Earlier this year I asked readers to send me their shortlists of great equations. I also asked them to explain why their nominations belonged on the list and why, if at all, the topic matters.

I received about 120 responses — including single candidates as well as lists — proposing about 50 different equations. They ranged from obvious classics to “overlooked” candidates, personal favourites and equations invented by the respondents themselves.

Several people inquired about the difference between formulae, theorems and equations — and which I meant. Generally, I think of a formula as something that obeys the rules of a syntax. In this sense, E = mc2 is a formula, but so is E = mc3. A theorem, in contrast, is a conclusion derived from more basic principles — Pythagoras’s theorem being a good example. An equation proper is generally a formula that states observed facts and is thus empirically true. The equation that describes the Balmer series of lines in the visible spectrum is a good example, as are chemical equations that embody observations about reactions seen in a laboratory.

However, these distinctions are not really so neat. Many classic physics equations — including E = mc2 and Schrödinger’s equation — were not conclusions drawn from statements about observations. Rather, they were conclusions based on reasoning from other equations and information; they are therefore more like theorems. And theorems can be equation-like for their strong empirical content and value.

It thus makes sense to classify both kinds as equations, which is exactly what respondent David Walton from the University of Manchester did. He distinguished between equations (such as F = ma) that comprise axiomatic models that “define the interrelationships between various observables for all circumstances” and equations that are approximate models (such as Hooke’s law), which define “the interrelationships between the various observables over a defined range and within a defined accuracy”. I therefore interpreted the term “equation” loosely.

More here.

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