Saturday, September 4, 2010

Effectiveness of mathematics

I ran across the above essay a few days ago. I read for content the first time, and detail the second. At the bottom of the essay it basically concludes that all 4 explanations, even if they were allowed to combine their evidence, would still not be enough to explain the “unreasonable effectiveness” of mathematics.
I saw no mention anywhere in the article that one must be a mathematician (or even mathematically inclined) to guess a solution to the “unreasonable effectiveness” of mathematics. So my guess goes as such.

Mathematics is built up from a set of givens. These givens are not proven true or false beforehand, they are simply givens. Laws of mathematics are built up by combining these givens. The more observably correct laws that can be built up from any set of givens, the more likely it is that some (or even all) of the givens are true. If you are ever able to construct a law that is provably false, and you could prove that all the laws that this provably false law reference are correct, then you know that some given that “builds up” to one of these laws must be incorrect.

Mathematics, is different from other science in the sense that it is extremely extensible. The language can be extended by simply creating a new syntax, and extending a particular law/property/invariant without breaking any existing laws/properties/invariants. If the new extension “fits” within the set of existing laws, then does that make it true? Well no, but if the new extension does not break anyone else, then there is a higher possibility of it being true.

My guess is that mathematics is so unreasonably effective because it is so amazingly extendable. We simply create new mathematics every time we need mathematics to cover something new. New distributions for new types of probabilities, moving from scalars to vectors to tensors for physics, it just keeps getting added on. Mathematics is effective because mathematics is based on symbolic manipulation, which is extremely flexible.

The interesting thing to note is how rarely we hit contradiction. Perhaps this points to just how limited (and hopefully not missguided) humanities mathematical understanding is. Picture the set of givens as the root of a tree, and the laws as branches coming out of the root (it has not trunk). Some laws are built using other laws, without even directly referencing a given.

As invariants are added, the chance that another invariant will cause a contradiction to some existing invariant increases. The fact that we can still “easily” add new invariants to the tree seems to indicate that we are not even close the the “final” number of invariants.

So my guess is that mathematics is effective because it is based on symbolic manipulation, which is probably the most extensible thing humanity has ever invented, and because the “tree” of mathematics is very small, and adding new “leaves” to it is still very easy.