try pondering this one. Why does 2 + 2 come out the same way each time? Never mind the question of why the laws of physics are stable - why is logic stable? Of course I can't imagine it being any other way, but that's not an explanation.

Do you have an answer which will be revealed in a later post?

My [uninformed] interpretation of mathematics is that it is an abstraction which does exist in this world, which we have observed like we might observe gravity. We then go on to infer things about these abstract concepts using proofs.

So we would observe numbers in many places in nature, from which we would make a model of numbers (which would be an abstract model of all the things which we have observed following the rules of numbers), and from our model of numbers we could infer properties of numbers (much like we can infer things about a falling ball from our model of gravity), and these inferences would be "proofs" (and thankfully, because numbers are so much simpler than most things, we can list all our assumptions and have perfect information about them, so our inferences are indeed proofs in the sense that we can be certain of them).

But it seems like a common view that mathematics has some sort of special place in the universe, above the laws of physics, and I don't really know what arguments people have for believing this. What are the arguments for this belief?

Edit: Reformulated my question to make it more specific.

Can anyone explain why epiphenomenalist theories of consciousness are interesting? There have been an awful lot of words on them here, but I can't find a reason to care.

Meditation:

If we can only meaningfully talk about parts of the universe that can be pinned down inside the causal graph, where do we find the fact that 2 + 2 = 4? Or did I just make a meaningless noise, there? Or if you claim that "2 + 2 = 4" isn't meaningful or true, then what alternate property does the sentence "2 + 2 = 4" have which makes it so much more useful than the sentence "2 + 2 = 3"?

It certainly looks like Harry is a horcrux in this universe, and Harry already thought of that possibility in different terms, yet the Sorting Hat says...

The exact phrasing of the Sorting Hat's statement was as follows:

...there is definitely nothing like a ghost - mind, intelligence, memory, personality, or feelings - in your scar. Otherwise it would be participating in this conversation, being under my brim.

Now, anyone that's read the sort of fairytale where riddles are important should immediately be able to come up with a half-dozen loopholes in that, but I think we can dismiss most of them out of hand given that the Sorting Hat has no particular incentive to be misleading. The most promising option that remains, by my reading, is that there's nothing separate about the Horcrux contents for the Hat to key off of -- they effectively are Harry, or part of him. He's probably tapping that part of himself when he has his Dark Side episodes, at the very least, but I don't think that's the full extent of the Horcrux's influence: at various points he asks himself or people around him why he doesn't think like other children, and narrative parsimony points rather strongly to the one unique trait we know he has.

The weakest point of this theory, as best I can tell, is the lack of any (obvious) memories from Voldemort; I think we can safely assume the Hat would have found them if they were locked away somewhere within him, but on the other hand it'd be a rather poor resurrection that resulted in an amnesiac personality-clone. Riddle's diary from Chamber of Secrets also argues along these lines. Unfortunately, we haven't seen any other Horcruces in MoR, so we have nothing in-universe to compare against, and canon may not be reliable. Perhaps the relevant memories got wiped out by infantile amnesia or something.

But the balance should probably be a lot further towards others' than it currently is.

Why? My should is different from your should. Who is to say that your should is better for me than mine?

And no, I don't accept your "idea that we have some obligation to try to help other people". I hate obligations. They piss me off. Whatever I do, I do because I want to, not because I owe it to others.

I don't understand how the first statement can be used to prove anything...

The second statement might be true for every statement, but it's not necessarily provable for every statement, which is required in the proof. In fact, the second statement is provable for "outputs(1)" by inspection of the program (because the program searches for proofs of "outputs(1)"), but not provable for "2==3" (because then "2==3" would be true, by Lob's theorem).

Edit: Wow, I really am an idiot. I assumed the second statement was true about every statement, but I just realized (by reading one extra comment after posting) that by Lob's Theorem that's not true. But my original idea, that the first statement is all that's required to prove anything, still seems to hold.

Okay, I can follow the first proof when I assume statement 1, but I don't quite understand how cousin_it got to 1. The Diagonal Lemma requires a free variable inside the formula, but I can't seem to find it.

In fact, I think I totally misunderstand the Diagonal Lemma, because it seems to me that you could use it to prove anything. If you replace "outputs(1)" by "2==3", the proof still seems to hold. Statement 2 would still be true with "2==3" (it is true about any statement, after all), and all the logic that follows from those two statements would be true. In fact, by an unclearly written chain of reasoning which I originally intended to post before realizing that it would be much simpler to just say this, all you seem to require is the first statement to be able to prove anything. If I am mistaken, which is probable, then I expect my error lies in the assumption that "outputs(1)" could be replaced by any string of code.

For my original unclear explanation of why the first statement in the proof seems to allow anything to be proven, in case anyone cares for it:

Also, I must be an idiot, but it seems to me that you can prove pretty much anything using 1. As far as I can tell, the "eval(outputs(1))" could be "2==3" or any other statement, and the only reason "eval(outputs(1))" is used is because it's useful in the proof. Given that statement 1 is proven, "eval(box)" must return true, because if it returns false, then "proves(box,n1)" cannot return true, and therefore "implies(proves(box, n1),eval(outputs(1)))" must return true, and false!=true. Assuming that my reasoning is correct (a very questionable assumption, I know), I have just proven that "eval(box)" is true, therefore "prove(box,n1)" is also true for some arbitrarily large n1. Therefore, the "implies" will return "eval(outputs(1))", which must be true for "eval(box)" to equal it. So given my earlier assumption (which I suspect is my error) that "eval(outputs(1))" can be replaced by anything, then I can replace it by any statement, which must return true to equal "eval(box)". So through this, I seem to have proven that "2==3".

Anyone care to point me to my mistake (or, to satisfy my wildest of dreams, say that I appear to be right ;) )?

For example, most people would think that needlessly hurting somebody else is wrong, just because. The claim doesn't need further elaboration, and in fact the reasons for it can't be explained, though people can and do construct elaborate rationalizations for why everyone should accept the claim.

I think this is a folk theory about how "moral intuitions" work, and I don't think that it is true, in the sense that it is a naive answer to a naive question that should have been dissolved rather than answered. For example, most people think everything "just because", and further elaboration is just confabulation unless you do something unusual.

Thinking that morality is a specialized domain (a separate magisterium?) leads to the idea of "debating morality" as though the actual real communication events that acquire that label are like other debates except about the specialized domain: engaged in for similar purposes, with similar actual end points, resolved according to similar rhetorical patterns, and so on. Compare and contrast variations on the terms: "ethical debates", "political debates", "scientific debates", "morality conversations", "morality dialogues", "political dialogues", etc. Imagine the halo of all such terms, and the wider halo of all communication events that match anything in the halo of terms, and then imagine running a clustering algorithm on those communication events to see if they are even distinct things, and if so what the real differences are.

I don't want to say "Boo!" here too much. I'm friendly to the essay. And given your starting assumptions it does pretty much lead to the open minded interpretation of moral debates you derived. I tend to like people who go a little bit meta on those communication events more then people who just participate in them by blind reflex, but I think that going meta on those communication events a lot (with tape recorders and statistics and hypothesis testing and a research budget and so on) would reveal a lot of really useful theory. You linked to Haidt... some of this research is being done. I suspect more would be worthwhile :-)

Edited to add: And I bet the researcher's "moral debating" performance and moral conclusions would themselves be very interesting objects of study. Imagine being a fly on the wall while Haidt, Drescher, and Lakoff tried to genuinely aumann updated on political issues of the day.