Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
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.
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
-- 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.
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.