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The fact that the laws of physics are not unchanged under a change of scale was discovered by Galileo. He realized that the strengths of materials were not in exactly the right proportion to their sizes, and he illustrated this property that we were just discussing, about the cathedral of matchsticks, by drawing two bones, the bone of one dog, in the right proportion for holding up his weight, and the imaginary bone of a ‘super dog’ that would be, say, ten or a hundred times bigger – that bone was a big, solid thing with quite different proportions. We do not know whether he ever carried the argument quite to the conclusion that the laws of nature must have a definite scale, but he was so impressed with this discovery that he considered it to be as important as the discovery of the laws of motion, because he published them both in the same volume, called ‘On Two New Sciences’One can easily see that Galileo's reasoning is not quite free of logical fallacies here: the scaling is not applied consistently at all levels of fractal structure of material universe. But let us take a look at Feynman’s own arguments against the principle of relativity of scale:
Suppose that we ask: ‘Are the physical laws symmetrical under a change of scale?’ Suppose we build a certain piece of apparatus, and then build another apparatus five times bigger in every part, will it work exactly the same way? The answer is, in this case, no! The wavelength of light emitted, for example, by the atoms inside one box of sodium atoms and the wavelength of light emitted by a gas of sodium atoms five times in volume is not five times longer, but is in fact exactly the same as the other. So the ratio of the wavelength to the size of the emitter will change... Today, of course, we understand the fact that phenomena depend on the scale on the grounds that matter is atomic in nature, and certainly if we built an apparatus that was so small there were only five atoms in it, it would clearly be something we could not scale up and down arbitrarily. The scale of an individual atom is not at all arbitrary – it is quite definite.From the logical perspective, Feynman’s reasoning is not much better than Galileo’s. Indeed, how does Feynman know that “the wavelength of light emitted, for example, by the atoms inside one box of sodium atoms and the wavelength of light emitted by a gas of sodium atoms five times in volume is not five times longer, but is in fact exactly the same as the other”? Did he, or someone else for that matter, conduct an experiment with enlarged five times atoms? He says: “matter is atomic in nature” and “scale of an individual atom is not at all arbitrary – it is quite definite”. How so? All our experience suggests otherwise. First we have discovered molecules, then atoms, then nucleus, then quarks (not entirely sure about quarks though - this could be a bogus theoretical construct). On what grounds is based Feynman’s belief that there is an end to that process? Does not Feynman know that the hypothesis of ‘elementary’ particles that occupy mere mathematical points in space without extended structures of any kind invariably leads to all kinds of nonsense and irreconcilable contradictions? It seems logically more consistent to conjecture that there is no such thing as ‘elementary’ particle in nature at all, that the distribution of matter is inherently fractal both up and down.
Scaling in turbulence. Taking a path-breaking intellectual step, Richardson 1922 adapted [Jonathan] Swift as follows
Big whorls have little whorls,
Which feed on their velocity
And little whorls have lesser whorls,
And so on to viscosity
(in the molecular sense).
The next step after Richardson was taken in Kolmogorov 1941. In a class only with Lévy, Kolmogorov (1903 – 1987) was the greatest probabilist of this century. I barely knew him personally, but greatly admired his extraordinary range of achievement. At the mathematical end of his range of interest in probability theory, Kolmogorov 1933 seemed to me too close to comfort to the work of the ultimate decorator who rearranges existing material. But Kolmogorov’s papers on turbulence were filled with novelty and daring.