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Comments (271)
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.
Yes. But in ZFC we can't talk about classes. We can construct predicates that describe classes, but one needs to prove that those predicates make sense. Can we in this context we can show that Plus(x,y,z) is a well-defined predicate that acts like we expect addition to act (i.e. associative, commutative and has 0 as an identity)?
In practice we tend to throw them around even when working in ZFC, on the understanding that they're just "syntactic sugar". For instance, if f(x,y) is a formula such that for all x there exists unique y such that f(x,y), and phi is some formula then rather than write "there exists y such that f(x,y) and phi(y)" it's much nicer to just write "phi(F(x))" even though strictly speaking there's no such object as F.
I think the proofs go through almost unchanged (once we prove 'finite replacement').
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.)