Quantum theory based on real numbers can be experimentally falsified
nature.com
nature.com
Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the time. Dirac figured he needed some 4×4 matrices in order to be able to take the square root of some operator at some point, which is how his equation predicted the existence of antimatter (because it implied the wavefunction had to be a vector with more components - some of the other components turned out to be antimatter).
But complex numbers happen to have the right properties that at least we can do without matrices and vectors for the lowest-level quantities in quantum mechanics: the state amplitudes or the values of wavefunctions of spinless, non-relativistic particles. This article is saying that you can't do away with these properties at the lowest level of quantum mechanics, whether or not you actually use complex numbers to represent them.
Because they encode phase information. Also because they come about in the solution of differential equations.
Physicists often talk about amplitudes, but I never hear them talk about phase. There was one paper that I can't find, complete with a diagram that suggested (to me) that phase was determining quite a bit.
The whole point of the notational soup that one finds e.g. in an extended field theory is that it correctly generates a lot of these details "automatically". It's obtuse and makes doing simple things hard, but makes showing non-trivial relationships that are true in general very much easier than the alternative ;-).
I haven't read through the paper, but this statement from the abstract confuses me:
> Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements.
Maybe I'm interpreting "real numbers" and "need" differently than the authors, since in my head complex numbers are basically just a structure containing two real numbers and some modified behavior.
If i remember, you don't get any other fields past this. No R^3,..,R^n. I also think this search for fields led to discovery of quaternion which doesn't have commutativity but is very close to being a field. So I find C as a field to be special (if not a useful distinction).
> basically just a structure containing two real numbers and some modified behavior
I think the issue might be that you're brushing away what is central to their utility and interestingness? Yes, you can take the view that it's a 2D euclidean vector space but it's not just. It's a 2D commutative algebra over the reals, an algebraically closed field and its algebraic properties and the addition of a notion of a rotation operation to our concept of number is what's of central importance.
Decades since I took Complex Analysis, but:
Not if you care about poles, zeros, residuals, cuts, conformal mappings, multiple layers of some sort overlaying the same point on the complex plane.
It goes way beyond "declaring two variables" vs. "declaring a struct containing two variables." Not just a representation issue.
polynominal in x == "flatland" view
I looked at the appendix of the paper and their claim seems to hinge on some form of non-decomposibility of a certain tensor product state? It is the paragraph below equation A2.
More generally I got a bit frustrated reading the paper because the axioms of real quantum mechanics did not seem to be properly formulated.
> "Quantum theory based on real numbers" means a specific thing -- quantum mechanics with real amplitudes (and real anything-else-that-would-follow-from-that)
I'm familiar with complex math as far as remedial DSP and electrical engineering goes, so this may be over my head. I'm not sure what a real amplitude is, since generally when I hear "complex number", my head thinks "compact way to represent a frequency, amplitude and phase", all of which are real. It seems like I was reeled in with familiar-sounding terminology that may in fact have a deeper meaning in this context.
The same is not true for QM: the wave part / the complex numbers are fundamental.
Also note, when people say "complex numbers" they refer to the algebraic field, which is the object composed of [a pair of reals, complex addition, complex multiplication]. In particular, complex multiplication is the key here, since it is what differentiates complex numbers from 2-vectors. To drive this point further, while 1-vectors, 2-vectors, 3-vectors, 4-vectors etc behave very similarly, only the reals and the complex numbers can form a field - there are no "numbers" in the same sense formed of 3-tuples or 4-tuples or other n>2-tuples of reals*.
This is why the discussion focuses on the complex numbers (meaning, again, not just a pair of reals, but also the particular +,-,*,/ operations for them that make them a field).
* to be fair, quaternions come pretty close - you can define a 4-tuple of reals + addition + multiplication that is almost a field, except that multiplication is anti-commutative instead of being commutative (p*q = -q*p).
edit: I'm still a bit unclear on what it means to form a field. Texts around this topic seem pretty dense and encyclopedic. Is there a straightforward explanation of what an algebraic field is?
The idea is that it's a collaborative game for three people, you are trying to work together to beat the rules of the game. Meanwhile the rules are trying to set you up so that one of the people betrays the other two. In 3 relativistically separated rooms (so they can’t communicate) they go, where they find a screen and buttons labeled 0 and 1. The screen displays a prompt, each teammate presses exactly one of the buttons once before time runs out, then the three numbers pressed get summed together into a number.
25% of the time we run a “control round,” everyone gets a prompt to make the sum of their numbers even, and they win if the sum is even. The easiest way is if everyone hits 0, 0+0+0 is even. But a team can also answer 0+1+1 or so and win. Otherwise we randomly choose one to be the traitor and send them the control prompt, to make the sum even. But we send the other two the prompt to make the sum odd! In this case the team will only win if their joint sum is odd.
Long story short, classical players of this game have a success probability bounded from above by 75%. This is the Bell inequality. But quantum capable players can walk in with a GHZ state,
|+++> + |–––>,
The more interesting thing, where i really matters, comes during the traitor rounds. Here you want to perform the phase rotation gate in the Hadamard basis,
|+> → |+>,
|–> → i |–>,
|+++> – |–––>,
I suspect that the inequality here is something similar, quantum mechanics but you can only form real-coefficient superpositions, and therefore you cannot take the square root of a unitary transformation just by doing it for half the time, per Schrödinger.
John Preskill has some lectures on quantum computing for the University of Waterloo I believe, also a Hans Bethe lecture at Cornell. If you are just looking for an hour's commitment to understand a little better, I would go with one of those.
If you wanted an actual textbook, Nielsen and Chuang is very popular... The other place I would look would be OpenCourseWare, you might be able to find some good problems to work on there. Sometimes video lectures can be good if you can pause the video right after a problem was introduced and try to solve it yourself before you get the answer from the professor.
The difficulty in being an autodidact is, listening to stories around a campfire is deep in our bones, it makes us feel good. But it's not a very efficient way to learn. So there is a mismatch where watching a TED talk feels like you have just changed everything, but then if I come to you a month later probably nothing has changed.
Text is a lot faster, as a medium. Way slower to write but seekable, skimmable, can contain links to previous sections... I'm pretty sure we also retain more of it. But that's not the main problem with videos/TED talks. Like, the text form of TED talks is someone telling you how monads are burritos and that makes it all better.
It's too clean?
Good learning is messy. A good abstraction allows you to clean up a mess of confusion in your head. This confused mess can only exist if you have created it. So you have to do lots of examples, exercises, memorize strange times when you have been wrong about things and your expectations don't align with the problem domain... If you think about learning a language, there is that phase where you don't know which thing to use when and your words are all out of order in the sentence... Per Ira Glass the only way to improve is to do lots of work, put yourself on a schedule, grind through mediocrity. The TED talk/monad tutorial fallacy is that we can give our children an easier time than we had it. It's BS. We can't. “I made so many mistakes, I will help you so that you don't have to deal with that pain” blithely unaware that the pain was how you learned it, that learning is pain.
Sorry, didn't mean to rant and now it seems awkward to delete it.
Instead of preparing |+++> + |–-->, you prepare the state (|+++>|x> + |–-->|x>) Here, |x> = |000>-|011>-|101>-|110>, and one qubit is sent to each player.
That is, you give each of Alice, Bob and Charlie an extra qubit. They can now measure in the computational basis on both qubits. And in the betrayal round two of the players can perform the orthogonal transformation id \otimes J, (controlled on having |->) where J = {{0,-1},{1,0}}. You can check that whenever exactly two players perform this operation on their systems you get back the state (|+++>-|--->)|x>, and thus your previous strategy works.
This simulation strategy for any full multipartite causal structure is described in arXiv:0810.1923. What OP has shown (roughly) is that three players connected as in A <-> B <-> C (where <-> is some shared randomness or quantum state) then this simulation breaks, and indeed there is a gap between what you can achieve in real and complex quantum mechanics.
To a computer programmer, the fundamental difference between complex numbers is they can express infinite repetition succinctly. Now I try to write down what that means precisely, it's hard. An example of the effect is the polynomial for sin() is infinite, or you can express it as sin(x) = (e^(ix) - e^(-ix))/2i using complex numbers.
To a computer programmer who makes his living from writing down formulas, the difference between having to write an infinite amount of code to express a concept or just use 20 characters above is profound. Different people find their profundity in different places, I guess - but this the sort of thing that drives us programmers to create entire new computer languages.
That's not to say the primary observation I see being made here is wrong. That observation is that there complex numbers bring nothing really new to the table. You can do everything they do in other ways, say with matrices and a few extra rules. And indeed, mathematicians have come up with numerous other ways of expressing iteration, ∫ and Σ springing to mind.
But nonetheless, the hyperoperations (counting, addition, multiplication, exponentiation, ...) are special. They are most heavily used mathematical operations, by a huge margin. They have one job - to capture the operation we programming nerds call iteration. But when restricted to real numbers, (I'm no mathematician, so this is conjecture), they can't capture self similarity. The result of function expressed as a finite number of hyperoperations operating on reals always flys off to infinity, or asymptotically approaches a constant, or is undefined over part of it's domain. When you add complex numbers you get another possibility: a possibly intricate pattern repeated infinitely.
So yes, complex numbers are basically just a structure containing two real numbers and some modified behaviour, and these is nothing special about that. But there is something profound in the way they allow the planets favourite mathematical operations to finitely express infinitely more behaviours - with no more lexical overhead.
Of course you can put real numbers at the base of nearly anything, but it the theory operates at a fundamental level on items that are more complex then real numbers, then it's not based on real numbers.
You might as well argue it's based on surreal numbers or Dedekind cuts at that point.
Somewhere Feynman has a quote that qm was the first theory that required complex numbers. Finding and understand that quote should explain this better.
In the same sense I did not see reference to General Relativity in the article -- as another fundamental ground in physics besides QM. Never learned manifolds stuff for GR, so I don't know if complex numbers become naturally essential there.
The Symmetrical Foundation of Measure, Probability, and Quantum Theories https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800...
The ABC of Physics https://www.mdpi.com/2673-9984/3/1/9/htm
Physics uses complex numbers in the first sense. There's really nothing too special about SO(2), there's an SO(n) for all n.
Whereas mathematics uses complex numbers in both senses. There is something rather special about complex numbers as the algebraic closure of the reals and it's what makes a lot of modern math tick.
Can you elaborate on this? What is an algebraic closure of the reals in higher dimensions?
The fluke is this: Euclidean space of dimension N has N(N-1)/2 rotational dimensions. If you plug 2 into that you get 2x1/2 which is 1 dimension. i.e. the rotations in 2D space look like a circle. If you add an extra dimension (the radius) you get the polar form of complex numbers.
In other dimensions this doesn't always work. In 3 dimensions we have 3x2/2 = 3 rotational dimensions, so we need a space with dimension 4 (the quaternions). In 4 dimensions we need a 6 dimensional rotation space. We just established that Cayley Dickson algebras only come in powers of 2, so it doesn't fit at all.
I'd say the same thing about any math that talks about "numbers" without defining which number system very explicitly, even at the Kindergarten level. The slop in these early abstractions is somewhat convenient but it erects serious mental barriers against more accurate abstractions.
Look at the sloppy way that many programming languages handle lossy casts for an example of the pernicious nature of the idea that "number" should have a highly intuitive meaning.
On the other hand, imagine all the nice notations/sugars we might invent that are highly intuitive and capture mathematical objects more elegantly than what we are using today. Representations are the interface between abstract concepts and brains evolved for eating, sex and lying.
Scott Aaronson has a more accessible (and humorous) article on it here[2].
Entanglement has been shown to be intimately linked to this[3] result, which is interesting given the experimental evidence[4] for entanglement.
Not my field, but I found this interesting at least.
[1]: https://arxiv.org/abs/quant-ph/0101012
[2]: https://www.scottaaronson.com/democritus/lec9.html
[3]: https://arxiv.org/abs/0911.0695
[4]: https://en.wikipedia.org/wiki/Quantum_entanglement#Notable_e...
> There's a cute little fact -- unfortunately I won't have time to prove it in class -- that the above equation has nontrivial integer solutions when n=1 or n=2, but not for any larger integers n.
I love this type of humour.
[1]: https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_...
I wonder if this has any consequence on their reasoning and conclusions.
This paper describes another set of inequalities similar to the Bell inequalities, but testing whether QM requires complex numbers or not, instead of whether there are hidden variables that can explain things like superposition.
The statement in the abstract I’m referring to is this:
> This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural.[3]
The reference [3] is to a letter by Einstein. Maybe I’m nitpicking words, but scientists tend to spend a lot of time getting the abstracts to their papers right. Any ideas why they would write it this way?
-- Apologies to Kronecker
See https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... for more.
Its a cool how few bits of pi are required to compute the circumference of the earth to within an atoms width accuracy. But equally cool how bad it gets if you try the same to compute the position of earth 2^64 years later.
Planetary orbits are chaotic. Long before your imprecision in pi is going to significantly mislead you, shifts in mass due to, for example, earthquakes and weather patterns are going to cause orbits to be impossible to predict.
There are theoretical systems where the exact value of pi matters. But no physical system is going to match that, and measurement error is going to quickly exceed calculation errors from pi.
If you define pi as a specific constant with limited precision - because "that's all physical systems need" - you lose insights into the web of relationships around it.
This is a bad thing and makes many kinds of math harder.
It's the conceptual equivalent of lossy data compression. You don't want to do it unless you really, really need to. And if you do it, you need to be aware that you're now using approximations instead of abstractions, and those are not the same thing.
The bigger problem is analyzis will not work, probably. Can you do analyzis with rationals? E.g. f(x)=x^2 isn’t continuous at f(x)=2/1.
I’d theorize that there is a way to do finite calc without giving up on “reals”, by using enough FT coefficients (and packing this complexity into sine waves), but it’s the same sort of cheating probably, unless pi is the “origin” number of all “physical” reals.
(I’m not a physicist nor math guy, so one can reframe my ideas as questions instead.)
Different basis have different advantages, same as different function basis. The classic example would be that in a standard basis, its easy to add and subtract, but more costly to multiply, divide or factorize, while in the prime basis the former is expensive as hell, but the latter is trivial.
As a result, rationals and pi in a sense disjunct domains. You cannot express either using less less than an infinite number of the other, and the same holds for combinations of a rational and pi. Numbers which behave this way relative to each other are more common than the rationals, and pi is just the most common example.
It does lead to a rather neat requirement for the fundamental physical constants though.
The reasoning goes like this, imagine that a model K2 of physics could be described using two constants, a, b. gravity and the speed of light say. Now lets say we managed to prove that a = 2b and therefore that everything predicted by model K2 can also be predicted by model K1, which just uses the coefficient b. K2 is equivalent to K1, sure, but only one constant would then be fundamentally required, and if K2 is sufficient to describe all of physics, physics would only have one fundamental constant. The same reasoning would hold if a=b^2, and so on. But, if the function required to express a as a function of b requires infinite information, this does not meaningfully apply, as this will always apply to every pair of numbers. Meaning that we know that if the fundamental constants of a model of physics does not lie in disjunct domains in the sense above, there is a simpler version which has fewer constants. For example, since pi has infinite information, if a=pi b, then the simplification cannot be meaningfully made without introducing pi as a fundamental. More generally this also fundamentally means that true physics cannot be expressed using finite precision if ideal grand unified theory as more than one fundamental constant.
Anyway, it’s a stupid layman’s theory. I’m pretty sure that a parallel/on-demand digit computation in R-based models is much easier than trying to get rid of philosophically infinite things which in practice do not matter that much.
There is no way to test our theories in practice that doesn't at some point involve comparing this finite precision prediction with a finite precision observation.
Pontificating about the failures of a discrete approximation to be able to be computationally accurate in a continuous world may be fun armchair philosophy, but CANNOT be useful scientifically. Because we can only measure and work with finite precision approximations to that hypothesized continuous reality.
In 2^64 years, Earth will be inside of a larger body. How many digits of Pi you need to predict that?
How do you know that in the first place? Maybe with rationals science it “will not”, so there is “no problem” at all.
I guess my argument is, since you can always just pick a rational approximation to Pi, you cannot prove empirically that we live in a universe where more than a finite number of digits of Pi matter. That is, the mathematical irrationality doesn't really matter, physically speaking, since no experiment could ever prove that every digit in Pi actually contributes to the result.
If the universe does have ways to do this, to mix an entire irrational number into a physical outcome, that means hypercomputation is probably possible, since Turing machines definitely can't.
Your simulation might need to ask for an increasingly tighter bound on the real value of Pi. You can totally do this with no more than the usual rational numbers, but it's not equivalent to "just picking some rational approximation" and running with it, because what accuracy/precision you pick is outcome-dependent and it's always possible to request more.
There must be some digit after which no computation will ever access, because it will require more negentropy than the entire universe has to even calculate. The digits after that don't matter to the universe.
Either way though, then doesn't your model of the universe just need an extra parameter, the number of digits to care about? Seems like everything else being equal, the fewer unmotivated parameters in your model, the better. Especially because this would rely on internal details of what happens in the universe, seems unlikely to be true unless this is a simulation.
In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal.
The stronger version of the argument is that the length of a steel beam cannot be more precise(-ish) than the radius of an iron atom, so only 10-12 decimal places (in meters) are required to fully describe a steel beam's length. Likewise an actual circle's area isn't a function of Pi, but is rather a 'really large number' regular polyhedron. Which could then be approximated by a fairly pedestrian number of decimal points of pi to atomic precision.
That said e.g. orbits are rather smooth, and could probably be considered to be fairly exact w.r.t. an arbitrary reference.
You start with doing something the most correct way possible on paper and then convert that into the fastest possible method within your allowable bounds on precision and/or convergence. Operational reordering to keep additions in floats with similar exponents is great but you save that concern until it’s time to crunch numbers. When you’re trying to build an entire theory on how something complex works you’ll have a much better time using the available abstractions to manage complexity without getting bogged down in implementation details.
Edit: addressing your point more directly, numerical computation itself must necessarily be done over fixed precision numbers but the tools we use to decide what and how to do that computation come out of theory done over the reals because of those specific properties of the reals. You can make things work over the rationals but the theory is tedious and the results of generally lower utility.
Any issues introduced by using a finite approximation to Pi will eventually be swamped by the uncertainty in the initial conditions. If there's no uncertainty in the initial conditions, there will still be some finite approximation to Pi that will give you results as accurate as you can measure...
Now we do have tools that overcome those limitations somewhat - like limits and stuff but doesn’t remove the need for better tools.
So I somewhat disagree with both of you - we need new, better numbers that are further from math abstraction and closer to actual reality.
Otherwise it is like this story with Poynting vector from Veritasium video - only confuses instead of explaining.
You might enjoy taking courses in real & complex analysis, the general purpose of which is to impart upon the receiver an understanding of why we’ve constructed those particular number systems and how despite the names they both describe things which are perfectly real in the philosophical sense.
Edit: recall that the rationals are simply defined as the set of numbers which can be represented in the form a/b where a is an integer and b is a nonzero integer. This isn’t some deep philosophical tie to an underlying reality, it’s just our first attempt at defining more useful numbers that lie between other useful numbers we already invented (the integers et al). The real and complex numbers are literally just the extension of that process, filling in holes between useful numbers with more numbers until the set is closed (i.e. there are no more holes, every operation between members of the set results in another member of the set). Closure, the real reason we care so much about the reals, is just a surprise tool that helps us later. For anyone who made it this far: if you find any of this interesting you should find a book or lecture on analysis. It’s not particularly difficult and presents deeper insights into the math you likely already learned.
Having implemented exact real computation to better understand reals, I think of a real number as a kind of machine that generates infinite streams. Operations on them instantiate new machines which query their real operands, computating until there's sufficient information to emit a next term of the stream. When the next term needs an infinite amount of information to decide what to spit out next, it results in an "unproductive" infinite loop.
Rational numbers are interesting, more realistic, because they always terminate. In the real world, measurement tolerances and physical limits means at some point having to extract a rational. When we work with reals we are really only working with rational approximations or symbols with associated properties and relations.
Reals are a powerful and elegant tool to rigorously reason about mathematical spaces and operations on algebraic objects but trying to work with them in reality in their exact form is a fun and visceral lesson on the nature of undecidability. It's hard to go two steps without tripping over a non-terminating loop (such as any operation that starts with an irrational and results in a rational or equality testing in general).
This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number).
I feel like this ties back into the distinction between theoretical and applied math.
The basis in reality for the integers is counting discrete objects with fingers, for the rationals it's (likely) an attempt to fill in the spaces between integers using known concepts (ratios / fractions). Rationals are great if you stick to numerical work where discontinuities below epsilon can be ignored, but the rationals don't actually map to what we think of when we consider a philosophically real number system -- a discontinuous set does not match our observed experience which is that you can have any number you want between two you already have. The construction of the reals varies depending on how you want to approach it but each is equivalent: you fill in the all the holes everywhere but at the infinities so that you have a continuous closed set, just like one would intuitively expect from an infinite set of numbers representing segments of reality.
There's nothing special about the rationals which ties them more closely to reality than the reals, the rationals are just our first attempt to rigorously define all of the numbers between other numbers using the tools we had at the time.
One could just as easily construct the set $ = {x#y for all x, y in Z+} and where a#b === a + the Riemann sum of 1/(a^n) from n = 0 ... b. This also fills in some of the gaps between integers, just not enough to be interesting or particularly useful.
The rationals are interesting and stuck around because they fill in almost enough gaps to allow you to conveniently construct useful things. They're not quite there though, which is why we eventually developed the reals. And then the imaginary numbers, because despite the name physical phenomena which can be modeled using square roots of negative numbers end up presenting a compelling use case for adoption. We don't have complex numbers because some math nerd thought they were cool, we have complex numbers because they are useful in describing observed physical phenomena succinctly and as such there's enormous utility in hacking an extension onto the reals to add them.
Pulling this back around, from a theoretical perspective real & complex numbers are as real as anything else in math and are very useful to boot. You only run into issues in applied circumstances where nothing is exact and half of the things end up nondeterministic for one reason or another. Applied math requires countless shortcuts and discretionary tactics to convert things with a guarantee of correctness on the theoretical side into things which can actually be computed albeit with a correctness only within specified bounds.
Mapping between theoretical and applied math is a decent example of a pseudo one-way function, all of applied math draws from the theoretical but insights from applied math don't really map back into anything useful on the theoretical side. Which is why when we do theory and build models, we use theoretical techniques since the ability to prove correctness is the entire point. If you need numerical computation you must in exchange give up absolute correctness, which is why it is only appropriate to use during numerical computation.
> This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number).
Well, let us know if you're able to develop a falsifiable experiment one way or another. That is definitely an interesting theory, unfortunately nobody has been able to figure out a way to poke that particular are-the-numbers-real-or-just-made-up bear.
It's just that (a) the real numbers work incredibly well as a "tool" or "model", with negligible shortcomings, and it's (b) extremely tedious to think of alternative number systems that are remotely as convenient as the real numbers. So it's not clear if alternative approaches are a waste of time, but that does not mean the reals are real!
If you want to learn more, check out the references in [1].
Now granted, there is an underlying assumption that when you need to use that number you'll select an appropriate algorithm to compute it to the degree of precision you need, much like how if you were instead considering the rational number 22/7 you would need an algorithm to numerically evaluate it. We don't quibble about whether or not the universe has enough space to hold that one though because we have a simple abstraction which lets us refer to it with infinite precision and evaluate it with arbitrary precision. Just. Like. π. Yes, literally none of the reals would fit in the universe no matter how small you wrote them if you want to represent them with full precision. That is literally the point of the reals, that they are an infinitely dense field. It doesn't matter, we wield the same tools we used to construct them and refer to them by their names or by their construction.
If your definition of "based in reality" means "can be explicitly written out with full precision" then literally none of the reals or rationals are "based in reality" because for otherwise finite numbers you can keep padding zeros to the right of the decimal place and a finite universe doesn't have enough space to hold infinite objects. Taking a definition of reality that provides actual utility, the reals are clearly based in physical reality by virtue of their construction being explicitly guided by the objective of modeling reality. Just like the rationals and integers before them and the complex numbers after. They were literally created to model reality. Imaginary numbers are based in reality too, despite it being equally impossible to own sqrt(-2) and π melons. At best I will concede that there is an additional layer of abstraction between whatever "reality" is and what the real numbers are, but that's not a very interesting distinction given that humans are already running a dozen intermediate layers of abstraction in order to process their surrounding reality and then overlay math on top of it.
You just provided a great argument that π, and many other real numbers, should be part of the 'alternative number system', because they can constructed, or because they represent a finite amount of information. I agree!
> That is literally the point of the reals, that they are an infinitely dense field.
You are arguing that an alternative number system should be 'infinitely dense', and I agree. But take e.g. the finite/constructive reals [1, 2], they are still 'infinitely dense'.
> They were literally created to model reality.
That's exactly my point. Maybe approaching it from the point of view of 'what are the limitations of this model?' is helpful. Also see the discussion in [2].
This is not an argument whether real numbers are useful, a good model, or interesting (there is not doubt they are all three).
[1] https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...
> You are arguing that an alternative number system should be 'infinitely dense', and I agree. But take e.g. the finite/constructive reals [1, 2], they are still 'infinitely dense'.
I'm only arguing that insofar as one can do useful things with holomorphic functions and the constructions we require to define them require the specific defining properties of the complex numbers (continuity, closure under important functions rather than clopensure, etc). If you want these properties then you're stuck with uncountable number systems and the baggage RE representation that comes along with them.
> That's exactly my point. Maybe approaching it from the point of view of 'what are the limitations of this model?' is helpful. Also see the discussion in [2]. > This is not an argument whether real numbers are useful, a good model, or interesting (there is not doubt they are all three).
Agreed, my argument is purely that the reals are "able" to "exist" in "reality" in the same way as the rationals. One of the more famous irrationals is Pi, which is of interest precisely because it is referenced by reality. The difference between the two is that the rationals are not continuous for useful definitions of continuity and so we fix that.
That "existence" is dependent on human interpretation of that word, and there's no particular reason to expect that we have the ability to see whatever the fundamental underlying reality[1] of our world even is. You could just as easily argue that negative integers do not exist because owing someone something generally requires a reference to the entity owed rather than just a indicating a lack of meaningful possession.
There are many intelligent species here on our planet which have internal models of reality, which we know are less than correct (e.g. good luck teaching anything beyond basic intuition of classical physics to a parrot), what makes us special? There are many humans who cannot handle the abstraction of charm and flavor and spin being very much real physical properties of the invisible objects underlying the reality we're able to observe.
You can argue forever over finitism and the holographic principle and what "really exists," but such discussions are fundamentally limited by the things having them. The thing we use for this analysis (logic) is itself a human construction and certainly not something which is even as real as the number 1. Our best understanding of the underlying structure of the universe right now is that it is probabilistic, which is almost antithetical to the idea of logic being the underlying set of rules by which it operates. I see the arguments people make regarding these, but what I do not see is a meaningful distinction between 1 in Z which maps to a human concept of possession of a singular instance of an object versus Pi in R which maps to a human concept of the ratio between specific properties of certain classes of objects. Both have their basis in reality grounded by human perception, both can be used to any precision you're able to, both are meaningless beyond the human constructions used to define them. The only reason we consider math to be a universal thing likely discovered by every sufficiently intelligent species is because it is of such high utility in constructing predictions which provide an evolutionary benefit.
[0] the constructability of some of the reals is likewise unimportant, they're in the set because we deemed their existence to be of future utility. there are infinitely many reals which can never and will never be specifically referenced, their purpose is no to be directly referenced but rather to be there in the background so that we can make useful assumptions concerning other numbers surrounding them. we do not have to directly reference, or even be able to directly reference for them to have value.
[1] which is under no obligation to even be the type of thing we consider to be an "underlying reality," that's just how it seems to present itself to us
And some thoughts on the limitations with respect to creating AGI: https://twitter.com/theshawwn/status/1446261451061145602
See section 5, "Can quantum systems be probabilistically simulated by a classical computer?"
> The probability that they match is eight-tenths, the probability that they mismatch is plus two-tenths; every physical probability comes out positive. But the original f's are not positive, and therein lies the great difficulty. The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative, and that we do not know, as far as I know, how to simulate. Okay, that's the fundamental problem. I don't know the answer to it, but I wanted to explain that if I try my best to make the equations look as near as possible to what would be imitable by a classical probabilistic computer, I get into trouble.
When I was younger and only slightly more foolish, I wanted to spend a lot of time researching this to see if there was a way around the problem. I quickly realized that perhaps I should focus on adding value in a field that I was good at. :) Maybe one of you can try, since the only alternative is to take Feynman at his word.
If it turns out that spacetime is discrete and not continuous then we should be able to simulate it. The possible states of some chunk of spacetime (given some maximum/known energy) would be non-infinite and even though it might take us years to calculate each time-step, we could still make progress.
I've seen the theory kicked around before, but why couldn't the universe basically consist of plank length sized voxels and a similarly small update timestep? It would still look plenty continuous to us.
Suppose the universe was a regular grid of voxels. You could run experiments to prove that. I don’t understand the details, but that was Feynman’s counter argument. (See the “messenger lectures” series on YouTube.)
If the grid isn’t regular, you run into other problems. But iirc at one point Feynman was toying with the idea that the grid might be randomly distributed.
https://royalsocietypublishing.org/doi/10.1098/rspa.2019.035...
There also a video where he explains the paper: https://www.youtube.com/watch?v=YglT09Korr0&t=2700s
There are some consequences by using Q instead of C that can be experimentally tested.
Real numbers in physics are no less bizzare than complex numbers.
Thanks for the clarification!
I wonder if the use of complex numbers in QM theories to describe a real world that only needed real numbers inspired Asimov's 1942 short story "The Imaginary"?
In that story psychology has been developed into a hard science. In some third rate college on some backwater planet some first year psychology students were doing a lab where they ran some animals through sequences of stimuli and observing the reactions and verifying they matched what the math said should happen.
One of the animals fell asleep, which was not what was supposed to happen. It was reproducible and very specific. You run through that exact sequence of stimuli, and as soon as you hit the last one it falls asleep. Vary the order and it doesn't sleep. Vary the timing by even a tiny amount, no sleep.
Word of this got back to the galactic federation's leading psychologist. Think the Einstein of psychology. He utterly could not explain it. Eventually though he came up with equations that worked, but they involved imaginary numbers. When applying these equations to the specific stimulus sequence all the imaginary quantities squared or cancelled out and you ended up with a real result, which was that the animal would sleep.
This was controversial and caused quite an uproar in psychological circles, and while the leading psychologist was away dealing with that a couple of his students found a case where the imaginary numbers did not get squared or cancelled out. The predicted real world reaction to the stimulus sequence involved an imaginary number, and they have no idea what the heck that even means.
They try it, and what it means turns out to be that some kind of slowly expanding radiation field gets created around the animal that kills other life that spends too long in the field.
The top psychologist is called back and is able to calculate further stimuli that will stop the expansion. That works and catastrophe is averted.
Asimov in 1942 would certainly have been aware of QM and the whole "complex number theory to make real number predictions" aspect of it.