Often, the kind of beauty that scientists, and especially physicists, look for in the equations that describe nature is taken to have simplicity as a primary component.
While simplicity is important, I wonder if we shouldn’t be careful not to overestimate its role. Consider two theories about some fundamental force F between particles with parameters α1 and α2 and distance r between them:
F = 0.8846583561447518148493143571151840833168115852975428057361124296α1α2/r2
F = 0.88465835614475181484931435711518α1α2/r2 + 2−64.
In both theories, the constants up front are meant to be exact and (I suppose) have no significantly more economical expression. By standard measures of simplicity where simplicity is understood in terms of the brevity of expression, (2) is a much simpler theory. But my intuition is that unless there is some special story about the significance of the 2 + 2−64 exponent, (1) is the preferable theory.
Why? I think it’s because of the beauty in the exponent 2 in (1) as opposed to the nasty 2 + 2−64 exponent in (2). And while the constant in (2) is simpler by about 106 bits, that additional simplicity does not make for significantly greater beauty.
4 comments:
The constant is not dimensionless (unless a1 and a2 have contrived dimensions). So unless you are using some sort of 'natural' units, its numerical value is meaningless.
Yes, and #1 has more "meaning" (it can be the solution to Maxwell's equation in a spherically symmetric geometry, for example).
I was assuming it's dimensionless, I guess.
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