Arbitrary Math Questions
post by ryan_b · 2017-11-21T01:18:47.430Z · LW · GW · 2 commentsPurpose: This post is so I can record math-ish questions I have, which I want answers to, but don't merit their own post or pestering the community about directly. My expectation is they will mostly yield to a little research. If you have the answer or a relevant source, please feel free to mention.
- Fixed-point theorems: mapping from a particular kind of set, onto a particular function, has a fixed point on that function.
- Is there a procedure for enumerating sets that map onto a function in this way, given just the function?
- If this function were a utility function, could we look at the sets we generated and test whether they are morphic with some other space we are concerned with: value space? mind space?
- How is it exactly we account for things we value which are non-fungible? I am only aware of trying to set a some equivalent by looking at how much we are willing to sacrifice to preserve it, but this fails to capture the dimensionality of the problem. Is my true utility function actually the multiplication of n utility functions, each of one parameter?
Note : I will update this periodically, both to add new questions, mark answered questions, and to break out anything that turns particularly interesting.
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comment by MrMind · 2017-11-21T17:00:50.020Z · LW(p) · GW(p)
Probably those questions needs to be polished and stated more clearly to receive a precise answer. I'll try to add something regarding the second point (the first I'm not sure I understand): from the point of view of VNM-rationality, which is the only guarantee that an agents has a utility function, you can only deduce that utility order-type is isomorphic to R, the set of reals. So in full generality, you cannot deduce anything about the dimensionality of the utility function before stating which actually it is.
Replies from: ryan_b↑ comment by ryan_b · 2017-11-21T17:14:50.806Z · LW(p) · GW(p)
Definitely true - when I say 'yield to research' I probably actually mean 'dissolve as meaningless' in a majority of cases.
On the subject of guarantees about a utility function - is there any notion of an approximate or a revealed utility function? In the same fashion as revealed preferences in economics?