How does this not come down to saying that people you consider rigorous, on average did more work on their texts than people you don't consider rigorous, and therefore they wrote less as a whole?
That is what it comes down to. I'm not trying to show any truth about the nature of rigorous thought.
Ok. In that sense I agree that this is likely to be the case, and would be the case more often than not with any educated person's assessment of who does rigorous work.
Now here is the weird and confusing part. If the above is a valid proof, then H will eventually find it. It searches all proofs, remember?
Fortunately, H will never find your argument because it is not a correct proof. You rely on hidden assumptions of the following form (given informally and symbolically):
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If φ is provable, then φ holds.
Provable(#φ) → φ
where #φ denotes the Gödel number of the proposition φ.
Statements of these form are generally not provable. This phenomenon is known as Löb's theorem - featured in Main back in 2008.
You use these invalid assumptions to eliminate the first two options from Either H returns true, or false, or loops forever. For example, if H returns true, then you can infer that "FF halts on input FF" is provable, but that does not contradict FF does not halt on input FF.
It is not that these statements are "not generally valid", but that they are not included within the axiom system used by H. If we attempt to include them, there will be a new statement of the same kind which is not included.
Obviously such statements will be true if H's axiom system is true, and in that sense they are always valid.
First of all, I'm not sure what measure you're using for "rigorous thought". Is it a binary classification? Are there degrees of rigor? I can infer from some of your examples what kind of pattern you might be picking up on, but if we're going to try and say things like "there's a correlation between rigor and volume of publication", I'd like to at least see a rough operational definition of what you mean by rigor.
The important thing is that I categorized people as rigorous or non-rigorous first, then found a difference between the groups. That suggests there's some relevant distinction in my mental model
If I'd made an operational definition, I'd have been testing the operational definition, not my mental model, and the definition might not have matched very well. Better to consult the oracle in my head.
I agree that what I'm saying would be more clear to you if I'd tried to define rigor afterwards. Certainly not being well-liked or influential. Zizek, Derrida, and Lacan are all well-liked and very influential today. Spinoza is not as influential as Nietzsche.
I consider Nietzsche not rigorous because he's upfront about not being rigorous, about not even considering it an issue. The Superman doesn't stop and try to figure out if he's correct. Nietzsche does philosophy by telling stories, not by defending propositions.
I consider Freud not rigorous because he made hypothesis but didn't test them (AFAIK). He told a lot of just-so stories, without contrasting them with alternative explanations. Similar thing with Marx. More a storyteller than a scientist.
I consider Lysenko not rigorous because instead of arguing with his opponents, he had them sent to Siberia and got a law passed saying it was illegal to argue with him.
I consider Hegel not rigorous because nobody can figure out what a lot of the stuff he wrote means, or if it means anything.
I also consider Stein not rigorous because nobody can figure out what she meant. She wrote like a stroke victim. Her book How to Write begins with 3 untranslated sentences in French, then says:
When he will see When he will see When he will see the land of liberty. The scene changes it is a stone high up against with a hill and there is and above where they will have time.
George Steiner is a curious case. He's very rigorous in considering the meanings and connotations of his words. But he doesn't believe reality is knowable, so he has no interest in whether anything he says is true.
I consider Spinoza rigorous because he wrote in the 17th century, and yet confined himself to meaningful statements and inferences that he could draw based on evidence. His reputation is not very high IMHO because he was so rigorous that he said mainly things we now take for granted and consider obvious.
I consider Robert Penn Warren rigorous because he looks at a story as something that communicates the author's opinions about life through the logical relationships between the different components of the story, and he illustrates this by going through dozens of stories and showing what the intended communication is and how the components interact to support it.
I consider I. A. Richards rigorous because he took stories, showed them to students, asked them to interpret them, and astounded everybody by demonstrating that university literature students understood much less of what was generally thought to be meant in those stories than literary critics believed the general public did.
Jon von Neumann was either rigorous in his thought, or magical.
I don't think I had any good justification for listing Minsky. I think I meant to contrast him with somebody else from MIT from some of the sloppy "look how cool my robot / simulation is" work done there, but was too lazy to put in the time to justify a choice.
Wiles wrote a humongous proof that has withstood (with some provisos that I don't understand) the scrutiny of many mathematicians.
Robert Frost is probably the most controversial. I listed him because his work is very tight. His poems have a surface meaning and one or more deeper meanings, and they communicate clearly enough to bring you to contemplate that deeper meaning, rather than (as most modernist poetry does) merely enough to let you contemplate what that deeper meaning might be.
How does this not come down to saying that people you consider rigorous, on average did more work on their texts than people you don't consider rigorous, and therefore they wrote less as a whole?
If we take a random (educated) person, and ask him to classify authors into rigorous and non-rigorous, something similar should be true on average, and we should find similar statistics. I can't see how that shows some deep truth about the nature of rigorous thought, except that it means doing more work in your thinking.
I agree that it does mean at least that, so that e.g. some author has written more than 100 books, that is a pretty good sign that he is not worth reading, even if it is not a conclusive one.
The program I specified is impossible to prove will halt. It doesn't matter what Turing machine, or human, is searching for the proof. It can never be found. It can't exist.
The paradox is that I can prove that. Which means I can prove the program searching for proofs will never halt. Which I just proved is impossible.
I looked at your specified program. The case there is basically the same as the situation I mentioned, where I say "you are going to think this is false." There is no way for you to have a true opinion about that, but there is a way for other people to have a true opinion about it.
In the same way, you haven't proved that no one and nothing can prove that the program will not halt. You simply prove that there is no proof in the particular language and axioms used by your program. When you proved that program will not halt, you were using a different language and axioms. In the same way, you can't get that statement right ("you will think this is false") because it behaves as a Filthy Liar relative to you. But it doesn't behave that way relative to other people, so they can get it right.
I feel like I might understand now. Can I represent your points as follows:
Assuming statement 1 is correct, without accepting a further premise I don't feel compelled to accept the second premise. It sounds like things which are logically impossible may in fact be equivelant to things which don't exist, and vice-versa. And that sounds intuitively compelling. If something was logically possible, it would happen. If it is wasn't possible, it's not going to happen. Or, the agent's modelling of the world is wrong.
Importantly, I don't accept premise 1, as I've indicated in another comment reply (something about how I find I'm wrong about the apparent impossibility of something, or possibility of something.)
I said "so the probability that a thing doesn't exist will be equal to or higher than etc." exactly because the probability would be equal if non-existence and logical impossibility turned out to be equivalent.
If you don't agree that no logically impossible thing exists, then of course you might disagree with this probability assignment.
I'm very confused about something related to the Halting Problem. I discussed this on the IRC with some people, but I couldn't get across what I meant very well. So I wrote up something a bit longer and a bit more formal.
The gist of it is, the halting problem lets us prove that, for a specific counter example, there can not exist any proof that it halts or not. A proof that it does or does not halt, causes a paradox.
But if it's true that there doesn't exist a proof that it halts, then it will run forever searching for one. Therefore I've proved that the program will not halt. Therefore a proof that it doesn't halt does exist (this one), and it will eventually find it. Creating a paradox.
Just calling the problem undecidable doesn't actually solve anything. If you can prove it's undecidable, it creates the same paradox. If no Turing machine can know whether or not a program halts, and we are also Turing machines, then we can't know either.
Also, there is definitely some objective fact where you cannot get the right answer:
"After thinking about it, you will decide that this statement is false, and you will not change your mind."
If you conclude that this is false, then the statement will be true. No paradox, but you are wrong.
If you conclude that this is true, then the statement will be false. No paradox, but you are wrong.
If you make no conclusion, or continuously change your mind, then the statement will be false. No paradox, but the statement is undecidable to you.
I'm very confused about something related to the Halting Problem. I discussed this on the IRC with some people, but I couldn't get across what I meant very well. So I wrote up something a bit longer and a bit more formal.
The gist of it is, the halting problem lets us prove that, for a specific counter example, there can not exist any proof that it halts or not. A proof that it does or does not halt, causes a paradox.
But if it's true that there doesn't exist a proof that it halts, then it will run forever searching for one. Therefore I've proved that the program will not halt. Therefore a proof that it doesn't halt does exist (this one), and it will eventually find it. Creating a paradox.
Just calling the problem undecidable doesn't actually solve anything. If you can prove it's undecidable, it creates the same paradox. If no Turing machine can know whether or not a program halts, and we are also Turing machines, then we can't know either.
There is no program such that no Turing machine can determine whether it halts or not. But no Turing machine can take every program and determine whether or not each of them halts.
It isn't actually clear to me that you a Turing machine in the relevant sense, since there is no context where you would run forever without halting, and there are contexts where you will output inconsistent results.
But even if you are, it simply means that there is something undecidable to you -- the examples you find will be about other Turing machines, not yourself. There is nothing impossible about that, because you don't and can't understand your own source code sufficiently well.
When I first saw the post, it was at +6. (I don't remember the % or how old it was.) It seems unlikely to me for something with a 38% approval rate to ever hit +6, although there are other hypotheses than Y_i_a sockpuppets. (E.g. sockpuppets used to downvote, or different demographics encountering it at different times.)
I've seen this kind of thing happen before, and I don't think it's a question of demographics or sockpuppets. Basically I think a bunch of people upvoted it because they thought it was funny, then after there were more comments, other people more thoughtfully downvoted it because they saw (especially after reading more of the comments) that it was a bad idea.
So my theory it was a question of difference in timing and in whether or not other people had already commented.
If I've understand correct, you're saying that the probability that x doesn't exist, can't less than the probabiltiy that x is logically impossible.
The reason that it can be true, is because I'm not smart enough to interpret that complicated proposition whether it's in symbolic form or even after I've managed to translate it into words.
Therefore, P(x doesn't exist) may very well be < P(x is logically impossible), I have no idea.
It is definitely true that this could be someone's subjective probability, if he he doesn't understand the statement.
But if you do understand it, a thing which is logically impossible doesn't exist, so the probability that a thing doesn't exist will be equal to or higher than the probability that it is logically impossible.
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I disagree with the idea that the desire to die is normal for humans.
The vast majority of humanity, spanning hunter-gatherers to information economy techies, believe in some form of consciousness which continues after the physical body as passed away. They believe this to the point that, if you disabuse them of this notion, they'll enter a spiritual crisis and begin to feel that life is meaningless. The older people get, the more enthusiastically they believe this.
If the collective fantasy common to our entire species doesn't reflect an extremely powerful human wish to live longer than we currently do, I don't know what does.
When the average person says they want to die at 80, what they really mean that they want to leave this world for another at 80. They don't want to continue things as they are, or re-live their youth - they want to move on to a different sort of existence.
But practically speaking, I think you might be right that getting someone interested in something worldly would encourage them to stay on in this world longer, and in the end that might be better than trying to explain that death means really death (once we actually have the option to stop true death which doesn't seem like a long shot, which realistically we really don't yet).
I don't think the belief in life after death necessarily indicates a wish to live longer than we currently do. I think it is a result of the fact that it appears to people to be incoherent to expect your consciousness to cease to be: if you expect that to happen, what experience will fulfill that expectation?
Obviously none. The only expectation that could theoretically be fulfilled by experience is expecting your consciousness to continue to exist. This doesn't actually prove that your consciousness will in fact continue to exist, but it is probably the reason there is such a strong tendency to believe this.
This article here talks about how very young children tend to believe that a mouse will have consciousness after death, even though they certainly do not hear this from adults: