Equality is one of the great issues of our age, but few people stop to wonder at its being an issue in politics at all. Yet it is surprising that a concept which has its natural habitat in the mathematical sciences should have taken root in our thinking about how we should be governed. We do not naturally think of society in terms of group theory, or rings or ﬁelds, and have long been aware of the diﬃculties in establishing any over-arching social or political order. But we unthinkingly assume that we can meaningfully ask, and reliably tell, whether people are, or ought to be, equal to one another even while admitting how diﬃcult it would be to say whether they were more or less than one another. Formal logic can help. Equality is an equivalence relation, that is to say one that is transitive, symmetric and reﬂexive. Equivalence classes pick out classes of people or things that are the same, or similar , in some respect or other. There are many such, and we need to specify in respect of what two things are or are not equivalent before we are saying, or asking, anything deﬁnite. I can be equivalent to you in respect of age, or height, or weight, and many equivalence classes— contemporaries, co-religionists, comrades—may be of great importance socially or politically. But equality is, in its original context, more than just an expression of sameness. It suggests also a possibility of being either more than or less than. I can be the same age as you, but if I were not, I should be either older than you, or younger. This is always the case in mathematics. The law of trichotomy holds, that if two things are not equal, then one is greater, in the relevant respect, than the other, and the diﬀerence can itself be measured. With human beings, however, there are rather few respects in which we can be properly measured. Age, height and weight apart, the ascription of most numerical measures is a dubious aﬀair. At one time psychologists were conﬁdent that they could measure intelligence, and economists still purport to measure wealth, but the ascriptions they actually make do not seem to deserve the conﬁdence called for..
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