In what relevant sense do imagenary numbers differ from real numbers?
Being a mathematical abstraction they are in no way required to help approximate phisical reality. That they do is a source of wonder to many, but that is beside the point.
Physicists today can happily make career by writing papers about things no one has ever observed, and never will observe. This continues to go on because there is nothing and no one that can stop it.
https://iai.tv/articles/why-physics-has-made-no-progress-in-...
While the real number 2.0 can represent the scaling of a vector by 2, the imaginary unit is the rotation of the same vector by a right angle.
Any oscillation is equivalent with the projection of a rotation on an axis. Because it is easier to make computations with rotations, all the calculations for electric circuits use rotations to model the oscillations, therefore they use imaginary numbers.
This name of "imaginary" numbers should better be abandoned, or at least their meaning should be much better explained in school, because they are not some mathematical abstraction used only in seldom cases, but it is almost impossible to design a device that does not use rotations either in the actual space or in an abstract space, thus needing imaginary numbers for mathematical modelling.
On the other hand, string theory and a few other theories that are explored by some physicists are completely artificial mathematical models about which nobody has proven yet that they have any relationship with the real world.
Not really. Imaginary numbers work really well for calculating real physics behaviour, string theory not at all pretty much. Unless n=0 in your analogy.
A lot of papers and books have been published about the string theory and all claim to calculate something.
As an interesting mathematical model, string theory certainly qualifies.
On the other hand, as a mathematical model usable in physics, a theory like string theory must pass 3 criteria:
1. It should be able to calculate some numerical values of physical quantities that can be measured.
2. Those measurements must be made and the results must be as predicted, with a reasonable accuracy.
3. It should not be possible to compute the same numerical values that have been validated by measurements using other simpler mathematical models.
I have browsed through many research papers and a few books about string theory and I have never seen any results even remotely approaching the fulfillment of these 3 criteria.
In classical electrodynamics it's the permittivity and permeability of the vacuum.
In Newton's theory is G.
In General Relativity is G and \lambda.
In QED is the fine-structure constant, \alpha.
And the Standard Model of particle physics has 20 free parameters that have to be set by hand using input from the experiment before being able to use it to make precise numerical computations and having any predictive power. If you don't fix these inputs you don't have a theory, you have a family of theories and you cannot discern which one is the correct one.
On top of that, Quantum Field Theory (QFT) the "theory" (continue reading to see why the word theory is a misnomer) that underlies the Standard Model is not constraining enough and you still need to pick the right gauge theories that model our universe. So QFT is better understood as a framework, from which we build a model of reality by hand-picking some theories that seem to suit our universe, then we fix their free parameters using experiments and only then we can predict everything else.
String Theory is much the same as QFT in this regard. It's a framework for building physical theories. It arguably has less free-parameters (it has only 1), the string tension. Buy you still need to pick the right models within the framework, here again String Theory is arguably more constraining but you still have an immense amount of options, commonly referred as the String Theory landscape (or vacua) and relate to how you compactify the extra dimensions.
So if you applied the same criteria to Quantum Field Theory you would come to the conclusion that is "an interesting mathematical model" but a "useless physical theory".
Disclaimer: I studied Theoretical Physics. I do not work in academia or in physics, for that matter. My sustenance does not depend on money flowing into High Energy physics.
It's even better than that. Working from the other side, we generally take the natural numbers N = {0, 1, 2, 3, ...} to be "real" in a philosophical sense as we have an implied mapping of x in N fundamentally describing possession of discrete quantities of goods. This get a little more tricky when you generalize it to the integers Z = {..., -2, -1, 0, 1, 2, ...}, but we still consider these physically grounded as you can "owe" a discrete quantity of goods to someone. This is an incomplete mapping to reality as possession of y goods where y is in Z- doesn't actually describe where your y goods go to, but it's useful enough that we ignore that.
Now that's all good and well for discrete quantities of goods, but what about fractional quantities? We need a new system to describe more numbers between the numbers we already have. Thus we defined the set of rational numbers Q = {n | n = p/q where p, q are in Z and q is not 0}. This lets us compactly describe almost any number we want between the existing integers. Most still consider these to be physically grounded, because we created these to describe concrete things in our reality which they do very compactly. You can claim that at least some of these are physically grounded in reality as using 1/3 a cup of flour or buying 1/2 a watermelon is certainly something one can very clearly and explicitly do.
The rationals have some issues though, namely that they still have holes in them. Suppose you want to describe the relationship between the diameter of a circle and its circumference. The constant you use to transform one to the other, π, does not exist in Q. You can get as close as you like, but you can't actually reach it. That's a bit of a problem for people whose job it is to make sure that the things math says are correct are actually fully correct. There are other problems, suppose you want to make a rectangular plot of land whose area is 2 square miles. How long does each side need to be? You can get as close as you like by using rational numbers, but the actual length of that side (sqrt(2)) is not in Q.
To fix this, we then very delicately construct the set of real numbers R = {x | where x is in {Q and all of the numbers described above which are missing from Q}}. This is where the physical grounding of the numbers starts to get ugly, because as it turns out that just like for some numbers in Q (consider 22/7, which does not evaluate to a fixed number of digits but rather goes on forever) these are uncountable and most of them unrepresentable, i.e. if you tried to write the number out completely on a piece of paper the universe isn't big enough to hold it (and in some cases, you can't even specifically refer to the number in constructive terms as with π). This turns into a whole philosophical debate which IMO is silly but some people do take pretty seriously.
But wait, there's more! The field of real numbers is closed under addition and multiplication (and thus subtraction and division), but it's _not_ closed under some other operations. Suppose you're an EE trying to represent physical signals that very much do exist in reality, and you need to take the root of a negative real number because that is a meaningful quantity in context of the still physically grounded thing you're modeling. Well you can take the root of a negative real number, but that number is not itself a real number. Thus we must define the imaginary numbers to hold those negative roots and then the complex numbers to join the imaginary numbers back to the reals.
In every single one of these steps, the new numbers were created to describe aspects of our observed physical reality. The break from the common definition of "numbers are real because I can go buy 24 tangerines and 24 is a number" happened way back at the integers where we added a reflection around the end of the natural numbers. From a perfectly reasonable perspective then, the real and imaginary numbers are both "real" in that they describe actual physical things that exist even though you can not in fact buy -1 apples or 22/7 cats or π bananas or sqrt(-1) movie tickets.
If I have two dogs, and you swap one of them out for a different dog; well, it's not the same and I'd notice.
What I find interesting about "24 apples" is that all of those apples are in fact also unique individuals - none are identical, they have different weights, percentage moisture, sugar content, number and positioning of seeds, etc. Their exterior markings will be as unique as fingerprints. None of them are really interchangeable.
Saying that "there are 24 of the same thing here" is an _abstraction_ of reality. It's our perception, our agreement that they can be considered "the same thing", but it is not a physical reality.
It might be real for of protons, but not of people, cats, apples or tangerines. Macroscopic objects are not numbers.
The base case of that mapping is natural numbers representing possession of quantities of fungible objects. That mapping is broken by the negative integers, literally the next step up from the natural numbers. If you're concerned about the reality of the reals you can ramble on about finitism or the holographic principle and how something can possibly be real if you can't name or explicitly define every version of it, or you can recognize that every number system from the natural numbers to the complex numbers shares the property that elements of this system of numbers map to elements of physical systems and thus they are all exactly as real as each other.
You have 24 somethings. The fact that they are not identical with regards to any physical quantity doesn't matter.
These 24 items are all unique, as are the excluded ones. The inclusion, the concept of set, the grouping, is not real, it is your criterion, mapping, your abstraction of reality.
but the right hand side contains more "information" than the left hand side, being not simply a quantity but a route to getting there from other quantities.
There is no such thing as physical circle.
Discrete quantities do seem to exist in as fundamental a sense as anything can, but on small scales and also once you get beyond that you're stuck with probabilistic descriptions. 22/7 is exactly as real as pi, you just use different algorithms to evaluate them to the required degree of precision.
When you try to examine the real world too closely using relative human terms like "exists," things get messy. It feels like I'm physically touching the keyboard keys as I type this but I'm not, it's field repulsion rather than physical contact (or rather, it turns out that what we call physical contact is not). It looks like I'm watching snowflakes swirl on the TV, but that's actually a series of still images exploiting meatspace processing limitations and not the continuous process of the real snowflakes falling outside my window - even though from here I can't really tell the difference. The delicious steaming mug of hot chocolate to my right doesn't actually taste like anything but warm river silt, the flavors are being processed by a different organ entirely. Our concept of reality is inexorably linked to how our brains process the signals around us, and every definition of "realness" I've seen based on human terms is weak.
To be fair, in practice you can't ever use exactly 1/3 cup of flour or buy exactly 1/2 a watermelon. So even with the fractionals we've started to wriggle away from physical reality in our effort to model it more precisely.
Dead wrong. Predicting protein folding has been a scientific holy grail for decades, and neural network-based AlphaFold is the leader. No non-neural approach comes within a country mile of its performance. They created a fold database of unprecedented coverage which is being used in research.
I think the string theory guys will stick with it while they can.