Just How Intrinsic Is It? (Second Response)

This is a continuing response to a post made here and the follow up posts here and here.

Bob, you write, “…he claims that math is completely a creative act on the part of humans.”

Not exactly. Certainly some things are entirely invented, such as notation and other things. But my position is a more evolutionary one. Consider a simple bacterium wandering on a two-dimensional substrate. It doesn’t really have any need for cognizance of the larger three-dimensional world because its motility is limited to that substrate. (And by cognizance I don’t mean consciousness, I just mean its ability to interact with its environment). And so, evolution plunks away giving the bacteria only what it needs to survive and reproduce. It may give it a way to sense the direction of polarized light, or a way to sense the chemicals of would-be competitors, or other beneficial attributes.

Now, we humans are in the enviable position of being conscious beings. This consciousness is, from an evolutionary standpoint, beneficial to our survival. You can think up ways of why that might be. For instance, being conscious allows us to freely access memories, letting us learn and plan for the future. It allows us to imitate our environments and the other organisms that live in those environments. We make sharp tools that imitate the claws of a saber-tooth. We build shelters that imitate caves. Of course, most of the goings on of our brains (as we know today) occurs subconsciously. We don’t realize most of what’s happening in our bodies. We don’t know how our brain, to use my previous example, calculates where to place our hand to catch that ball we’ve just thrown. We don’t know how our brains coordinate our complex bodies.

So my point is, mathematics really is intrinsic to how we work, because our brains obviously utilize it at some level. But the caveat to that is that this mathematics is still just an approximation. It’s the best solution given finite resources that evolution has provided to us. Like the bacterium, we, being three-dimensional creatures, only need a certain kind of software to operate in that three-dimensional world. So when I say mathematics isn’t intrinsic to the universe, I’m saying that the logical system that evolution provided to our brains to preform those subconscious computations is one based upon the trials and errors of our ancestors.

Of course, that’s not the whole story. We are, by nature, social creatures, and we have this ability to change socially much faster than we could ever evolve biologically. This is where our creativity comes to bear in mathematics. We build up these axiomatic systems, we find contradictions, and we puzzle out where the problems arise. For example, we looked at Euclid’s postulate about parallel lines and deduced that it wasn’t necessarily true. This led to a geometry that, though completely unintuitive, was nonetheless consistent. But as you can see, this all depends on what we take as axiom. And, as Godel and Turing showed us, axioms can be interesting things. You can’t ever prove your axioms are consistent using those same axioms themselves. You must invent a hierarchy of axioms, each system proving the consistency of the last.

This, I think, solidifies for most mathematicians the notion that mathematics just isn’t that perfect logical system they wished it could be. At the end of the day, it’s based upon certain carefully chosen axioms. Any consistent logical system will work no matter what the universe might look like. You could construct a consistent, and yet completely impractical and stupid, logical system that doesn’t look anything like reality. It would still work and be consistent, and you could derive things from it, but it would be useless.

P.S.

I changed the presentation of my blog, which is why the name disappeared. I hadn’t even realized it until you pointed it out. So that’s all that happened. You can call me by my name if you want.

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