What is Simplicity?

It is commonly accepted that ceteris paribus the simplest explanation is the most likely one. But what is simplicity? In this blog entry I’ll borrow heavily from Richard Swinburne’s excellent Simplicity as Evidence of Truth. I’ve talked a bit about this before but in this post I’ll go into more detail. Because I’m talking about the sort of simplicity that Richard Swinburne has in mind, I’ll refer to this sort of simplicity as “Swinburnian simplicity.”

Other Explanatory Virtues

In explaining what simplicity is it is helpful to contrast it with what simplicity isn’t. Below are some explanatory virtues (things that help make an explanation good) apart from simplicity. Richard Swinburne notes two a posteriori factors (1 and 2) and one a priori one (3):

1. Yielding the data. This category covers both explanatory scope (how much data the theory explains) and explanatory power (the probability of expecting the data if the explanation were true).
2. Fitting in with background knowledge. For example, “The hypothesis that John stole the money is rendered more probable if we know [due to our background knowledge] that John has stolen on other occasions and comes from a social group among whom stealing is widespread.”[1] The likelihood of such background beliefs being true plays a role in our judgments. As I explained and wrote about earlier, part of how well a theory fits background knowledge depends on simplicity.
3. Content. The greater the content of the hypothesis, the less likely it is to be true. In this context, “content” refers to how much a theory “claims.” For example, the claim “at least one swan is white” has less content then “most swans are white.” The claim “at least swan is white” makes no claim as to whether or not most swans are white, whereas “most swans are white” contains the claim that at least one swan is white and that this whiteness holds for the majority of swans. As Swinburne explains, “The more claims you make, the greater the probability that your claims will contain some falsity, and so be as a whole false.”[2]

The next a priori factor is simplicity.

Factors of Simplicity

Swinburne also says that “One theory is simpler than another if and only if the simplest formulation of the former is simpler than the simplest formulation of the latter.” He also delineates several facets of simplicity. Here are some of them (note that since the topic of his book is largely philosophy of science, some of these factors have to do with physical laws):

1. Number of entities. The theory that postulate fewer entities is simple than if it postulated more entities. As Swinburne notes, “The application of this facet in choosing theories is simply the use of Ockham’ razor.”[3]
2. Number of kinds of things. A theory that postulates fewer different kinds of entities is simpler than if it postulated many different kinds of entities, e.g. a theory that postulates fewer different kinds of quark is simpler than a theory that postulates more of them.
3. Fewer separate laws is simpler than many separate laws. All else held constant, a theory is simpler than another if it contains fewer laws.
4. A theory in which individual laws have fewer variables is simpler than a theory where the laws have more variables ceteris paribus. For example, suppose we have two theories that have physical laws yielding the data equally well; one set of laws uses three variables to yield the data and the other uses seven. All else held constant, the theory which uses only three variables is simpler.

The above three are more quantitatively ontological in nature, i.e. having more to do with quantity with respect the nature of reality: fewer laws, fewer variables in how laws work, fewer entities, and fewer kinds of entities. The next ones are more conceptual.

5. A theory that uses a term T1 that can be grasped only by people who grasp some other term T2 (whereas T2 can be understood without grasping T1) is less simple than if the theory otherwise just used T1. For example, if someone defined “grue” in terms of green, blue, and time (say, something is grue if and only if it was green before 2000 CE but blue after 2000 CE) is less simple than the predicate “green.” Swinburne notes that the “general force of this requirement is of course, other things being equal, to lead us to prefer predicate designating the more readily observable properties rather than ones a long way distant from observations.[emphasis mine]” Swinburne also says this “facet of simplicity says: do not postulate underlying theoretical properties, unless you cannot get a theory which yields the data equally well without them.”[4]
6. A theory with a mathematically simpler formulation is simpler than it would otherwise be. Two facets of mathematical simplicity:
   1. Fewer terms. For example, y = x is simpler than y = x + 2x2.
   2. Simpler mathematical entities or relations to one another. Let S (for simple) and C (for complex) be placeholders for mathematical entities/relations. A mathematical entity/relation S is simpler than another one C if S can be understood by someone who does not understand C but C cannot be understood who does not understand S. For example, 0 and 1 (S) are simpler than 2 (C), 2 is simpler than 3, and so forth; you cannot understand the notion of 2 rocks (C) without first understanding the notion of 1 rock (S).So 1 of something is simpler than 2 of something. For this reason, y=z+x is simpler than y=z+2x.
      1. Consequently, multiplication is simpler than addition (you need to understand addition to understand multiplication); power is less simple than multiplication (you need to understand multiplication to understand power) e.g. y=x is simpler than y=5x2, vector addition is less simple than scalar addition.
      2. Swinburne says that an infinitely large quantity is graspable by someone who hasn’t grasped a very large number. “One does not need to know what a trillion is in order to understand what is the infinitely long or lasting or fast. It is because infinity is simple in this way that scientists postulate infinite degrees of quantities rather than very large degrees of quantities, when both are equally consistent with the data. The medieval postulated an infinite velocity of light, and Newton postulated an infinite velocity for the gravitational force, when in each case large finite velocities would have been equally compatible with the data then available measured to the degree of accuracy then obtainable.”

And there you have it, six factors illustrating what simplicity is—or at least Swinburnian simplicity (the sort of simplicity that Swinburne has in mind). Not all philosophers have the same precise conception of simplicity when they use the term. For example, one could have an idea of simplicity that unlike Swinburnian simplicity includes content, where a theory that contains fewer assumptions is considered simpler.

Applications

To give an example of math simplicity, suppose we had these two equations (physics nerds will recognize these as equations for gravity) that are empirically identical to each other as far as our precision was able to determine:

F = G

m1m2 r2

F = G

m1m2 r2.000...(100 zeroes)...0001

A person needs to understand whole numbers (1, 2, 3...) before he understands decimals, and so by criterion 6 if nothing else we should prefer the first equation all else held constant. In my Simplicity and Theism article, I explain how the principle of simplicity can be used in the service of theism.

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[1] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 18.

[2] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 18.

[3] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 29.

[4] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 31.