Why would a purely physical universe need imaginary numbers?

In an excerpt from his new book, The art of more: how mathematics created civilization, Science writer Michael Brooks offers the intriguing idea that the modern world was born out of imaginary numbers:

Imaginary numbers are not imaginary at all. The truth is, they’ve impacted our lives far more than anything truly imaginary ever could. Without imaginary numbers and the vital role they have played in powering homes, factories and internet server farms, the modern world would not exist.

Michael Brooks“Imaginary numbers are reality” at Nautilus (February 9, 2022)

Imaginary numbers, remember from school, are the square roots of minus numbers. Two plus numbers, multiplied, give a plus number. But the same goes for the two minus numbers. The square roots of the minus numbers must exist, but they exist in an “imaginary” sense. And yet they matter:

Students who might complain to their math teacher that it’s pointless for someone to learn to use imaginary numbers should put down their phones, turn off their music, and pull the wires out of their broadband router.

Michael Brooks“Imaginary numbers are reality” at Nautilus (February 9, 2022)

While acknowledging that “A complete mathematical description of nature” requires the existence of imaginary numbers, Brooks considers the term “imaginary” numbers to be of no use as he considers all numbers to be imaginary. Towards the end of his essay he asserts: “So what we discover here is not some deep mystery about the universe, but a clear and useful set of relationships which are a consequence of defining numbers in different ways .”

But what does his assertion that numbers are “not a deep mystery about the universe” leave us with? Recent studies have shown that imaginary numbers – which we can’t really represent by objects, like we can represent natural numbers by objects – are necessary to describe reality. The pioneers of quantum mechanics have made not like them and found ways around them:

In fact, even the founders of quantum mechanics themselves thought the implications of having complex numbers in their equations were disturbing. 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: the use of complex numbers. Ψ is surely fundamentally a real function.

Ben Turner“Imaginary numbers may be needed to describe reality, new studies show” at Live Science (December 10, 2021)

But studies from the scientific journals Nature and Physical Review Letters have shown, via a simple experiment, that the mathematics of our universe requires imaginary numbers.

Theoretical physicist Sabine Hossenfelder offers a common sense view:

If you are willing to modify quantum mechanics, so that it becomes even more nonlocal than it already is, you can still create the necessary entanglement with real-valued numbers.

Why is this controversial? Well, if you belong to the camp of silence and calculation, then this conclusion is completely irrelevant. Because there’s nothing wrong with complex numbers in the first place. That’s why you have half the people saying “what’s the point” or “why all the fuss”. If you, on the other hand, are in the camp of those who think there is something wrong with quantum mechanics because it uses complex numbers that we will never be able to measure, then you are now caught between the hammer and the anvil. Embrace complex numbers or accept that nature is even more non-local than quantum mechanics.

Sabine Hossenfelder, “Do complex numbers exist?” at ReturnRe(Action) (March 6, 2021)

This is not the right time for mere materialism.


You can also read:

Why the unknowable number exists but is incalculable. To feel that a computer program is “elegant” requires discernment. Mathematically proving that it is elegant is, Chaitin shows, impossible. Gregory Chaitin guides readers through his proof of unknowability, which is based on the law of non-contradiction.

Most real numbers are not real, or not as you think. Most of the real numbers contain an encoding of all of the United States Library of Congress books. Infinity only exists as an idea in our mind. Therefore, oddly enough, most real numbers are not real. (Robert J. Marks)

and

Can we add new numbers to math? We can work with hyperreal numbers using conventional methods. Surprisingly, yes. It all started when the guy who discovered irrational numbers was, we are told, thrown into the sea. (Jonathan Bartlett)

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