Analogy in Science

The discovery (in a realistic sense; the subjectivist would say “formulation”) of laws and their manifestation are the essential tasks of the physical sciences: physics, chemistry, astronomy, geology, biology… Those laws which are continuously affirmed and reaffirmed by experiment and/or observation comprise the standard theories of their science. Much of normal science (in Thomas Kuhn’s sense) is the elaboration of theory, especially definition of the domain of its application. Thus, Newton’s law of gravitation finds application within the range of observation capabilities of the late Seventeenth Century; in modern terms this means objects moving at relative velocities significantly less than the speed of light and minimal effects due to other forces (electromagnetic, Strong, and Weak). At high relative velocities, Einstein’s theory of relativity comes into play, while the non-gravitational forces are significant at molecular and smaller scales.


Whether expressed in words (the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance separating them) or algebraically (G =  m1 m2 / r2), the words and alphanumerical characters are analogs for the inferred law, and, are literally symbolic. Yet, while symbols in mathematical context, do not have the same symbolic depths or meaning as religious and mythic symbols (such as the Cross or the grail), in non-scientific contexts, “Newtonian gravitation” does evoke symbolic depths, especially in a Calvinistic mechanical worldview. Hierarchies of causation are implied by the equality character (=).


Even before the onset of relativity and quantum mechanics, mathematicians and physicists began to recognize that pure equality is not observed in nature. So, for example, measured gravity is only approximated by Newton’s law:

where +/- e means plus or minus “error”. The so-called error term in any physical equation can incorporate several effects. If a measurement differs significantly from that predicted by theory (e is large), (1) other phenomena might be contributing to the observed effect, (2) the measurement device is poorly designed, or (3) the theory might be wrong or at least inappropriately applied. For example, (1) measured gravity on the surface of the earth is affected by the distribution of mass within the earth, rotation of the earth, lunar and solar effects (“earth tides”, due to the masses of the moon and sun), and planetary gravitational effects (to a much lesser extent than lunar and solar). For (2) early pendulum “gravity meters” were big, awkward, and imprecise; more precise and accurate meters have been developed in more recent years. And for (3), at high velocities, relativity must be taken into account; Newton’s law is inadequate.


Another expression of inexactitude, instead of the error term, could involve the “approximately equal” character (≈). A physicist would be reluctant to interpret the character as meaning “similar”, but it is close. The gravitational constant term, , in Newton’s law, is the proportionality factor that could also be thought of as a similarity coefficient. The reluctance of a scientist to use “similarity” in either case is largely due to the fact that Newton’s law produces a scalar (that is, single-valued). Where similarity more comfortably comes into play is in the comparison of multi-valued objects or data sets. For example, there are measures for comparing two digital electronic signals (ordered sets of numbers) – cross correlation, semblance, and coherency. Sometimes, a signal (or ordered set) possesses some sort of self-similarity, either in a repetitive, constant frequency pattern (for example, a musical note) or a structure that replicates itself at a range of different scales.


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