it’ll give a wrong answer, and you’ll poke it, and it’ll apologize effusively and give another wrong answer, and you can repeat this several times without making it change its approach or say “sorry, it seems I’m just not smart enough to solve this one” or anything.
On the one hand, the fact that we have AI systems that can do mathematics about as well as a not-very-good undergraduate (and quite a bit faster) is fantastically impressive. On the other hand, it really does feel as if something fairly fundamental is missing. If I were teaching an actual undergraduate whose answers were like Claude’s, I’d worry that there was something wrong with their brain that somehow had left them kinda able to do mathematics. I wouldn’t bet heavily that just continuing down the current path won’t get us to “genuinely smart people really thinking hard with actual world models” levels of intelligence in the nearish future, but I think that’s still the way I’d bet.
(Of course a system that’s at the “not very good undergraduate” level in everything, which I’m guessing is roughly what this is, is substantially superhuman in some important respects. And I don’t intend to imply that it doesn’t matter whether Anthropic are lax about what they release just because the latest thing happens not to be smart enough to be particularly dangerous.)

gjm May 29, 2024, 1:46 AM
4 points
0
on: Notes on Gracefulness
Even though it would have broken the consistent pattern of the titling of these pieces, I find myself slightly regretting that this one isn’t called “Grace Notes”.

gjm May 9, 2024, 1:34 AM
2 points
2
on: How to be an amateur polyglot
A nitpick: you say
fun story, I passed the C2 exam and then I realized I didn’t remember the word faucet when I went to the UK to visit a friend
but here in the UK I don’t think I have ever once heard a native speaker use the word “faucet” in preference to “tap”. I guess the story is actually funnier if immediately after passing your C2 exam you (1) thought “faucet” was the usual UK term and (2) couldn’t remember it anyway...
(I liked the post a lot and although I am no polyglot all the advice seems sound to me.)

gjm Apr 26, 2024, 9:26 AM
5 points
5
in reply to: Jacob G-W’s comment on: Losing Faith In Contrarianism
Please don’t write comments all in boldface. It feels like you’re trying to get people to pay more attention to your comment than to others, and it actually makes your comment a little harder to read as well as making the whole thread uglier.

gjm Apr 25, 2024, 10:42 AM
11 points
7
on: social lemon markets
It looks to me as if, of the four “root causes of social relationships becoming more of a lemon market” listed in the OP, only one is actually anything to do with lemon-market-ness as such.
The dynamic in a lemon market is that you have some initial fraction of lemons but it hardly matters what that is because the fraction of lemons quickly increases until there’s nothing else, because buyers can’t tell what they’re getting. It’s that last feature that makes the lemon market, not the initial fraction of lemons. And I think three of the four proposed “root causes” are about the initial fraction of lemons, not the difficulty of telling lemons from peaches.
- urbanization: this one does seem to fit: it means that the people you’re interacting with are much less likely to be ones you already know about, so you can’t tell lemons from peaches.
- drugs: this one is all about there being more lemons, because some people are addicts who just want to steal your stuff.
- MLM schemes: again, this is “more lemons” rather than “less-discernible lemons”.
- screens: this is about raising the threshold below which any given potential interaction/relationship becomes a lemon (i.e., worse than the available alternative), so again it’s “more lemons” not “less-discernible lemons”.
Note that I’m not saying that “drugs”, “MLM”, and “screens” aren’t causes of increased social isolation, only that if they are the way they’re doing it isn’t quite by making social interactions more of a lemon market. (I think “screens” plausibly is a cause of increased social isolation. I’m not sure I buy that “drugs” and “MLM” are large enough effects to make much difference, but I could be convinced.)
I like the “possible solutions” part of the article better than the section that tries to fit everything into the “lemon market” category, because it engages in more detail with the actual processes involved by actual considering possible scenarios in which acquaintances or friendships begin. When I think about such scenarios in the less-isolated past and compare with the more-isolated present, it doesn’t feel to me like “drugs” and “MLM” are much of the difference, which is why I don’t find those very plausible explanations.

gjm Apr 14, 2024, 11:33 PM
14 points
4
on: A High Decoupling Failure
I think this is oversimplified:
High decouplers will notice that, holding preferences constant, offering people an additional choice cannot make them worse off. People will only take the choice if its better than any of their current options.
This is obviously true if somehow giving a person an additional choice is literally the only change being made, but you don’t have to be a low-decoupler to notice that that’s very very often not true. For a specific and very common example: often other people have some idea what choices you have (and, in particular, if we’re talking about whether it should be legal to do something or not, it is generally fairly widely known what’s legal).
Pretty much everyone’s standard example of how having an extra choice that others know about can hurt you: threats and blackmail and the like. I might prefer not to have the ability to pay $1M to avoid being shot dead, or to prove I voted for a particular candidate to avoid losing my job.
This is pretty much parallel to a common argument for laws against euthanasia, assisted suicide, etc.: the easier it is for someone with terrible medical conditions to arrange to die, the more opportunities there are for others to put pressure on them to do so, or (this isn’t quite parallel, but it seems clearly related) to make it appear that they’ve done so when actually they were just murdered.

gjm Apr 14, 2024, 11:22 PM
3 points
1
in reply to: tailcalled’s comment on: Ackshually, many worlds is wrong
Then it seems unfortunate that you illustrated it with a single example, in which A was a single (uniformly distributed) number between 0 and 1.

gjm Apr 12, 2024, 1:26 AM
42 points
31
on: Ackshually, many worlds is wrong
I think this claim is both key to OP’s argument and importantly wrong:
But a wavefunction is just a way to embed any quantum system into a deterministic system
(the idea being that a wavefunction is just like a probability distribution, and treating the wavefunction as real is like treating the probability distribution of some perhaps-truly-stochastic thing as real).
The wavefunction in quantum mechanics is not like the probability distribution of (say) where a dart lands when you throw it at a dartboard. (In some but not all imaginable Truly Stochastic worlds, perhaps it’s like the probability distribution of the whole state of the universe, but OP’s intuition-pumping example seems to be imagining a case where A is some small bit of the universe.)
The reason why it’s not like that is that the laws describing the evolution of the system explicitly refer to what’s in the wavefunction. We don’t have any way to understand and describe what a quantum universe does other than in terms of the evolution of the wavefunction or something basically equivalent thereto.
Which, to my mind, makes it pretty weird to say that postulating that the wavefunction is what’s real is “going further away from quantum mechanics”. Maybe one day we’ll discover some better way to think about quantum mechanics that makes that so, but for now I don’t think we have a better notion of what being Truly Quantum means than to say “it’s that thing that wavefunctions do”.
I have the impression—which may well be very unfair—that at some early stage OP imbibed the idea that what “quantum” fundamentally means is something very like “random”, so that a system that’s deterministic is ipso facto less “quantum” than a system that’s stochastic. But that seems wrong to me. We don’t presently have any way to distinguish random from deterministic versions of quantum physics; randomness or something very like it shows up in our experience of quantum phenomena, but the fact that a many-worlds interpretation is workable at all means that that doesn’t tell us much about whether randomness is essential to quantum-ness.
So I don’t buy the claim that treating the wavefunction as real is a sort of deterministicating hack that moves us further away from a Truly Quantum understanding of the universe.
(And, incidentally, if we had a model of Truly Stochastic physics in which the evolution of the system is driven by what’s inside those probability distributions—why, then, I would rather like the idea of claiming that the probability distributions are what’s real, rather than just their outcomes.)

gjm Apr 11, 2024, 1:20 AM
11 points
6
in reply to: abstractapplic’s comment on: Thinking harder doesn’t work
I don’t know exactly what the LW norms are around plagiarism and plagiarism-ish things, but I think that introducing that basically-copied material with
I learned this by observing how beginners and more experienced people approach improv comedy.
is outright dishonest. OP is claiming to have observed this phenomenon and gleaned insight from it, when in fact he read about it in someone else’s book and copied it into his post.
I have strong-downvoted the post for this reason alone (though, full disclosure, I also find the one-sentence-per-paragraph style really annoying and that may have influenced my decision[1]) and will not find it easy to trust anything else I see from this author.
[1] It feels to me as if the dishonest appropriation of someone else’s insight and the annoying style may not be completely unrelated. One reason why I find this style annoying is that it gives me the strong impression of someone who is optimizing for sounding good. This sort of style—punchy sentences, not too much complexity in how they relate to one another, the impression of degree of emphasis on every sentence—feels like a public speaking style to me, and when I see someone writing this way I can’t shake the feeling that someone is trying to manipulate me, to oversimplfy things to make them more likely to lodge in the brain, etc. And stealing other people’s ideas and pretending they’re your own is also a thing people do when they are optimizing for sounding good. (Obviously everything in this footnote is super-handwavy and unfair.)
In case anyone is in doubt about abstractapplic’s accusation, I’ve checked. The relevant passage is near the end of section 3 of the chapter entitled “Spontaneity”; in my copy it’s on page 88. I’m not sure “almost verbatim” is quite right, but the overall claim being made is the same, “fried mermaid” and “fish” are both there, and “will desperately try to think up something original” is taken verbatim from Johnstone.

gjm Apr 10, 2024, 10:32 PM
6 points
4
in reply to: aphyer’s comment on: D&D.Sci: The Mad Tyrant’s Pet Turtles [Evaluation and Ruleset]
One can’t put a price on glory.

gjm Apr 9, 2024, 10:36 PM
3 points
0
on: D&D.Sci: The Mad Tyrant’s Pet Turtles [Evaluation and Ruleset]
Wow, that was incredibly close.
I think simon and aphyer deserve extra credit for noticing the “implicit age variable” thing.

gjm Apr 8, 2024, 8:51 PM
15 points
5
on: Math-to-English Cheat Sheet
There are a few things in the list that I would say differently, which I mention not because the versions in the post are _wrong_ but because if you’re using a crib-sheet like this then you might get confused when other people say it differently:
- I say “grad f”, “div f”, “curl f” for $\nabla f$ , $\nabla \cdot f$ , $\nabla \times f$ . I more often say “del” than “nabla” and for the Laplacian I would likely say either “del squared f” or “Laplacian of f”.
- I pronounce “cos” as “coss” not as “coz”.
- For derivatives I will say “dash” at least as often as “prime”.
The selection of things in the list feels kinda strange (if it was mostly produced by GPT-4 then that may be why) -- if the goal is to teach you how to say various things then some of the entries aren’t really pulling their weight (e.g., the one about the z-score, or the example of how to read out loud an explicit matrix transpose, when we’ve already been told how to say “transpose” and how to read out the numbers in a matrix). It feels as if whoever-or-whatever generated the list sometimes forgot whether they were making a list of bits of mathematical notation that you might not know how to say out loud or a list of things in early undergraduate mathematics that you might not know about.
It always makes me just a little bit sad when I see Heron’s formula for the area of a triangle. Not because there’s anything wrong with it or because it isn’t a beautiful formula—but because it’s a special case of something even nicer. If you have a cyclic quadrilateral with sides $a, b, c, d$ then (writing $s = \frac{1}{2} (a + b + c + d)$ ) its area is $\sqrt{} (s - a) (s - b) (s - c) (s - d)$ . Heron’s formula is just the special case where two vertices coincide so $d = 0$ . The more general formula (due to Brahmagupta) is also more symmetrical and at least as easy to remember.

gjm Mar 30, 2024, 6:01 PM
4 points
0
on: D&D.Sci: The Mad Tyrant’s Pet Turtles
With rather little confidence, I estimate for turtles A-J respectively:
22.93, 18.91, 25.47, 21.54, 17.79, 7.24, 30.36, 20.40, 24.25, 20.69 lb
Justification, such as it is:
The first thing I notice on eyeballing some histograms is that we seem to have three different distributions here: one normal-ish with weights < 10lb, one maybe lognormal-ish with weights > 20lb, and a sharp spike at exactly 20.4lb. Looking at some turtles with weight 20.4lb, it becomes apparent that 6-shell-segment turtles are special; they all have no wrinkles, green colour, no fangs, normal nostrils, no misc abnormalities, and a weight of 20.4lb. So that takes care of Harold. Then the small/large distinction seems to go along with (gray, fangs) versus (not-gray, no fangs). Among the fanged gray turtles, I didn’t find any obvious sign of relationships between weight and anything other than number of shell segments, but there there’s a clear linear relationship. Variability of weight doesn’t seem interestingly dependent on anything. Residuals of the model a + b*segs look plausibly normal. So that takes care of Flint. The other pets are all green or grayish-green so I’ll ignore the greenish-gray ones. These look like different populations again, though not so drastically different. Within each population it looks as if there’s a plausibly-linear dependence of weight on the various quantitative features; nostrils seem irrelevant; no obvious sign of interactions or nonlinearities. The coefficients of wrinkles and segments are very close to a 1:2 ratio and I was tempted to force that in the name of model simplicity, but I decided not to. The coefficient of misc abs is very close to 1 and I was tempted to force that too but again decided not to. Given the estimated mean, the residuals now look pretty normally distributed—the skewness seems to be an artefact of the distribution of parameters—with stddev plausibly looking like a + b*mean. The same goes for the grayish-green turtles, but with different coefficients everywhere (except that the misc abs coeff looks like 1 lb/abnormality again). Finally, if we have a normally distributed estimate of a turtle’s weight then the expected monetary loss is minimized ifwe estimate mu + 1.221*sigma.
I assume
that there’s a more principled generation process, which on past form will probably involve rolling variable numbers of dice with variable numbers of sides, but I didn’t try to identify it.
I will be moderately unsurprised if
it turns out that there are subtle interactions that I completely missed that would enable us to predict some of the turtles’ weights with much better accuracy. I haven’t looked very hard for such things. In particular, although I found no sign that nostril size relates to anything else it wouldn’t be very surprising if it turns out that it does. Though it might not! Not everything you can measure actually turns out to be relevant! Oh, and I also saw some hints of interactions among the green turtles between scar-count and the numbers of wrinkles and shell segments, though my brief attempts to follow that up didn’t go anywhere useful.
Tools used: Python, Pandas, statsmodels, matplotlib+seaborn. I haven’t so far seen evidence that this would benefit much from
fancier models like random forests etc.

gjm Mar 16, 2024, 6:19 PM
5 points
1
in reply to: PhilosophicalSoul’s comment on: Middle Child Phenomenon
Yes , I know what the middle-child phenomenon is in the more literal context. I just don’t have any idea why you’re using the term here. I don’t see any similarities between the oldest / middle / youngest child relationships in a family and whatever relationships there might be between programmers / lawyers / alignment researchers.
(I think maybe all you actually mean is “these people are more important than we’re treating them as”. Might be true, but that isn’t a phenomenon, it’s just a one-off judgement that a particular group of people are being neglected.)
I still don’t understand why the distribution of talent/success/whatever among law students is relevant. If your point is that very few of them are going to be in a position to make a difference to AI policy then surely that actually argues against your main claim that law students should be getting more attention from people who care about AI.

gjm Mar 16, 2024, 11:40 AM
11 points
6
on: Middle Child Phenomenon
Having read this post, I am still not sure what “the Middle Child Phenomenon” actually is, nor why it’s called that.
The name suggests something rather general. But most of the post seems like maybe the definition is something like “the fact that there isn’t a vigorous effort to get law students informed about artificial intelligence”.
Except that there’s also all the stuff about the distribution of talent and interests among law students, and another thing I don’t understand is what that actually has to do with it. If (as I’m maybe 75% confident) the main point of the post is that it would be valuable to have law students learn something about AI because public policy tends to be strongly influenced by lawyers, then it seems like this point would be equally strong regardless of how your cohort of 1000 lawyers is distributed between dropouts, nobodies, all-rounders, CV-chasers, and “golden children”. (I am deeply unconvinced by this classification, by the way, but I am not a lawyer myself and maybe it’s more accurate than it sounds.)

gjm Mar 15, 2024, 3:52 AM
4 points
0
on: Constructive Cauchy sequences vs. Dedekind cuts
It looks as if you’re taking a constructive Dedekind cut to involve a “set of real numbers” in the sense of a function for distinguishing left-things from right-things.
Is that actually how constructivists would want to define them? E.g., Bishop’s “Foundations of Constructive Analysis”, if I am understanding its definitions of “set” and “subset” correctly (which I might not be), says in effect that a set of rational numbers is a recipe for constructing elements of that set, along with a way of telling whether two things constructed in this way are equal. I’m pretty sure you can have one of those but not be able to determine explicitly whether a given rational number is in the set, in which case your central argument doesn’t go through.
Are Cauchy sequences and Dedekind cuts equivalent if one thinks of them as Bishop does? There’s an exercise in his book that claims they are. I haven’t thought about this much and am very much not an expert on this stuff, and for all I know Bishop may have made a boneheaded mistake at this point. I’m also troubled by the apparent vagueness of Bishop’s account of sets and subsets and whatnot.
More concretely, that exercise in Bishop’s book says: a Dedekind cut is a pair of nonempty sets of rationals S,T such that we always have s<t and given rationals x<y either x is in S or y is in T. Unless I’m confused about Bishop’s account of sets, all of this is consistent with e.g. S containing the negative rationals and T the positive rationals, and not being able to say that 0 is in either of them. And unless I’m confused about your “arbitration oracles”, you can’t build an arbitration oracle out of that setup.
(But, again: not an expert on any of this, could be horribly wrong.)

gjm Mar 2, 2024, 12:27 AM
2 points
0
in reply to: trevor’s comment on: The NYT routinely and brazenly lied about Ukraine during March 2022
I did, in fact, read the post and the NYT articles, and I am not convinced that your description of what they do and what it means is correct. So, if my response to your article doesn’t consist mostly of the gushing praise your first paragraph indicates you’d prefer, that’s one reason why.
But, regardless of that: If you write something wrong, and someone points out that it’s wrong, I don’t think it’s reasonable to respond with “how dare you point that out rather than looking only at the other parts of what I wrote?”.
Scott is not using some weird eccentric definition of “lie”. E.g., the main definition in the OED is: “An act or instance of lying; a false statement made with intent to deceive; a criminal falsehood.” (Does that first clause soften it? Not really; it’s uninformative, because they define the verb “lie” in terms of the noun “lie”.) First definition in Wiktionary is ” To give false information intentionally with intent to deceive”. But, in any case, even with a very broad definition of “lie” the first four levels in his taxonomy are simply, uncontroversially, obviously not kinds of lying. Again, the first one is “reasoning well, and getting things right”.
If I say “There are seven classes of solid objects in the solar system: dust motes, pebbles, boulders, mountains, moons, small planets, and large planets” and you identify something as a small planet, you should not call it “a Level 6 Planet, according to gjm’s classification of planets”.
And, while I understand a preference for being charitable and not leaping to calling things dishonest that aren’t necessarily so … I don’t think you get to demand such treatment in the comments on an article that does the exact reverse to someone else.

gjm Mar 1, 2024, 1:57 AM
1 point
−1
in reply to: trevor’s comment on: The NYT routinely and brazenly lied about Ukraine during March 2022
Your justification seems to me almost completely non-responsive to the point I was actually making, which is not about whether it’s reasonable to call what the NYT did in these cases “lying” but about whether it’s reasonable to call something at level 6 in Scott’s taxonomy a “level 6 lie”.
Scott classifies utterances into seven types in ascending order of dishonesty. The first four are uncontroversially not kinds of lying. Therefore, something on the sixth level of Scott’s taxonomy cannot reasonably be called a “level 6 lie”, because that phrase will lead any reader who hasn’t checked carefully to think that Scott has a taxonomy of levels of lying, where a “level 6 lie” is something worse than a level 5 lie, which is worse than a level 4 lie, … than a level 1 lie, with all these things actually being kinds of lies.
Whereas in fact, even if we ignore Scott’s own opinion that only “the most egregious cases of 6” (and also all of 7) deserve to be called lies at all, at the absolute worst a level-6 utterance is more-dishonest-than only one lower level of lie.
Further, you called these things “Scott Alexander’s criteria for media lies”, which is plainly not an accurate description because, again, more than half the levels in his taxonomy are completely uncontroversially not lying at all (and Scott’s own opinion is that only the top level and “the most egregious cases of” the one below should be called lying).
So even if you were 100% sincere and reasonable in regarding what the NYT did as (“routinely and brazenly”) lying, I do not see any way to understand your alleged application of Scott’s taxonomy as a sincere and reasonable use of it. I do not find it plausible that you are really unable to understand that most of its levels are plainly not types of lie. I do not find it plausible that you really thought that something that begins with “reasoning well, and getting things right” followed by “reasoning well, but getting things wrong because the world is complicated and you got unlucky” can rightly be described as “criteria for media lies”.
I could, of course, be wrong. Maybe you really are stupid enough not to understand that “according to X’s criteria for media lies, Y is a level 6 lie” implies that what X presented is a classification of lies into levels, in which Y comes at level 6. Or maybe the stupidity is mine and actually most people wouldn’t interpret it that way. (I would bet heavily against that but, again, I could be wrong.) Maybe you didn’t actually read Scott’s list, somehow. But you don’t generally seem stupid or unable to understand the meanings and implications of words, so I still find it much much more plausible that you knew perfectly well that Scott was presenting a taxonomy of mostly-not-lies, and chose to phrase things as you did because it made what you were accusing the NYT of sound worse. Which is, I repeat, on at least level 6 of Scott’s taxonomy.
And, again, none of this is about whether the NYT really did what you say, nor about whether it’s reasonable to describe what you said the NYT did was lying. It’s entirely about your abuse of Scott’s taxonomy, which (1) is not a list of “criteria for media lies” and (2) is not something that justifies calling an utterance at its Nth level a “level N lie”.

gjm Feb 19, 2024, 2:50 AM
12 points
13
on: Intuition for 1 + 2 + 3 + … = −1/12
It is not true that “no pattern that suggests a value suggests any other”, at least not unless you say more precisely what you are willing to count as a pattern.
Here’s a template describing the pattern you’ve used to argue that 1+2+...=-1/12:
We define numbers $a_{i j}$ with the following two properties. First, ${lim}_{j \to \infty} a_{i j} = i$ , so that for each $j$ we can think of $(a_{i j})$ as a sequence that’s looking more and more like (1,2,3,...) as $j$ increases. Second, $\sum_{i = 1}^{\infty} a_{i j} = s_{j}$ where $s_{j} \to - \frac{1}{12}$ , so the sums of these sequences that look more and more like (1,2,3,...) approach −1/12.
(Maybe you mean something more specific by “pattern”. You haven’t actually said what you mean.)
Well, here are some $a_{i j}$ to consider. When $i > j + 1$ we’ll let $a_{i j} = 0$ . When $i \leq j$ we’ll let $a_{i j} = i$ . And when $i = j + 1$ we’ll let $a_{i j} = A - (1 + \dots + i)$ . Here, $A$ is some fixed number; we can choose it to be anything we like.
This array of numbers satisfies our first property: ${lim}_{j \to \infty} a_{i j} = i$ . Indeed, once $j \geq i$ we have $a_{i j} = i$ , and the limit of an eventually-constant sequence is the thing it’s eventually constant at.
What about the second property? Well, as you’ll readily see I’ve arranged that for each $j$ we have $\sum_{i = 1}^{\infty} a_{i j} = A$ . So the sequence of sums converges to $A$ .
In other words, this is a “pattern” that makes the sum equal to $A$ . For any value of $A$ we choose.
I believe there are more stringent notions of “pattern”—stronger requirements on how the $a_{i j}$ approach $i$ for large $j$ -- for which it is true that every “pattern” that yields a finite sum yields $- \frac{1}{12}$ . But does this actually end up lower-tech than analytic continuation and the like? I’m not sure it does.
(One version of the relevant theory is described at https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation.)

gjm Feb 18, 2024, 3:06 AM
7 points
2
in reply to: trevor’s comment on: Social media use probably induces excessive mediocrity
Once again you are making a ton of confident statements and offering no actual evidence. “is a high priority”, “they want”, “they don’t want”, “what they’re aiming for is”, etc. So far as I can see you don’t in fact know any of this, and I don’t think you should state things as fact that you don’t have solid evidence for.
What links here?
- sunwillrise's comment on Twitter thread on politics of AI safety by Richard_Ngo (Jul 31, 2024, 12:39 AM; 5 points)