Imaginary Numbers May Be Essential for Describing Reality

Eh, because of quantisation you could argue only integers exist ;)

The phrase "mosts [sic] scales of reality" is doing an incredible amount of heavy lifting here, though, since the most reliable mathematics describing three of what appear to be the four fundamental forces of nature are indeed quantized.

It's more accurate to say that only integers appear to be relevant in describing the nature of the nuclear forces and electromagnetism/electroweak force.

And it seems that, at least for the moment, we're headed in a direction of quantization of gravity and a Planck unit of time as well.

One of the most influential physicists of all time, John Wheeler, was of the philosophical opinion that physical laws must only be describable in terms of a finite integers [1]. This is an ideal that at least some other physicists hope for. Recently, the "It from Qubit" program in physics [2], attempts to understand reality by reducing everything to computation on qubits, and has led to some interesting insights.

[1] Information, physics, quantum: The search for links, John Wheeler 1990

[[2] The Simon's institute has formally funded such a program, though the movement within physics is much larger https://www.simonsfoundation.org/mathematics-physical-scienc...

Edit: fixed citation 1. And removed reference to infinities, as indicated below.

"God made the integers; all else is the work of man." Kronecker

(I assume you meant natural numbers, as "-1" is not a useful cardinality for a set)

The construction you refer to is due to Frege and Russell but it has its own kinds of subtleties. In particular it does not work in ZFC which we often base upon, so we usually use Peano's axioms instead to define them (depending on your context of course). While they are not all too surprising, I wouldn't call them absolutely obvious.

If you split one apple in two, you get two apples?

> if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2.

The artificiality is high here -- would you trade your bag of "2" apples for someone else's bag of two apples without asking about the size and quality of the apples? While counting numbers are wonderful things for examining things like "sets" it seem to me to be kind of silly to assign any "reality" to them.

Discrete identical things are pretty much nonexistent at human scales. Continuous identical things, however, are numerous, where continuous here is not strict, but practical.

This is not realistic as a foundation for mathematics. But I think the idea of using propositional logic and set theory was a mistake that we are only now starting to correct. Intuitionalists and category theorists (and finitists, frankly) are leading us in a better direction now.

How do the apples get into the bag?

Do they magically show up?

Or do they transition from not in the bag to in the bag in some unit?

What is the minimum possible unit of Apple change? Is it discrete or continuous?

Maybe a X plank units of apple transitions every second.

Another commenter linked this post by Scott Aaronson: https://www.scottaaronson.com/blog/?p=4021

The money quote (which leads a few more paragraphs on the details) is this:

> But, while I remain less than 100% satisfied about “why the complex numbers? why not just the reals?,” there’s one conclusion that my recent circling-back to these questions has made me fully confident about. Namely: quantum mechanics over the quaternions is a flaming garbage fire, which would’ve been rejected at an extremely early stage of God and the angels’ deliberations about how to construct our universe.

Imagine going back to Kronecker and their contemporaries and telling them that a spinning top weighs more than a top at rest. They would have laughed you out of the room, but a few years later the arrogant scientific confidence of the 19th century was completely crushed by quantum mechanics and general relativity.

Science is not a body of knowledge, it is a frontier of ignorance. I like to "feel" that complex numbers are part of the fabric of the universe, if only because it would be more beautiful.

"Clouds are not spheres, Mountains are not cones" - Mandelbrot

Are there any integers in the physical world? To me they all seem in my head.

If I have 3 match sticks, they are each made of billions of molecules, atoms, electrons, quarks, gluons, etc. So there is no integer count, just in my perception.

Complex numbers themselves can be explained with real numbers only.

Absolutely agree.

What's your preferred terminology? I like "Forward" and "Reverse" for positive and negative number and "Lateral" for the imaginary unit, since it is just perpendicular to the Forward and Reverse numbers.

Do you have a suggestion for an alternative to "real number"?

In quantum physics complex numbers are used to represent circular symmetry, so they are more like modulus and phase.

> They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.

Sorry to be blunt, but your comment is horrible. There is far more going on here than you realize. Real numbers have an isomorphism with our reality. We thought we didn't need imaginary numbers to describe reality. They were inteded to be used more as a mathematical tool.

Another good example is euclidean vs non-euclidean geometry. They are both incredibly useful, but the former one was though to be "real" and the latter "imaginary". But it turns, physics is okay with us living in a non-euclidean universe (https://en.wikipedia.org/wiki/Shape_of_the_universe).

Mathematicians have largely abandoned the term "imaginary number" and use "complex number" instead. I never used the term "imaginary number" when I was teaching (except to point out that it wasn't very good terminology).

But the old term still survives in some areas. For instance, the real and imaginary parts of a complex number:

  re (3 + 4 i) = 3  and  im (3 + 4 i) = 4.

There are 2d values that aren't imaginary numbers.

And real numbers are ∞-d, over rationals.

I think about it this way: the complex-ness of the number is really just a representational trick. What it means, intuitively, depends on what the question is really asking and why the behaviors of math on complex numbers is well-suited to representing the answer, and that reason might change based on the context.

You can (almost). You can do the same thing again to get quaternions, and again to get octonions, and so on.

https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...

That's different from extending 1d to 2d, 3d, etc as vectors.

Representing complex numbers as 2-dim. real vectors is just one way of constructing complex numbers. It is convenient because you can draw 2-dim. vectors, and it makes complex numbers feel more tangible.

Something like the original motivation (which you might find less "arbitrary") goes like this. We know how to solve some quadratic equations like x^2 = 9: We get x = 3 and x = -3 (though it was a nontrivial thing to realize those negative solutions are there). What about x^2 = 2? We can "see" the solution should be the length of the hypotenuse of a right triangle whose legs are both 1, so maybe encouraged by the geometry we don't feel too bad about having solutions (+/- sqrt(2)) which are, however, not rational numbers.

(Notice that we keep "extending" our idea of "number". Well, if we didn't have feature creep no one would buy the new version.)

Since we now have negative numbers, we ask about solutions to x^2 = -1. The first response is there are "no solutions", because there are no real numbers that square to -1. Rather than give up, we decide to "extend" the real numbers to provide solutions to this equation.

The idea is actually pretty easy if you know some modular arithmetic. For instance, mod 6 arithmetic means that you take the integers and consider things that differ by multiples of 6 to be "the same". As a consequence, 17 = 5, 4 = -14, and (importantly) 6 = 0.

Thinking in the same way, to get solutions to x^2 = -1, we want x^2 + 1 = 0. That's like "6 = 0", so "x^2 + 1" should be our "modulus", but what "number system" do we "mod out of" (instead of the integers, with the integers mod 6)? Since x^2 + 1 is a real polynomial, it looks like our "base number system" should be R[x], the (ring of) polynomials with real coefficients (stuff like -17 x^3 + 1.374 x^2 - 31/42 and so on).

So we should take real polynomials, but agree that two polynomials differing by a multiple of x^2 + 1 are "the same". We conclude that the complex numbers "should be"

  R[x]  mod  <x^2 + 1>

where the "<x^2 + 1>" means "all (polynomial) multiples of x^2 + 1".

For instance,

  x^3 + 4 x + 1 = (x^3 + x) + (3 x + 1) = x(x^2 + 1) + (3 x + 1)
    = 3 x + 1 (mod x^2 + 1)  (the multiple of x^2 + 1 is 0)

(To agree that some things should be regarded as "the same" is to construct an equivalence relation, and the "bags" of "same things" are equivalence classes. I'm omitting the details from abstract algebra, but this construction is called "forming the quotient ring R[x]/<x^2 + 1>".)

By this construction, "3 + 4 i" appears as

  3 + 4 x (mod x^2 + 1)  or  (3 + 4 x) + <x^2 + 1>.

In the construction of the complex numbers via 2-dim. real vectors, "3 + 4 i" appears as (3, 4).

You can also construct the complex numbers as a subring of the 2 x 2 real matrices, so "3 + 4 i" appears as

  3 [1 0]  +  4 [0  1]
    [0 1]       [-1 0] .

Notice that

  [0  1]^2 = [-1 0] = - [1 0]   (like i^2 = -1)
  [-1 0]     [0 -1]     [0 1]

So those are three ways of constructing the complex numbers, and each has pros and cons. No one of them "is" the complex numbers. "The" complex numbers are really an isomorphism class of rings, and we're just giving different representatives from that class.

They do match. The thing is that the description is much more complex, nothing else.

What I see from the preprint (notice that the authors have not discussed it with Quanta, because it is under peer-review) and bothers me is that they "add a real qubit" (...) which should be done very carefully because the two real qubits are not independent (they are linked by the hamiltonian).

I have not read the whole preprint, though.

To a certain extent, you could probably say the same about the time term in the flat spacetime metric in relativity.

There might be ways around it, but I'd wager that anything you come up with can be massaged a bit to make it look like time is an imaginary dimension of space where it's contribution to distance along the spacetime manifold is negative when squared.

Sorry because I do not understand your wording very well (English is not my FL), so the answer to your question is yes, you can:

e^(-at)*(A cos(bt) + C sin(bt))

where b is the frequency and a the dampening.

(But you might not be asking for an answer).

as I was writing that matrix out from memory I was thinking there was something funny... Thank you for correcting me.

I have read the introduction (only cursorily). What bothers me is that they may be conflating

C as C-vector space, which is R (as R-vector space) tensor C (complex dim 1).

with

R^2 as R-vector space, tensored by C (complex dim 2).

Because when they "add a real qubit", they seem to be adding a degree of freedom which you do not have when you have just one C-qubit.

But -I repeat- I have only looked over the intro.

That's the point, in mathematics you can model the same object in many different ways save for trivial transformations.