## Posts

## Comments

**sil-ver**on Open & Welcome Thread - November 2019 · 2019-12-03T09:07:43.707Z · score: 1 (1 votes) · LW · GW

I did. It's hard to quantify how much, but I'm falling asleep in < 20 min more reliably and am less tired during the day. I also remember dreams much more frequently, which is a welcome change.

**sil-ver**on Effect of Advertising · 2019-11-26T19:11:07.230Z · score: 2 (2 votes) · LW · GW

Advertisement is a component of the cost of a product, right? Some percentage of the total cost associated with producing and selling a product is ads. If they're no longer allowed, that component disappears.

I'm not saying this leads to a net decrease in cost, but it is a factor which leads to some decrease in cost, so if you want to argue that a net increase in cost takes place, you have to argue why the decreased competition matters more than the direct savings.

**sil-ver**on Open & Welcome Thread - November 2019 · 2019-11-24T17:19:38.221Z · score: 1 (1 votes) · LW · GW

I've gotten my hands on melatonin now, inspired by your comment. I've found a site that actually shipped it from Poland, which I believe isn't even illegal and definitely won't get me into trouble. The cost is still minimal. So far, it seems to help. Thanks again.

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-14T18:10:48.639Z · score: 1 (1 votes) · LW · GW

That doesn't seem to follow, actually. You could easily have a very large universe that's almost entirely empty space (which does "repeat"), plus a moderate amount of structures that only appear once each.

Yeah, nonemptiness was meant to be part of the assumption in the phrase you quoted.

And as a separate argument, plenty of processes are irreversible in practice. For instance, consider a universe where there's a "big bang" event at the start of time, like an ordinary explosion. I'd expect that universe to never return to that original intensely-exploding state, because the results of explosions don't go backwards in time, right?

We're getting into territory where I don't feel qualified to argue – although it seems like that objection only applies to some very specific things, and probably not to most Sleeping Beauty like scenarios.

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-14T10:09:59.541Z · score: 1 (1 votes) · LW · GW

But by number of bits, which is what you need to avoid repetition.

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-14T10:04:27.941Z · score: 1 (1 votes) · LW · GW

You mean P(monday)? In that case it would be different although have some similarity. Why is the description length of the monday observer moment longer than the tuesday one?

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-14T10:01:04.016Z · score: 1 (1 votes) · LW · GW

I actually knew about UDT. Enough to understand how it wins in Transparent Newcomb, but not enough to understand that it extends to anthropic problems.

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-14T09:59:41.884Z · score: 1 (1 votes) · LW · GW

The solution to this kind of thing should be a wiki, I think. If the LessWrong wiki were kept up to date enough to have a page on anthropics, that would have solved the issue in this case and should work for many similar cases.

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-14T00:30:30.759Z · score: 1 (1 votes) · LW · GW

The ASSA is the Absolute Self Selection Assumption. It is a variant on the Self Selection Assumption (SSA) of Nick Bostrom. The SSA says that you should think of yourself as being a randomly selected conscious entity (aka "observer") from the universe. The Absolute SSA extends this concept to "observer moments" (OMs). An observer moment is one moment of existence of an observer's consciousness. If we think of conscious experience as a process, the OM is created by dividing this process up into small units of time such that no perceptible change occurs within that unit. The ASSA then says that you should think of the OM you are presently experiencing as being randomly selected from among all OMs in the universe.

This is what I'm doing. I haven't read the entire thing yet, but this paragraph basically explains the key idea of my model. I was going to address how to count instances eventually (near the end), and it bottoms out at observer moments. The full idea, abbreviated, is "start with a probability distribution over different universes, in each one apply the randomness thing via counting observer moments, then weigh those results with your distribution". This gives you intuitive results in Doomsday (no update), P/P (some bias towards larger universe depending on how strongly you believe in other universes), Sleeping Beauty (basically 1/3) and the "how do we update on X-risk given that we're still alive" question (complicated).

It appears that I independently came up with ASSA, plus a different way of presenting it. And probably a weaker formalism.

I'm obviously unhappy about this, but thank you for bringing it to my attention now rather than later.

One reason I was assuming there couldn't be other theories I was unaware of is that Stuart Armstrong was posting about anthropics and he seemed totally unaware.

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-14T00:02:00.742Z · score: 1 (1 votes) · LW · GW

I have answers to all of these questions! I just haven't posted them yet. If I present an entirely new theory in one super long post, then obviously no-one reads it. In fact, it would be irrational to read it because the prior that I'm onto something is just too low to invest the time. A sequence of short posts where each post makes a point which can be understood by anyone having read up to that post – that's not optimal, but how else could you do it? This is a completely genuine question if you have an answer.

So the structure I've chosen is to first state the distinction, then lay out the model that deals with randomness only (because that already does some stuff which SIA and SSA can't), then explain how to deal with ignorance, which makes the model complete, and *then* present a formalized version. The questions you just listed all deal with the ignorance part, the part that's still in the pipeline.

Well, and I didn't know I was competing with UDASSA, because I didn't know it existed. For some reason it's sitting at 38 karma, which makes it easy to miss, and you're the first to bring it up. I'll read it before I post anything else.

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-13T22:49:48.503Z · score: 2 (2 votes) · LW · GW

I think the relevant difference is that, in the set of integers, each element is strictly more complex than the previous one, but in the universe, you can probably upper bound the complexity (that's what I'm assuming, anyway). So eventually stuff should repeat, and then anything that has a nonzero probability of appearing will appear arbitrarily often as you increase the size. For example, if there's an upper bound to the complexity of a planet, then you can only have that many planets until you get a repeat.

**sil-ver**on Insights from the randomness/ignorance model are genuine · 2019-11-13T22:27:58.804Z · score: 3 (2 votes) · LW · GW

Yeah, I wrote this assuming people have the context.

So there's a class of questions where standard probability theory doesn't give clear answers. This was dubbed anthropics or anthropic probability. To deal with this, two principles were worked out, SSA and SIA, which are well-defined and produce answers. But for both of them, there are problems where their answers seem absurd.

I think the best way to understand the problem of anthropics is by looking at the Doomsday argument as an example. Consider all humans who will ever live (assuming they're not infinitely many). Say that's many. For simplicity, we assume that there are only two cases, either humanity goes extinct tomorrow, in which case is about sixty billion – but let's make that for simplicity – or humanity flourishes and expands through the cosmos, in which case is, say, . Let's call the hypothesis that humans go extinct, and the hypothesis that they don't (that's for "short" and "long" human history). Now we want to update on given the observation that you are human number (so will be about 30 billion). Let's call that observation . Also let be your prior on , so .

The Doomsday argument now goes as follows. The term is , because if is true then there are a total of people, each position is equally likely, so is just the chance to get your particular one. On the other hand, is , because if is true there are only people total. So we simply apply Bayes on the observation , and then use the law of total probability in the demonimator to obtain

If , this term equals about 0.00989. So even if you were very confident that humanity would make it, you should still assign just below 1% on that after updating. If you want to work it out yourself, this is where you should pause and think about what part of this is wrong.

So the part that's problematic is the probability for . There is a hidden assumption that you had to be one of the humans who was actually born. This was then dubbed the Self-Sampling Assumption (SSA), namely

All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class.

So SSA endorses the Doomsday argument. The principled way to debunk this is the Self-Indexing Assumption (SIA), which says

All other things equal, an observer should reason as if they are randomly selected from the set of allpossibleobservers.

If you apply SIA, then and hence . Updating on no longer does anything.

So this is the problem where SSA gives a stupid anwer. The problem where SIA gives the stupid answer is the Presumptuous Philosopher problem: there are two theories of how large the universe is, according to one it's times as large as it is according to the other. If you apply the SIA rule, you get that the odds for living in the small universe is (if the prior was on both).

There is also Full Non-indexical Conditioning which is technically a different theory, and it argues differently, but it outputs the same as SIA in every case, so basically there are just the two. And that, as far as I know, is the state of the art. No-one has come up with a theory that can't be made to look ridiculous. Stuart Armstrong has made a bunch of LW posts about this recently-ish, but he hasn't proposed a solution, he's pointed out that existing theories are problematic. This one, for example.

I've genuinely spent a lot of time thinking really hard about this stuff, and my conclusion is that the "reason as if you're randomly selected from a set of observers" thing is the key problem here. I think that's the reason why this still hasn't been worked out. It's just not the right way to look at it. I think the relevant variable which everyone is missing is that there are two fundamentally different kinds of uncertainty, and if you structure your theory around that, everything works out. And I think I do have a theory where everything works out. It doesn't update on Doomsday and it doesn't say the large universe is times as likely as the small one. It doesn't give a crazy answer anywhere. And it does it all based on simple principles.

Does that answer the question? It's possible that I should have started the sequence with a post that states the problem; like I just assumed everyone would know the problem without ever thinking about whether that's actually the case.

**sil-ver**on The randomness/ignorance model solves many anthropic problems · 2019-11-12T19:34:45.926Z · score: 1 (1 votes) · LW · GW

I don't think you need to claim that there are different kinds of uncertainty to solve these. If you clearly specify what predicted experiences/outcomes you're applying the probability to, both of these examples dissolve.

This implies that everyone arguing about the correct probability in Sleeping Beauty is misguided, right?

I definitely think it is essential to differentiate between the two. I think there are cases where the question is the same and meaningful but the answer changes as the nature of uncertainty changes. Presumptuous Philosopher is such a case.

I argue more that the results of this model are meaningful in the next post.

**sil-ver**on The randomness/ignorance model solves many anthropic problems · 2019-11-11T18:15:19.967Z · score: 3 (2 votes) · LW · GW

Yes. It's much harder to find clear cases for ignorance than to find clear cases for randomness. That makes it telling that the P/P problem is about a case where it's at least arguable.

**sil-ver**on Open & Welcome Thread - November 2019 · 2019-11-11T16:33:51.550Z · score: 3 (2 votes) · LW · GW

Usually bread and/or cereal.

Your point that it might make sleep less effective even if it reduces falling asleep time is well taken.

**sil-ver**on Open & Welcome Thread - November 2019 · 2019-11-11T14:25:43.494Z · score: 1 (1 votes) · LW · GW

Well, the disclaimer here is that I wrote that without giving it conscious thought, so I have to examine why I believe it retroactively. I notice that I'm pretty confident.

I think the biggest reason is just that it feels that way. The second is that I sometimes intent to go to bed, notice that I'm hungry, then eat something and go back to bed, and I seem to sleep better than average on those cases.

I've only tried to use it deliberately once. It was recently; I failed to fall asleep for about an hour, at which point I usually give up and just stay awake longer. Instead I got up, deliberately ate too much and went back to bed. It actually worked.

**sil-ver**on Open & Welcome Thread - November 2019 · 2019-11-11T13:53:21.113Z · score: 1 (1 votes) · LW · GW

Thanks!

The meditation one in particular sounds interesting. I'll try it. flux is something I've already experimented with, and I do exercise regularly.

I've had the opposite experience with eating – if I eat a lot right before going to bed, it seems to help. But I guess it's not too surprising if that one differs from person to person.

**sil-ver**on Open & Welcome Thread - November 2019 · 2019-11-11T13:49:43.709Z · score: 1 (1 votes) · LW · GW

Thanks. That is very useful to know.

**sil-ver**on Open & Welcome Thread - November 2019 · 2019-11-10T18:16:38.227Z · score: 1 (1 votes) · LW · GW

:(

I've already been convinced by gwern that melatonin is worth taking, but it's prescription-only where I live. Still, you've given me a better sounding reason why I need it. I'm definitely going to ask for it the next time I see a doctor.

**sil-ver**on Open & Welcome Thread - November 2019 · 2019-11-09T10:49:03.339Z · score: 2 (2 votes) · LW · GW

Is there anything one can do for shortening the amount of time needed to fall asleep, and making it more robust? Currently I will be unable to fall asleep if I've overslept on the previous day or engaged with something too stimulating before going to bed. This is still true even though I've followed a strict schedule for a fairly long time. It's pretty annoying.

Something I already do is sleep with white noise, partially to make it less likely that I'll wake up from unexpected sounds.

**sil-ver**on Randomness vs. Ignorance · 2019-11-08T13:54:22.243Z · score: 4 (3 votes) · LW · GW

It could be argued that it's all ignorance. The die will roll the way that physics demands, based on the velocity, roll, pitch, yaw of the die, and the surface properties of the felt. There's only one possible outcome, you just don't know it yet. If you roll a die in an opaque cup, the uncertainty does not change in kind from the time you start shaking it to the time you slam it down - it's all the same ignorance until you actually look.

You can, if you like, believe that there is unknowability at the quantum level, but even that doesn't imply true randomness, just ignorance of which branch you'll find your perceptive trail following.

I'm not going to argue for unknowability at the quantum level, but I will argue (in the next post) that you are not sufficiently smart to differentiate precisely enough between the different possible situations, and that's why you have to group a bunch of different situations together, and that's how you get what I call randomness. I'm not arguing for or against any kind of "true" randomness. I agree that you can argue it's all ignorance, but (I claim) not doing so will solve a lot of problems

**sil-ver**on We tend to forget complicated things · 2019-10-25T10:32:50.261Z · score: 1 (1 votes) · LW · GW

If learning complicated things isn't hard, then what's the bottleneck on learning a new field?

**sil-ver**on Open & Welcome Thread - October 2019 · 2019-10-21T20:15:49.413Z · score: 1 (1 votes) · LW · GW

Has anyone written a summary of all organizations that work on AI alignment? If not, what is the best way to keep track of that?

**sil-ver**on We tend to forget complicated things · 2019-10-21T09:09:08.208Z · score: 2 (2 votes) · LW · GW

Yeah, your goal is not always to deeply understand the material you're looking at, and the post only applies when it is.

**sil-ver**on We tend to forget complicated things · 2019-10-21T09:03:48.897Z · score: 1 (1 votes) · LW · GW

Yes, great correction. I've modified the post to state that it only applies for Deep Things You Truly Want To Understand.

I'm often confronted with the difference you're describing but haven't ever articulated it as you just have.

**sil-ver**on Insights from Linear Algebra Done Right · 2019-07-17T19:07:00.464Z · score: 2 (3 votes) · LW · GW

I honestly don't think the tradeoff is real (but please tell me if you don't find my reasons compelling). If I study category theory next and it does some cool stuff with the base map, I won't reject that on the basis of it contradicting this book. Ditto if I actually use LA and want to do calculations. The philosophical understanding that matrix-vector multiplication isn't ultimately a thing can peacefully coexist with me doing matrix-vector multiplication whenever I want to. Just like the understanding that the natural number 1 is a different object from the integer number 1 peacefully coexists with me treating them as equal in any other context.

I don't agree that this view is theoretically limiting (if you were meaning to imply that), because it allows any calculation that was possible before. It's even compatible with the base map.

**sil-ver**on Insights from Linear Algebra Done Right · 2019-07-17T18:59:06.996Z · score: 1 (1 votes) · LW · GW

I wouldn't be heartbroken if it was defined like that, but I wouldn't do it if I were writing a textbook myself. I think the LADR approach makes the most sense – vectors and matrices are fundamentally different – and if you want to bring a vector into the matfrix world, then why not demand that you do it explicitly?

If you actually use LA in practice, there is nothing stopping you from writing . You can be 'sloppy' in practice if you know what you're doing while thinking that drawing this distinction is a good idea in a theoretical text book.

**sil-ver**on Insights from Linear Algebra Done Right · 2019-07-16T19:18:59.515Z · score: 1 (1 votes) · LW · GW

That looks like it also works. It's a different philosophy I think, where LADR says "vectors and matrices are fundamentally different objects and vectors aren't dependent on bases, ever" and your view says "each basis defines a bijective function that maps vectors from the no-basis world into the basis-world (or from the basis1 world into the basis2 world)" but it doesn't insist on them being fundamentally different objects. Like if then they're the same kind of object, and you just need to know which world you're in (i.e. relative to which basis, if any, you need to interpret your vector to).

II don't think not having matrix-vector multiplication is an issue. The LADR model still allows you to do everything you can do in normal LA. If you want to multiply a matrix with a vector , you just make into the n-by-1 matrix and then multiply two matrices. So you multiply rather than . It forces you to be explicit about which basis you want the vector to be relative to, which seems like a good thing to me. If is the standard basis, then will have the same entries as , it'll just be written as rather than .

**sil-ver**on No nonsense version of the "racial algorithm bias" · 2019-07-15T16:20:15.824Z · score: 2 (2 votes) · LW · GW

Afaik, in ML, the term bias is used to describe any move away from the uniform / mean case. But in common speech, such a move would only be called a bias if it's inaccurate. So if the algorithm learns a true pattern in the data (X is more likely to be classified as 1 than Y is) that wouldn't be called a bias. Unless I misunderstand your point.

**sil-ver**on Insights from Linear Algebra Done Right · 2019-07-14T22:56:36.507Z · score: 2 (2 votes) · LW · GW

Ow. Yes, you do. This wasn't a typo either, I remembered the result incorrectly. Thanks for pointing it out, and props for being attentive enough to catch it.

Or to be more precise, you only need one scalar, but the scalar is for not , because isn't given. The theorem says that, given and , there is a scalar and a vector such that and is orthogonal to .

**sil-ver**on Insights from Linear Algebra Done Right · 2019-07-14T21:02:41.828Z · score: 1 (1 votes) · LW · GW

I wonder, what do you think about the chapter about dual spaces, dual maps, annihilator, etc.?

Nothing, because it wasn't in the material. I worked through the second edition of the book, and the parts on duality seem to be new to the third edition.

I believe when mathematicians say that in general P(x) holds, they mean that for any x in the domain of interest P(x) holds. Perhaps you want to youtypicalinstead ofgeneralhere. E.g. there is a notion calledtypicaltensor rank of tensors of given shape, which means a tensor rank which occurs with non-zero probability when a random tensor of given shape is sampled.

Thanks for that, I changed it.

**sil-ver**on Contest: $1,000 for good questions to ask to an Oracle AI · 2019-07-04T14:27:11.833Z · score: 1 (1 votes) · LW · GW

It seems to me that one of the qualities of a good question for the LBO is that a correct answer has high utility, while another is the ability to judge the oracle's answer. If we take only the intersection between the two sets of questions that meet 1 and 2, it'll be a relatively small set. But if there is a set of questions where one might or might not be able to judge the answer but which are high utility, one could ask the LBO a set of such questions. Then, once the episode is over, give reward on those where one can evaluate the answers, and give null reward on the others.

Not really a submission, just thinking out loud.

**sil-ver**on Contest: $1,000 for good questions to ask to an Oracle AI · 2019-07-02T19:20:47.403Z · score: 2 (2 votes) · LW · GW

Submission for LBO:

Input a corpus of text (could be multiple posts) describing technical approaches to align a powerful AI. Split this into a finite number of items that are relatively short (such as paragraphs). Ask the oracle to choose the part that is most worth spending more time on. (For example, there might be a paragraph with a dangerous hidden assumption in an otherwise promising approach, and thinking more about it might reveal that and lead to conceptual progress.)

Have a team of researches look into it for an adequate amount of time which is fixed (and told to the oracle) in advance (maybe three months?) After the time is over, have them rate the progress they made compared to some sensible baseline. Use this as the oracle's reward.

Of course this has the problem of maximizing for apparent insight rather than actual insight.

**sil-ver**on On pointless waiting · 2019-06-11T09:32:13.421Z · score: 4 (2 votes) · LW · GW

Yes. Well put. This is related (though not identical) to the excellent Rest in Motion post from Nate Soares.

**sil-ver**on Drowning children are rare · 2019-05-29T21:44:39.504Z · score: 5 (2 votes) · LW · GW

Either charities like the Gates Foundation and Good Ventures are hoarding money at the price of millions of preventable deaths

My assumption before reading this has been that this is the case. Given that, does a reason remain to update away from the position that the GiveWell claim is basically correct?

For the rest of this post, let's suppose the true amount of money needed to save a life through GiveWell's top charities is 50.000$. I don't think anything about Singer's main point changes.

For one, it's my understanding that decreasing animal suffering is at least an order of magnitude more effective than decreasing human suffering. If the arguments you make here apply equally to that (which I don't think they do), and we take the above number, well that's 5000$ for a benefit-as-large-as-one-life-saved, which is still sufficient for Singer's argument

Secondly, I don't think your arguments apply to existential risk prevention and even if they did and we decrease effectiveness there by one order of magnitude, that'd also still validate Singer's argument if we take my priors.

I notice that I'm very annoyed at your on-the-side link to the article about OpenAI with the claim that they're doing the opposite of what the argument justifying the intervention recommends. It's my understanding that the article, though plausible at the time, was very speculative and has been falsified since it's been written. In particular, OpenAI has pledged not to take part in an arms race under reasonable conditions, which directly contradicts one of the points of that article. Quote:

Therefore, if a value-aligned, safety-conscious project comes close to building AGI before we do, we commit to stop competing with and start assisting this project. We will work out specifics in case-by-case agreements, but a typical triggering condition might be “a better-than-even chance of success in the next two years.”

That, and they seem to have an ethics board with significant power (this is based on deciding not to release the full version of GPT). I believe they also said that they won't publish capability results in the future, which also contradicts one of the main concerns (which, again, was reasonable at the time). Please either reply or amend your post.

**sil-ver**on Rationalist Vipassana Meditation Retreat · 2019-04-21T17:00:29.110Z · score: 1 (1 votes) · LW · GW

I'll also be attending the full 10 day version. I've only been meditating for a couple of months so the prospect of such a long retreat feels fairly threatening, but looking at the mean outcome, I think it's the correct call.

**sil-ver**on Open Thread April 2019 · 2019-04-06T09:09:29.732Z · score: 10 (6 votes) · LW · GW

What is the best textbook on analysis out there?

My goto source is Miri's guide, but analysis seems to be the one topic that's missing. TurnTrout mentioned this book which looks decent on first glance. Are there any competing opinions?

**sil-ver**on Humans Who Are Not Concentrating Are Not General Intelligences · 2019-03-28T22:50:19.852Z · score: 7 (4 votes) · LW · GW

I’ve noticed that I cannot tell, from casual conversation, whether someone is intelligent in the IQ sense.

I can't really do anything except to state this as a claim: I think a few minutes of conversation with anyone almost always gives me significant information about their intelligence in an IQ sense. That is, I couldn't tell you the exact number, and probably not even reliably predict it with an error of less than 20 (maybe more), but nonetheless, I know significantly more than zero. Like, if I talked to 9 people evenly spaced within [70, 130], I'm pretty confident that I'd get most of them into the correct half.

This does not translate into and kind of disagreement wrt to GPT's texts seeming normal if I just skim them. Or to Robin Hanson's thesis.

**sil-ver**on Ask LW: Have you read Yudkowsky's AI to Zombie book? · 2019-03-23T10:37:46.041Z · score: 3 (2 votes) · LW · GW

No, but I've read almost all of the sequences on website, I think. I didn't do it systematically, so it's almost a guarantee that I missed a few, but not many. Read some stuff twice, but again, not systematically.

I think they're amazing, and they've had a profound impact on me.

**sil-ver**on [Question] Tracking accuracy of personal forecasts · 2019-03-21T09:22:29.838Z · score: 1 (1 votes) · LW · GW

I do have a spreadsheet where I keep track of predictions, though only tracking the prediction, my confidence, and whether it came true or false. It's low effort and I think worth doing, but I can't confidently say that it has improved my calibration.

**sil-ver**on What kind of information would serve as the best evidence for resolving the debate of whether a centrist or leftist Democratic nominee is likelier to take the White House in 2020? · 2019-02-02T21:01:31.224Z · score: 4 (3 votes) · LW · GW

This does not answer the question, but it seems plausible to me that the leftist-centrist axis only has a very small impact on who is likely to win, which would be consistent with PredictIt's estimates.

**sil-ver**on Drexler on AI Risk · 2019-02-01T17:36:55.916Z · score: 13 (4 votes) · LW · GW

6.7 Systems composed of rational agents need not maximize a utility functionThere is no canonical way to aggregate utilities over agents, andgame theory shows that interacting sets of rational agents need not achieve even Pareto optimality.

Is [underlined] true? I know it's true if you have agents following CDT, but does it still hold if agents follow FDT? (I think if you say 'rational' it should not mean 'CDT' since CDT is strictly worse than FDT).

**sil-ver**on Topological Fixed Point Exercises · 2018-12-04T13:00:24.452Z · score: 1 (1 votes) · LW · GW

is defined just for one particular graph. It's the first edge in that graph such that . (So it could have been called ). Then for the next graph, it's a different . Basically, looks at where the first graph skips over the zero mark, then picks the last vertex before that point, then looks at the next larger graph, and if that graph skips later, it updates to the last vertex before that point in that graph, etc. I think the reason I didn't add indices to was just that there ar ealready the with two indices, but I see how it can be confusing since having no index makes it sound like it's the same value all throughout.

**sil-ver**on How rapidly are GPUs improving in price performance? · 2018-11-25T22:52:51.044Z · score: 4 (3 votes) · LW · GW

That makes perfect sense. Thanks.

**sil-ver**on How rapidly are GPUs improving in price performance? · 2018-11-25T21:47:55.077Z · score: 7 (5 votes) · LW · GW

When trying to fit an exponential curve, don't weight all the points equally.

Um... why?

**sil-ver**on Topological Fixed Point Exercises · 2018-11-24T23:23:25.357Z · score: 9 (2 votes) · LW · GW

Ex 5 (fixed version)

Let denote the triangle. For each , construct a 2-d simplex with nodes in , where the color of a point corresponds to the place in the disk that carries that point to, then choose to be a point within a trichromatic triangle in the graph. Then is a bounded sequence having a limit point . Let be the center of the disc; suppose that . Then there is at least one region of the disc that doesn't touch. Let be the distance to the furthest side, that is, let . Since the sides are closed regions, we have . Using continuity of , choose small enough such that . Then choose large enough so that (1) all triangles in have diameter less than and (2) . Then, given any other point in the triangle around in , we have that , so that . This proves that the triangle in does not map points to all three sides of the disc, contradicting the fact that it is trichromatic.

Ex 6

(This is way easier to demonstrate in a picture in a way that leaves no doubt that it works than it is to write down, but I tried to do it anyway considering that to be part of the difficulty.)

(Assume the triangle is equilateral.) Imbed into such that , , . Let be continuous. Then given by is also continuous. If then . Let be the circle with radius 2 around ; then because (it is in fact contained in the circle with radius 1, but the size of the circle is inconsequential). We will use exercise 5 to show that maps a point to the center, which is , from which the desired result follows. For this, we shall show that it has the needed properties, with the modification that points on any side may map precisely into the center. It's obvious that weakening the requirement in this way preserves the result.

Rotate the disk so that the red shape is on top. In polar coordinates, the green area now contains all points with angles between and , the blue area contains those between and , and the red area those between and and those between and . We will show that does not intersect the red area, except at the origin. First, note that we have

Since both and are convex combinations of finitely many points, it suffices to check all combinations that result by taking a corner from each. This means we need to check the points

and and and and and .

Which are easily computed to be

and and and and and

Two of those are precisely the origin, the other four have angles and and and . Indeed, they are all between and .

Now one needs to do the same for the sets and , but it goes through analogously.

**sil-ver**on Topological Fixed Point Exercises · 2018-11-23T21:22:41.367Z · score: 13 (4 votes) · LW · GW

I'm late, but I'm quite proud of this proof for #4:

Call the large triangle a graph and the triangles simply triangles. First, note that for any size, there is a graph where the top node is colored red, the remaining nodes on the right diagonal are colored green, and all nodes not on the right diagonal are colored blue. This graph meets the conditions, and has exactly one trichromatic triangle, namely the one at the top (no other triangle contains a red node). It is trivial to see that this graph can be changed into an arbitrary graph by re-coloring finitely many nodes. This will affect up to six triangles; we say that a triangle has changed iff it was trichromatic before the recoloring but not after, or vice versa, and we shall show that re-coloring any node leads to an even number of triangles being changed. This proves that the number of trichromatic triangles never stops being odd.

Label the three colors , and . Let be the node being recolored, wlog from to . Suppose first that has six neighbors. It is easy to see that a triangle between and two neighbors has changed if and only if one of the neighbors has color and the other has color or . Thus, we must show that the number of such triangles is even. If all neighbors have color , or if none of them do, then no triangles have changed. If exactly one node has color , then the two adjacent triangles have changed. Otherwise, let and denote two different neighbors of color . There are two paths using only neighbors of between and . Both start and end at a node of color . By the 1-D Sperner Lemma (assuming the more general result), it follows that both paths have an even number of edges between nodes of color and ; these correspond to the triangles that have changed.

If is a node on one of the graph's boundaries changing color from to , it has exactly 4 neighbors and three adjacent triangles. The two neighbors that are also on the boundary cannot have color , so either none, one, or both of the ones that aren't do. If it's none, no triangle has changed; if it's one, the two neighboring triangles have changed; and if it's both, then the two triangles with two nodes on the graph's boundary have changed.

**sil-ver**on Topological Fixed Point Exercises · 2018-11-22T20:12:15.427Z · score: 10 (3 votes) · LW · GW

Ex 1

Let and . Given an edge , let denote the map that maps the color of the left to that of the right node. Given a , let . Let denote the color blue and the color green. Let be 1 if edge is bichromatic, and 0 otherwise. Then we need to show that . We'll show , which is a striclty stronger statement than the contrapositive.

For , the LHS is equivalent to , and indeed equals if is bichromatic, and otherwise. Now let and let it be true for . Suppose . Then, if , that means , so that , and if , then , so that . Now suppose . If , then , so that , and if , then , so that . This proves the lemma by induction.

Ex 2

Ordinarily I'd proof by contradiction, using sequences, that can neither be greater nor smaller than 0. I didn't manage a short way to do it using the two lemmas, but here's a long way.

Set . Given , let be a path graph of vertices , where . If for any and we have , then we're done, so assume we don't. Define to be 1 if and have s different sign, and 0 otherwise. Sperner's Lemma says that the number of edges with are odd; in particular, there's at least one. Let the first one be denoted , then set .

Now consider the sequence . It's bounded because . Using the Bolzano-Weierstrass theorem, let be a convergent subsequence. Since for all , we have . On the other hand, if , then, using continuity of , we find a number such that . Choose and such that , then for all , so that and then for all , so that , a contradiction.

Ex 3

Given such a function , let be defined by . We have . If either inequality isn't strict, we're done. Otherwise, . Taking for granted that the intermediate value theorem generalizes to this case, find a root of , then .

**sil-ver**on Preschool: Much Less Than You Wanted To Know · 2018-11-21T18:03:21.591Z · score: 3 (3 votes) · LW · GW

Mostly agreed. But I think the obvious counter point is that you're arguing for a slightly different standard. Like, if the question is 'does pre-school basically make sense' then you're right, it doesn't, and the black box approach is weird. But if the question is 'should you send your children to pre-school' then the black box approach seems solid. Even if you could come up with something better in five minutes, you can't implement it, so the standard for it being worthwhile might be really low.

**sil-ver**on Diagonalization Fixed Point Exercises · 2018-11-18T23:11:24.616Z · score: 13 (4 votes) · LW · GW

Ex 4

Given a computable function , define a function by the rule . Then is computable, however, because for , we have that and .

Ex 5:

We show the contrapositive: given a function halt, we construct a surjective function from to as follows: enumerate all turing machines, such that each corresponds to a string. Given a , if does not decode to a turing machine, set . If it does, let denote that turning machine. Let with input first run halt; if halt returns , put out , otherwise will halt on input ; run on and put out the result.

Given a computable function , there is a string such that implements (if the turing thesis is true). Then , so that is surjective.

Ex 6:

Let be a parametrization of the circle given by . Given and , write to denote the point , where . First, note that, regardless of the topology on , it holds true that if is continuous, then so is for any , because given a basis element of the circle, we have which is open because is continuous.

Let be a continuous function from to . Then is continuous, and so is the diagonal function . Fix any , then given by is also continuous, but given any , one has and thus . It follows that is not surjective.

Ex 7:

I did it in java. There's probably easier ways to do this, especially in other languages, but it still works. It was incredibly fun to do. My basic idea was to have a loop iterate 2 times, the first time printing the program, the second time printing the printing command. Escaping the " characters is the biggest problem, the main idea here was to have a string q that equals " in the first iteration and " + q + " in the second. That second string (as part of the code in an expression where a string is printed) will print itself in the console output. Code:

package maths;public class Quine{public static void main(String[]args){for(int i=0;i<2;i++){String o=i==1?""+(char)34:"";String q=""+(char)34;q=i==1?q+"+q+"+q:q;String e=i==1?o+"+e);}}}":"System.out.print(o+";System.out.print(o+"package maths;public class Quine{public static void main(String[]args){for(int i=0;i<2;i++){String o=i==1?"+q+""+q+"+(char)34:"+q+""+q+";String q="+q+""+q+"+(char)34;q=i==1?q+"+q+"+q+"+q+"+q:q;String e=i==1?o+"+q+"+e);}}}"+q+":"+q+"System.out.print(o+"+q+";"+e);}}}