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Are you sure that by "one-to-one" Halmos means "bijective"? A more common usage is for it to mean "injective". (But I don't have NST and maybe he has an unusual idiom.)
In response to
Book Review: Naive Set Theory (MIRI research guide)
There is a convention according to which a one-to-one function is injective, while a one-to-one correspondence is an injective function that is also surjective, ie, a bijection. (I don't know whether Halmos uses this convention.)
In response to
Book Review: Naive Set Theory (MIRI research guide)
When I first worked through this book, it didn't result in long-term retention of the material (I'm sure some people will be able to manage, just not me, not without meditating on it much longer than it takes to work through or setting up a spaced repetition system). In that respect, Enderton's Elements of Set Theory worked much better. Enderton's book goes into more detail, giving enough time to exercise intuition about standard proofs. At the same time, it's an easier read, which might be helpful if Halmos's text seems difficult.
In general, reading about the same subject from a different author is a great way to learn and retain the material better. This is true even if neither author is objectively "better" than the other. Something about recognizing the same underlying concept expressed in different words helps to fix that concept in the mind.
It's possible to exploit this phenomenon even when you have only one text to work with. One trick I use when working through a math text is to willfully use different notation in my notes next to the text. Using a different notation forces me to make sure that I'm really following the details of the argument. Expressing the same logic in different symbols makes it easier to see through those symbols to the underlying logic.
Just a minor point:
"when you have the statement refer to itself, you get a paradox" is not necessarily true. For example, the statement "this statement has five words" is self-referential and true. No paradox. Even a self-referential statement that includes its own truth value can be non-paradoxical: "This statement is true and has two words" is merely false.
By the way, this leads me to consider Prior's resolution as somewhat problematic:
"This statement is true and has eight words" "This statement has eight words"
The first statement is true and the second false, hence they cannot be equivalent. Nevertheless, adding "This statement is true and " to any statement should not change the statement's truth value if we accept that every statement implicitly states its own truth.
That's not really a problem with Prior's resolution. Rather, it's a different problem with self-reference, which appears whether we adopt Prior's resolution or not.
Compare: "P" and "P and P" are usually equivalent. But
"This sentence has five words." and "This sentence has five words and this sentence has five words."
don't have the same truth value. The problem seems to be that the meaning of "this sentence" isn't the same in the two ostensibly equivalent sentences. Whatever your favorite solution of this problem is, it seems that Prior could just graft that solution onto his own.
Prior's solution to the liar paradox needn't solve all paradoxes of self-references. As long as his solution is compatible with other solutions to other paradoxes, Prior has still contributed something of value.
In response to
Entropy and Temperature
This is a good article making a valuable point. But this —
Temperature is sometimes taught as, "a measure of the average kinetic energy of the particles," because for an ideal gas U/N = (3/2) kBT. This is wrong, for the same reason that the ideal gas entropy isn't the definition of entropy.
— is a confusing way to speak. There is such a thing as "the average kinetic energy of the particles", and one measure of this thing is called "temperature" in some contexts. There is nothing wrong with this as long as you are clear about what context you are in.
If you fall into the sun, your atoms will be strewn far and wide, and it won't be because of something "in the mind". There is a long and perfectly valid convention of calling the relevant feature of the sun its "temperature".
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Is arguing worth it? If so, when and when not? Also, how do I become less arrogant?
(Here's an example of my doing this.)
Thank you for providing an example. By the way, it looks to me like lukeprog never actually clarified for you what he meant by "mathematicians succeed and fail on this issue in a wide range of degrees"
Agreed?
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Is arguing worth it? If so, when and when not? Also, how do I become less arrogant?
Yes.
As I added in my reply to him, his reply did help me with other parts of his argument. But I needed more iterations of questions and clarifications before I could understand that particular phrase better.
This doesn't seem to me like wasted effort, though, because I expect that what he did clarify would have helped me to understand that particular phrase, had we continued to discuss it. So, while I can't explain that particular phrase better than I could before, I expect that I am closer to understanding it. Certainly, partial illumination of the argument surrounding a specific sentence is normally the preamble to full illumination of that specific sentence, if this full illumination ever happens.
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Is arguing worth it? If so, when and when not? Also, how do I become less arrogant?
Notice that "re-express your target’s position clearly" was not the entirety of Rapaport's advice, or even of that line of his advice.
I agree. Still, that part of his advice -- to state the other person's position very clearly -- is not going to work out in practice a lot of the time. Before you can state someone's position "clearly vividly and fairly" you have to understand it. And the person has to actually have a position which can be stated "clearly vividly and fairly."
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Is arguing worth it? If so, when and when not? Also, how do I become less arrogant?
When I find someone else's argument puzzling, it is often for a reason that they didn't anticipate. Because they didn't anticipate that I would find a particular step puzzling in a particular way, they didn't explain this step, at least not in a way that I understood.
Thus, I need them to (1) be willing to do the work of understanding which step I found puzzling and why, and (2) be willing to do the work of addressing my idiosyncratic confusion. (They will perceive my confusion as idiosyncratic, because this is the first time that they are encountering it.*)
Both of those steps require some work on their part. Moreover, they need to do this work to bridge a step that seemed obvious to them, and hence which seemed like it could be missed only by someone who is, in a certain sense, unusually stupid. This automatically puts me under suspicion of being "not worth the time", either because I'm stupid or because I'm asking in bad faith. (See Expecting Short Inferential Distances.)
So, most people aren't willing to undertake this work unless they have some sympathy for me. The other lines of Rapaport's advice serve to build this sympathy, so they should happen before I attempt the "re-express clearly and vividly" stage.
When I do attempt a "re-expression" as part of my process of understanding their argument, my first attempt is accompanied by something like "Here is my attempt to restate what you are saying, but I know that it is probably wrong. This attempt is just to give you something to work with as you address the error in my understanding of your meaning." (Here's an example of my doing this.)
This may seem overly humble or deferential, but, in my experience, it is effective and literally true. This kind of expression really does make people more willing to attempt a helpful reply, and their replies really do fill in gaps in my understanding of their position. (Again, see the above example. I didn't entirely resolve my confusion, but I did come way understanding my interlocutor's position better.)
* However, if I continue to profess confusion over this step, and I haven't made myself sympathetic by following the rest of Rapaport's advice, then my professions won't be chalked up to idiosyncratic confusion, but rather to willful stupidity or bad faith.
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Is arguing worth it? If so, when and when not? Also, how do I become less arrogant?
You should attempt to re-express your target’s position so clearly
When I debate people on the internet, I find that much of the time the other fellow does not have a clear position. That when I gently and politely ask questions aimed at clarifying the person's position, I get evasions, ad hominems, strawmanning, weaseling, and attempts to change the subject.
Of course I usually limit my debates to topics where I've though pretty carefully about the subject. And I'm most interested in topics where popular views are questionable.
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Is arguing worth it? If so, when and when not? Also, how do I become less arrogant?
Notice that "re-express your target’s position clearly" was not the entirety of Rapaport's advice, or even of that line of his advice.
When I read "i believe people are nicer than they really are" I got the impression her meaning was along the lines of "people are nicer inside than their actions. On reflection, this might be because that's what I believe. It ties in to fundamental attribution error. Peoples actions are based so much on environment and circumstance that if you had a way to truly look into a person I think you'd see a better person than you would have guessed if you only looked at their actions. Most people don't see themselves as evil. They do things we see as evil but in their heads they are doing what they think is good.
Id be interested in hearing what exactly she said that brought on your analysis Eliezer. I realise it was a long time ago, and im not likely to get a reply anyway, but it seems likely to me her statemen came from an intuitive belief in fundamental attribution error. I know I held that belief long before I encountered it first in HPMoR, so its possible for her too.
I think that there's a better chance that he'll see your comment if you reply directly to the post rather than to another comment. At least, I think that that's how it works.
That's a great explanation (the non-relative, non-indexical now, and "who is" analogy - not the "fundamental" talk, which just makes me cringe). But that's A-theory, not presentism, which is being explained, right? This paper <pdf> claims there's a distinction, at least in that one can be a presentist without endorsing A-theory.
But that's A-theory, not presentism, which is being explained, right? This paper <pdf> claims there's a distinction
Yes. One can certainly be an A-theorist without being a presentist. Some people really have subscribed to so-called "moving spotlight" theories. (Hermann Weyl was an example.)
I'm less convinced that anyone was ever a presentist but not an A-theorist. The paper you cite doesn't convince me for at least the following reasons.
First, the paper doesn't even argue that any non-A-theorist presentists have ever actually existed. Rather, the paper attempts to show that such a theory is, as it were, technically possible.
Second, I don't buy that the paper succeeds even at this. The author constructs the theory in Section 4. But the constructions essentially depends on a loophole: A-theories must posit A-properties, he says, but existence is not a property. Then, in Section 5.3, he deals with what seems to me to be the obvious reply. He allows that maybe A-theories only require A-facts, and not necessarily A-properties. If existence is a fact, then his construction fails. His reply is that "it is still possible to be a presentist without being an A-theorist: we need simply deny the existence of facts. ... If there are no facts at all then there are no existence facts. ... This is not an unreasonable view. There are metaphysical systems that do not posit facts—versions of substance theory, bundle theory, and so on."
I find this unconvincing. I don't know enough about these other theories to know how they get by without facts. But I suspect that they introduce some kind of things, call them faks, that do the work of facts. I suspect that the A-theory could just as well be held to require only that there are A-faks.
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Reading this was a bit annoying:
It is just communicating badly https://xkcd.com/169/ . In a common parse, Ace is more likely to occur. It would be more likely to be parsed as you intended if you had said
(like you did on the next question!)
I'd guess that getting this question "correct" almost requires having been trained to parse the problem in a certain formal way — namely, purely in terms of propositional logic.
Otherwise, a perfectly reasonable parsing of the problem would be equivalent to the following:
On this reading, Ace is most probable.
Indeed, this "algorithmic" reading seems like the more natural one if you're used to trying to model the world as running according to some algorithm — that is, if, for you, "learning about the world" means "learning more about the algorithm that runs the word".
The propositional-logic reading (the one endorsed by the OP) might be more natural if, for you, "learning about the world" means "learning more about the complicated conjunction-and-disjunction of propositions that precisely carves out the actual world from among the possible worlds."