Occam's razor

An assumption that does no work

Cookies keep disappearing from the jar in a shared apartment. One roommate admits he takes one most nights, and the number missing matches. Another roommate suggests there might also be a mouse.

Both explanations account for the missing cookies, but the second adds a mouse that explains nothing the first doesn’t: no droppings, no crumbs, no extra cookies gone. The razor says not to believe in the mouse until something needs it. That isn’t a proof there’s no mouse. If droppings turn up, the mouse starts doing explanatory work, and it earns its place.

Simple, but it doesn’t fit

A small shop’s website loses about 40% of its visitors starting one Tuesday. The owner says, “People are just busy this week. That’s the simplest explanation.” The logs show the drop is entirely in visitors arriving from search engines, and it began within an hour of a site update that changed the settings telling search engines which pages they may index.

“People are busy” has fewer moving parts, but it doesn’t explain why only search visitors disappeared, or why the drop began at that hour. The explanation involving the update, the settings file and the search engines has more parts, and it fits everything. The razor only chooses between explanations that fit the evidence equally well, so it gives no support to the “simpler” one here. Treating “simplest” as the tiebreaker before checking the fit is the most common misuse.

The philosopher Alan Baker distinguishes the slogan form, “entities are not to be multiplied beyond necessity”, from the version most philosophers defend today, which adds a condition: other things being equal, it’s rational to prefer the theory that commits you to less. He notes that this version is “considerably weaker” than the slogan, since it’s compatible with simplicity being a fairly minor consideration. In this form the razor is a tiebreaker: once one explanation fits the evidence clearly better, other things aren’t equal.

Statistics adds a wrinkle. When models are compared by trading simplicity off against fit, a model that fits the data slightly better only because it has many more adjustable parts can lose to a simpler one (see overfitting, below). But the trade runs both ways: Baker notes that always picking the simplest model, “regardless of its fit to the data, cuts the model free from any link to observation or experiment.”

That condition is also why the razor’s reach is narrower than it seems. Baker quotes the biologist Kent Holsinger: because it should be invoked only when hypotheses “explain the same set of facts equally well, in practice its domain will be very limited”.

So the razor doesn’t say:

• That the simplest explanation is true. It tells you which explanation to prefer or to work with for now. The preferred one can still be wrong, and the more complicated rival can still be right. “Simpler, therefore true” isn’t a valid inference: the premise can be true while the conclusion is false.
• That simple beats better-fitting. An explanation that leaves some of the evidence unexplained isn’t a simpler account of the same facts. It’s an account of fewer facts.
• That the more complex explanation is refuted. Baker observes that when philosophers invoke the razor against a rival view, the aim often seems to be “shifting the burden of proof”, not refuting the rival outright.

The razor is named after William of Ockham (c. 1287–1347), an English philosopher and theologian, but the famous wording isn’t his. The Stanford Encyclopedia of Philosophy’s entry on Ockham says that although the sentiment “is certainly Ockham’s, this particular formulation is nowhere to be found in his texts.” It describes his version as “merely a cautionary methodological principle”: don’t accept a claim that requires more entities than a competing claim, unless you have grounds for the extra entities. In his fullest formulation, the grounds he accepted were reason, experience, and the authority of Scripture.

The idea is older and broader than Ockham. Baker traces versions of it to Aristotle, who preferred demonstrations that derive “from fewer postulates or hypotheses”, and to Thomas Aquinas. Later, Isaac Newton made a version of it the first of his rules of reasoning in the Principia (1687): “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.” Baker also notes that modern formulations are connected “only very tenuously” to Ockham himself.

• Simpler hypotheses are easier to test. Karl Popper, as Baker reports, argued that a hypothesis with fewer adjustable parts is more falsifiable, since it rules out more possible results, and so should be the default. A theory with many adjustable parts can be tuned to fit almost anything, which is the problem described on the Unfalsifiability page.
• Simpler models are less prone to overfitting. When you fit a curve to data, a curve with enough adjustable parameters can pass through every point, noise included. Baker describes the danger of “overfitting” the model “to accidental discrepancies unrepresentative of the broader regularity”. Parsimony acts as a counterweight: the extra wiggles that match this data set’s noise tend to predict the next data set badly.
• Each extra assumption is another chance to be wrong. When one explanation is just another one plus an additional assumption, the longer one can’t be more probable, because both parts have to be true (the rule broken in the Conjunction fallacy). Real rivals are rarely nested this neatly, but it’s the cleanest case for the razor.

An ad hoc assumption is one added only to rescue a theory from a result that would otherwise count against it (see Unfalsifiability). Every such patch makes the theory less simple, so the razor and the worry about ad hoc rescue often point the same way. Baker’s example is the physics that Einstein’s special relativity replaced: to “save the phenomena”, the older Lorentz–Poincaré theory had accumulated “a number of ad hoc and physically unmotivated patches”, and relativity did without them.

The razor doesn’t do all the work, though. An ad hoc rescue is objectionable mainly because its patch can’t be checked independently of the failure it explains, not just because it’s one more assumption. Baker also points out that relativity had several advantages besides parsimony, which is why the episode doesn’t show that parsimony alone decided the matter.

• It’s a heuristic, not a law. Philosophers disagree about why it works at all. One question is epistemic: are simpler theories more likely to be true? Another is practical: is it just more useful to work with simpler ones? Some, like Richard Swinburne, hold that simplicity is evidence for truth; others, like Elliott Sober, argue that parsimony has no single justification and that what makes it reasonable depends on the subject matter in each case.
• “Simple” is hard to define. Baker separates elegance (fewer and more concise principles) from parsimony (fewer kinds of things), and notes that the two “typically pull in different directions.” Postulating Neptune before it could be observed added an entity, but it explained irregularities in the orbits of the known planets without complicating the laws of celestial mechanics. Which theory was simpler depends on which kind of simplicity you count.
• Simplicity can depend on how you describe things. A hypothesis that is short to state in one vocabulary can be long in another, so measures based on how a theory is written raise the question of which vocabulary is the right one. Baker reports that statistical approaches can at least partly answer this worry.
• No agreed exchange rate. Statistical methods for balancing simplicity against fit disagree about how much fit a gain in simplicity is worth, and Baker notes that Thomas Kuhn thought the weight scientists give simplicity was ultimately a matter of taste.
• There are counter-principles. Ockham’s contemporary Walter of Chatton proposed a counter-principle: “if three things are not enough to verify an affirmative proposition about things, a fourth must be added”. Kant held that “the variety of entities should not be rashly diminished.” The razor warns against assuming too much; these warn against explaining too little.

So “that’s not the simplest explanation” is a reason to ask what the extra assumptions buy, not a verdict that the explanation is false.