WE ARE SO USED to considering the abstraction of physical objects and phenomena into numbers "data" or "coordinates" (not to mention words and stories) and thinking of the numbers and the varying values of numbers, pairs and sets of numbers that come from manipulating equations, indeed turning those back into abstracted "objects" in the form of lines and curves on graphs that are held to be real in themselves, that we never really think that what is being done is an act of abstraction removed from the physical reality of the objects and phenomena and not the things themselves.
It seems extremely strange and insanely nit-picky when someone points out what we do, constantly. But that becomes important the more we want things from the more attenuated scientific practice of doing that, of course, including the question of God, the meaning of things beyond the everyday, the possibility of an afterlife. For my purposes as a political blogger, those issues of morality that are essential to my political ideology of egalitarian governance. And this is important.
Consider how conservatives are ready to destroy all programs for the
elevation of poor families, especially on the basis of race on the say
so of a bunch of racist academics, journalists, stink tank hacks on the
basis of an ideologically based bell curve, one of the most persistent
obvious uses of this kind of thing for evil purposes sold on the repute
of science. If people understood what was being done, the habits of thought that we've developed through modern mathematics and their use in science, we'd all be a lot less vulnerable to that kind of manipulation.
I think that's what Eddington is doing in this section in so far as science and its position in human life, pointing out what we are doing, starting with the assignment of symbols for "things" which could turn into an infinite regression even before anything potentially useful is done. As to my use of what he says, that may or may not be successful. Though I think it's important.
To introduce mathematics we must somehow put a stop to the infinite regression of symbols. Such a termination will be reached if we find that the X, Y, Z, . . . are not new operations, but are already contained in the first set of operations P, Q, R, . . . that we introduced, that is to say, if we find that the same operation which changes one entity into another will also change one operation into another.
As an example,[ and only one such unrelated example] consider the operations of duplicating, triplicating, quadruplicating, etc. If these are taken as P, Q, R, . . . , we have next to consider, say, the operation Y which changes duplicating into quadruplicating. Quadruplication consists of two operations of duplication, i.e. of duplicating duplication. Thus the operation Y is duplicating, and has already been introduced as P. More generally if the set P, Q, R, . . . denotes all possible operations of multiplication, fractional as well as integral, the operations of changing P into Q, P into R, Q into R, etc. are also operations of multiplication, and therefore no new symbols are required.
I think the confusing part of Eddington's examples in this section are that he is using trios of letters to represent larger groups of operations, no doubt a mathematician of his enormous sophistication and the group of students and faculty he was addressing would understand that a lot more readily than we mere lay people. And they would have been far more familiar with the categories and operations he was talking about than those of us who use a little math from time to time and go for decades without thinking about even second year Algebra. At least that's the most obvious of the problems I had when I read it the first time and now, too and I occasionally have tutored people in math. Remember this lecture dealt with the concept of structure, it gets really interesting later in the chapter when he uses that to make some crucial points about the relationship of our consciousness to the physical world external of us.
And, as you read the following, forget about the example he just gave of the TYPES of abstraction that mathematics is often used for in science, certainly at times imposing structure on reality and not merely finding what's there. Sometimes, I would assert, making things up with equations and graphs of equations. Though not always, by any means. Especially not when there is an actual, observable, definable thing to keep the abstraction and creativity in check.
As another example, suppose that the initial entities A, B. C,. . . . are points on a sphere. The operation of changing one point on a sphere into another is a rotation of the sphere; thus the operations P, Q, R, are rotations. If P and Qu are rotations through equal angles in different planes, the one plane is changed into the other, and therefore P into Q, by another rotation, say R. If P and Q are rotations through unequal angles, one can be changed into the other by a combination of the operations of rotation and multiplication. Grouping together all possible operations of rotation and multiplication, no further operations are introduced in comparing one rotation with another.
We see therefore that there exist "terminable sets of operations" which do not lead to a regression of nomenclature of ever-increasing complexity. It is only through such terminable sets that mathematical thoughts can be introduced. To the extent to which the various portions of our experience can be related to one another in terms of these operations the form material for mathematical treatment . The full development of th idea, here briefly indicated, is contained in the Theory of Groups.*
* An elementary account of the theory of groups, and of the part it plays in the foundation of theoretical physics, is given in New Pathways in Science, Ch. XII
The chapter in New Pathways in Science, page 251 on the PDF, is worth looking at, especially the first couple of pages where Eddington used a fragment of The Jabberwocky taken as representing an unknown universe (or time) to more easily understood and somewhat more entertaining effect. I think it's more compelling because it more closely matches our world of sense, the reason that Lewis Carroll's doggerel is more interesting than the as absurd mathematical description of a successful marriage posted yesterday.
I think there are a number of problems with the ideal presentation of things in which scientists and those who understand them always make the fine distinctions between real things and the abstracted structural mathematical descriptions of them is too abstract in itself because scientists are constantly presenting their abstractions as the more real reality, indeed as I pointed out, some, and some of the most well regarded cosmologists these days, largely due to the quantum theory that Eddington points out this kind of thing is so useful for, insist that all of their abstract universes must be real. To the extent that they seem to believe such things as Boltzmann's brains are real enough to introduce into their heated brawls over multiverses only sometimes admitted to having been invented by ideological cosmologists as a means of getting rid of God. And Boltzmann invented them before quantum theory had really precipitated out of turn-of-the-20th century physics out of some of the most insane mathematical speculation that overlooked little things like the impossibility of mathematically producing a description of "a brain" sufficient to plug its possible existence into a scheme of probability. Who knows how improbable they are in an abstract model of thermodynamic flux? But they still use them in arguing.
Yet they wonder why people have such a hard time taking their claims seriously, including those which are, actually, serious and important. Though, as in the case of The Bell Curve, they take another type of the same thing as far too important.
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