More π, Please

I’d like to elaborate on Myk’s interesting post on π-Day, continuing the path that π is fundamentally so much more important than its digits. (Warning: This post will follow a philosophical turn, so it will deal with foolhardy mental constructs with the intent to change how you view the world and, more importantly, change it.)

Stephen Hawking wrote an anthology called God Created the Integers. The title refers to mathematician Leopold Kronecker’s quote, “numbers were created by God, everything else is the work of men”. His meaning, if I may be so bold, is that numbers are more basic, or more true, than our impressions of the world around us. Numbers transcend. 

Indeed, an early Neanderthal picked up a stone or stared at the moon and recognized the unit, 1. The moon or the stone seemed perfectly demarcated as a “1”, which could stand on its own—epistemologically fundamental. 

However, (there’s always “however”) this little essay attempts to show that “epistemologically fundamental” is not metaphysically or ontologically fundamental. Numbers, despite Kronecker, Hawking, or God, are more likely constructs of man, not God, and may (merely may, madam) not be transcendent. Put better, you can most likely claim a good mate, secure food, and think highly of yourself by putting your faith in numbers, but there may be a better mate available to you, if you don’t.

Let’s take a closer look at the crucial stone in our Neanderthal’s hand. We glance, as he has, at a large, hungry saber-toothed cat he has just spotted. And then another! and another! The world certainly appears to contain numbers. The rock now contains sweat from his hand, but it is still one rock, or his tightening grip has broken a piece from the stone. Yet, it is still unity. One of the approaching cats brushes against a tree leaving a large clump of fur, but it is still unity.  How much can be altered and remain one object? Is half a rock still a rock? Is half a cat still a cat? Can we divide matter to an atom unit? Have we found our “one”? Is "1" really transcendent? 

I’m not sure, but it seems to me that what we have here is “the ineluctable modality of the visible”. We humans, in order to survive, necessarily need modality, from which numbers follow. If we want to use the rock or run from hungry cats, we had better perceive each as a unity. But, in a less stressful time, when we really look at the world, we see π — the relationship between the circumference and its diameter, an irrational number compared to the rational, distinct numbers of our modally viewed world.

The abstract “1” is too clear and clean. The real world is not. The atom is the basic unit of matter. Oh, well, perhaps the quark, or fermions and bosons or a hypothetical particle, like the graviton or the chameleon. Numbers provide us with an indispensable method for dealing with the world. Without numbers and mathematics, we would be lost in knowledge back to the Neanderthal who understood “1”, and “2”, and “3” rocks and tigers. Our world works well knowing precisely how many rocks are needed for how many tigers, and what is the precise position and direction of each. 

However, I submit that the real reality is vague, not distinct, not cleanly modal enough to really have a “1” or a “2” or a “3”. They are indispensable tools, but, perhaps, will never be able to describe the real world no matter how abstract and complex we forge them. The world is more the relationship between the circumference and the diameter, something that is irrational to the human concept of numbers. Numbers are almost perfect in modeling the world, but not exactly, as π shows.

Perhaps, as useful as they are, numbers will always fall short of letting us describe the world. Perhaps they aren't from God after all. Perhaps we need a new vocabulary to really understand it.