J_Thomas2
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Larry, you have not proven that 6 would be a prime number if PA proved 6 was a prime number, because PA does not prove that 6 is a prime number.
The theorem is only true for the phi that it's true for.
The claim that phi must be true because if it's true then it's true, and if it's false then "if PA |- phi then phi" has an officially true conclusion whenever PA does not imply phi, is bogus.
It's simply and obviously bogus, and I don't understand why there was any difficulty about seeing it.
Caledonian, it's possible to care deeply about choices that were made in a seemingly-arbitrary way. For example, a college graduate who takes a job in one of eight cities where he got job offers, might within the year care deeply about that city's baseball team. But if he had taken a different job it would be a completely different baseball team.
You might care about the result of arbitrary choices. I don't say you necessarily will.
It sounds like you're saying it's wrong to care about morals unless they're somehow provably correct? I'm not sure I get your objection. I want to point out that usually when we have a war, most of the... (read more)
I haven't read Roko's blog, but from the reflection in Eliezer's opposition I find I somewhat agree.
To the extent that morality is about what you do, the more you can do the higher the stakes.
If you can drive a car, your driving amplifies your ability to do good. And it amplifies your ability to do bad. If you have a morality that leaves you doing more good than bad, and driving increases the good and the bad you do proportionately, then your driving is a good thing.
True human beings have an insatiable curiousity, and they naturally want to find out about things, and one of the things they like is to find... (read more)
"You should care about the moral code you have arbitrarily chosen."
No, I shouldn't. Which seems to be the focal point of this endless 'debate'.
Well, you might choose to care about a moral code you have arbitrarily chosen. And it could be argued that if you don't care about it then you haven't "really" chosen it.
I agree with you that there needn't be any platonic absolute morality that says you ought choose a moral code arbitrarily and care about it, or that if you do happen to choose a moral code arbitrarily that you should then care about it.
We are born with some theorems of right (in analogy to PA).
Kenny, I'd be fascinated to learn more about that. I didn't notice it in my children, but then I wouldn't necessarily notice.
When I was a small child people claimed that babies are born with only a fear of falling and a startle reflex for loud noises. I was pretty sure that was wrong, but it wasn't clear to me what we're born with. It takes time to learn to see. I remember when I invented the inverse square law for vision, and understood why things get smaller when they go farther away. It takes time to notice that parents have their own desires that need to be taken into account.
What is it that we're born with? Do you have a quick link maybe?
Larry, one of them is counterfactual.
If you draw implications on a false asumption then the result is useful only to show that an assumption is false.
So if PA -> 1=2 then PA -> 1<>2. How is that useful?
If PA -> 6 is prime then PA also -> 6 is not prime.
Once you assume that PA implies something that PA actually implies is false, you get a logical contradiction. Either PA is inconsistent or PA does not imply the false thing.
How can it be useful to reason about what we could prove from false premises? What good is it to pretend that PA is inconsistent?
Honestly I do not understand how you can continue calling Eliezer a relativist when he has persistently claimed that what is right doesn't depend on who's asking and doesn't depend on what anyone thinks is right.
Before I say anything else I want you to know that I am not a Communist.
Marx was right about everything he wrote about, but he didn't know everything, I wouldn't say that Marx had all the answers. When the time is ripe the proletariat will inevitably rise up and create a government that will organize the people, it will put everybody to work according to his abilities and give out the results according to the needs, and... (read more)
But Larry, PA does not actually say that 6 is prime, and 6 is not prime.
You could say that if PA proved that every theorem is false then every theorem would be false.
Or what would it mean if PA proved that Lob's theorem was false?
It's customary to say that any conclusion from a false premise is true. If 6 is prime then God's in his heaven, everything's right with the world and we are all muppets. Also God's in hell, everything's wrong with the world, and we are all mutant ninja turtles. It doesn't really matter what conclusions you draw from a false premise because the premnise is false.
Your argument about what conclusion we could draw if PA said that 6 is prime is entirely based on a false premise. PA does not say that 6 is prime.
Let me try to say that clearer.
Suppose that A is false.
How the hell are you going to show that if PA proves A true then A will be true, when A is actually false?
If you can't prove what would happen if PA proved A true when A is actually false, then if you can prove that if PA proves A is true then A has to be true, it must be that A is true in the first place.
If this reasoning is correct then there isn't much mystery involved here.
One more time. If PA proves you are a werewolf, then you're really-and-truly a werewolf. PA never proves anything that isn't actually true.... (read more)
I thought of a simpler way to say it.
If Hillary Clinton was a man, she wouldn't be Bill Clinton's wife. She'd be his husband.
Similarly, if PA proved that 6 was prime, it wouldn't be PA. It would be Bill Clinton's husband. And so ZF would not imply that 6 is actually prime.