Thursday, July 5, 2012

Simplicity as Evidence of Truth: Justifying Ockham’s Razor

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This part 1 of a series on simplicity being evidence of truth.
  1. Simplicity as Evidence of Truth: Justifying Ockham’s Razor
  2. Simplicity as Evidence of Truth: Theories Tying Into Background Knowledge
  3. Simplicity as Evidence of Truth: How Do We Know It?
In one blog entry where I argued that objective morality could be used as evidence for theism, I used what I called the argument from ontological simplicity. In that blog entry I mentioned the principle of rationality that, all else held constant, the simplest explanation is the best one—which incidentally is one of the formulations of Ockham’s razor (also known as Occam’s razor) named after 14th century philosopher William of Ockham. It’s also been called (perhaps more accurately) as the law of parsimony. A formulation of Ockham’s razor that is closer to what the philosopher originally stated—and one facet of simplicity—is to not multiply explanatory entities beyond necessity. Though both versions of Ockham’s razor are often used in science, philosophy, and everyday life, is it really true that ceteris paribus the simplest explanation is the one most likely to be true?

A Quick Justification for Simplicity

Much of this entry (and its sequels) will be taken from Richard Swinburne’s excellent little book Simplicity as Evidence of Truth, including the following illustration. Suppose we investigate a new area of scientific research for which we have no background knowledge to tell us which theory is more probable and we study two variables: x and y. We have the following data:

Data Set 1
 
x0123456
y024681012


Let’s call the above situation the “Data Set 1 Scenario.” An equation presents itself for predicting the other values of x and y: y = 2x (we’ll call this equation 1), but that isn’t the only formula that fits the data. As Swinburne points out, all formulas of the following form (which we’ll call equation 2) yield the same data as well.[1]

Equation 2
 
y = 2x + (x − 1)(x − 2)(x − 3)(x − 4)(x − 5)(x − 6)z


Where z can be any constant or function of x. Although they agree with the data, the two equations may make very different predictions of unobserved data. For example, we can let z be x720 to get the following equation that predicts unobserved data differently from equation 1:

Equation 3
 
y = 2x + (x − 1)(x − 2)(x − 3)(x − 4)(x − 5)(x − 6)(x720)


Equation 1 predicts that when x = 9, y will be 18. Equation 3 predicts that when x = 9, y will be 270. If we were forced to go with either equation 1 or equation 3 to predict further data, which one would we choose? Obviously equation 1, and the reason seems clear: simplicity. There are literally infinitely many equations fitting Data Set 1 yielding infinitely many different y values for x = 9 (if nothing else, there are infinitely many numbers one could use for z), yet if one were forced to correctly predict what y would be for x = 9 and the consequences were sufficiently dire for predicting incorrectly (say, upon pain of ignominious death that one strongly wants to avoid), we would think it quite irrational to give any answer other than y = 18 for x = 9.

Examples like this strongly suggest that simplicity is among the tools of rationality. For equations in science (e.g. the multitude of equations in physics) there are literally infinitely many possible equations perfectly fitting the observed empirical data that give different predictions of unobserved data, including wild ones like that of equation 2 (where (x − 1)(x − 2)... are added so that when x is 1, 2, etc. the data will come out right) but out of the multitude of equations that fit the observed data, scientists rationally prefer the simpler ones when it comes to making new predictions, even if they don’t do so consciously. If simplicity isn’t a guide for truth, why is it that if the stakes were sufficiently high it would be irrational to go with anything other than y = 18 for x = 9?

Objections and Rebuttals

Objection: We don’t need simplicity; we just assume that the future is like the past to make the next prediction. That is how we can favor equation 1 over equation 3.

Rebuttal: Both equations assume the future resembles the past, e.g. both equations say that for any time in the future when x = 4, we will get y = 8. Swinburne notes that the criterion, “Choose the theory which postulates that the future resembles the past” is empty (infinitely many theories that are inconsistent with each other do that), and that to have real content we should change it to “Choose the theory which postulates that the future resembles the past in the simplest respect.”[2] But in that case we have the simplicity criterion in action.

Objection: We can test and eliminate alternate theories with further testing rather than relying on simplicity. For example, equation 3 predicts that when x = 7, y would be 21 whereas equation 1 predicts y would be 14. So if we observe that y = 14 when x = 7, then we’ve confirmed equation 1 over equation 3.

Rebuttal: Even with this approach infinitely many theories will always remain. In philosophy of science, that there are countless theories that fit the empirical data is sometimes described as empirical data underdetermining theories. For example, suppose it is true we observe that y = 14 when x = 7 such that we get Data Set 2 below:

Data Set 2
 
x01234567
y02468101214


We can see that there will be infinitely many equations fitting Data Set 2 by noting equation 4 below, where z stands for any constant or function of x.

Equation 4
 
y = 2x + (x − 1)(x − 2)(x − 3)(x − 4)(x − 5)(x − 6)(x − 7)z


Thus the “just keep on testing” approach just isn’t enough when trying to eliminate the alternatives in the Data Set 1 scenario. Of course, it is possible that the proposed simplest theory might later shown to be false with later observations, but even when that occurs, if we are to choose between empirically identical theories, all else held constant we are rational to prefer the simplest theory as the most likely one.

Objection: There are reasons why we choose simpler theories that don’t have to do with simpler theories being more likely to be true; e.g. it is more convenient to work with simpler theories.

Rebuttal: Convenience is nice to have, but it’s still the case that we in practice rely on simpler theories as being true ceteris paribus. If we want to know whether e.g. bridge will be able to withstand trucks driving over it, we want a theory that gives us the truth, not one that is merely more convenient to work with. In practice, when seeking the truth we go with the simplest theories ceteris paribus. To illustrate further, consider the following case. A scientific experiment has gone horribly wrong and part of the building will be destroyed. You are trapped inside the building and have the option of being either in region #1 of the building or region #2 by the time the explosion occurs, but one of those regions will annihilated and these are your only two options. Which region will be destroyed will depend on which theory about the experiment is true: theory S (the simpler theory) or theory C (the more complicated theory). Both theories are equal in explanatory power, explanatory scope, how well they tie in with background knowledge etc. Theory S says that region #1 will be destroyed, and theory C says region #2 will be destroyed. The rational thing to do would be to go with the simpler theory and move to region #2—not because the simpler theory is easier and more convenient to use, but because the simpler theory is more likely to be true.

To make things more concrete, suppose it came down to the Data Set 1 scenario, where the data set was this:

Data Set 1
 
x0123456
y024681012


Suppose guessing the right non-destroyed region depended on predicting the right answer for what y would be when x = 9. Again, it would be irrational to guess anything other than y = 18, even if it meant going to region #2 and even if the travel would be somewhat burdensome (e.g. you would have to rush up some flights of stairs as opposed to sitting on a comfortable couch).





[1] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 22.

[2] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 23

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