A friend of mine recently commented, in The Secrets of Pi, a post on this blog, that pi, like the circle, like plane geometry, and like all math, are, of course, and obviously, inventions made up by mathematicians. This is a frequently encountered belief which I think I shared, myself, at one time. In fact, I think I recall being taught in elementary school that mathematics, precisely because it is an abstraction, has no real dimensionality to it, and the perfect sphere, such as a soap bubble does not and cannot exist. We thus, in one swell foop, categorize mathematics as dreaming, and mental phenomena as unnatural and unreal. In a feat of mental gymnastics, we then go on to physics and other sciences to apply math to our problems, and stand back in amazement that the wedding of real, factual phenomena, and mathematics, which has nothing to do with reality, seems to work out so well. It must be a coincidence.
There is some psychological or cultural problem at the root of this misconception, but I’m not sure what it is. There’s even a possibility that it stems back to the mediaeval battle between cartesians and the emerging rationalists inspired by Francis Bacon. In this debate, we have to make a marked separation between mental and spiritual phenomena on the one hand, all of which are seated wholly in our mental imaginations, and true physical realities on the other hand, which are rooted in sensory experience and the operation of natural laws. The first set get categorized as imaginary so that empiricism can have a more free reign to resolve all problems.
This ancient strategy is wrong, as any programmer who has struggled with computers has learned at great pain. Computers implement logic. This logic cannot be bent, cannot be negotiated, cannot be remodeled or revised. And this has nothing to do with human design of computers. It has to do with the same logic that scientists believe underlies natural phenomena, and that allows us to speculate on the farthest regions of the Universe on the basis of what we observe here: There are universal laws that apply everywhere, all the time. This belief (it is a belief, since natural laws cannot be directly observed) is based on a very old greek idea called Logos, sometimes translated as logic, but originally meaning something much more like order; the fundamental order of the universe which underlies all things.
I first addressed this concept of Logos explicitly in my post, Logic: A Principle of Order.
John Searle introduced a rule of thumb to distinguish subjective phenomena from external phenomena. He was discussing social constructions like property, money, functionally defined objects like chair and bathroom, and governments, and his point was that, while these objects are not physical, they aren’t subjective, either, because you can’t change them just by changing your mind. Taxation isn’t something you dreamed up, and you can’t undream it, as much as we’d like to. So he concluded that socially constructed phenomena have a type of reality because they have a direct impact on us which can be measured, often, in dollars, a very concrete kind of reality in itself.
Searle’s concept applies to the question of logic, mathematics, and natural law, just as much as to social realities. A state legislature in the United States at one time seriously considered passing a law (I think the state government in question was that of Illinois) to define pi as exactly 3.15. They said this would simplify life for children in school, and all sorts of engineering and science. Luckily the bill wasn’t passed, because it would have been impossible to enforce. Pi appears in science and engineering in many places, not just in the calculation of circumferi of circli. Apparently these legistlators thought, since it’s only mathematics, it’s an invention of minds and hence subjective, and we can easily change our minds about subjective choices. That was their mistake: given the definitions of circle and radius, the concept of the ratio pi falls out without making any choices at all: it’s a logical implication of the definitions. Because Pi is involved in the calculus, it also appears in electrical engineering applications — but what, you might wonder, does plane geometry and circles have to do with electricity?
I am sure many people feel you can still trace all these matters back to fundamental definitions, and if we were just willing to give up our definitions of circles, we could toss Pi into the oblivion of nonexistence from which it came. But you can’t undo geometry. Not even plane geometry, because plane geometry is implicate in solid geometry, and solid geometry is essential to how we conceptualize and perceive the “real” world. Eventually, it dawned on some German Jew that geometry was at the root of gravity and many of our basic physical truths. Einstein’s general relativity remains a mainstay of our understanding of the universe to this day, because the world is organized in an orderly way, and the mathematics he used is an expression of this.
Which brings me to my final point. Newton and Leibniz both “invented” the calculus. Their inventions, however, were not without their differences. The concepts put forward by the two mathematicians were equivalent, and so we do not now and never did have two different concepts of calculus. Even though they worked independently, they both discovered the same inherent properties of change and movement. But the two men used different notations. Ironically, it’s Newton’s publication of the laws of calculus which are taken as the mark of invention, but it’s Leibniz’s notational system which we use today. This tells us which parts are a fundamental law, and which parts are subjective: the notation is human-made. But the ordering concepts were discovered, not made.
You wouldn’t say Galileo invented gravity; he only discovered some of the mathematically consistent properties of it. He did, however, invent some tests to demonstrate the behavior of gravity, and some simple laws of proportion to describe its effects on matter. It is now conventional to consider gravity an external reality, so we say Galileo discovered rather than invented it. But not all discoveries involve dropping military hardware from towers of Pizza. Mathematians are involved in discovering what consequences must be true, if our basic assumptions are true. The inventions are the notation and the axioms, but the conclusions are locked into the way we have to think. Have to, because the universe is not negotiable.
The fundamental underlying order of the universe is reflected in mathematics. The expressions of the mathematics are the inventions, but the order is not. And that’s why airplanes fly.
