I started thinking about correlation again, and it dawned on me, that when we do math and science, we’re looking for computable relationships between sets of numbers. The most basic example is a function that describes the behavior of a system. Ideally, we have exact descriptions, but we often settle for probabilistic descriptions, or descriptions that are exact within some tolerance. I’d say that the search for computable relationships between sets of numbers is in fact science. But when you consider the full set of relationships between two sets of numbers, the set we’re interested in is therefore minuscule in terms of its density. In the extreme case, the set of computable functions has a density of zero compared to the set of non-computable functions. Yet, science and mathematics have come to the point where we can describe the behaviors of far away worlds, and nanoscopic terrestrial systems.

Therefore, it’s tempting to think that the computable is everything, and the balance of relationships are nonsense. For physical intuition, consider a Picasso painting represented as an RGB image, and compare that to the full set of random RGB images of the same size. What’s strange, is that the set in question will contain other known masterpieces, and new unknown masterpieces, if the image is large enough and the pixel size is small enough. And while I haven’t done the math, I’d wager the density of masterpieces is quite small, otherwise we’d all be famous painters, and since we’re not, I don’t think you can gamble your way into a masterpiece.

Similarly, if I have two sets of random numbers, and I simply connect them with strings, you’d probably think I’m a lunatic, though I’ve just defined a function. Whereas if I point out that I can inject the set of integers into the set of even integers, you’d swear I’m a genius. This might seem like unscientific thinking, but it isn’t. It’s arguably all the same, in that humans have a clear preference for computability, and that translates into a preference for symmetry over asymmetry. Personally, I listen to a lot of strange, highly random music, and enjoy Gregory Chaitin’s work as well, but in all seriousness, are we missing valuable information in the form of more complex functions, and perhaps tragically in the form of non-computable functions, assuming no non-computable machine exists?