pcalau12i

We all have an intuitive grasp of 0, it's just when you define what it looks like for something to change in a particular situation, i.e. you define what x>0 or x<0 looks like, and then x=0 is just when it hasn't changed at all.

I feel like this discussion is getting too philosophical. My point isn't that deep, it is just to keep complex numbers intuitive and tied to physical reality. We shouldn't treat imaginary numbers like some out there, almost mystical thing we should just accept at face value. When we realize that they are mathematically equivalent to a set of operations on a vector of two real numbers, we can then get an intuitive understanding of what they actually represent in the real world. You can visualize a complex number as a vector representing on a plane (called the complex plane), and then visualize operations on the complex numbers as manipulations of that vector.

The Fourier transform has complex numbers in it. This isn't mysterious, it's just that the Fourier transform deals with waves, and waves are two-dimensional, so they need to be described by a vector of two numbers. The Fourier form effectively wraps the wave around a circle, and if the rate of wrapping is different from the wavelength of the wave, then every time you complete a revolution of the circle, you will either have overshot or undershot a complete cycle of the wave, causing your second wrapping to be off-center, and if you repeat this indefinitely, then all the off-center wrappings will cancel each other out, giving you 0 in the limit. But if the rate of wrapping is equivalent to the wavelength, then a revolution around the circle would exactly correspond to a cycle of the wave, so it would you would not get this cancelling and it would blow up to infinity in the limit.

You can get very intuitive mental images of what complex numbers are actually doing when you recognize this. There shouldn't be a layer of mystery put on top of them. People often act like they are something so mysterious we just have to accept at face-value, and others will even justify this by pointing out that they're used in quantum mechanics, and quantum theory is "weird," therefore we should just accept this weird thing at face-value and not question it.

All I am trying to point out is that complex numbers are not "weird," they have clear meaning you can visualize them and get an intuition for them, and the reason they show up in certain equations always has very good and intuitive explanations for it. I am not making a deep philosophical point here. I am only arguing against the notion of obfuscating the meaning of imaginary numbers. The term "imaginary" is honestly not a good name. Complex numbers probably should just be called 2D numbers, with the real and imaginary components called the X and Y component or something like that. They are just a way of concisely representing something that is two-dimensional. There are also quaternions which are 4D numbers.

What point is there to "prove"? Your argument now is just that we defined them differently therefore they are different. Which suggests a straw man to my original point as I never once implied or suggested that in mathematics, real and complex numbers don't have different definitions, that's not relevant to anything.

Because your arguments are just bizarre. Imaginary numbers do not have a priori definitions. Humans have to define imaginary number and define the mathematical operations on them. There is no "hostile confusion" or "flaw," there is you making the equivalent of flat-earth arguments but for mathematics. You keep claiming things that are objectively false and so obviously false it is bizarre how anyone could even make such a claim. I do not even know how to approach it, how on earth do you come to believe that complex numbers have a priori definitions and they aren't just humans defining them like any other mathematical operation? There are no pre-given definitions for complex numbers, their properties are all explicitly defined by human beings, and you can also define the properties on vectors. You at first claim that supposedly you can only do certain operations on complex numbers that you cannot on vectors, I point out this is obviously false and you can't give a single counter-example, so now you switch to claiming somehow the operations on complex numbers are all "pre-given." Makes zero sense. You have not pointed out a "flaw," you just ramble and declare victory, throwing personal attacks calling me "confused" like this is some sort of competition or something when you have not even made a single coherent point. Attacking me and downvoting all my posts isn't going to somehow going to prove that you cannot decompose any complex-valued operations into real numbers, nor is it going to prove that complex numbers somehow don't have to have their properties and operations on them postulated just like real numbers.

And you can also just write it out using real numbers if you wish, it's just more mathematically concise to use complex numbers. It's a purely subjective, personal choice to choose to use complex-valued notation. You are trying to argue that making a personal, subjective, arbitrary choice somehow imposes something upon physical reality. It doesn't. There isn't anything wrong with the standard formulation, but it is a choice of convention, and conventions aren't physical. If I describe my losses in a positive number, and then later change convention and describe my winnings with a negative number, the underlying physical reality has not changed, it's not going to suddenly transmute into something else because of a change in convention in how I describe it.

The complex numbers in quantum theory are not magic. They are also popular in classical mechanics as well, and are just quite common in wave mechanics in general (classical or quantum). In classical wave mechanics, in classical computer science, we use the Fourier transform a lot which is typically expressed as a complex number. It's because waves have two degrees of freedom, and so you could describe them using a vector of two real numbers, or you could describe them using complex numbers. People like the complex-valued notation because it's more concise to write down and express formulas in, but at the end of the day it's just a convention, a notation created by human beings which many other mathematically equivalent notations can describe the same exact thing.

I am having genuine difficulty imagining in your head how you think you made a point here. It seems you're claiming that given if two vectors have the same symbols between them, they should have identical output, such as (a,b) * (c,d) should have the same mathematical definition as (a+bi) * (c+di), or complex numbers are not reducible to real numbers.

You realize mathematical symbols are just conventions, right? They were not handed down to us from Zeus almighty. They are entirely human creations. I can happily define the meaning of (a,b) * (c,d) to be (ac-bd,ad+bc) and now it is mathematically well-defined and gives identical results.

Negative numbers are just real numbers with a symbol attached. Yes, that's literally true. In computer code we only ever deal with 0s and 1s. We come up with a convention to represent negative numbers, they are still ultimately zeros and ones but we just say "zeros and ones in this form represent a negative number," usually just by having the most significant bit 1. They are no physical negative numbers floating out there in the world like in a Platonic sense. What we call "negative" is contextual. It depends upon how we frame a problem and how we interpret a situation. You can lose money at a casino and say your earnings are now negative, or you can say your losses are now positive. Zeus isn't going to strike you down for saying one over the other. There is nothing physically dictating what convention you use. You just use which convention you find most intuitive and mathematically convenient given the problem you're trying to describe.

Yes, when we are talking about how computers work, we are talking about how numbers actually manifest in objective, physical reality. They are not some magical substance floating out there in the Platonic realm. Whenever we actually go to implement complex numbers or even negative in the real world, whenever we try to construct a physical system that replicates their behavior and can perform calculations on a physical level, we always just use unsigned real numbers (or natural numbers), and then later establish signage and complexity as conventions combined with a set of operations on how they should behave.

I'm not sure your point about fractional numbers. If you mean literally a/b, yes, there is software that treats a/b as just two natural numbers stitched together, but it's actually a bit mathematically complicated to always keep things in fractional form, so that's incredibly rare and you'd only see it in very specialized math software. Usually it's represented with a floating point number. In a digital computer that number is an approximation as it's ultimately digital, but I wouldn't say that means only digital numbers are physical, because we can also construct analogue computers that can do useful computations and are not digital. Unless we discover that space is quantized and thus they were digital all along, then I do think it is meaningful to treat real numbers as, well, physically real, because we can physically implement them.

uh... broski... you do realize a vector of two real numbers can be rotated... right? Please give me a single example for a supposed impossible operation to do on a vector of two real numbers that you can do on complex numbers. I can just define v² where v is a vector (a,b) as (a,b)²=(a²-b²,2ab). Okay, now I've succeeded in reproducing your supposedly mathematically impossible operation. Give me another one.

A complex number is just two real numbers stitched together. It's used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.

In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn't, then no one would use complex numbers, because computers can't process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.

A lot of people go into physics because they want to learn how the world works, but then are told that is not only not the topic of discussion but it is actively discouraged from asking that question. I think, on a pure pragmatic standpoint, there is no problem with this. As long as the math works it works. As long as the stuff you build with it functions, then you've done a good job. But I think there are some people who get disappointed in that. But I guess that's a personal taste. If you are a pure utilitarian, I guess I cannot construct any argument that would change your mind on such a topic.

I'm not sure I understand your last question. Of course your opinion on physical reality doesn't make any different to reality. The point is that these are different claims and thus cannot all be correct. Either pilot wave people are factually correct that there are pilot waves or they are wrong. Either many worlds people are factually correct that there is a multiverse or they are wrong. Either objective collapse people are factually correct that there is an objective collapse or they are wrong (also objective collapse theories make different predictions, so they are not the same empirically).

If we are not going to be a complete postmodernist, then we would have to admit that only one description of physical reality is actually correct, or, at the very least, if they are all incorrect, some are closer to reality than others. You are basically doing the same thing religious people do when they say there should be no problem believing a God exists as long as they don't use that belief to contradict any of the known scientific laws. While I see where they are coming from, and maybe this is just due to personal taste, at the end of the day, I personally do care whether or not my beliefs are actually correct.

There is also a benefit of having an agreement on how to understand a theory, which is it then becomes more intuitive. You're not just told to "shut up and calculate" whenever someone asks a question. If you take a class in general relativity, you will be given a very intuitive mental picture of what's going on, but if you take a class in quantum mechanics, you will not only not be given one, but be discouraged from even asking the question of what is going on. You just have to work with the maths in a very abstract and utilitarian sense.

No, it's the lack of agreement that is the problem. Interpreting classical mechanics is philosophical as well, but there is generally agreement on how to think about it. You rarely see deep philosophical debates around Newtonian mechanics on how to "properly" interpret it. Even when we get into Einsteinian mechanics, there are some disagreements on how to interpret it but nothing too significant. The thing is that something like Newtonian mechanics is largely inline with our basic intuitions, so it is rather easy to get people on board with it, but QM requires you to give up a basic intuition, and which one you choose to give up on gives you an entirely different picture of what's physically going on.

Philosophy has never been empirical, of course any philosophical interpretation of the meaning of the mathematics gives you the same empirical results. The empirical results only change if you change the mathematics. The difficulty is precisely that it is more difficult to get everyone on the same page on QM. There are technically, again, some disagreements in classical mechanics, like whether or not the curvature of spacetime really constitutes a substance that is warping or if it is just a convenient way to describe the dispositions of how systems move. Einstein for example criticized the notion of reifying the equations too much. You also cannot distinguish which interpretation is correct here as it's, again, philosophical.

If we just all decided to agree on a particular way to interpret QM then there wouldn't be an issue. The problem is that, while you can mostly get everyone on board with classical theories, with QM, you can interpret it in a time-symmetric way, a relational way, a way with a multiverse, etc, and they all give you drastically different pictures of physical reality. If we did just all pick one and agreed to it, then QM would be in the same boat as classical mechanics: some minor disagreements here and there but most people generally agree with the overall picture.

it isn't scientifically accurate...

There are plenty of simple ways to understand QM on a more ontological level than just the maths. The literature is filled to the brim with them these days. The problem is not so much that it's difficult, but that there is no agreement. So discussions regarding it just lead to arguments that can't be settled, and so professors get tired of it and tell people to just shut up and calculate.