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It's all about the deep questions.

This is the first post in the philosophical project motivated by the “What it is to Know” blog post. (If you haven’t read it, you should probably start with that post.)

Imagine all of the possible metaphysical theories (corresponding to possible worlds) that account for (are consistent with) the observations of a particular phenomenon. Now line them up according to their ontological commitments on a ray. The beginning of the ray will be the theory with the least ontological commitments, with precisely none. In other words, the phenomenon can be explained with recourse to the ontological commitments we already have prior to philosophizing on the phenomenon. Just above the beginning point, the theories with the smallest and simplest ontological commitments will lie. Continuing on, the distance from the beginning point will correspond to the complexity of the theory.

Now, for any specified level of ontological commitment (except for the zero level, in which case there will only be one) there will be multiple metaphysical theories with the same rough level of complexity. To represent this, the ray’s width at a particular distance from the beginning will correspond to the number of metaphysical theories that are observationally adequate at that specific level of complexity. Now, I take a stronger statement of the first crucial step in the argument to be intuitively clear: at more austere levels of simplicity in ontology fewer metaphysical theories are ideally imaginable, and those levels of ontology with increasing complexity have “more to work with” so to say, and therefore have more ideally imaginable metaphysical theories at their disposal. Here, an example might serve to elucidate things. In constructing a theory of the world, it may or may not be true that God sustains every event that happens and he is the “grounding” for causation and physical law and such. However, one can easily increase the complexity of this picture. One can say there are two gods sustaining the world; one is in charge of events of type A and the other of events of type B. And one can do this for as high a number as one wishes. Also, one can give the Gods certain psychological traits, names and so on, so that the number of possible metaphysical theories of the grounding of the universe would increase exponentially as complexity is increased. These theories would all furthermore be consistent with all observations. However, for this argument to work, it is more than sufficient to claim that the higher levels of complexity have at least an equal number of potential theories than the simpler ones. This claim makes up one kernel of the argument.

To come to the conclusion of the argument a mathematical idealization works well. If my previous claim, that the ray’s width will vary directly with the distance from the distance from the ray’s beginning is true, then what this ray is is actually an “infinite triangle” (or for the weaker claim an “infinite rectangle”). Note that there is no limit to the complexity of a theory, for any finite degree of complexity of a theory, an even more complex theory is possible, since the number of ontological posits for a theory, in principle, may be infinite (for example, positing infinitely many gods). Consequently, if one were to draw this ray, along with its thickness, on the coordinate plane, with “zero ontology” at the origin and increasing complexity running along the positive y-axis, then a very rough approximation of this drawing would be something like f(x)= IxI. Now the question becomes: which is the “expected” level of complexity of the actual world? Now, if the ray, instead of being a ray was a line segment, than the problem can be answered by subtracting the integral of the function from -c to c, where the greatest width of the line segment is 2c, from 2ch, where h is the height of the line segment with increasing width (this would get the area of the triangle if the function we were using was f(x)=IxI). Then, divide this value by h. The resulting answer would be the answer for the expected value of the complexity of the world. Applying this same method for the infinite ray, taking the limit as the height apporaches infinity (and, if desired, for the stronger claim as c approaches infinity), one gets an expected value of complexity of infinity. So, in fact the exact opposite of Occam’s razor is true, we should expect the actual world to be infinitely complex.

In summary, the conclusion of the argument depends on 1) there is no limit to how complex a metaphysical theory can be 2) there is a limit on how simple a metaphysical theory can be (precisely, one that has zero ontological commitments) 3)It is not the case that number of ideally imaginable metaphysical theories decrease as complexity increases, at least not so much so that the aforementioned mathematical calculation produces a finite (albeit high) complexity. I take all three claims to be necessary a priori truths.

So what exactly does this argument show? For example, if two theories were on the table, one complex and one simple, this argument doesn’t show that one should pick the complex one; in this case it is dissimilar to an anti-Occam’s Razor. Because any particular complex theory is just as likely as any particular simple theory, it just goes to show that overrall level of complexity that we should expect from the universe is infinite. And that, is a substantial claim.

Now one semi-obvious rebuttal that a proponent of Occam’s Razor can say is that this argument really does not falsify Occam’s Razor at all, and in fact may even be consistent with it by designating weighted values to possible worlds lower down on the ray. My conclusion, however, would probably go through even if one does hold on to a weak form (or maybe even a strong form) of Occam’s Razor since the number of possible metaphysical theories increases exponentially as complexity levels are allowed to increase. However, if one holds to a very strong form of Occam’s Razor which greatly prioritizes simpler theories, my conclusion may not hold. Again, however, this is all a matter of degree, and I have not gone as much into detail about the math of my argument as I would have liked. In the end, however, I really didn’t set out to rebut Occam’s Razor since there isn’t anything to rebut. No attempt at a justification for Occam’s razor has been successful. That’s why I’m pretty much baffled why the criterion of simplicity is used so often in metaphysics if it has no grounding whatsoever. In the absence of any positive justification to give completely arbitrary weights to simpler theories (I could imagine someone giving arbitrary weights to theories that invoke fairies with just as much justification), it is most rational to believe that the ontology underlying the universe is one that is infinitely complex. In this sense, I take my argument to falsify the thought behind Occam’s Razor, that we should expect to be in a world with smaller and simpler ontological commitments.

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Hi there again,

I thought you might want to take a look at the paper below, published in the journal Science and Education, which provides a probabilistic justification for Occam’s razor (in particular, see Section 8).

Does Science Presuppose Naturalism (Or Anything at All)?

http://link.springer.com/article/10.1007/s11191-012-9574-1

The paper can be downloaded here:

https://sites.google.com/site/maartenboudry/teksten-1

Thanks again for the papers! I’ve always wanted to get around to reading formal epistemology papers about justifications for Occam’s razor and such (and whether it can be justified in metaphysics as well as empirical science). As a caveat to some of the posts in this blog, they’re ruminations of my high-school self a few years ago, so several of my ideas have changed. (Still need to get back to you on your other comment – busy with upcoming finals in college and stuff but I believe I have a good response!)

No problem- take your time and good luck with the finals! In fact, I have a response to my own comment but I’m curious to see what you have to say first…