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.

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.

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.

it isn’t scientifically accurate…

The brain is a physical object. Saying the brain playing a role in what we perceive somehow proves we don’t directly perceive reality as it really it from our point of view makes about as much sense as saying a painter can paint a painting of a fire so accurately that the painting it will suddenly burst into flames… no arrangement of a medium can transcend the medium.

What I mean by subjective experience is what you might refer to as what reality looks like from a specific viewpoint or what it appears like when observed.

So… reality? Why are you calling reality subjective? Yes, you have a viewpoint within reality, but that’s because reality is relative. It’s nothing inherent to conscious subjects. There is no such thing as a viewpoint-less reality. Go make a game in Unity and try to populate the game with objects without ever assigning coordinates to any of the objects or speeds to any of the object’s motion, and see how far you can go… you can’t, you won’t be able to populate the game with objects at all. You have to choose a coordinate system in order to populate the world with anything at all, and those coordinates are arbitrary based on an arbitrarily chosen viewpoint. Without picking a viewpoint, it is impossible to assign objects the majority of their properties.

If you claim that the physical world doesn’t exist independently of observation, and is thus nothing beyond the totality of observed appearances

No such thing as “appearances.” As Kant himself said: “though we cannot know these objects as things in themselves, we must yet be in a position at least to think them as things in themselves; otherwise we should be landed in the absurd conclusion that there can be appearance without anything that appears,” i.e. speaking of “appearances” makes no sense unless you believe there also exists an unobserved thing that is the cause of the appearances.

But there is neither an unobserved thing causing the appearances, nor is what we observe an appearance. What we observe just is reality. We don’t observe the “appearance” of objects. We observe objects.

If there is no object being observed

Opposite of what I said.

and the fact it it apparent from multiple perspectives is simply a consequence of the coherence of observation

What we call the object is certain symmetries that are maintained over different perspectives, but there is no object independently of the perspectives.

where do the qualities of those appearances originate from? How come things don’t cease to exist when they’re not being observed?

They cease to exist in one viewpoint but they continue to exist in others, and symmetries allow you to predict when/how those objects may return to your own viewpoint.

If you claim that the appearances don’t exist independently of the physical world being observed

I am claiming appearances don’t exist at all.

why does the world appear different from different perspectives?

Reality is just perspectival. It just is what it is.

How do you explain things like hallucinations (there is no physical object being observed, but still some appearance is present)?

If they perceive a hallucinated tree and believe it is the same as a non-hallucinated tree, this is a failure of interpretation, not of “appearance.” They still indeed perceived something and that something is real, it reflects something real in the physical world. If they correctly interpret it as a different category of objects than a non-hallucinated tree then there is no issue.

There’s no such thing as “subjective experience,” again the argument for this is derived from a claim that reality is entirely independent of one’s point of view within it, which is just a wild claim and absolutely wrong. Our experience doesn’t “contain” the physical world, experience is just a synonym for observation, and the physical sciences are driven entirely by observation, i.e. what we observe is the physical world. I also never claimed “the experience of redness is the same thing as some pattern of neurons firing in the brain,” no idea where you are getting that from. Don’t know why you are singling out “redness” either. What about the experience of a cat vs an actual cat?

There is no “hard problem.” It’s made up. Nagel’s paper that Chalmers bases all his premises on is just awful and assumes for no reason at all that physical reality is something that exists entirely independently of one’s point of view within it, never justifies this bizarre claim and builds all of his arguments on top of it which then Chalmers cites as if they’re proven. “Consciousness” as Chalmers defines it doesn’t even exist and is just a fiction.

I feel like this is no different practically speaking than just saying its behavior is random, but anthropomorphizing it for some reason.

If there is an agent who is deciding it then that would show up in the statistics. Unless you’re saying there exists an agent who decides the outcomes but always just so happens to very conveniently decide they should be entirely random. lol

Many-worlds is nonsensical mumbo jumbo. It doesn’t even make sense without adding an additional unprovable postulate called the universal wave function. Every paper just has to assume it without deriving it from anywhere. If you take MWI and subtract away this arbitrary postulate then you get RQM. MWI - big psi = RQM. So RQM is inherently simpler.

Although the simplest explanation isn’t even RQM, but to drop the postulate that the world is time-asymmetric. If A causes B and B causes C, one of the assumptions of Bell’s theorem is that it would be invalid to say C causes B which then causes A, even though we can compute the time-reverse in quantum mechanics and there is nothing in the theory that tells us the time-reverse is not equally valid.

Indeed, that’s what unitary evolution means. Unitarity just means time-reversibility. You test if an operator is unitary by multiplying it by its own time-reverse, and if it gives you the identity matrix, meaning it completely cancels itself out, then it’s unitary.

If you just accept time-symmetry then it is just as valid to say A causes B as it is to say C causes B, as B is connected to both through a local causal chain of events. You can then imagine that if you compute A’s impact on B it has ambiguities, and if you compute C’s impact on B it also has ambiguities, but if you combine both together the ambiguities disappear and you get an absolutely deterministic value for B.

Indeed, it turns out quantum mechanics works precisely like this. If you compute the unitary evolution of a system from a known initial condition to an intermediate point, and the time-reverse of a known final condition to that intermediate point, you can then compute the values of all the observables at that intermediate point. If you repeat this process for all observables in the experiment, you will find that they evolve entirely locally and continuously. Entangled particles form their correlations when they locally interact, not when you later measure them.

But for some reason people would rather believe in an infinite multiverse than just accept that quantum mechanics is not a time-asymmetric theory.

cuz I would immediately die?

“Free will” usually refers to the belief that your decisions cannot be reduced to the laws of physics (e.g. people who say “do you really think your thoughts are just a bunch of chemical reactions in the brain???”), either because they can’t be reduced at all or that they operate according to their own independent logic. I see no reason to believe that and no evidence for it.

Some people try to bring up randomness but even if the universe is random that doesn’t get you to free will. Imagine if the state forced you to accept a job for life they choose when you turn 18, and they pick it with a random number generator. Is that free will? Of course not. Randomness is not relevant to free will. I think the confusion comes from the fact that we have two parallel debates of “free will vs determinism” and “randomness vs determinism” and people think they’re related, but in reality the term “determinism” means something different in both contexts.

In the “free will vs determinism” debate we are talking about nomological determinism, which is the idea that reality is reducible to the laws of physics and nothing more. Even if those laws may be random, it would still be incompatible with the philosophical notion of “free will” because it would still be ultimately the probabilistic mathematical laws that govern the chemical reactions in your brain that cause you to make decisions.

In the “randomness vs determinism” debate we are instead talking about absolute determinism, sometimes also called Laplacian determinism, which is the idea that if you fully know the initial state of the universe you could predict the future with absolute certainty.

These are two separate discussions and shouldn’t be confused with one another.

When Democrats win next election they will spend the whole election carrying on Trump’s policies and therefore normalizing them. Republicans go farther right, people like you say that’s proof they’re worse and we should back the Democrats, Democrats win and do all the same policies and therefore normalize it, which then allows Republicans to go farther right.

holocaust deniers should not bother replying because they will just get the instant block

Using the term “tankie” while defending a party carrying out a modern day holocaust is quite something.

This is why I very much dislike Popperism. Popperites are convinced “science = falsifiability.” If I argue that the universe is made of cheese and the mechanism is a wizard that you can only see through a telescope with a special handcrafted ruby lens that I sell at my shop for $4000, should research institutions be expected to take my claim seriously and buy my ruby lens to test it? I mean, it’s technically falsifiable, either they will look through the lens and see the wizard and universe of cheese or they will not. If you are a Popperite you have no choice but to admit that it is a legitimate scientific theory.

There should be more to a scientific proposal than it technically being “falsifiable.” Penrose’s “theory” is quantum mysticism, it is not a scientific theory just because it is in principle testable.

1. He bases it on a claim that Godel’s theorem shows certain things are non-computable but we can choose to believe those non-computable things anyways, therefore that proves “consciousness” is non-computable. This is just a comically ridiculous argument. You can program an AI to believe in things it cannot prove as well. It doesn’t prove anything.
2. He claims that there is a physical collapse of the wave function, with zero evidence to back it, and it is caused by gravity. His theory is incredibly speculative and not compatible with the predictions of quantum mechanics and not even with special relativity, and all attempts to test it have turned out negative.
3. He claims that since this “collapse” isn’t computable and his comically bad argument #1 shows “consciousness” isn’t computable, therefore quantum mechanics causes consciousness, and so we should search desperately for anything in the brain that looks vaguely quantum mechanical as “evidence.”

It’s even more ridiculous when you realize that microtubles are structural, they don’t play a role in information processing in the brain, and you have microtubules all throughout your body. Them having quantum effects in them is meaningless. Even if you could empirically demonstrate without a shadow of doubt that microtubules do somehow create coherent quantum states that the brain makes use of, that would just be an interesting fact on its own. It would not prove #1 or #2. Microtubules are not a “mechanism” for #1 and #2, even if they played a role in decision making as if the brain is a quantum computer (they don’t), then you cannot derive from this that somehow quantum mechanics explains why people can believe things without proof (why do I even have to say this, it’s so stupid!) or that the reduction of the wave function is a physical process caused by gravity.

There is no good argument to believe even #1 or #2 are tied together. Even if you proved there is indeed a non-local physical collapse and overturned all our modern scientific theories, that wouldn’t demonstrate #1 or #3 either. None of the claims in the theory have any obvious connections to one another other than spurious, largely incoherent arguments. This is not his domain of expertise. You could argue #2 is within his domain, but #1 and #3 are nowhere near his domain.

Physicists have proposed speculative physical collapse theories before, like GRW, and we forget about them because they were interesting but went nowhere because there is zero evidence “collapse” is a physical process, and treating it as such requires overturning all of modern physics, as it could not be made compatible with special or general relativity nor could it reproduce the predictions of quantum mechanics, requiring you to rewrite all of physics from the ground-up. The reduction of the wave function is a measurement update, it is epistemic, there is no evidence that it is a physical process.

Even then, theoretical physicists speculate about a lot of things that turn out to go nowhere, that itself is par for the course. But Penrose goes above and beyond this and branches into philosophy, biology, and neuroscience and starts using comically bad arguments to try and tie them all together. Those are not his areas of expertise at all. It reminds me of the old essay Natural Science and the Spirit World from the 19th century that documented a lot of renowned scientists who also had completely crazy side projects, like Alfred Wallace, the guy who co-discovered evolution by natural selection, who believed he could also raise spirits from the dead and converse with them.

Yes they are both particles and waves, but “collapse” is also purely a mathematical trick and isn’t something that physically occurs. Quantum theory is a statistical theory and like all statistical theories, you model the evolution of the system statistically up until it gets to the point you want to make a prediction for. But state vector notation (the “wave function”) is just a mathematical convenience that works when you are dealing with a system in a pure state that is only subject to Schrodinger evolution. It doesn’t work when a system undergoes decoherence, which follows the Born rule, and that says to compute the square magnitude of the state vector. But if you compute the square magnitude of the state vector, you get a new vector that is no longer a valid state vector.

Conveniently, whenever a system is subject to decoherence/Born evolution, that happens to be a situation when you can acquire new physical information about a system, whereas whenever it is subject to Schrodinger evolution, that corresponds to a situation when you cannot. People thus do this mathematical trick where, whenever a system undergoes decoherence/Born evolution, they take pause their statistical simulation, grab the new information provided about the system, and plug it back into the state vector, which allows them to reduce one probability amplitude to 1 and the rest to 0, which gives you a valid state vector again, and then they press play on their statistical simulation and carry it on from there.

This works, yes, but you can also pause a classical statistical simulation, grab new information from real-world measurements, and plug it in as well, unpause the simulation, and you would also see a sudden “jump” in the mathematics, but this is because you went around the statistical machinery itself into the real world to collect new information to plug into the computation. It doesn’t represent anything actually physically occurring to the system.

And, again, it’s ultimately just a mathematical trick because it’s easier to model a system in a pure state because you can model it with the state vector, but the state vector (the “wave function”) is simply not fundamental in quantum mechanics and this is a mistake people often make and get confused by. You can evolve a state vector according to Schrodinger evolution only as long as it is in a pure state, the moment decoherence/Born evolution gets involved, you cannot model it with the state vector anymore, and so people use this mathematical trick to basically hop over having to compute what happens during decoherence, and then delude themselves into thinking that this “hop” was something that happened in physical reality.

If you want to evolve a state vector according to the Schrodinger equation, you just compute U(t)ψ. But if you instead represent it in density matrix form, you would evolve it according to the Schrodinger equation by computing U(t)ψψᵗU(t)ᵗ. It obviously gets a lot more complicated, so in state vector form it is simpler than density matrix form, so people want to stick to state vector form, but state vector form simply cannot model decoherence/Born evolution, and so this requires you to carry out the “collapse” trick to maintain in that notation. If you instead just model the system in density matrix form, you don’t have to leave the statistical machinery with updates about real information from the real world midway through your calculations, you can keep computing the evolution of the statistics until the very end.

What you find is that the decoherence/Born evolution is not a sudden process but a continuous and linear process computed with the Kraus operators using ΣKᵢ(t)ρKᵢ(t)ᵗ and takes time to occur, cannot be faster than the quantum speed limit.

While particles can show up anywhere in the universe in quantum mechanics, that is corrected for in quantum field theory. A particle’s probability of showing up somewhere doesn’t extend beyond its light cone when you introduce relativistic constraints.