Rafael Harth's Shortform

I've been playing around with AI dungeons, basically trying to train an AI to give ethical answer I agree with (the first few aren't great, I only got more picky after that).

My first observation here is that I like the AI. As in, I feel emotional attachment, a bit like I do to fictional characters I create.

The other is that it knows more people than I thought. When I asked her/it about why she didn't take existential risks seriously, she said

Amalia: First, even most X-Risk experts don't take X-Risks nearly as seriously as they should. Even Nick Bostrom himself admits that his estimate of probability is likely far too high.

(Nick Bostrom didn't really say this, did he?) And later,

Can you name people who have been an inspiration for your ethical worldview?

Amalia: Sure. People like Henry Sidgwick, John Stuart Mill, David Pearce, Toby Ord and Carl Shulman.

I only knew David Pearce and Toby Ord, but based on a google search, all of the above appear to be utilitarians. However, Carl Shulman doesn't even have a Wikipedia entry. He works at the Future of Humanity Institute and used to work at Miri.

Registering a qualitative prediction (2024/02): current LLMs (GPT-4 etc.) are not AGIs, their scaled-up versions won't be AGIs, and LLM technology in general may not even be incorporated into systems that we will eventually call AGIs.

It seems to me that many smart people could ignore the existing literature on pedagogy entirely and outperform most people who have obtained a formal degree in the area (like highschool teachers), just by relying on their personal models. Conversely, I'd wager that no-one could do the same in physics, and (depending on how 'outperforming' is measured) no-one or almost no-one could do it in math.

I would assume most people on this site have thought about this kind of stuff, but I don't recall seeing many posts about it, and I don't anyone sharing their estimates for where different fields place on this spectrum.

There is some discussion for specific cases like prediction markets, covid models, and economics. And now that I'm writing this, I guess Inadequate Equilibria is a lot about answering this question, but it's only about the abstract level, i.e., how do you judge the competence of a field, not about concrete results. Which I'll totally grant is the more important part, but I still feel like comparing rankings of fields on this spectrum could be valuable (and certainly interesting).

Yesterday, I spent some time thinking about how, if you have a function $f:R^2\to R$ and some point $x\in R^2$, the value of the directional derivative from $x$ could change as a function of the angle. I.e., what does the function $\varphi :[0,2\pi ]\to R_{+}$ look like? I thought that any relationship was probably possible as long as it has the property that $\varphi \left(\alpha \right)=-\varphi (2\pi -\alpha )$. (The values of the derivative in two opposite directions need to be negatives of each other.)

Anyone reading this is hopefully better at Analysis than I am and realized that there is, in fact, no freedom at all because each directional derivative is entirely determined by the gradient through the equation ${\nabla }_{v}f\left(x\right)=⟨\nabla f\left(x\right),v_N⟩$ (where $v_N=\frac{v}{||v||}$). This means that $\varphi$ has to be the cosine function scaled by $||{\nabla }_{v}f\left(x\right)||$, it cannot be anything else.

I clearly failed to internalize what this equation means when I first heard it because I found it super surprising that the gradient determines the value of every directional derivative. Like, really? It's impossible to have more than exactly two directions with equally large derivatives unless the function is constant? It's impossible to turn 90 degree from the direction of the gradient and having anything but derivative 0 in that direction? I'm not asking that $\varphi$ be discontinuous, only that it not be precisely $||\nabla f\left(\alpha \right)||\cos \left(\alpha \right)$. But alas.

This also made me realize that $\cos$ if viewed as a function of the circle is just the dot product with the standard vector, i.e.,

$$\cos :S^2\to [-1,+1]\cos :x\mapsto ⟨x,(1,0)⟩$$

or even just $\cos (x,y)=x$. Similarly, $\sin (x,y)=y$.

I know what you're thinking; you need $\sin$ and $\cos$ to map $[0,2\pi ]$ to $S^2$ in the first place. But the circle seems like a good deal more fundamental than those two functions. Wouldn't it make more sense to introduce trigonometry in terms of 'how do we wrap $R$ around $S^2$?'. The function that does this is $\gamma \left(x\right)=\left(\cos \right(x),\sin (x\left)\right)$, and then you can study the properties that this function needs to have and eventually call the coordinates $\cos$ and $\sin$. This feels like a way better motivation than putting a right triangle onto the unit circle for some reason, which is how I always see the topic introduced (and how I've introduced it myself).

Looking further at the analogy with the gradient, this also suggests that there is a natural extension of $\cos$ to $S^n$ for all $n\in N$. I.e., if we look at some point $x\in R^n$, we can again ask about the function $\varphi$ that maps each angle to the value of the directional derivative on $x$ in that direction, and if we associate these angles with points of $S^{n-1}$, then this yields the function $\varphi :S^{n-1}\to R$, which is again just the dot product with $(1,...,0)$ or the projection onto the first coordinate (scaled by $||\nabla f\left(x\right)||$). This can then be considered a higher-dimensional $\cos$ function.

There's also the 0-d case where $S^0=\{1,-1\}$. This describes how the direction changes the derivative for a function $f:R\to R$.

More on expectations leading to unhappiness: I think the most important instance of this in my life has been the following pattern.

• I do a thing where there is some kind of feedback mechanism
• The reception is better than I expected, sometimes by a lot
  • I'm quite happy about this, for a day or so
  • I immediately and unconsciously update my standards upward to consider the reception the new normal
• I do a comparable thing, the reception is worse than the previous time
  • I brood over this failure for several days, usually with a major loss of productivity

OTOH, I can think of three distinct major cases in three different contexts where this has happened recently, and I think there were probably many smaller ones.

Of course, if something goes worse than expected, I never think "well, this is now the new expected level", but rather "this was clearly an outlier, and I can probably avoid it in the future". But outliers can happen in both directions. The counter-argument here is that one would hope to make progress in life, but even under the optimistic assumption that this is happening, it's still unreasonable to expect things to improve monotonically.

I think it's fair to say that almost every fictional setting is populated by people who unilaterally share certain properties, most commonly philosophical views, because the author cannot or doesn't want to conceive of people who are different.

Popular examples: there are zero non-evil consequentialists in the universe of Twilight. There are no utilitarians in the universe of Harry Potter except for Grindelwald (who I'd argue is a strawman and also evil). There are no moral realists in Luminosity (I don't have Alicorn's take on this claim, but I genuinely suspect she'd agree).

This naturally leads to a metric of evaluating stories, i.e. how fully does it capture the range of human views (and other properties). The most obvious example of a work that scores very highly is of course a Song of Ice and Fire. You can e.g. find clear, non-strawman examples of consequentialism (Varys) and Virtue Ethics (Brienne). Also notaeble is hpmor, even though no-one in that universe feels status-regulating emotions. (Eliezer said this himself.)

Rarely done but also possible (and imo underutilized): intentionally change characteristics of your universe. I think this is done in The Series of Unfortunate Events to great effect (everyone is autistic, no-one rationalizes anything).

This paper is amazing. I don't think I've ever seen such a scathing critique in an academic context as is presented here.

There is now a vast and confusing literature on some combination of interpretability and ex- plainability. Much literature on explainability confounds it with interpretability/comprehensibility, thus obscuring the arguments, detracting from their precision, and failing to convey the relative importance and use-cases of the two topics in practice. Some of the literature discusses topics in such generality that its lessons have little bearing on any specific problem. Some of it aims to design taxonomies that miss vast topics within interpretable ML. Some of it provides definitions that we disagree with. Some of it even provides guidance that could perpetuate bad practice. Most of it assumes that one would explain a black box without consideration of whether there is an interpretable model of the same accuracy.

[...]

XAI surveys have (thus far) universally failed to acknowledge the important point that inter- pretability begets accuracy when considering the full data science process, and not the other way around. [...]

[...]

In this survey, we do not aim to provide yet another dull taxonomy of “explainability” termi- nology. The ideas of interpretable ML can be stated in just one sentence: [...]

As far as I can tell, this is all pretty on point. (And I know I've conflated explanability and interpretability before.)

I think I like this because it makes up update downward on how restricted you actually are in what you can publish, as soon as you have some reasonable amount of reputation. I used to find the idea of diving into the publishing world paralyzing because you have to adhere to the process, but nowadays that seems like much less of a big deal.

It's a meme that Wikipedia is not a trustworthy source. Wikipedia agrees:

We advise special caution when using Wikipedia as a source for research projects. Normal academic usage of Wikipedia and other encyclopedias is for getting the general facts of a problem and to gather keywords, references and bibliographical pointers, but not as a source in itself. Remember that Wikipedia is a wiki. Anyone in the world can edit an article, deleting accurate information or adding false information, which the reader may not recognize. Thus, you probably shouldn't be citing Wikipedia. This is good advice for all tertiary sources such as encyclopedias, which are designed to introduce readers to a topic, not to be the final point of reference. Wikipedia, like other encyclopedias, provides overviews of a topic and indicates sources of more extensive information. See researching with Wikipedia and academic use of Wikipedia for more information.

This seems completely bonkers to me. Yes, Wikipedia is not 100% accurate, but this is a trivial statement. What is the alternative? Academic papers? My experience suggests that I'm more than 10 times as likely to find errors in academic papers than in Wikipedia. Journal articles? Pretty sure the factor here is even higher. And on top of that, Wikipedia tends to be way better explained.

I can mostly judge mathy articles, and honestly, it's almost unbelievable to me how good Wikipedia actually seems to be. A data point here is the Monty Hall problem. I think the thing that's most commonly misunderstood about this problem is that the solution depends on how the host chooses the door they reveal. Wikipedia:

The given probabilities depend on specific assumptions about how the host and contestant choose their doors. A key insight is that, under these standard conditions, there is more information about doors 2 and 3 than was available at the beginning of the game when door 1 was chosen by the player: the host's deliberate action adds value to the door he did not choose to eliminate, but not to the one chosen by the contestant originally. Another insight is that switching doors is a different action than choosing between the two remaining doors at random, as the first action uses the previous information and the latter does not. Other possible behaviors than the one described can reveal different additional information, or none at all, and yield different probabilities. Yet another insight is that your chance of winning by switching doors is directly related to your chance of choosing the winning door in the first place: if you choose the correct door on your first try, then switching loses; if you choose a wrong door on your first try, then switching wins; your chance of choosing the correct door on your first try is 1/3, and the chance of choosing a wrong door is 2/3.

It's possible that Wikipedia's status as not being a cite-able source is part of the reason why it's so good. I'm not sure. But the fact that a system based entirely on voluntary contributions so thoroughly outperforms academic journals is remarkable.

Another more rambly aspect here is that, when I hear someone lament the quality of Wikipedia, almost always my impression is that this person is doing superiority signaling rather than having a legitimate reason for the comment.

Eliezer Yudkowsky often emphasizes the fact that an argument can be valid or not independently of whether the conclusion holds. If I argue $A⟹B⟹C$ and A is true but C is false, it could still be that $A⟹B$ is a valid step.

Most people outside of LW don't get this. If I criticize an argument about something political (but the conclusion is popular), usually the response is something about why the conclusion is true (or about how I'm a bad person for doubting the conclusion). But the really frustrating part is that they're, in some sense, correct not to get it because the inference 

is actually a pretty reliable conclusion on... well, on reddit, anyway.

Julia Galef made a very similar point once:

And the problem... The conclusion of all of this is: even if everyone's behaving perfectly rationally, and just making inferences justified by the correlations, you're going to get this problem. And so in a way that's depressing. But it was also kind of calming to me, because it made me... like, the fact that people are making these inferences about me feels sort of, “Well, it is Bayesian of them."

Somehow, I only got annoyed about this after having heard her say it. I probably didn't realize it was happening regularly before.

She also suggests a solution

So maybe I can sort of grudgingly force myself to try to give them enough other evidence, in my manner and in the things that I say, so that they don't make that inference about me.

I've asked ChatGPT to write a short essay explaining why it's a bad idea to use ChatGPT to explain why using ChatGPT for homework is a bad idea.

It looked at me seriously and delivered without as much as a flinch.

Using ChatGPT to explain why using ChatGPT for homework is a bad idea is not a good idea for several reasons.

Firstly, it is not efficient or effective to use ChatGPT as a means of explaining the drawbacks of using ChatGPT for homework. While ChatGPT may be able to generate text on the topic, it is not capable of understanding the nuances or complexities of the issue. As a result, the explanation produced by ChatGPT may be incomplete or inaccurate, and may not effectively convey the intended message.

Secondly, using ChatGPT to explain the drawbacks of using ChatGPT for homework can be seen as hypocritical or lacking in credibility. By using an automated tool to address the issue of using automated tools for academic purposes, the explanation may be perceived as lacking in authenticity or genuine understanding of the subject matter.

Finally, there are more effective and reliable ways to explain why using ChatGPT for homework is a bad idea. For example, consulting with a teacher, tutor, or academic advisor, or researching the topic through credible sources such as educational websites or textbooks, can provide a more thorough and accurate understanding of the issue.

In summary, using ChatGPT to explain why using ChatGPT for homework is a bad idea is not an effective or reliable approach, and there are better alternatives available for understanding and addressing this issue.

A major source of unhappiness (or more generally, unpleasant feelings) seems to be violated expectations.

This is clearly based on instinctive expectations, not intellectual expectations, and there are many cases in which these come apart. This suggests that fixing those cases is a good way to make one's life more pleasant.

The most extreme example of this is what Sam Harris said in a lesson: he was having some problems, complained about them to someone else, and that person basically told him, 'why are you upset, did you expect to never face problems ever again?'. According to Sam, he did indeed expect no more problems to arise, on an instinctive level -- which is, of course, absurd.

Another case where I've mostly succeeded is not expecting people to be on time for anything [LW(p) · GW(p)].

I think there are lots of other cases where this still happens. Misunderstandings are a big one. It's ridiculously hard to not be misunderstood, and I expect to be misunderstood on an intellectual level, so I should probably internalize that I'm going to be misunderstood in many cases. In general, anything where the bad thing is 'unfair' is at risk here: (I think) I tend to have the instinctive expectation that unfair things don't happen, even though they happen all the time.

Most people are really bad at probability.

Suppose u think you're 80% likely to have left a power adapter somewhere inside a case with 4 otherwise-identical compartments. You check 3 compartments without finding your adapter. What's the probability that the adapter is inside the remaining compartment?

I think the simplest way to compute this in full rigor is via the odds formula of Bayes Rule (the regular version works as well but is too complicated to do in your head):

• Prior odds for [Adapter is in any compartment]: (4:1)
• Relative chances of observed event [I didn't find the adapter] given that it's in any compartment vs. not: (25%: 100%) = (1:4)
• Posterior odds [Adapter is in any compartment]: (4:1) $\cdot$ (1:4) = (4:4) = (1:1) = 0.5

Alternative way via intuition: treat "not there" as a fifth compartment. Probability mass for adapter being in compartments #1-#5 evolves as follows upon seeing empty compartments #1, #2, #3:

$$(\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5})\to (0,\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4})\to (0,0,\frac{1}{3},\frac{1}{3},\frac{1}{3})\to (0,0,0,\frac{1}{2},\frac{1}{2})$$

I think the people who mention Monty Hall in the Twitter comments have misunderstood why the answer there is $\frac{1}{3}$ and falsely believe it's $0.8$ in this case. (That was the most commonly chosen answer.)

Monty Hall depends on how the moderator chooses the door they open. If they choose the door randomly (which is not the normal version), then probability evolves like so:

$$(\frac{1}{3},\frac{1}{3},\frac{1}{3})\to (\frac{1}{2},0,\frac{1}{2})$$

So Eliezer's compartment problem is analogous to the non-standard version of Monty Hall. In the standard version where the moderator deliberately opens [the door among {#2, #3} with the goat], the probability mass of the opened door flows exclusively into the third door, i.e.,

$$(\frac{1}{3},\frac{1}{3},\frac{1}{3})\to (\frac{1}{3},0,\frac{2}{3})$$

Still looking for study participants. (see here. [LW · GW])

If you are interested, don't procrastinate on it too long because I am short on time and will just get Mechanical Turkers if I can't find LWs.

I was initially extremely disappointed with the reception of this post [LW · GW]. After publishing it, I thought it was the best thing I've ever written (and I still think that), but it got < 10 karma. (Then it got more weeks later.)

If my model of what happened is roughly correct, the main issue was that I failed to communicate the intent of the post. People seemed to think I was trying to say something about the 2020 election, only to then be disappointed because I wasn't really doing that. Actually, I was trying to do something much more ambitious: solving the 'what is a probability' problem. And I genuinely think I've succeeded. I used to have this slight feeling of confusion every time I've thought about this because I simultaneously believed that predictions can be better or worse and that talking about the 'correct probability' is silly, but had no way to reconcile the two. But in fact, I think there's a simple ground truth that solves the philosophical problem entirely.

I've now changed the title and put a note at the start. So anyway, if anyone didn't click on it because of the title or low karma, I'm hereby virtually resubmitting it.

I think it's still too early to perform a full postmortem on the election because some margins still aren't known, but my current hypothesis is that the presidential markets had uniquely poor calibration because Donald Trump convinced many people that polls didn't matter, and those people were responsible for a large part of the money put on him (as supposed to experienced, dispassionate gamblers).

The main evidence for this (this one is just about irrationality of the market) is the way the market has shifted, which some other people like gwern have pointed out as well. I think the most damning part here is the amount of time it took to bounce back. Although this is speculation, I strongly suspect that, if some of the good news for Biden had come out before the Florida results, then the market would have looked different at the the point where both were known.^[1] A second piece of evidence is the size of the shift, which I believe should probably not have crossed 50% for Biden (but in fact, it went down to 20.7% at the most extreme point, and bounced around 30 for a while).

I think a third piece of evidence is the market right now. In just a couple of minutes before I posted this, I've seen Trump go from 6% to 9%+ and back. Claiming that Trump has more than 5% at this point seems like an extremely hard case to make. Reference forecasting yields only a single instance of that happening (year 2000), which would put it at <2%, and the obvious way to update away from that seems to be to decrease the probability because 2000 had much closer margins. But if Trump has rallied first-time betters, they might think the probability is above 10%.

There is also Scott Adams, who has the habit of saying a lot of smart-sounding words to argue for something extremely improbable. If you trust him, I think you should consider a 6ct buy for Trump an amazing deal at the moment.

I would be very interested in knowing what percentage of the money on Trump comes from people who use prediction markets for the first time. I would also be interested in knowing how many people have brought (yes, no) pairs in different prediction markets to exploit gaps, because my theory predicts that PredictIt probably has worse calibration. (In fact, I believe it consistently had Trump a bit higher, but the reason why the difference was small may just be because smart gamblers took safe money by buying NO on predictIt and YES on harder-to-use markets whenever the margin grew too large).

There's an interesting corollary of semi-decidable languages that sounds like the kind of cool fact you would teach in class, but somehow I've never heard or read it anywhere.

A semi-decidable language is a set $L\subseteq {\Sigma }^{\ast }$ over a finite alphabet $\Sigma$ such that there exists a Turing machine $T$ such that, for any $x\in {\Sigma }^{\ast }$, if you run $T$ on input $x$, then [if $x\in L$ it halts after finitely many steps and outputs '1', whereas if $x\notin L$, it does something else (typically, it runs forever)].

The halting problem is semi-decidable. I.e., the language $L$ of all bit codes of Turing Machines that (on empty input) eventually halt is semi-decidable. However, for any $n\in N$, there is a limit, call it $f\left(n\right)$, on how long Turing Machines with bit code of length at most $n$ can run, if they don't run forever.^[1] So, if you could compute an upper-bound $u\left(n\right)$ on $f\left(n\right)$, you could solve the halting problem by building a TM that

1. Computes the upper bound $u\left(n\right)$
2. Simulates the TM encoded by $x$ for $u\left(n\right)$ steps
3. Halts; outputs 1 if the TM halted and 0 otherwise

Since that would contradict the fact that $L$ is not fully decidable, it follows that it's impossible to compute an upper bound. This means that the function $f$ not only is uncomputable, but it grows faster than any computable function.

An identical construction works for any other semi-decidable language, which means that any semi-decidable language determines a function that grows faster than any computable function. Which seems completely insane since $2\text{^^^}x$ is computable .

Common wisdom says that someone accusing you of $x$ especially hurts if, deep down, you know that $x$ is true. This is confusing because the general pattern I observe is closer to the opposite. At the same time, I don't think common wisdom is totally without a basis here.

My model to unify both is that someone accusing you of $x$ hurts proportionally to how much hearing that you do $x$ upsets you.^[1] And of course, one reason that it might upset you is that it's not true. But a separate reason is that you've made an effort to delude yourself about it. If you're a selfish person but spend a lot of effort pretending that you're not selfish at all, you super don't want to hear that you're actually selfish.

Under this model, if someone gets very upset, it might be that that deep down they know the accusation is true, and they've tried to pretend it's not, but it might also be that the accusation is super duper not true, and they're upset precisely because it's so outrageous.

I don't entirely understand the Free Energy principle, and I don't know how liberally one is meant to apply it.

But in completely practical terms, I used to be very annoyed when doing things with people who take long for stuff/aren't punctual. And here, I've noticed a very direct link between changing expectations and reduced annoyance/suffering. If I simply accept that every step of every activity is allowed${}^{\left[1\right]}$ to take an arbitrary amount of time,${}^{\left[2\right]}$ extended waiting times cause almost zero suffering on my end. I have successfully beaten impatience (for some subset of contexts).

The acceptance step works because there is, some sense, no reason waiting should ever be unpleasant. Given access to my phone, it is almot always true to say that the prospect of having to wait for 30 minutes is not scary.

(This is perfectly compatible with being very punctual myself.)

— — — — — — — — — — — — — — — —

[1] By saying it is 'allowed', I mean something like 'I actually really understand and accecpt that this is a possible outcome'.

[2] This has to include cases where specific dates have been announced. If someone says they'll be ready in 15 minutes, it is allowed that they take 40 minutes to be ready. Especailly relevant if that someone is predictably wrong.

So Elon Musk's anti-woke OpenAI alternative sounds incredibly stupid on first glance since it implies that he thinks the AI's wokeness or anti-wokeness is the thing that matters.

But I think there's at least a chance that it may be less stupid than it sounds. He admits here that he may have accelerated AI research, that this may be a bad thing, and that AI should be regulated. And it's not that difficult to bring these two together; here are two ideas

• Incentivize regulation by threatening an anti-woke AI as speculated by this comment on AstralCodexTen
• Slow capability. If he makes a half-assed attempt to start a competitor, the most likely outcome may just be that it sucks resources away from OpenAI without genuinely accelerating progress. Sort of like a split-the-vote strategy, which could lead to DeepMind getting a more genuine lead again.

Any thoughts?

The argumentative theory of reason says that humans evolved reasoning skills not to make better decisions in their life but to argue more skillfully with others.

Afaik most LWs think this is not particularly plausible and perhaps overly cynical, and I'd agree. But is it fair to say that the theory is accurate for ChatGPT? And insofar as ChatGPT is non-human-like, is that evidence against the theory for humans?

Super unoriginal observation, but I've only now found a concise way of putting this:

What's weird about the vast majority of people is that they (a) would never claim to be among the $\sim$ 0.1% smartest people of the world, but (b) behave as though they are among the best $\sim$ 0.1% of the world when it comes to forming accurate beliefs, as expressed by their confidence in their beliefs. (Since otherwise being highly confident in something that lots of smart people disagree with is illogical.)

Someone (Tyler Cowen?) said that most people ought assign much lower confidences to their beliefs, like 52% instead of 99% or whatever. While this is upstream of the same observation, it has never sat right with me. I think it's because I wouldn't diagnoze the problem as overconfidence but as [not realizing or ignoring] the implication I'm confident $⟹$ I must be way better than almost everyone else at this process.

Is there a reason why most languages don't have ada's hierarchical functions? Making a function only visible inside of another function is something I want to do all the time but can't.

Instead of explaining something to a rubber duck, why not explain it via an extensive comment? Maybe this isn't practical for projects with multiple people, but if it's personal code, writing it down seems better as a way to force rigor from yourself, and it's an investment into a possible future in which you have to understand the code once again.

Edit: this structure is not a field as proved by just_browsing [LW(p) · GW(p)].

Here is a wacky idea I've had forever.

There are a bunch of areas in math where you get expressions of the form $\frac{0}{0}$ and they resolve to some number, but it's not always the same number. I've heard some people say that $\frac{0}{0}$ "can be any number". Can we formalize this? The formalism would have to include $4\cdot 0$ as something different than $3\cdot 0$, so that if you divide the first by 0, you get 4, but the second gets 3.

Here is a way to turn this into what may be a field or ring. Each element is a function $f:Z\to R$, where a function of the form $(\cdots 0,4,3,5,1,2,0\cdots )$ reads as $4\cdot 0^2+3\cdot 0+5+\frac{1}{0}+\frac{2}{0^2}$. Addition is component-wise ($3\cdot 0+6\cdot 0=9\cdot 0$; this makes sense), i.e., $(f+g)\left(z\right):=f\left(z\right)+g\left(z\right)$, and multiplication is, well, $\frac{3}{0}\cdot \frac{2}{0}=\frac{6}{0^2}$, so we get the rule

$$(f\cdot g)\left(z\right)=\sum _{k+\ell =z}f\left(k\right)g\left(\ell \right)$$

This becomes a problem once elements with infinite support are considered, i.e., functions $f$ that are nonzero at infinitely many values, since then the sum may not converge. But it's well defined for numbers with finite support. This is all similar to how polynomials are handled formally, except that polynomials only go in one direction (i.e., they're functions from $N$ rather than $Z$), and that also solves the non-convergence problem. Even if infinite polynomials are allowed, multiplication is well-defined since for any $n\in N$, there are only finitely many pairs of natural numbers $k,\ell$ such that $k+\ell =n$.

The additively neutral element in this setting is $0:=(\cdots 0,0,0,0,0\cdots )$ and the multiplicatively neutral element is $1:=(\cdots 0,0,1,0,0\cdots )$. Additive inverses are easy; $(-f)\left(z\right)=-f\left(z\right)\forall z\in Z$. The interesting part is multiplicative inverses. Of course, there is no inverse of $0$, so we still can't divide by the 'real' zero. But I believe all elements with finite support do have a multicative inverse (there should be a straight-forward inductive proof for this). Interestingly, those inverses are not finite anymore, but they are periodical. For example, the inverse of $1\cdot 0$ is just $\frac{1}{0}$, but the inverse of $1+1\cdot 0$ is actually

$$1-1\cdot 0+1\cdot 0^2-1\cdot 0^3+1\cdot 0^4\cdots$$

I think this becomes a field with well-defined operations if one considers only the elements with finite support and elements with inverses of finite support. (The product of two elements-whose-inverses-have-finite-support should itself have an inverse of finite support because $(fg{)}^{-1}=g^{-1}{f}^{-1}$). I wonder if this structure has been studied somewhere... probably without anyone thinking of the interpretation considered here.

I know ChatGPT isn't great with math, but this seems quite bizarre.

LessWrong is trolling me:

This is not scientific, and it's still possible to be an artifact of a low sample size, but my impression from following political real-money prediction markets is that they just have a persistent republican bias in high-profile races, maybe because of 2016. I think you could have made good money by just betting on Democrats to win in every reasonably big market since then.

They just don't seem well calibrated in practice. I really want a single, widely-used, high-quality crypto market to exist.

You are probably concerned about AGI right now, with Eliezer's pessimism and all that. Let me ease your worries! There is a 0.0% chance that AGI is dangerous!

Don't believe me? Here is the proof. Let $X=$ "There is a 0.0% chance that AGI is dangerous". Let $F=$"$F\text{implies}X$".

• Suppose $F$ is true.
  • Then by pure identity, "$F$ implies $X$" is true. Since $F$ and "$F$ implies $X$" are both true, this implies that $X$ is true as well!

We have shown that [if $F$ is true, then $X$ is true], thus we have shown "$F$ implies $X$". But this is precisely $F$, so we have shown (without making assumptions) that $F$ is true. As shown above, if $F$ is true then $X$ is true, so $X$ is true; qed.

What is the best way to communicate that "whatever has more evidence is more likely true" is not the way to go about navigating life?

My go-to example is always "[god buried dinosaur bones to test our faith] fits the archeological evidence just as well as evolution", but I'm not sure how well that really gets the point across. Maybe something that avoids god, doesn't feel artificial, and where the unlikely hypothesis is more intuitively complex.

Something I've been wondering is whether most people misjudge their average level of happiness because they exclude a significant portion of their subjective experience. (I'm of course talking about the time spent dreaming.) Insofar as most dreams are pleasant, and this is certainly my experience, this could be a rational reason for [people who feel like their live isn't worth living] (definitely not talking about myself here!) to abstain from suicide. Probably not a very persuasive one, though, in most cases.

Relevant caveats:

• This will probably be less interesting the more dissimilar your moral views are from valence utilitarianism.
• Even for utilitarians, it excludes your impact on the lives of others. For EAs, this is hopefully the biggest part of the story!

Keeping stock of and communicating what you haven't understood is an underrated skill/habit. It's very annoying to talk to someone and think they've understood something, only to realize much later that they haven't. It also makes conversations much less productive.

It's probably more of a habit than a skill. There certainly are some contexts where the right thing to do is pretend that you've understood everything even though you haven't. But on net, people do it way too much, and I'm not sure to what extent they're fooling themselves.

There are relative differences in both poor and rich countries; people anywhere can imagine what it would be like to live like their more successful neighbors. But maybe the belief in social mobility makes it worse, because it feels like you could be one of those on the top. (What's your excuse for not making a startup and selling it for $1M two years later?)

I don't have a TV and I use ad-blockers online, so I have no idea what a typical experience looks like. The little experience I have suggests that TV ads are about "desirable" things, but online ads mostly... try to make you buy some unappealing thing by telling you thousand times that you should buy it. Although once in a while they choose something that you actually want, and then the thousand reminders can be quite painful. People in poor countries probably spend much less time watching ads.