Sunday, November 2, 2014

The importance of parsimony

A fractal pattern is present in this and every picture in this entry. 

As I have previously discussed here, when comparing potential explanations, all else being equal, the simpler one is better because it's more likely to be true.

You may already be familiar with the principle of parsimony, or Occam's Razor. The eponymous idea is that explanatory "entities must not be multiplied beyond necessity" though surprisingly, those words aren't found in any of Occam's writing. The earliest attribution may actually be to Aristotle (384-322 BC), who wrote that "we may assume the superiority, all things being equal, of the demonstration which derives from fewer postulates or hypotheses".

Probability theory (Bayes' theorem, in particular)  mathematically proves that the more simple explanation is the most likely one (see chapter 28), but intuition is helpful here, too. If an explanation requires 6 steps, then for that explanation to be true, each step must be true. All else equal, another explanation requiring only 1 step is therefore, more likely to be true.

But what exactly are explanatory entities or steps; what precisely is meant by simplicity? If just the fewest items is what counts, then isn't solipsism a better explanation for my experiences than realism because the former only requires that I exist while the latter requires that everything in the real world exists? And isn't one God a better explanation for existence than many brute natural laws and particles interacting?

The problem with both of these examples is the failure to think of how much information is required by each potential explanation to do the job. Consider the picture on the left (or above if you're reading this on your phone). Imagine that you wanted to perfectly reproduce it and sell copies. One way to do that would be to store the 24-bit color of each pixel in the image, which would require 1.62 million bits of information (as the Wikipedia page on Kolgomorov complexity explains). On the other hand, a small computer program can perfectly reproduce these 1.62 million bits using just the definition of the Mandelbrot set (a repeating equation) and the coordinates of the corners of the image. When you consider these 2 ways of reproducing the picture, it's clear that the latter requires much less information to describe and do the same job and is therefore, much simpler. 

Returning to the question of solipsism versus realism, if the former is true, then my mind is the only thing that really exists and it's fabricating my experiences for me. But notice that my mind must then contain all of the information required to plan and create the coherent world that I experience down to every last detail. In solipsism, the information must be in my mind, while in realism, it's all out there, in the world. But the amount of information required by both explanations is the same, so neither explanation is simpler. In fact, one could argue that realism is simpler, because solipsism requires the existence of an extra mysterious component of my mind that's doing all the work of fabricating my experiences for me, including the information required for that component of my mind to function in this way.

Similarly, if the particles and natural laws (quantum field theory, actually, or something like it) can explain the universe without needing to postulate God, then one God as an explanation isn't actually any simpler because that God still would have to contain the information required to have made the universe precisely that way with just those particles and laws. God as an explanation is actually less simple because in addition to the particles and laws, God is required, and there is nothing simple about God. From an information perspective, his omniscient mind doesn't just contain all of the information required to plan and make the universe just so, it contains an infinite amount of information.

When no better explanation for the apparent complexity of creation existed, God was all that our primitive ancestors could come up with. Postulating God was really just an argument from ignorance, and let's face it, saying that the folks who wrote the Bible were as ignorant as modern day Afghan warlords is an insult to the latter. But today, there are ways of explaining that apparent complexity without having to resort to God. Natural selection working on random genetic variations does just fine over geological time scales. Fractals like the Mandelbrot set that produced the image above, explain many stunning features of nature from the horns of mountain goats, to airways and blood vessels, and even your own heartbeat and the structure of your brain.

Of course, many questions remain, including ones about the origins of our universe, but the search for answers over the centuries has lead us away from supernatural answers. It didn't have to be that way. If supernatural forces were at work (like ESP, mind reading, water dousing, fortune telling, etc) science could have discovered them just as easily as it has falsified so many. This randomized clinical trial of intercessory prayer could have shown improved outcomes among patients prayed for, but it didn't. Had it been a positive trial, further research could have shown, for example, that only prayer by Christians improves outcomes in patients previously baptized as Christians, or some similar denominational finding. Consider all of the supernatural explanations that have been superseded by natural ones. Can you think of a single, natural, scientific explanation that has been supplanted by a better supernatural one?

Still, one can always add an extra metaphysical explanatory layer onto whatever natural understanding we arrive at. As long as God can be conveniently hidden and mysterious when needed, He's always going to be compatible with our most honest findings, so there's nothing earth-shattering in Pope Francis recently emphasizing that modern cosmology and biology are compatible with Catholicism, especially if you're prepared to jettison a straightforward reading of the Bible. But if that's ok, why remain keen to keep carrying such extra explanatory baggage when it's merely possible? Given what we know now, it seems that a universe without God is just more probable than one with Him, and one of the important reasons that that is so is that a universe without God is a simpler one.

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