Our computers and the entire modern world depend on them, says science writer Michael Brooks in an excerpt from his new book:
But what does his claim that the numbers are “not some deep mystery about the universe” leave us? Recent studies have shown that imaginary numbers — which we can’t really represent by objects, the way we can represent natural numbers by objects — are needed to
describe reality. Quantum mechanics pioneers did not like them and worked out ways around them:
In fact, even the founders of quantum mechanics themselves thought that the implications of having complex numbers in their equations was disquieting. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrödinger — the first person to introduce complex numbers into quantum theory, with his quantum wave function (ψ) — wrote, “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Ψ is surely fundamentally a real function.”Ben Turner, “Imaginary numbers could be needed to describe reality, new studies find” at LiveScience (December 10, 2021)
Why would a purely physical universe need imaginary numbers?” at Mind Matters News (February 16, 2022)
Takehome: The most reasonable explanation is that the universe, while physical, is also an idea, one that cannot be reduced to its physical features alone.
You may also wish to read:
Why the unknowable number exists but is uncomputable. Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is, Chaitin shows, impossible. Gregory Chaitin walks readers through his proof of unknowability, which is based on the Law of Non-contradiction.
Most real numbers are not real, or not in the way you think. Most real numbers contain an encoding of all of the books in the US Library of Congress. The infinite only exists as an idea in our minds. Therefore, curiously, most real numbers are not real. (Robert J. Marks)
Can we add new numbers to mathematics? We can work with hyperreal numbers using conventional methods. Surprisingly, yes. It began when the guy who discovered irrational numbers was—we are told—tossed into the sea. (Jonathan Bartlett)