This post will be the first in a short series on Paul Adrien Maurice Dirac and his scientific methodology—what I call in my dissertation his “epistemic strategies.” Dirac was famous for his emphasis on the primary role of mathematics in physical theory. According to Dirac, when crafting a new theory, the mathematics should come first. What sort of mathematics? Pretty, or beautiful mathematics. I’m going to use these posts to try to think through some of the historiographical issues here, and would love feedback with links to good sources on the aesthetics of science.
Dirac’s first public expression of his scientific methodology and advice for fellow researchers came in 1931. It was contained in his famous paper that explored the properties of magnetic monopoles (Dirac 1931), and also, as a side note, predicted the existence of the anti-electron.* Dirac began his methodological thoughts by writing:
The steady progress of physics requires for its theoretical formulation a mathematics that gets continually more advanced. This is only natural and to be expected. What, however, was not expected by the scientific workers of the last century was the particular form that the line of advancement of the mathematics would take, namely, it was expected that the mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundations and gets more abstract.
Dirac may have been thinking of David Hilbert’s famous problems for mathematicians, introduced in 1900. The sixth problem (as published in 1902) was the axiomatization of physics. As examples, Dirac considered
Non-euclidean geometry and non-commutative algebra, which were at one time considered to be purely fictions of the mind and pastimes for logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and that advance in physics is to be associated with a continual modification and generalisation of the axioms at the base of the mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.
Here Dirac expressed the opinion that Hilbert’s programme would be fruitless—rather than searching for a stable axiomatization for physics, one should be constantly modifying and generalizing the axioms upon which physics was based. He characterized this as a process of increasing abstraction. It is important to note here that Dirac emphasized that these mathematics were “very necessary for the description of general facts of the physical world.” He was not (at this stage at least) a pythagorean, believing that the universe was itself made of mathematics.** He was not a mathematician and did not wish to be one.
Building on this foundation, he offered guidance for his fellow researchers at the forefront of quantum theory:
The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalise the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities (by a process like Eddington’s Principle of Identification).
Dirac presented his 1930 theory of electrons and protons as an example of this work. This is the first thing to emphasize: Dirac was giving advice by explicating his own method.
Starting in 1928 Dirac had published a relativistic theory of the electron that accounted for its newly-discovered spin and magnetic moment. This was almost universally regarded as a great success in the physics community. Prior to the late 1920s quantum mechanics had not been consistent with relativity at all—that is to say its fundamental equations were not invariant with respect to Lorentz transformations, and space and time variables were not treated on the same footing. In 1926 many authors including Oskar Klein and Walter Gordon proposed a relativistic generalization of the Schrödinger equation, now called the Klein-Gordon equation. The K-G equation had a problem. It could not be interpreted in the same way as standard quantum mechanical formulae. Under Born‘s interpretation of QM the square modulus of the wave equation was to be interpreted as a probability. Probabilities, of course, can only run from zero to 1. And here was the problem with the K-G equation: it gave the possibility for negative probabilities. It could not be interpreted according to Dirac’s “general transformation” formalism of quantum mechanics. In 1931 Dirac pointed to his success in formulating a relativistic theory that gave only positive probabilities as “employing all the resources of pure mathematics in attempts to perfect and generalise the mathematical formalism that forms the existing basis of theoretical physics.”
What of the second part of Dirac’s methodological dictum? (“[A]fter each success in this direction, to try to interpret the new mathematical features in terms of physical entities (by a process like Eddington’s Principle of Identification).”) One important factor is the order of presentation, as noted above: first comes the mathematics, then comes the physical interpretation. But more specifically, Dirac points to Eddington’s Principle of Identification. What’s that?
Arthur Stanley Eddington is best known for bringing Einstein’s general theory of relativity to Britain, and performing the famous eclipse experiments that were taken as proof of the theory in 1919. He was a proponent of Einstein’s theory, and a scientific popularizer as well. Eddington expressed his philosophy of science in both his scientific and popular works, in 1920 and 1923. Eddington was something of an idealist—he strongly suggested that the mind structures the world.
But the conclusion is that the whole of those laws of nature which have been woven into a unifed scheme—mechanics, gravitation, electrodynamics and optics—have their origin, not in any special mechanism of nature, but in the workings of the mind (1920, 180).
This may be surprising coming from Britain in the 1920s where empiricism dominated the philosophical discourse, but Helge Kragh (1982) has charted the territory of this “British rationalism” in astrophysics and cosmology. Within this broader philosophical outlook, Eddington’s “principle of identification” was a methodological dictum. According to Olivier Darrigol (1992), Eddington thought “the mathematics of a physical theory had to be developed at an a priori level before the identification of physically accessible quantities took place.” “Identification” here means just that: pointing to a mathematical expression and asserting that it represented the world.
Was Dirac Eddingtonian? Darrigol argues “yes.” As the earliest evidence for this perspective he cites an undated manuscript, probably from 1924, in which Dirac used the word “beauty” parenthetically:
The modern physicist does not regard the equations he has to deal with as being arbitrarily chosen by nature…. In the case of gravitational theory, for instance, the inverse square law of force is of no more interest—(beauty)?—to the pure mathematician than any other inverse power of distance. But the new law of gravitation has a special property, namely its invariance under any coordinate transformation, and being the only simple law with this property it can claim attention from the pure mathematician.
Darrigol’s emphasis in analyzing this quote is on the first sentence, claiming that it shows Dirac shared Eddington’s belief in the necessity of physical law, with some qualification: “it is not the search of the mind for permanence but its predilection for mathematical beauty which enforces the necessity of the laws” (302). But while this is surely an example of Dirac’s use of the word, what was he really saying here about beauty? That it may sit alongside interest for pure mathematicians. Here Dirac was motivating his presentation of the theory to a group of mathematicians.*** The gravitational law of Newton may be of no more interest/beauty to mathematicians than any other law of the same form, but Einstein’s gravitational theory was different (it was invariant under any coordinate transformation).
Thus, while this is undoubtably Dirac writing on the theme of beauty, it seems to me that there is less reason for ascribing Dirac to Eddington’s metaphysical rationalism here. This is especially true if the dating of this document to 1924 is correct—this would be just at the time of Dirac’s exposure to Bohr’s correspondence principle and prior to his exposure to Heisenberg’s positivist writing. We are then left with Dirac’s own methodological writing in 1931 to follow “a process like” Eddington’s methodological principle (hardly a full throated endorsement of Eddington’s metaphysics). Dirac’s view expressed in this quote was not a rationalism but an aesthetics; not the view that the human mind determined the form of natural laws, but the view that nature’s laws could be identified by their beauty.
Does it make sense to call Dirac Eddingtonian? On balance, I would say “no.” It unduly associates Dirac with Eddington’s rationalism, and there is nowhere in Dirac that suggests he privileged the human mind in the construction of reality. Even while arguing for the preeminence of mathematics in theoretical physics, he kept the fact that the mathematics was required by physical facts in view.
The next post in the series will look at Dirac’s views on mathematical beauty, and continue engaging with Kragh and Darrigol’s pictures of Dirac.
Thanks for reading!
*In 1930 Dirac had predicted that the proton would be what we call the anti-particle partner of the electron (Dirac 1930).
**See Ian Hacking, The Lure of Pythagoras. Ilyyun: The Jerusalem Philosophical Quarterly 61: 1-26.
***Henry Frederick Baker’s Cambridge tea parties, see Darrigol 1992, 295–96.