SIR KARL RAIMUND POPPER, SIR HAROLD JEFFREYS, AND HEURISTIC SIMPLICITY IN SCIENCE

Scientists know Karl Popper (1902-1994) for his Logic of Scientific Discovery, published first in German (1935, Vienna) and much later in English (1959, London). Geophysicists know Harold Jeffreys (1891-1989) for his Earth, which went through six editions (1924-1976). Chapter V in Popper (1935) deals with simplicity (Einfachheit) in science, Popper developing ideas of Hermann Weyl (1885-1955). Popper later discovered that Wrinch and Jeffreys (1921) anticipated Weyl (1927). In his (1959) footnotes, Popper gives Jeffreys credit for ideas on simplicity. Popper uses deduction idealistically to falsify scientific theories, and Jeffreys uses "common-sense" induction, which is logically untenable in Popper's view, to construct scientific theories. Popper considers the simpler equation to have lower probability, but Jeffreys observes that scientists choose as better the simpler of two equations describing the same phenomenon, even where the more complex equation fits the data better.

A simple equation has the dependent variable, a physical quantity, as the left hand side of the equation and a function of n independent variables, also physical quantities, as the right hand side, where n is 0, 1, 2, 3, or 4. Each variable has k dimensions, where k is 0, 1, 2, or 3. These are Bridgman's L, T, M dimensions. Where the dependent variable equals a constant, n equals 0, but that constant has the dimensions of the dependent variable. Where an independent variable is dimensionless, k equals 0.

Stokes Law, which describes the gravity-driven movement of a sphere in a fluid, is a type simple equation of mechanics. The dependent variable is velocity (L/T), so each additive term on the RHS must combine physical quantities to produce L/T as its dimension. Possible variables on the RHS are limited by relevant physics to the sphere diameter, the fluid viscosity, and the buoyancy. Stokes Law is valid over a specific range of the variables, as are most simple laws. The Reynolds Number combines both dependent and independent variables into one dimensionless product to provide a range. Given n possible variables, k possible dimensions, and range constraints, the simple equations of mechanics are logically required.