Friday, October 28, 2022

Simplicity and gravity

I like to illustrate the evidential force of simplicity by noting that for about two hundred years people justifiably believed that the force of gravity was Gm1m2/r2 even though Gm1m2/r2 + ϵ fit the observational data better if a small enough but non-zero ϵ. A minor point about this struck me yesterday. There is doubtless some p ≠ 2 such that Gm1m2/rp would have fit the observational data better. For in general when you make sufficiently high precision measurements, you never find exactly the correct value. So if someone bothered to collate all the observational data and figure out exactly which p is the best fit (e.g., which one is exactly in the middle of the normal distribution that best fits all the observations), the chance that that number would be 2 up to the requisite number of significant figures would be vanishingly small, even if in fact the true value is p = 2. So simplicity is not merely a tie-breaker.

Note that our preference for simplicity here is actually infinite. For if we were to collate the data, there would not just be one real number that fits the data better than 2 does, but a range J of real numbers that fits the data better than 2. And J contains uncountably many real numbers. Yet we rightly think that 2 is more likely than the claim that the true exponent is in J, so 2 must be infinitely more likely than most of the numbers in J.

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