In 1939, Paul Dirac observed that “the physicist, in his study of natural phenomena, has two methods of making progress”: experiment and observation, and mathematical reasoning. Although he said, “there is no logical reason why the second method should be possible,” nevertheless it works, and to great effect. The key, Dirac felt, was beauty, leading him to his principle that successive theories of nature are characterized by increasing mathematical beauty. The results of this were rich and included some predictions not confirmed until after Dirac’s death. Nevertheless, the powerful guidance Dirac found in mathematics did sometimes lead him astray, as he rejected the principle of “renormalization,” developed by Feynman, Schwinger, and Tomonaga, to remedy the nonphysical infinities that kept cropping up in Dirac’s equations for quantum electrodynamics. Even as other physicists accepted it, Dirac never did, saying it was “just not sensible mathematics.” Nevertheless, it was powerful physics.He followed with this interview:
Why do the atoms have those properties? Because they’re made of quarks and electrons. What about the electron? What properties does it have? And the cool thing is, all the properties that electrons have are purely mathematical. It’s just a list of numbers. So in that sense, an electron is a purely mathematical object. In fact, there’s no evidence right now that there’s anything at all in our universe that is not mathematical.Yes, there is evidence. There is no purely mathematical description of an electron because, according to the rules of quantum mechanics, measurements on an electron can depend on measurements of a possibly-distant entangled electron.
The usual explanation is that the two electrons have a joint mathematical description as a spinor wave function in a tensor product Hilbert space. This leads to paradoxes of relativity and causality.
There are different interpretations of quantum mechanics. The electron is never just a list of numbers. All attempts to reduce the electron to some lcoal set of numbers have failed. We can predict measurements of an alectron based on previous measurement of that electron as well as any entangled electrons. But we cannot reduce the electron to numbers.
Dirac got Einstein's disease, and was unproductive once he got the idea that physical theories could be predicted by mathematical beauty.
Tegmark says he is writing a book, and will promote it on his Facebook page.