One could argue that all numbers are an abstraction; they don't "exist" in any physical form, even integers. Therefore, why is it such a surprise that complex numbers are sometimes also needed to model reality?
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
Well, I think instantaneous velocity is real, and I think I have a convincing argument. Imagine you have a piece of gum stuck to your bike tire. As your tire spins, the piece of gum spins with it, and its velocity is constantly changing. But when the piece of gum flies off your tire, it flies off in a straight line, following the instantaneous velocity at the moment of separation. Seeing as the gum can fly off in a straight line, its instantaneous velocity at that moment must be well-defined.
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.
The answer will have to grapple with both the philosophy of physics and the properties of these various fields.
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.
The proper question is not "do I like it" but is it necessary. The answer apparently is No for the quaternions.
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.
In some sense (because of the wavefunction) quantum mechanics is less discrete than classical mechanics, a qubit system which would be classicaly described by a single bit is described by a state that sits on a continuous manifold (the Bloch sphere).
However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?
I don't know what this means, but there is this idea that quantum mechanics leads to integer valued stuff because the first models studied were things like the hydrogen atom and the simple harmonic oscillator which both have a natural integer valued observable (the energy). This is not generally the case, however and in quantum mechanics there are many observables which are not integer valued, for example the kinetic energy of a free particle can be any non-negative real.
It turns out that a lot of interesting physics these operators have physically interesting discrete spectrum.
[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.
I think it's a quite interesting crossroad.
Either reals don't exist, and everything is at most turing complete, in which case the universe is detemernistic and predictable, yet there are very hard boundaries to what can be known and proven.
Or reals DO exist in which case the church turing thesis is false, as the universe (with it's operations on reals) is a reified example of a super turing system, in which case we can know a lot more things than in the discrete case, but some of that knowledge can only be grasped intuitively, because the moment you try to articulate it into formulas, it's discretized and thus limited to your sub-universe/brain powerfull turing machine.
Non-computable reals are also perfectly capable of modeling "analog reality" (because for all intents and purposes, reality could be discrete and finitely describable very small scales).
Quantization would like a word with you.
So any part of reality that you can reduce to an efficient computation on a quantum computer, admits a description using the rationals only. But I might be out of my depth here.
Then you define all the operations between integers as operations of cardinality between set. I really enjoyed that semester when I was getting my degree on Maths, by the way, although I use the knowledge I got there as a curiosity at parties, because it didn't prove useful (not that it is bad).
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.
In the beginning there were only Natural numbers.
-1 appeared the moment one decided to define complementary operation to addition. -1 in reality is only a shorthand for "0-1", 0 comes here as neutral element of addition. Addition is well defined on Natural numbers. Subtraction generates Zahl numbers.
Similarly multiplication is well defined on Zahl numbers. It's the complementary operation - division - that generates Quotient/rational numbers.
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.
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.
All of those are part of Peano and ZFC axiomatic systems, which were thought to be enough to explain all of physics (albeit not easily). Matrices and imaginary numbers are not part of those formal systems.
Just ask yourself. If you are right, why didn't Schrodinger or the researches that are publishing this study just think of your idea of using matrices?