Radians are a ratio of lengths (specifically, arc length to radius) whereas degrees are the same ratio multiplied by an arbitrary constant (180/pi). We could imagine that halfradians (the ratio of arc length to diameter) might also be a natural unit, and then we’d have to go into calculus to make a case for radians, but degrees and arc-minutes are right out.
Lengths offer one degree of freedom because they lack units but not an origin (all lengths are positive, and this pinpoints a length of 0). For utilities, we have two degrees of freedom. One way to convert such a quantity to a dimensionless one is to take (U1 - U2)/(U1 - U3), a dimensionless function of three utilities.
This is more or less what you’re doing in your “dimensionless utility” section. But it’s important to remember that it’s a function of three arguments: 0.999 is the value obtained from considering Satan, paperclips, and whales simultaneously. It is only of interest when all three things are relevant to making a decision.
Incidentally, there’s a typo in your quote about Re: 103 should be 10^3.
We could imagine that halfradians (the ratio of arc length to diameter) might also be a natural unit, and then we’d have to go into calculus to make a case for radians
I was actually thinking of diameter-radians when I wrote that, but I didn’t know what they were called, so somehow I didn’t make up a name. Thanks.
For utilities, we have two degrees of freedom. One way to convert such a quantity to a dimensionless one is to take (U1 - U2)/(U1 - U3), a dimensionless function of three utilities.
This is more or less what you’re doing in your “dimensionless utility” section. But it’s important to remember that it’s a function of three arguments: 0.999 is the value obtained from considering Satan, paperclips, and whales simultaneously. It is only of interest when all three things are relevant to making a decision.
Ok good, that’s what I was intending to do, maybe it should be a bit clearer?
Incidentally, there’s a typo in your quote about Re: 103 should be 10^3.
Shamelessly ripped from wikipedia; their typo. 10^3 does seem more reasonable. Thanks.
I was actually thinking of diameter-radians when I wrote that, but I didn’t know what they were called, so somehow I didn’t make up a name.
For the record, I was also making up a name when I said “halfradians”. And now that I think about it, it should probably be “twiceradians”, because two radians make one twiceradian. Oops.
Radians are a ratio of lengths (specifically, arc length to radius) whereas degrees are the same ratio multiplied by an arbitrary constant (180/pi). We could imagine that halfradians (the ratio of arc length to diameter) might also be a natural unit, and then we’d have to go into calculus to make a case for radians, but degrees and arc-minutes are right out.
Lengths offer one degree of freedom because they lack units but not an origin (all lengths are positive, and this pinpoints a length of 0). For utilities, we have two degrees of freedom. One way to convert such a quantity to a dimensionless one is to take (U1 - U2)/(U1 - U3), a dimensionless function of three utilities.
This is more or less what you’re doing in your “dimensionless utility” section. But it’s important to remember that it’s a function of three arguments: 0.999 is the value obtained from considering Satan, paperclips, and whales simultaneously. It is only of interest when all three things are relevant to making a decision.
Incidentally, there’s a typo in your quote about Re: 103 should be 10^3.
I was actually thinking of diameter-radians when I wrote that, but I didn’t know what they were called, so somehow I didn’t make up a name. Thanks.
Ok good, that’s what I was intending to do, maybe it should be a bit clearer?
Shamelessly ripped from wikipedia; their typo. 10^3 does seem more reasonable. Thanks.
For the record, I was also making up a name when I said “halfradians”. And now that I think about it, it should probably be “twiceradians”, because two radians make one twiceradian. Oops.