The ethical question of fairness, of whether to treat people equally, does not depend on whether or not people are in fact equal. I get irate at how people confuse these two. To treat unequal people equally is an ethical decision, not an expression of belief in pre-existing equality. The expense of this moral choice, however, IS proportional to the extent to which equality is truly there.

Discrimination laws restrict the data which can be used to make certain decisions.

Fuzzy points: how is it that women getting cheaper car insurance than men is not sexual discrimination? To collect data on race for that same purpose would clearly be illegal. What about lower insurance for high GPA students?

In health insurance, the improvement of predictive diagnostics forces this ethical confrontation, where newly unequal individuals are to be treated equally.

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Personally, the lack of discrimination is a question of consistency. The bigger the groups which one treats uniformly are, the more one is able to develop and expose rules of reward and retaliation.

From City Slickers, as seen at Coding Horror:

Curly: Do you know what the secret of life is?

Curly: This. [holds up one finger]

Mitch: Your finger?

Curly: One thing. Just one thing. You stick to that and the rest don't mean shit.

Mitch: But what is the "one thing?"

Curly: [smiles] That's what you have to find out.

Let's say you have a box full of particles, and each of those particles is independently in one of N states. Each state has a probability of occuring. Let us further suppose that none of these probabilities are the same - if so, then there is one state with the highest probability. If there are then properties E and V which are diagonal and have a 1-1 correspondence to each other amongst all the states, then the most common E and most common V definitely are related to each other by E=MV^2/2.

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Examine the Speed and Energy distributions of the ideal gas. The modal values of these two distributions do not correspond to each other via the E = MV^2/2 relation. Paradox?

resolution:

The modal value of P(v) is the value of v for which the integral from v to v+dv is maximal for any sufficiently small dv. Imagine sliding a small window of width dv over P(v) and finding the interval which contained the most probability density - this interval always includes the modal value of P(v), and that value can be found by taking dv to zero. An identical process applies to the modal energy.

The resolution of the paradox lies in the fact that dv and dE are both the wrong measures to use. The correct measure is number of quantum states, which classically corresponds to phase space. The correct procedure would be to find P in terms of phase space, then to find the values of E and v associated with the modal state. E definitely is the same as mv^2/2 in that case.

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Homework:

Explicitly prove this by showing how the phase space approach derives the correct values of E and V corresponding to using the quantum partition function with the classical limit of particle in a box.