Mathematical proof debunks idea the universe is a computer simulation(phys.org)
phys.org
Mathematical proof debunks idea the universe is a computer simulation
https://phys.org/news/2025-10-mathematical-proof-debunks-idea-universe.html
26 comments
It's also very hard to verify the sources for some claims: I would expect the snag to be that many model theory results we have (Such as Gödel theorems) require quantifying over an infinite set, but that seems plausibly not possible to model in the physical universe. I quickly found this quote from the paper:
> Arithmetic expressiveness; LQG can internally model the natural numbers with their basic operations. This is important as quantum gravity should reproduce calculations used for amplitudes, curvature scalars, entropy, etc in appropriate limits. Both string theory [34, 37] and LQG [35, 38] satisfy this by reproducing GR and QM in appropriate limits
Here the citations are four entire books. How am I supposed to very that LQG can model N with that?
> Arithmetic expressiveness; LQG can internally model the natural numbers with their basic operations. This is important as quantum gravity should reproduce calculations used for amplitudes, curvature scalars, entropy, etc in appropriate limits. Both string theory [34, 37] and LQG [35, 38] satisfy this by reproducing GR and QM in appropriate limits
Here the citations are four entire books. How am I supposed to very that LQG can model N with that?
Here's a dumber argument: suppose you simulate a Newtonian universe in a computer. We do this at a coarse scale all the time. Now, suppose we dedicate a few percentage of solar output to this project and out pops functioning artificial life that can think more or less like we do. Such an "organism" would be able to discover Gödel incompleteness just as well, and thus eventually conclude via the same chain of logic as this paper that the simulation hypothesis is false. While inside a simulation.
Sure, I'm assuming here that nothing Gödel's brain did is fundamentally non-computable, but that's a pretty easy lift I think. Math is hard but it's not that hard.
Sure, I'm assuming here that nothing Gödel's brain did is fundamentally non-computable, but that's a pretty easy lift I think. Math is hard but it's not that hard.
For what it's worth, while I find this "obvious" as well, given the Church-Turing Thesis etc, Nobel Prize-winning physicist and philosopher Roger Penrose famously does think that human brains require access to non-computable insights to do math.
appear to the a-life Moses with tablets with the non-simulation proof written in Lean. They don't need insights to verify that proof, just a computer running inside the simulation.
Or just start simulating QM on a limited basis, just inside their brains. You might need to run evolution for another few million years until they start taking advantage of whatever Penrosian effects there are.
Or just start simulating QM on a limited basis, just inside their brains. You might need to run evolution for another few million years until they start taking advantage of whatever Penrosian effects there are.
He, however, seems to hold the minority view under Nobel prize winning physicists on this subject.
This is not really necessary tho; it only requires that the mathematical model has a certain arithmetic complexity. The usual demo is Robinson Arithmetic, which is addition, multiplication on the natural numbers, and a successor operation.
Godel then latches onto that to create an alphabet of the symbols which then are mapped to numbers; thus formulas are even bigger numbers, and derivations are even bigger bigger numbers. So for any statements there should be a derivation that prove the statement is true or a derivation that proves the statement is false. Of course most statements will be false, but even then there will be a derivation showing so.
Then Godel does some clever manipulation to show that there will be some statements for which there can be no such derivation in either way. But that does not need the physics theory to express things about itself. It only requires to be mathematically complex enough (it'd be weird if a theory of everything was simpler than Robinson Arithmetic) and that it has rules of derivation of its statements (ie, that mechanical math can be applied to deduce the truth of the matter from the first principles of the theory).
Of course, the actual undecidable godel number and the associated physical proposition would be immensely complex. But that is only cause nobody has tried to improve on Godel's methodology of assigning numbers to propositions. He used what was simpler, prime factorization, cause it was easy to reason about, but results in astronomical numbers. But there is no reason a better, less explosive way of encoding propositions could be found that made an undecidible Godel number to be translated into something comprehensible.
But this is largely unnecessary; Godel proof forces the mathematical system to speak about itself and then abuses this reflection to create a contradiction. It means the system is not complete, that there are statements in the system that cannot be proven from its first principles and derivation rules; the fact that the one Godel showed to exist is self referential does not mean all the undecidable propositions _are_ self referential. There well could be other, non self referential undecidable propositions, that could very well have a comprehensible physical interpretation.
And, regardless of the universe being a simulation or not, the physical theory will ultimately need to deal with this incompleteness.
Godel then latches onto that to create an alphabet of the symbols which then are mapped to numbers; thus formulas are even bigger numbers, and derivations are even bigger bigger numbers. So for any statements there should be a derivation that prove the statement is true or a derivation that proves the statement is false. Of course most statements will be false, but even then there will be a derivation showing so.
Then Godel does some clever manipulation to show that there will be some statements for which there can be no such derivation in either way. But that does not need the physics theory to express things about itself. It only requires to be mathematically complex enough (it'd be weird if a theory of everything was simpler than Robinson Arithmetic) and that it has rules of derivation of its statements (ie, that mechanical math can be applied to deduce the truth of the matter from the first principles of the theory).
Of course, the actual undecidable godel number and the associated physical proposition would be immensely complex. But that is only cause nobody has tried to improve on Godel's methodology of assigning numbers to propositions. He used what was simpler, prime factorization, cause it was easy to reason about, but results in astronomical numbers. But there is no reason a better, less explosive way of encoding propositions could be found that made an undecidible Godel number to be translated into something comprehensible.
But this is largely unnecessary; Godel proof forces the mathematical system to speak about itself and then abuses this reflection to create a contradiction. It means the system is not complete, that there are statements in the system that cannot be proven from its first principles and derivation rules; the fact that the one Godel showed to exist is self referential does not mean all the undecidable propositions _are_ self referential. There well could be other, non self referential undecidable propositions, that could very well have a comprehensible physical interpretation.
And, regardless of the universe being a simulation or not, the physical theory will ultimately need to deal with this incompleteness.
Godel's proof relies on the self-referential nature of the Godel sentence; without that, his theorem does not apply. Generally you need arithmetic, but also (something equivalently expressive to) universal quantification. Physical theories do not need to include that.
Note Godel's proof is mechanically exactly analogous to Turing's proof of the undecidability of the halting problem, because ultimately it's the same thing (Curry-Howard, Prolog, and all that). So you can bypass arithmetic, but you can't really bypass self-reference; just like programming languages need some looping or recursion (or equivalent expressiveness) to be Turing-complete, mathematical theories need universal quantification to be subject to Godel's Incompleteness Theorem.
Of course, you can have a physical theory that _is_ Turing-complete, say the Newtonian billiard ball model (and, y'know, we can build computers); but that doesn't mean the theory will necessarily tell you, as a static, measureable physical fact, whether a particular physical process (say, an n-body system) will ever halt or loop, or go on forever with ever-increasing complexity; so you could (in principle, in Newtonian mechanics) build some (mechanical!) physical system that simulates the Goldbach conjecture, or looks for solutions to an arbitrary Diophantine equation, but if there are no integer solutions you'll never actually be able to show it; the theory is incomplete in the mathematical sense, but just as complete a description of reality's rules.
Note Godel's proof is mechanically exactly analogous to Turing's proof of the undecidability of the halting problem, because ultimately it's the same thing (Curry-Howard, Prolog, and all that). So you can bypass arithmetic, but you can't really bypass self-reference; just like programming languages need some looping or recursion (or equivalent expressiveness) to be Turing-complete, mathematical theories need universal quantification to be subject to Godel's Incompleteness Theorem.
Of course, you can have a physical theory that _is_ Turing-complete, say the Newtonian billiard ball model (and, y'know, we can build computers); but that doesn't mean the theory will necessarily tell you, as a static, measureable physical fact, whether a particular physical process (say, an n-body system) will ever halt or loop, or go on forever with ever-increasing complexity; so you could (in principle, in Newtonian mechanics) build some (mechanical!) physical system that simulates the Goldbach conjecture, or looks for solutions to an arbitrary Diophantine equation, but if there are no integer solutions you'll never actually be able to show it; the theory is incomplete in the mathematical sense, but just as complete a description of reality's rules.
They have some explicit examples of physics explainable by quantum gravity that resolve but are undecidable, n-body thermalization being one. Of course that’s given a sort of hand wavey understanding of quantum gravity, I guess one that they say should tell us whether a system thermalizes.
EDIT: I should also mention the idea that reality can tell us if a statement about a theory is true, given that the theory is an accurate description of reality. So if there’s an accurate Turing complete theory of reality, and we see some process that’s supposed to encode a decision on an undecidable statement being resolved (I guess in a non-probabilistic way as well), then we can conclude that reality is deciding undecidable statements in some nontrivial way.
EDIT: I should also mention the idea that reality can tell us if a statement about a theory is true, given that the theory is an accurate description of reality. So if there’s an accurate Turing complete theory of reality, and we see some process that’s supposed to encode a decision on an undecidable statement being resolved (I guess in a non-probabilistic way as well), then we can conclude that reality is deciding undecidable statements in some nontrivial way.
Note that in general, a physical instantiation of an undecidable problem must be specified/realized to _infinite_ precision; that is, for any such system S, and for any eps>0, there is a perturbation p with distance d<eps (eg, move a billiard ball an arbitrarily small amount) that is provable; this is analogous to the fact that existence of solutions to Diophantine equations is undecidable, but the theory of real closed fields is decidable, which means that the only undecidable case is when an equation has solutions _arbitrarily close_ to integers, but never quite an integer. I am not a physicist, but I don't believe any physics actually cares about infinitely-precise setups.
Integers exist in quantum physics (e.g. electron charge, spin), which is why I think quantum gravity is important to this argument. Spacetime ends up being discretizable and we can end up having rational valued physical phenomena.
> integers exist
Mostly as an abstraction on top of a continuous wavefunction/quantum field
> Spacetime ends up being discretizable
As far as I know this is speculative and usually assumed by physicists to be false; it's definitely not a required feature of quantum mechanics per se, and as far as I know not of any other well-accepted theory.
Mostly as an abstraction on top of a continuous wavefunction/quantum field
> Spacetime ends up being discretizable
As far as I know this is speculative and usually assumed by physicists to be false; it's definitely not a required feature of quantum mechanics per se, and as far as I know not of any other well-accepted theory.
> Integers are fundamental in quantum mechanics, particularly as quantum numbers that define the discrete properties of particles, such as energy levels, angular momentum, and spin.
> Quantum mechanics dictates that certain properties, like energy and angular momentum, are quantized, meaning they can only exist in discrete packets or "quanta".
This was from a cursory google search.
> Quantum mechanics dictates that certain properties, like energy and angular momentum, are quantized, meaning they can only exist in discrete packets or "quanta".
This was from a cursory google search.
The universe is a simulation, question is whether we're running on original hardware or not.
Seems at best they may have proved you can't simulate the universe on hardware that exists within this universe, which is a bit of a no-duh kinda thing.
Imagine running a simulation in our universe and using a hardware random generator. And AI mathematicians inside your simulation proclaiming confidently that it would be impossible for them to be in a simulation because all randomness must be algorithmic and thus impossible to generate such randomness.
Seems at best they may have proved you can't simulate the universe on hardware that exists within this universe, which is a bit of a no-duh kinda thing.
Imagine running a simulation in our universe and using a hardware random generator. And AI mathematicians inside your simulation proclaiming confidently that it would be impossible for them to be in a simulation because all randomness must be algorithmic and thus impossible to generate such randomness.
A simulation to what purpose ? What's the difference between a simulation and reality ? What is more "real" about the original hardware than about us ?
It doesn't make obvious sense to waste such vast amount of energy for absolutely no purpose that we can observe: surely you could add a grain of doubt in your absolute statements no ?
The universe is probably nothing very interesting, and reality depressingly obvious once we figure it out. It's like all these astrological sky maps they were doing to predict the mood of the gods above, before we realized we were a rock turning around an hydrogen ball, itself turning around the galactic center, completely non-special, like every other block of rock out there.
Being in a simulation is just a way for you to replace god with a machine equivalent. You want a purpose, a father figure, an observer and a designer. Sadly, I think you and I are as precious as an ant, completely not part of any simulation, having no purpose and barely any effect on the universe. We're here because we could and we'll disappear because we must, with barely a blip, while everything in the universe is turning around senselessly.
It doesn't make obvious sense to waste such vast amount of energy for absolutely no purpose that we can observe: surely you could add a grain of doubt in your absolute statements no ?
The universe is probably nothing very interesting, and reality depressingly obvious once we figure it out. It's like all these astrological sky maps they were doing to predict the mood of the gods above, before we realized we were a rock turning around an hydrogen ball, itself turning around the galactic center, completely non-special, like every other block of rock out there.
Being in a simulation is just a way for you to replace god with a machine equivalent. You want a purpose, a father figure, an observer and a designer. Sadly, I think you and I are as precious as an ant, completely not part of any simulation, having no purpose and barely any effect on the universe. We're here because we could and we'll disappear because we must, with barely a blip, while everything in the universe is turning around senselessly.
> "Therefore, no physically complete and consistent theory of everything can be derived from computation alone. [...]"
It wasn't stated why all truths need to be provable though. Perhaps the paper goes into this detail that I'd like explained.
It wasn't stated why all truths need to be provable though. Perhaps the paper goes into this detail that I'd like explained.
I think you nailed it. There are true statements which cannot be proven, yes. But those could be true statements also in the simulated universe. The simulation does not need to calculate the correct ANSWER to every question.
The incompleteness theorem doesn't even say that correct answers can't be computed, only that they can't be proven to be correct using the axioms of a consistent system.
Good point. But you can say there is always the question "Can this be proven with our rules of logic?"
It's a curious thing, I wonder how to precisely define "proving without logic". Proof by "meta-thinking"?
It's a curious thing, I wonder how to precisely define "proving without logic". Proof by "meta-thinking"?
Feels like the Halting Problem all over again...
Would be fascinating if true, but I'd be very curious what kind of model of "reality" they actually have.
The paper itself [1] seems quite compact and extremely high level, so I'm sure some heavy hitters would try to reformulate it. Would be the most unintuitive thing to happen since Bell's theorem [2].
[1] https://jhap.du.ac.ir/article_488.html
[2] https://en.wikipedia.org/wiki/Bell%27s_theorem
The paper itself [1] seems quite compact and extremely high level, so I'm sure some heavy hitters would try to reformulate it. Would be the most unintuitive thing to happen since Bell's theorem [2].
[1] https://jhap.du.ac.ir/article_488.html
[2] https://en.wikipedia.org/wiki/Bell%27s_theorem
Even if they prove our universe can’t be simulated in a computer built the way we build them, how can they prove there aren’t other ways to build computers?
Yea I mean a more generic version of the simulation theory is just that there is an "outside world" within which our universe exists in containment. Seems probably impossible to disprove (or prove) that for the same reason that proofs about the existence of God are hard.
But, making proofs about the capabilities of the exact types of computation we currently use can still be interesting.
But, making proofs about the capabilities of the exact types of computation we currently use can still be interesting.
The way we build computers can’t simulate quantum fields. Just means our computers are limited.
Doesn’t mean the universe isn’t a simulation.
Everything you perceive is through the brain. Brain could be in a jar receiving the same neuron signals, it wouldn’t be able to know if it is in a simulation or not.
There is no way for a program to know if it’s inside a virtual machine or not.
Doesn’t mean the universe isn’t a simulation.
Everything you perceive is through the brain. Brain could be in a jar receiving the same neuron signals, it wouldn’t be able to know if it is in a simulation or not.
There is no way for a program to know if it’s inside a virtual machine or not.
There is no disagreement about that. But the simulation hypothesis claims that is is more likely that what we perceive as reality is a simulation, run by someone else, than that we are in "real" world.
I think it is worse hypothesis than even Drake's equation - there, at least we know all factors, we just have no idea what their value is and with sample size of 1, the current uncertainly is like 40+ orders of magnitude. That still brings us closer to "is there inteligent life in universe" answer than we ever got with "is this universe a simulation".
I think it is worse hypothesis than even Drake's equation - there, at least we know all factors, we just have no idea what their value is and with sample size of 1, the current uncertainly is like 40+ orders of magnitude. That still brings us closer to "is there inteligent life in universe" answer than we ever got with "is this universe a simulation".
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The central claim in particular is not proven because a physical theory P need not be able to express statements like "there exists a number G, which, when interpreted as the text of a theory T, essentially states that the theory T itself is unprovable in the broader physical theory P" as an empirical physical fact.