In a Flight of Starlings by Giorgio Parisi

Author:Giorgio Parisi [Parisi, Giorgio]
Language: eng
Format: epub
Publisher: Penguin Publishing Group
Published: 2023-07-11T00:00:00+00:00

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Before getting to the physics, let’s try to understand the modification that was needed from a mathematical point of view.

To make the replica method work, I had to “extend” it. The possibility of extending a mathematical method is based on a very old idea. The French bishop, mathematician, physicist, and economist Nicola d’Oresme was probably the first to use it, in the mid-fourteenth century. D’Oresme was an incredible figure, and provides conclusive proof, if proof were needed, that the medieval period was not the dark age for science that our school textbooks might have us believe. To choose just one of many examples, he was the author of a book on how the positions of stars are distorted by atmospheric refraction, written around 1360. Since it is in Latin, I cannot claim to have read it through. It is nevertheless obvious that his reasoning was remarkably sound from a conceptual point of view. He probably got the idea while observing the sun “crushing” the horizon at sunset, suggesting that there must be a distortion. Calculating the degree of distortion is vital if we are to make precise astronomical observations, as the apparent measurement of the stars needs to be corrected by as much as two or three degrees.

Returning to our problem, d’Oresme was the first to realize that raising a number by ½ was equivalent to finding its square root. This seems now like a banal enough truth, of the kind that we all learn in high school—but this fails to recognize the logical leap that d’Oresme made by extending the properties of powers to fractional numbers. These were properties that had been reserved exclusively for integers.

The idea of raising a number to a power is very simple. Squaring a number, or raising it to the power of two, means multiplying the number by itself. Cubing a number, or raising it to the power of three, means multiplying the number by itself, then by itself again. And so on. To raise to the power of ½ seems absurd: What does “multiplying a number half times” mean? D’Oresme’s idea was to extend a property of exponentiation: that exponents must be multiplied when raising to a power a number that has been raised to a power. For example, 22 raised to the power of 3 is equivalent to 26 (i.e., 64, or 43).

If squaring a number raised to the power of ½ results in the number with which we began (since 2 times ½ is 1), then raising to the power of ½ is equivalent to extracting the square root: the square root of a number squared is that number itself.

These properties are formally derived, since multiplying a number half times makes no sense; the formal properties, however, guarantee a coherent result. Nicola d’Oresme went beyond the original point of view, beyond immediate understanding, but by maintaining the formal properties, he obtained a very simple method for solving even complex operations. After d’Oresme, mathematics often advanced by extending properties in a formally correct way in new conditions, broadening their scope.

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