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## Main Question or Discussion Point

The concept of naturalness as dimensionless ratios of parameters of order unity has recently come under criticism, most obviously because Sabine Hossenfelder wrote a book

Very recently however, Peter Shor and Lee Smolin had a discussion about this over at Peter Woit's blog, in which Smolin gave an explanation on 'why the order of unity' is important for the practice of physics, going back to Fermi and Feynman:

*(Lost In Math)*criticizing it.Very recently however, Peter Shor and Lee Smolin had a discussion about this over at Peter Woit's blog, in which Smolin gave an explanation on 'why the order of unity' is important for the practice of physics, going back to Fermi and Feynman:

Peter Shor said:@Lee:

When you say that “any pure dimensionless constants in the parameters of a physical theory that are not order unity require explanation,” you are implicitly putting a probability distribution on the positive reals which is sharply peaked at unity.

Doesn’t this assumption also require explanation? Why should the range of numbers between 1 and 2 be any more probable than the range between 10^10 and 10^20? Aren’t there just as many numbers in the range between 10^10 and 10^20? as there are between 1 and 2? (Uncountably many in each.)

From Smolin's historical explanation, i.e. that naturalness is essentially a heuristic tool based on dimensional analysis making it a good strategy for quickly solving Fermi problems, I would say that adherence to naturalness is a pretty strong criteria for 'doing good physics'.Lee Smolin said:Dear Peter Shor,

Yes, exactly, and let me explain where that expectation for dimensionless ratios to be order unity comes from.

Part of the craft of a physicist is that a good test of whether you understand a physical phenomena-say a scattering experiment-is whether you can devise a rough model that, with a combination of dimensional analysis and order of magnitude reasoning, gets you an estimate to within a few orders of magnitude of the measured experimental value. People like Fermi and Feynman were masters at this, a skill that was widely praised and admired.

The presumption (rewarded in many, many cases) was that the difference between such rough estimates and the exact values (which were by definition dimensionless ratios) were expressed as integrals over angles and solid angles, coming from the geometry of the experiment, and these always gave you factors like 1/2pi or 4pi^2, which were order unity.

Conversely, if your best rough estimate does not get you within a few orders of magnitude of the measured value, then you don’t understand something basic about your experiment.

Seen from the viewpoint of this craft, if your best estimate for a quantity like the energy density of the vacuum is 120 orders of magnitude larger than the measured value, the lesson is that we don’t understand something very basic about physics.

Thanks,

Lee