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JoshuaZ comments on The Amazing Virgin Pregnancy - Less Wrong
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Comments (271)
First, that book only has the elegant proofs. Second this totally misses the point: whether a statement is a theorem of a given formal system is objectively true or false is a distinct claim from the claim that some set of axioms is objectively a set of axioms that is somehow worth paying attention to. Even if two mathematicians disagree about whether or not one should include the Axiom of Choice in set theory, they'll both agree that doing so is equivalent to including Zorn's Lemma.
You aren't just claiming that there are "theorems" from some set of moral axioms, but seem to be claiming that some sets of axioms are intrinsically better than others. You keep making this sort of leap and keep giving it no substantial justification other than the apparent reasoning that you want to be able to say things like "Gandhi was good" or "genocide is bad" and feel that there's objective weight behind it. And we all empathize with that desire, but that doesn't make those statements more valid in any objective sense.
I haven't said anything is intriniscally better: I have argued that the choice of basic principles in maths, morlaity, etc is constrained by what we expect to be able to do with those things.
If you vary the way games work too much, you end with useless non-games (winning is undefinable, one player always wins...) If you vary the way rationality works too much, you end up with paradox, quodlibet etc. If you vary the rules of meta ethics too much, you end up with anyone being allowed to do anything, or nobody being allowed to do anything.
Yes. I do want to be able to say murder is wrong. I should want to be able to say that. It's a feature. not a bug.. What use is a new improved rationalised system of mathematics which can't support 2+2=4?
Peter, how do you reconcile this statement with your statement such as the one's here where you say that
I don't see the problem. What needs reconciling with what?
How can you say someone has a glitch if they simply aren't adopting your system which you acknowledge is arbitrary?
Yet again: I never said morality was arbitrary.
I said he has a glitch if he can't see that murder is wrong. I didn't say he had to arrive at it the way I arrived at it.. I am selling a meta ethical theory. I am not selling 1-st order system of morality like Roman Catholicism or something. I use core intuitions, common to all 1st order systems, as test cased. If you can't get them out of your metaethical principles, you are doing something wrong.
What use is a new improved rationalised system of mathematics which can't support 2+2=4?
So morality is like chess, but there's some sort of grounding for why we should play it? I am confused as to what your position is.
I'm not sure what you mean by that. If I'm following your analogy correctly then this is somewhat wrong. Any reasonable general philosophy of metamathematics would tell you that 2+2=4 is only true in certain axiomatic systems. For example, if I used as an axiomatic system all the axioms of ZFC but left out the axiom of infinity and the axiom of replacement, I cannot then show that + is a well-defined operation. But this is an interesting system which has been studied. Moreover, nothing in my metamathematics tells me that that I should be more interested in ZFC or Peano Arithmetic. I am more interested in those systems, but that's due to cultural and environmental norms. And one could probably have a whole career studying weak systems where one cannot derive 2+2=4 for the most natural interpretations of "2", "+","=" and "4" in that system.
To return to the original notion, just because a metaethical theory has to support that someone within their more and ethical framework has "murder is wrong" doesn't mean that the metaethical system must consider that to be a non-arbitrary claim. This is similar to just because our metamathetical theory can handle 2+2=4 doesn't mean it needs to assert that 2+2=4 in some abstract sense.
I know this is a sidetrack, but I don't think that's right, unless we're omitting the axiom of pairing as well. Can't we use pairing to prove the finite version of replacement? (This needs an induction, but that doesn't require the axioms of replacement or infinity.) Hence, can't we show that addition of finite ordinals is well-defined, at least in the sense that we have a class Plus(x,y,z) satisfying the necessary properties?
(Actually, I think it ought to be possible to show that addition is well-defined even without pairing, because power set and separation alone (i.e. together with empty set and union) give us all hereditarily finite sets. Perhaps we can use them to prove that {x,y} exists when x and y are hereditarily finite.)
If we don't have the axiom of infinity then addition isn't a function (since its domain and range aren't necessarily sets).
Sure, in the sense that it's not a set. But instead we can make do with a (possibly proper) "class". We define a formula Plus(x,y,z) in the language of set theory (i.e. using nothing other than set equality and membership + logical operations), then we prove that for all finite ordinals x and y there exists a unique finite ordinal z such that Plus(x,y,z), and then we agree to use the notation x + y = z instead of Plus(x,y,z).
This is not an unusual situation in set theory. For instance, cardinal exponentiation and 'functions' like Aleph are really classes (i.e. formulas) rather than sets.
By arbitrarily declaring what qualifies as a glitch. (Which is only partially arbitrary if you have information about typical or 'intended' behaviour for an agent.)