The Principle of Indifference

Definition

If two observers describe the same phenomenon using different vocabularies or symbols, a genuine law relating their observations must give the same answer regardless of which observer's notation is used to compute it. Weinberg states it plainly: a valid relationship "should not depend on a particular choice of notation." It functions as a fast filter for spotting formulas and models that look rigorous but have secretly encoded an arbitrary choice as if it were a fact about the world.

In the Book

Weinberg demonstrates the principle by debunking a real formula a systems researcher had proposed to measure the difficulty of a selection task, defined as D = R² where R is the percentage of items rejected. Applied to separating 10 sheep from 90 goats, the formula gives D = 0.81; simply reframing the identical task as separating 90 goats from 10 sheep gives D = 0.01 — an absurdity, since renaming which group is "selected" cannot change how hard the physical task actually is. A formula that obeys the principle, D = R² + S², gives the same value (0.82) either way. He generalizes the point with the "If you call a tail a leg, how many legs has a dog?" riddle attributed to Lincoln — still four, because calling a tail a leg doesn't make it one — and extends it to boundary-drawing: by the Principle of Indifference, either side of a line could be labeled "the system" and the other "the environment," since that choice is a matter of notation, not of the world.

Why It Matters

The principle gives a fast, near-instant sanity check on any proposed metric, model, or law: does its value change under a relabeling that changes nothing real? If yes, the model has smuggled in an arbitrary choice disguised as a fact. This generalizes well past Weinberg's sheep-and-goats example — it is the same failure mode as a metric that flips sign depending on which category is coded "positive," or a comparative analysis whose conclusion depends on which group was arbitrarily chosen as the baseline.