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asr comments on Open thread, Aug. 03 - Aug. 09, 2015 - Less Wrong
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Cardinal numbers for utilons?
I have a hunch.
Trying to add up utilons or hedons can quickly lead to all sorts of problems, which are probably already familiar to you. However, there are all sorts of wacky and wonderful branches of non-intuitive mathematics, which may prove of more use than elementary addition. I half-remember that regular math can be treated as part of set theory, and there are various branches of set theory which can have some, but not all, of the properties of regular math - for example, being able to say that X < Y, but not necessarily that X+Z > Y. A bit of Wikipedia digging has reminded me of Cardinal numbers, which seem at least a step in the right direction: If the elements of set X has a one-to-one correspondence with the elements of set Y, then they're equal, and if not, then they're not. This seems to be a closer approximation of utilons than the natural numbers, such as, say, if the elements of set X being the reasons that X is good.
But I could be wrong.
I'm already well past the part of math-stuff that I understand well; I'd need to do a good bit of reading just to get my feet back under me. Does anyone here, more mathematically-inclined than I, have a better intuition of why this approach may or may not be helpful?
(I'm asking because I'm considering throwing in someone who tries to follow a cardinal-utilon-based theory of ethics in something I'm writing, as a novel change from the more commonly-presented ethical theories. But to do that, I'd need to know at least a few of the consequences of this approach might end up being. Any help would be greatly appreciated.)
It's a tempting thought. But I think it's hard to make the math work that way.
I have a lovely laptop here that I am going to give you. Suppose you assign some utility U to it. Now instead of giving you the laptop, I give you a lottery ticket or the like. With probability P I give you the laptop, and with probability 1 - P you get nothing. (The lottery drawing will happen immediately, so there's no time-preference aspect here.) What utility do you attach to the lottery ticket? The natural answer is P * U, and if you accept some reasonable assumptions about preferences, you are in fact forced to that answer. (This is the basic intuition behind the von Neumann-Morgenstern Expected Utility Theorem.)
Given that probabilities are real numbers, it's hard to avoid utilities being real numbers too.
If we are going into VNM utility, it is defined as the output of the utility function and the utility function is defined as returning real numbers.
I could try to rescue the idea by throwing in units, the way multiplying distance units by time units gives you speed units... but I'd just be trying to technobabble my way out of the corner.
I think the most that I can try to rescue from this failed hunch is that some offbeat and unexpected part of mathematics might be able to be used to generate useful, non-obvious conclusions for utilitarian-style reasoning, in parallel with math based on gambling turning out to be useful for measuring confidence-strengths more generally. Anybody have any suggestions for such a subfield which won't make any actual mathematicians wince, should they read my story?