Depends entirely on Cybercab. A driverless car can be made cheaper for a variety of reasons. If the self-driving tech actually works, and if it’s widely legal, and if Tesla can mass produce it at a low price, then they can justify that valuation. Cybercab is a potential solution to the problem that they need to introduce a low priced car to get their sales growing again but cheap electric cars is a competitive market now without much profit margin. But there’s a lot of ifs.
transhumanist_atom_understander
Karma: 364
Yeah, just went through this whole same line of evasion. Alright, the Collatz conjecture will never be “proved” in this restrictive sense—and neither will the Steve conjecture or the irrationality of √2—do we care? It may still be proved according to the ordinary meaning.
The pilot episode of NUMB3RS.
The tie-in to rationality is that instead of coming up with a hypothesis about the culprit, the hero comes up with algorithms for finding the culprit, and quantifies how well they would have worked applied to past cases.
It’s really a TV episode about computational statistical inference, rather than a movie about rationality, but it’s good media for cognitive algorithm appreciators.
Alright, so Collatz will be proved, and the proof will not be done by “staying in arithmetic”. Just as the proof that there do not exist numbers p and q that satisfy the equation p² = 2 q² (or equivalently, that all numbers do not satisfy it) is not done by “staying in arithmetic”. It doesn’t matter.
Not off the top of my head, but since a proof of Collatz does not require working under these constraints, I don’t think the distinction has any important implications.
We can eliminate the concept of rational numbers by framing it as the proof that there are no integer solutions to p² = 2 q²… but… no proof by contradiction? If escape from self-reference is that easy, then surely it is possible to prove the Collatz conjecture. Someone just needs to prove that the existence of any cycle beyond the familiar one implies a contradiction.
Yes, the proof that there’s no rational solution to x²=2. It doesn’t require real numbers.
Yeah, I think this is the distinction I’m struggling with. To start with proofs in Euclid, why can I prove that the square root of two is irrational? Why can I prove there are infinite prime numbers? If I saw that they are escaping this self-reference somehow, maybe I’d get the point. And without that, I don’t see that I can rule out such an escape in Collatz.
2024 was the year of the big battery, and what that means for solar power
Yes, de Finetti’s theorem shows that if our beliefs are unchanged by exchanging members of the sequence, that’s mathematically equivalent to having some “unknown probability” that we can learn by observing the sequence.
Importantly, this is always against some particular background state of knowledge, in which our beliefs are exchangeable. We ordinary have exchangeable beliefs about coin flips, for example, but may not if we had less information (such as not knowing they’re coin flips) or more (like information about the initial conditions of each flip).
In my post on unknown probabilities, I give more detail on how they are definedc which turns out to involve a specific state of background knowledge, so they only act like a “true” probability relative to that background knowledge. And how they can be interpreted as part of a physical understanding of the situation.
Personally, rather than observing that my beliefs are exchangeable and inferring an unknown probability as a mathematical fiction, I would rather “see” the unknown probability directly in my understanding of the situation, as described in my post.
Well, usually I’m not inherently interested in a probability density function, I’m using it to calculate something else, like a moment or an entropy or something. But I guess I’ll see what you use it for in future posts.
This two-point distribution is important as the distribution where Markov’s inequality is an equality, so it’s cool to have it visualized as part of the proof.
is the correction to the probability density function really what you want, and are other deviations from Gaussianity expressible with cumulants? All I can think of is that the Gaussian is the maximum entropy distribution so maybe there’s a formula for how far below the maximum entropy you are. I don’t know what it’d be good for though.
I’m in the process of turning the ideas in a stack of my notebooks into what I hope will be a short paper, which is just one illustration of what I think was the real trade-off, which is between conciseness and time spent writing. Or for another, see the polished 20-page papers on logical decision theory. Though it’s not the same, they cover much of the same ground as the older expositions of timeless decision theory and updateless decision theory. There was a long period where these kinds of decision theories were only available through posts, and then through Eliezer Yudkowsky’s long TDT paper. That period could not have been skipped, and could only have been shortened in the sense that the same work could have been done faster at the expense of other work. See also this exchange on Twitter, though they’re not talking about being concise specifically:
Miles Brundage: I can’t speak to the details of those experiments but I at least read a much higher fraction of your paper outputs than your blog post outputs. Possibly I’m a minority here but I am certainly not the only one.
Eliezer Yudkowsky: Yeah, I tried it and it was way too fucking time-expensive. My guess is that 100x the output you like less ends up having the larger impact on the world.
Yes, and I’m realizing I went into a digression that wasn’t really relevant to my original point. In this particular post I just wanted to discuss the first principles calculation, that tells you that the sunlight hitting a relatively small area can supply all our electricity needs. The fact that just the area on roofs even makes a dent is one of the things that makes sense from this perspective, since roof area is not that large. Where to put solar panels is an economic question that doesn’t particularly matter for any of the points I’m going to make in this sequence, although I do want to get into the economics of batteries in some detail in the next two posts because that’s one of the things that limits how much solar capacity you can install. And, yes, the other big limitations are transmission and permitting—that’s a relevant point and I see now that you were trying to communicate how these other limitations can be addressed. I won’t really be getting to transmission and permitting, because this sequence was prompted by considering how I should update on battery storage exceeding expectations.
I’m replying to a post that said they lacked energy despite iron supplementation. Maybe it wasn’t iron deficiency, or maybe it could have been solved by raising the supplement dose, I don’t know, but if it was iron deficiency and the supplements weren’t helping then it’s good to know you can get iron in the same form as in meat from Impossible burgers.
Yeah, this is a US-centric perspective of mine but there’s no shortage of land here. This sounds to me like classic thriftiness about that which is not scarce, which isn’t real thriftiness. I mean, “effective use of both farmland and rooftops”… rooftops? What’s scarce here in the US is labor, not land. Why have all these people climbing up on rooftops? An interesting contrast is Texas (mostly utility solar) vs California (lots of rooftop solar). The interesting number, which I don’t know off the top of my head, is how many people are employed in the solar sector per unit capacity installed. I seem to remember California employs a lot more people, disproportionately to how much more solar it has.
The heme iron in meat is absorbed better than the non-heme iron in iron supplements, but Impossible Burger has heme iron. It’s very frustrating that this biotech advance of in vitro heme production is so far only used for this specific brand of meat substitutes but that’s the situation. I’m not sure why iron supplements didn’t work for you, as that same paper shows that even non-heme iron is absorbed well when blood iron is low, but maybe it depends on the individual? In any case, and I promise the company isn’t paying me to say this, I recommend Impossible burgers to vegans. It has to be this specific brand, they have the patent. There was another company with a heme production patent but they recently shut down.
The natural gas generation capacity that you need to cover for solar when it’s cloudy is, of course, less than what is required to make up for loss of solar after sundown.
After reading this post, I handed over $200 for a month of ChatGPT pro and I don’t think I can go back. o1-pro and Deep Research are next level. o1-pro often understands my code or what I’m asking about without a whole conversation of clarifying, whereas other models it’s more work than it’s worth to get them focused on the real issue rather than something superficially similar. And then I can use “Deep Research” to get links to webpages relevant to what I’m working on. It’s like… smart Google, basically. Google that knows what I’m looking for. I never would have known this existed if I didn’t hand over the $200.