I have been informed that it’s inappropriate to talk about your criticisms of someone that originated on one forum on a different forum where the target of your criticisms cannot respond. So I’m retracting the parent comment, and apologizing to Said Achmiz for it.
Dacyn
Karma: 620
Chemistry nit: the molecule composed of hydrogen and chlorine is HCl (hydrochloric acid), not H2C.
But in this case, look, as I mentioned in §II.3, Said has been compiling a collection of his best comments. I think that if you read it, it’s just very obviously good stuff by the usual standards that rationalists use to evaluate our stuff.
I mostly agree with this, but want to highlight an exception: Said linked to his explanation of the solution to the Monty Hall problem as an example of one of his best comments. This started a thread on DSL about how accurate his purported solution was. The consensus by the end of it was that Said was somewhere between confused and crackpottish. Said never retracted anything he said though. Just wanted to add that as a data point,
OK, sounds reasonable.
I think that although most people are probably “CEV-good”, there are also quite many “CEV-monsters”, i.e. people who value suffering for the sake of suffering (of others).
Just to be clear, you are labelling as “CEV-monsters” people who value justice for its own sake, even if/when justice involves some amount of suffering? I don’t think the “monster” label is appropriate, even if you disagree with the position.
Isn’t this the fallacy of the gray? Yes consensuses can be wrong, but they’re probably better than nothing, which makes them evidence in a Bayesian sense, a valid argument even if not a decisive one.
I see. And I agree that there’s a place for such an attitude, that it can be interesting to explore the formal consequences of a system such as ZFC even if you don’t necessarily think it’s “true” in any Platonic sense. (Although maybe you want to at least be able to hope that the system is consistent; no one proves things in naive set theory anymore even though it is an extremely generative system!)
Anyway, part of my point was that systems like PA that are weaker and more intuitive than ZFC can be nearly equally “interestingly generative”, with some narrow exceptions. So even if that is your primary criterion for choosing an axiom system, you could still come to the conclusion that PA is a better foundational system if you still care at all about an abstract notion of truth.
Regarding the von Neumann quote, presumably the math you are encountering has already been vetted by other mathematicians, so you can assume it is likely to be true even without understanding it. In such a situation, it makes sense to allow yourself to get used to it. It is different if you want to study the foundations of math; since people disagree about it you will need to form your own opinion.
I’m getting the feeling you want to make a point but you’re not looking at the context of my comment.
Actually I was trying to make three separate unrelated points. I think we agree sufficiently on the first and third points that it would be unproductive to discuss them further. Regarding my second point:
This feels like a plug or advertisement for PA not an actual point.
Well I do like to plug for PA when I can (it’s a good foundation for math!), but I think I do have an actual point as well.
Truth
I mean, sure, there are a lot of unintuitive things that you can prove from true axioms. But with the axioms themselves, the only basis we could have for asserting that the axioms are true is that they are intuitive. I mean, why else would you think they are true? (In mathematics at least, of course in other fields you could have empirical evidence of something unintuitive.)
Multisets are defined as a tuple of a set
Multisets can be defined in terms of sets, and conversely sets can be defined in terms of multisets. I don’t see either of them as “more fundamental” than the other necessarily, except in terms of which one we’ve chosen to give a more prominent place to.
ZFC was not built to be intuitive. It was built to unify and formalize all of mathematics. It is not supposed to be some kind of universal human way of seeing the world through a mathematical lens. It’s just a set of axioms we build on, and we can choose different sets of axioms at will.
If ZFC is not intuitive (and my opinion is that it is not), then we shouldn’t assume that any theorems it proves are actually true, and we should if possible rely on a weaker system which is intuitive, such as the Peano axioms. Peano arithmetic is in fact powerful enough to deal with almost all of modern mathematics (except for abstract set theory), it’s just that people are so used to using ZFC as the foundation that they rarely check to see whether their theorems can be proven in PA. See for example this book where the author develops a “strong undergraduate curriculum” using a system conservative over PA as the foundation.
Your example of a water molecule assumes we do not distinguish between the hydrogen atoms with indexing
On the atomic level, quantum effects prevent us from distinguishing
I think E(blue_pressers|blue_wins) should be replaced by N/2, where N is the number of players. If more than 50% of people are already voting Blue, then your vote makes no difference to anything.
I will also quote my comment from DataSecretsLox (https://www.datasecretslox.com/index.php/topic,15775.msg787363.html#msg787363):
If you are perfectly altruistic and assume all lives are equal, then the question can be modelled mathematically as follows:
>50% Blue: It doesn’t matter which button you press. 50% exactly: If you press red you are killing N/2 people where N is the number of people playing the game. You survive either way. >50% Red: You survive if you press red, die if you press blue. Your decision has no impact on anyone else’s fate.
So you should press Blue if and only if N/2 times Prob(50% exactly) is greater than Prob(>50% Red). Which is probably true if you expect most people to choose Blue, and is probably false if you expect most people to choose Red.
So basically it is a Keynesian beauty contest, with a bias towards Red coming from the fact that not everyone is perfectly altruistic (an understatement). In my opinion, that makes Red the correct answer even for someone who is perfectly altruistic, and even if most people are perfectly altruistic, as long as the deviations from altruism tend towards selfishness.
Of course this mathematical model is simplifying a lot, but I think it is enough to undercut the idea that any sufficiently altruistic person should pick Blue.
Never mind, for some reason I thought you were being offered a 2:1 payout as well as lopsided odds; that doesn’t appear to be the case.
Which is maximised when f = 0.5
I calculate f = 5⁄8, not 1⁄2.
The changes I’ve made for this version may seem trivial
Well, in one version you are being extorted for money, whereas in the other version you are merely being bribed. If you buy Eliezer’s theory that you should pay up for bribes but not for extortions (because paying up for bribes increases the probability that people will try to bribe you, which is good, but paying up for extortion increases the probability that people will try to extort you, which is bad), then the difference matters.
Why can’t player 1 just make a really bad move, then switch with player 2 no matter what he plays? That seems like it would give player 1 (who then becomes player 2) a huge advantage.
It’s not right to say that the Copenhagen interpretation means that “only quantum mechanics” is aleatory. First of all, QM describes all physical phenomena so presumably what you meant was “only microscopic phenomena”. But this is not right either, as chaotic dynamical systems send microscopic differences to macroscopic differences and therefore send microscopic aleatory randomness to macroscopic aleatory randomness. It’s possible that there’s even enough chaos in a coin flip to make it aleatorily random.
My conception of mathematics is that you start with a set of axioms and then explore the implications of them. There are infinitely many possible sets of starting axioms you can use [1] .
This is a popular view but in my opinion it is wrong. My conception of math is that you start with a set of definitions and the axioms only come after that, as an attempt to formalize the definitions. For example:
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The natural numbers are defined as the objects that you get by starting with a base object “zero” and iterating a “successor operation” arbitrarily many times. Addition and multiplication on the natural numbers are defined recursively according to certain basic formulas. The axioms of Peano arithmetic can then be viewed as simply a way of formalizing these definitions: most of the axioms are just the recursive definitions of addition and multiplication, and the induction schema is an attempt to formalize the fact that all natural numbers result from repeatedly applying the successor operation to 0.
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The universe of sets is defined as the collection you get by starting with nothing, and repeatedly growing the collection by at each stage replacing it with the set of all its subsets (i.e. its powerset). The axioms of Zermelo-Fraenkel set theory are an attempt to state true facts about this universe of sets.
Of course, it’s possible to claim that the definitions in question are not valid—they are not “rigorous” in the sense of modern mathematics, i.e. they do not follow from axioms because they are logically prior to axioms. This is particularly true for the definition of the universe of sets, which in addition to being vague has the issues that it presupposes the notion of a “subset” of a collection while we are currently trying to define the notion of a set, and that it’s not clear when we are supposed to “stop” growing the collection (it’s not at “infinity”, because the axiom of infinity implies that we are supposed to continue on past infinity). But Peano arithmetic doesn’t have those problems, and in my opinion is therefore on an epistemologically sound basis. And to be honest much (most?) of modern mathematics can be translated into Peano arithmetic; people use ZFC for convenience but it’s actually not necessary much of the time.
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also because sharing the planet with a slightly smarter species still doesn’t seem like it bodes well. (See humans, neanderthals, chimpanzees).
From what I can tell from a quick Google search, current evidence doesn’t show that neanderthals were any less smart than humans.
Yes. If f and g are in the original category and are inverses of each other, the same will be true of any larger category (technically: any category which is the codomain of a functor whose domain is the original category).
OK, maybe if we look at some other definitions of equality we can get a grip on it? In set theory, you say that two sets are equal if they’ve got the same elements. How do you know the elements are the same i.e. equal? You just know.
You are misunderstanding the axiom of extensionality, which states that two sets A and B are equal if both (1) every element of A is an element of B and (2) every element of B is an element of A. This does not require any nebulous notion of “they’ve got the same elements”, and is completely unrelated to the concept of equality at the level of elements of A and B.
By the way, the axiom of extensionality is an axiom rather than a definition; in set theory equality is treated as an undefined primitive, axiomatized as a notion of equality as in first order logic. This is important because if A and B are equal according to the axiom of extensionality, then that axiom implies that A is in some collection of sets C if and only if B is in C.
But if you enrich the category with some more discriminating maps, say distance preserving ones, then the sphere and cube are no longer equal. Conversely, if you reduce the category by removing all the isomorphisms between the sphere and the cube, then they are no longer equal.
Actually you have just described the same thing twice. There are actually fewer distance-preserving maps than there are continuous ones, and restricting to distance-preserving maps removes all the isomorphisms between the sphere and the cube.
Another consideration regarding suicide paternalism: the way we do things now, people considering whether to commit suicide are disincentivized to talk it through with friends and therapists, since it may lead them to being involuntarily committed to a mental hospital (which can sometimes lead to other negative events like their being fired from their job). If on the other hand people had the chance to talk through their suicidal tendencies without fear from these repercussions, many of them might not end up attempting suicide after all,