I would like to observe here something about mathematics that I touched on in my SoloHQ article “Errors of Modern Science—A Philosophical Magic Act.” (Of course, the “article” is only a transcription of an entertainment, not an argument presented.) Namely, that although mathematics can be regarded as a science, since it can discover certain facts about the world of numbers (such as whether there is a highest prime), it is primarily a technology, a method.
The basic material of that method is the series of “natural” or counting numbers (1, 2, 3, etc. without the zero), which is based on the human ability to regard concrete things as units (members of an open-ended class). And that is the extent of math’s connection between numbers and the world of entities. Everything else in math represents ways of usefully mapping the natural numbers upon complex realities existing in the universe.
In “Errors” I mention two basic cases on the lowest level of math—the zero and negative quantities:
I say the error here is: mathematics
Does not apply directly to the real,
But only through swift mental acrobatics
That practice and accomplishment conceal.
For instance, we use zero—nothing to it!
But when you seek for naught, you misconstrue it.
For zero can’t be found—it’s zilch, it’s zip.
When it’s a temperature, you catch the grippe;
And, on the Kelvin scale, you can’t get colder.
How do we use it? As a mere place-holder.
In tennis, love’s declared to mark a space
During the match till someone scores an ace.
And minus one’s a sheer impossibility.
Apart from someone’s debt, it lacks utility:
You have to run up some kind of account
To make sense of a negative amount.
Such things as “imaginary” and “complex” numbers follow the same principle: they are ways of applying basic arithmetic to complex states of affairs using “mental acrobatics.” And the goal is always to end up with results with natural-number implications. Sometimes, through sheer logic, mathematics can adumbrate new discoveries of science, but it must always be remembered that a mathematical process is only as good as the mapping to reality of its assumptions, definitions, and procedures.
One thing that contributes to the mistaken reification of math may be the term “numbers.” When we solve an equation and find out that a barrel contains about 100 apples, we know that that number corresponds to a reality. Now suppose the equation’s solution is 0. Right away, we have left the natural numbers and are into the class of non-negative integers—that is, the result has immediately ceased to correspond directly to reality (since “nothing” is not a thing). A negative integer solution, such as –21, carries us even further away from concrete reality into the realm of human method. Yet the integers are still referred to as “numbers.” Why? Because they have so many properties exactly like those of the counting numbers—and in fact include the counting numbers—that it is preferable to expand the meaning of the term, and make the counting numbers into a subdivision, than to invent an entirely new idea. (Here I disagree with Introduction to Objectivist Epistemology: Rand refers to such new number classes as a subdivision of the concept of a number; I see them as an expansion of the term, and in a special sense an expansion of the concept.)