Saturday, May 3, 2014

What Do You Mean, "Equal"?

Mathematics makes use of the terms equal, equivalent, similar and congruent quite frequently—we could stretch ourselves out a little further and include analogous. They all have something in common: they refer to some kind of "sameness."  But they are not interchangeable.

By similar, when speaking of shapes in basic geometry, we mean that all of the angles between adjacent lines are the same between the two shapes being considered and those angles can be construed as being in the same order (say that three times fast).  We have to sharpen our pencils to deal with curved shapes, but we can in fact speak precisely about similarity between curves.  We might say, A is similar to B if and only if there is some function f that can be applied to A to turn it into B which is (strictly) a composition of translations, rotations, and (universal) scaling.

By congruent, when speaking of shapes in geometry, we are referring to similarity, but nix the scaling. Scaling not allowed. This is a more restrictive requirement. Two things might be similar, but not congruent.

The word "equivalent" is often used in the context of logic. (Logic, by the way, is what many math geeks really like about math—not mainly numbers! Mind you, it is generally hard to argue with the "logic" of numbers when they have been handled rightly. But numbers aren't really useful without logic being applied to their interpretation.) Two propositions are equivalent to each other if they have the same "truth table". When we say propositions P and Q are equivalent we mean that whenever P is true, Q is true, when Q is true, P is true, when P is false Q is false, when Q is false, P is false.

Equivalence is defined for other applications as well.  Equivalence classes are an overarching concept that can be applied in describing similarity, congruence, and various types of equality.  What constitutes equivalent has to be spelled out either by context, convention, or explicit definition for clear communication to occur.

Suppose I said triangle A is equal to triangle B? If you've taken a bit of geometry, you know that you are supposed to frown when people say rubbish like that.
  • Do you mean the triangles have equal area?
  • Do you mean the triangles have equal perimeter?
  • Do you mean the angles are all equal? (similar)
  • Do you mean the corresponding angles and sides are equal? (congruent)
When we talk about equality in whatever form, we should take some thought to what we mean. In well established fields, this is communicated by using the relevant term which has been defined in that field for the "kind of sameness" you wish to refer to. However, when the field is not so well established—or if the audience might not have the same understanding of the meaning, it's best to be explicit.


What, for example, does it mean that "all men are created equal"? I do not intend to discuss the matter at length (as I do not pretend to have a complete answer) but only to point out that our discussions and claims of the properness of equality amongst human beings are fraught with the difficulty of differing understandings of what "equal" and "equality" refer to in such a discussion. For the time being I will content myself with one great, confusing ideal prevalent in my culture (western): meritocracy. (I mean the term in a loose sense—there probably is no such a thing, in reality, on planet earth.)

You may well be shocked at my so (apparently) deriding such a principle. To judge by merits is, in many circumstances, to be contrasted with partiality or prejudice. I heartily recommend the phrase "all men are create equal", but on what basis? Surely merit will not undergird this recommendation. Merit is a basis for distinguishing, which I also recommend!

Perhaps you do not believe that the two issues (merit vs inherent equality) are commonly confounded. I am satisfied that they are. When I hear the merits of representatives of one group (demarcated by ethnicity, gender, or whatever) touted for the vindication of that group, I normally listen quietly, but I do not need such things. Nor do I have much appreciation for the comparing of statistical averages of various metrics in an effort to prove equality. I realize such things are sometimes done in an effort to silence and defend against bigotry, and I do not deride that intent. But I have also seen the defended group touted implicitly as superior (better at this, better at that), in which case I take it as a betrayal of a pretense—appearing to seek equality but with dominance of some kind as the true objective. I do not see inherent equality as based on merit. My key word here, as you may guess, is "inherent". There may be value in recognizing differences, but merit is the "poor man's" rubric for deciding the matter of inherent equality.

Whatever you think of these things, you'll have to deal in your own mind with what you mean by "equal" and what your basis for it is. But I'll tell you where I start:
So God created man in his own image, in the image of God created he him; male and female created he them. (Genesis 1:27)
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