I'm decoding logarithm expressions in my 1958 book. The problem sets have sneaked in e without telling me that they were going to do this; e hasn't been mentioned in the text yet. As usual with my math learning, all this is puzzle-solving with no context whatsoever. But I give things context even when they don't have any, because I'm too imaginative for my own good. Logarithms, you see, are about the very basic patterns of reality. The logarithm world is also rich in evocative words.

Much earlier in the book, when the text is explaining exponents, I find these lines:

The process of extracting roots is known as evolution. However, special methods for extracting square and cube roots were developed independently of any general theory of evolution.
While the process of raising any number to any integral power, known as involution, leads to a single result, evolution does not necessarily lead to a single result… if b is positive, it has two square roots, numerically equal but opposite in sign. (Page 76)

I have not found these words used this way in any other of my math textbooks. The "process of extracting roots" sounds painful, but I must remember that numbers feel no pain, and they never protest no matter what you do to them. But what other word would you use to describe getting the square, cube, etc. root of a number? If you said "rooting," it conjures up images of trained pigs and dogs finding truffle fungus under trees. Or perhaps a gardener taking cuttings of some plant and sticking the stems in potting soil, hoping that they will sprout. But that's propagation, another branch (or offshoot) of mathematics.

And what about squaring or cubing a number, or "raising to a power?" This sounds much more high and mighty than "rooting." I myself would love to be raised to a power; it sounds rather Nietzschean. Here it involves multiplying a number by itself, any number of times. The word multiplication itself comes from the Latin for "many folds" and so you are folding something again and again when you are multiplying it. This especially seems true for squaring and cubing and so forth, where it gets folded on itself over and over. The word exponentiation may cover it, but I haven't found that in a math text either. Somehow (and this is a big "somehow," for you philosophers of science out there) the process of multiplying something by itself, or folding it, is a basic part of the world, and shows up in all sorts of monumental formulas revealed by Pythagoras and Newton and Einstein and other heroes, including Hero (of Alexandria).

The book's definition of "involution" and "evolution," to me, seems counter-intuitive. "Involution" should be the rooting, not the raising, and "evolution" should be the raising. We associate "evolution" with old-fashioned upward-striving and reaching toward "higher" forms of life, whatever they might be. But according to the 1958 book, "evolution" or getting the square/cube root makes you smaller (unless you are less than or equal to 1) and "involution" makes you larger (unless you are less than or equal to 1). Throw i into the picture and you get all sorts of mirror involutions and evolutions. Go ask Alice! It is no accident that the creator of "Alice in Wonderland" was a mathematician.

So when you take a square root, are you "evolving?" If you exponentiate, are you "involving?" Fractional exponents are even more convoluted (convolving?) because they do involution and evolution at the same time! A fractional exponent will have you square the cube root of a number, or cube the square root of a number, or any sort of root-raise combination that doesn't even out to 1 or an identity. Evidently, if you do this enough (and even exponents have exponents, and their exponents have other exponents, just like big fleas having little fleas upon their backs to bite'em) you get a logarithm. I eagerly await further clarification on this, as well as….e.

To make things even more complicated, "involution" and "evolution" were taken up and used as metaphysical terms by Madame H.P. Blavatsky, the founder of the nineteenth-century esoteric philosophy known as Theosophy. For Theosophy, "involution" describes the motion of "spirit descending into the material world" and "evolution" describes its opposite, the evolution of matter back into spirit. If you take the time to look into Theosophy, you will find that it is a vast swirling stew of pseudo-science and jumbled religions and philosophies, which continues assimilating anything it can get, even up to the present time. The pure-minded science and math devotee should turn away immediately; the purest-minded of them don't even know about it. But since I come from the netherworld of art, I am familiar with Blavatsky and her proliferating multiverse. When I see all these words and concepts, which mean so many different things depending on where you wander or wonder, it's difficult to separate myth from math.

Posted at 3:32 am | link