transhumanist_atom_understander

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Prospects for Solartopia

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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.

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.

Answer by transhumanist_atom_understander10

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.

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.

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