Just How Intrinsic Is It?

Sorry for the delay between posts. I’ve gotten into sort of a summer lull where I feel like wasting my time doing mundane and useless things. Anyways..

This is a response to this post about math’s relation to nature.

While I’m unsympathetic toward any “Goddidit”-type answer, I certainly understand the drive of your post. The elegance and sheer predictive power of mathematics seems to suggest that it’s tied intimately to how the universe actually works. One must always remember, I think, that abstractions, no matter how accurate, are still abstractions. They’re not real. They don’t actually exist.

Mathematics is, in a sense, an artifact of our finite brains. Those finite brains can never comprehend reality perfectly. They make predictions that, while imperfect, can approximate the world around us to a very high degree of accuracy. Consider, for example, an equilateral triangle drawn on a piece of paper with a pen. It’s obviously a very crude approximation to what we consider such a triangle to actually be. Its shape depends on the pressure applied to the pen, on the molecular structure of the ink, on the adhesive properties of the paper, etcetera. If we examined such a drawing closely, we would of course discover the extent of its imperfection. We would discover ink molecules strewn about haphazardly, piled one atop the other at varying heights, like a box of carrots dumped onto the floor. The “lines” would actually be very complicated zigzags, almost fractal in nature, but much more chaotic.

So consider the same drawing but on a computer screen. Assume the pixels line up perfectly. Still the “lines” of the triangle do not match the mathematical theory. Pixels have width and height. If you’re on a Liquid Crystal Display (LCD), you get into the same problem you did with the pen ink. If you’re on a back-lit CRT monitor, the lines depend on how the electrons fired from the cathode ray hit the screen. You get into a probabilistic problem, then.

So what is an equilateral triangle? Such a thing can never really physically exist, at least not by our present scientific reckoning. When we see that triangle printed out nice and neatly on a piece of paper, our brains really are just fooling us. Why? Because they don’t need to know about the minuscule goings-on of atoms and molecules for us to function. After a certain accuracy, the triangle is just “good enough”. We’re from “middle world”, as they say, and we need only concern ourselves with things that are about average in size. In other words, the concept of the perfect triangle is just an easy first approximation to what triangles really are in this universe. The differences between the abstract and physical triangle may be negligible for all practical purposes, but the two are still different.

So does \pi exist in the real world? Well, no, because circles don’t exist in the real world. But if they did, it would, and this is something our brains have somehow worked out throughout our evolutionary history. The same pattern-finding behaviors we use unconsciously to approximate where to place our hand in order to catch a ball we’ve just thrown into the air can also be explored consciously. That’s what I view mathematics as, exploring those automatic pattern-seeking behaviors that exist already in our brain. They may not be perfect ever, but they’re “good enough” to let us do the things average-sized creatures need to do in a complicated universe.

So when you say “God made 1, and all else is the work of man”, I would disagree. I would say that even the concept of “1” is man-made. Why? The same reason; because our brains like to approximate things. An apple is “one apple” if it has a certain approximate shape and color. But no two apples are the same, in reality. That apple is really just billions and billions of plant cells organized haphazardly. But those cells are really just billions and billions of atoms organized haphazardly. And those atoms are even smaller particles organized haphazardly. The differences between two apples may be billions and billions of atoms; something that, from an atom’s point of view, is not arbitrary in the least! From a middle world creature’s perspective, of course, the differences may be negligible.

So we’re left with the following dilemma: does one of anything really exist? It seems to me that when we talk about “one” of something, we’re necessarily forced to neglect more complicated structures for ease of discussion. Saying there exists “one” of something is just a convenient mnemonic. When we say our solar system has “one sun”, we’re classifying implicitly the totality of “sun-like” objects. We’re saying, “things that have a certain look, and preform in a certain general way, will here forth be referred to as [blank]”. But that’s just giving a name to something. It’s an abstraction, just like triangle. It would take an infinite amount of time to perfectly describe our sun at even a single moment. And so we approximate.

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3 responses to this post.

  1. […] 7th, 2007 at 8:25 pm (Math) In Just How Intrinsic Is It?, I find that there are actually people who are willing to read the philosophical dronings of people […]

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  2. […] Just How Intrinsic Is It?, Jon McKenzie sides with many, or most possibly, philosophically inclined mathematicians; he claims […]

    Reply

  3. […] Mark Chu-Carroll, Math) It’s just as well that I didn’t get far. Jon first says in Just How Intrinsic Is It?: Mathematics is, in a sense, an artifact of our finite […]

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