Computation as a universal and fundamental concept
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Computation is a metaphysically universal and fundamental concept, since metaphysics is (tautologically) the domain of humans and we use symbolic communication. So of course very general theories of symbolic processes (e.g. Turing machines) are pertinent to the symbolic methodology we use to understand scientific processes.
But it is a fundamental mistake to jump from that to saying computation extends to a law of the universe. Computation reflects laws of the universe, but only in the exact same way that scientific and mathematical human speech do. The mystery (still totally unsolved) is how humans are able to intuitively understand space / time / causality / etc in order to define coherent symbolic rules that reflect real processes. That computers can seemingly always implement these rules having been given the symbols is of philosophical/scientific interest, but it's solipsistic to say it's a fundamental concept of the universe.
Information is actually tangible. It’s not just an analogy or a coincidence that the word “entropy” is a word used in both physics and computer science (information theory). Thermodynamics, mind you, is perhaps THE most fundamental elements of physics and how the universe works.
That said, you are still making the same mistake I pointed out, elevating human symbolic information to a higher plateau than it deserves. I think it's because you're being too vague about the connection, when it's fairly mundane. The "fundamental connection" is that physical quantities are information, and on the other hand information is always a large collection of semi-independent semi-stochastic physical objects and can be profitably modelled by some sort of statistical mechanics. Information theory is relevant all over physics because human agents collect all sorts of physical information. The universe "doesn't care" about information in and of itself. Shannon entropy and Boltzmann entropy have similar formulas because they measure precisely the same thing; put another way, a goofy but formally equivalent way to model a gas would be a noisy radio channel communicating each molecule's kinetic energy.
The problem with the black hole information paradox isn't that information is destroyed per se, but that it appears to be destroyed in a way that violates quantum mechanics (destroying quantum state without a measurement). The theoretically predicted destruction of information points to a more general problem.
The Laundauer limit is no more fundamental to the universe than "a mechanical crane cannot violate the laws of pulleys." It says that no matter how you design your binary (or whatever) computer, it must involve an ensemble of binary states, and statistical mechanics puts an absolute floor on how little heat is required to alter such states. Whether these states are gas molecules or the written symbols "0" and "1" is immaterial.
Shannon information, sure.
However, algorithmic information (Kolmogorov complexity, etc.) is based on computation.
Information and order are effects of human perception and preference. They exist as abstractions in the mind and not in reality.
No information without an interpretation. The "amount" of information is completely dependent on the observer.
You'd think people who work constantly with abstraction wouldn't fall prey to reifying abstractions but they actually seem more susceptible to it than anyone else.
The more I learn about the fundamental nature of the electron, probability in quantum mechanics, and the wave function in general... the more information being fundamental substrate makes sense.
I'm not saying it is... just that it makes more sense the deeper you get.
Claiming that "information" is the underlying basis of the real is the same as effectively saying that "stuff" is the underlying basis of existence. It's essentially Platonism. It is not informative. What is informative is the operational use of the mathematical concept of information as defined by Shannon to understand real things or to structure theories. But this is a very different thing than identifying a substance that exists, independently of us in the universe. This "making real" of what is ultimately a mathematical abstraction is precisely reification.
Physical reality is a slippery thing. Yes, at the level of our daily life it all seems very solid and obvious, but inspecting that foundation revealed a bewildering world of relativity and uncertainty. And one of the most intriguing ways out of this bewilderment seems to be the "it from bit" trend in physics, which makes information the fundamental reality.
I personally don't feel the need to tie the notion of reality to physical phenomena. I prefer something more along the lines of Philip K. Dick's "reality is that which doesn't go away when you stop believing in it", or, as you put it, "something which exists independently of us". The fact that there are an infinite number of primes, is such a fact. Surely any alien civilization would know this fact. Maths doesn't go away when we look away, and neither does the notion of computation. Sure, Turing machines are a human construct, and there can be others as well. But the Church-Turing thesis speaks about how these constructs all ultimately describe the same underlying reality.
Bits and information are the same thing. They are a system of measurement to express relationships of difference. Difference and change as physical phenomena exist, but bits do not.
So, yes, pointing out that the limits of computation give us some useful facts about the way things change in reality is a useful perspective. Going further and saying something like "computation is the basis of reality" or similar extremes is too reductive and too abstract to really be a useful piece of knowledge in my view. Finally, to claim that information is somehow the important fundament of existence is patently absurd to me, since it would've similar to arguing that miles have some very special epistemic and ontological status.
Yes, separating symbols/descriptions from the physics is a common conflation, rightfully avoided and called out here, but I think you might be overcorrecting unconflating the concepts too much.
Quantum information is strange precisely because the theory treats information-like relationships as physically consequential.
Entanglement, path dependent stuff, cloning theorum, black hole information paradox... There are so many areas you run into where its information-centric descriptions appear to fit unreasonably well.
Again, I'm not saying the symbolic representation is "real" per se (not saying it's not either, I simply don't know what it is).
I agree bits are just nomenclature, but I don't agree that "bits" and "information" are the same thing. Like distance, "information" has a deeper and general architecture to it.
And it's also weirdly physical in places that I wouldn't expect it to be, for reasons I simply don't understand at the moment.
From Shannon's theory, it's abundantly clear that "information" is just a mathematical description of difference and expectation. I can frame anything as information. If I know who my parents are, a statement about who they are is not informative to me because it constitutes no difference. The entropy and expectation/surprise is low. If someone approaches me with a compelling argument that no, actually, my parents are not my parents I was adopted and my real parents are so and so, this is informative. The entropy is high.
Crucially, this all depends on me my personal history, and my status as an observer. It is in this sense that I find information to be a communications concept (which is how it started!) or semiotic concept more than a useful abstraction for organizing our knowledge about other parts of reality. Yes, the abstraction might still be useful if e.g. quanta end up behaving in some way according to shannon's laws, that is interesting, but to me uh at is not an argument that turns information into some kind of "universal substrate" of the real—it is just a surprising applicability of laws in one domain to another. Is it surprising that empirical data fit the model? Maybe. Idk, lots of empirical data can be made to fit all kinds of models (there are ways in which hI can reasonably take a Newtonian view of various aspects of my life and the data generally won't contravene the laws).
That's a very narrow and technical definition of information. It's what people refer to by the word when they discuss things like: communication, medium, surprise, compression, etc. in a technical context. Otherwise one should refer to the dictionary definition, so we don't end up talking past each other.
If you'd like to get a better intuitive sense of what "it from bit" points at, check out David Deutsch's Constructor Theory and the Stephen Wolfram Physics Project. Also, Bernardo Kastrup, on the philosophy front.
As the sibling comment says, no, it is like arguing that distance/space/extension has a special ontological status — which it does. "It from bit" is just using the word "bit" for the rhyme, the theory is quite profound and compelling, at least to me.
Then there's also Max Tegmark's "Mathematical Universe" idea, which makes the symbolic world the fundamental one. It goes a bit far for my taste, but it has its own internal coherence.
A rock rolling downhill has state-dependent future behavior you could describe informationally, but nothing is gained by modeling it with automata theory instead of Newtonian mechanics, and it isn't computing in any substantive sense.
Shannon asked von Neumann what to call it, and von Neumann said "You should call it entropy, for two reasons: In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage."
There is, however, a direct equivalence between them derived from the 2nd Law of Thermodynamics. https://en.wikipedia.org/wiki/Landauer's_principle
But “computation is a fundamental aspect of the universe”, in the way it's being understood in this thread (as opposed to the article) does not remain within any of those frameworks. It makes a claim about reality as a whole.
Once you make that kind of universal claim, all the assumptions built into “computation” - about identity, distinct states, lawful transitions, causation, logic, etc. - become part of the claim. You cannot use those assumptions silently and then present the conclusion as framework-independent.
And those frameworks come with consequences. Using the language of computation, they come with "lossiness" wrt modelling "reality".
(see Aristotle Metaphysics -> Kant CoPR -> later Wittgenstein)
Given the quantum fields were collapsed in the early universe, I’m now wondering how information theory looked like during the different phases on the universe
> metaphysics is (tautologically) the domain of humans
which is frankly incoherent.
I don't think your claim is possible to prove statistically - what does "pure reason" mean? And of course it's "pure reason + empirical observations": even if they're not running reproducible experiments they still see natural phenomena.
Pure reason is conjecture with little empirical input (and big intuitive or memey input) and little verification of output, i.e. a large deviation from empiricism.
> There is no scientific proof that the scientific method is valid.
The scientific method is provably effective in a lawful world. That the scientific method isn't provably effective is because the world is not provably lawful ... it's logically possible for the "laws" of physics--which are simply regularities that we have discovered--to suddenly change, oscillate, be random, etc. But as long as they don't the scientific method works ... and we can't do better. (Not in this world, anyway ... in some other world there might be oracles (aka gods or bibles) that always have the right answer and we could simply query them. Of course, such oracles are also not provably correct.)
The attitude of "we can't do better" is precisely the kind of attitude that would have hampered the creation of various sciences and their methods in the first place, and the production of scientific thought is often not as pure or logically consistent as all the methodological purists or Popperians seem to believe it is. Feyerabend, Quine, Von Foerster, Dataon and Gallison and several others have made this pretty clear and make the argument pretty forcefully in my opinion.
All sciences carry an ontology and value system along with them. Religion is also a rich and highly effective epistemic structure when your underlying ontological assumptions, values, and goals differ from those established and implicit in the scientific (predominantly utilitarian) Western enlightenment value system. Science is not able to found itself on some irrefutable bedrock "truth" correspondence any more than religion is. There is no basis by which we can actually solidify empiricism as somehow more privileged and more capital T true if we admit the possibility of human error.
This lie is so tiresome.
It's notable that science has a history of effectively producing confirmable fact whereas religion produces only falsehoods.
There is a sense in which all of my observations hence will be "theory laden". When I seek "proof" i'm likely to do so within the constraints of fabulashinaism.
Science proceeds in the same way. Only crises, which are unpredictable and may have much more than simple "new data collected" behind them actually drive theory change. Beyond the theoretical assumptions, many of which are not proven but are constraining hypotheses, there are the social value assumptions I mentioned as well. As someone else said in the thread, science cannot prove science. You can be of the opinion that it's been more effective than alternatives, and I would agree, but there is no way to prove this because it is ultimately a question of theoretical assumptions and of values.
The belief is like a cheat code: somehow our tinkering and 9-5 job somehow grants us, the computer nerds, a deep understanding of life, the universe and everything. Deeper than scientists, philosophers, etc. It’s epistemic catnip and pretty tasty at that!
His website also hosts a bunch more work as well as various lecture notes and exercises: https://timroughgarden.org/
Tim's lectures helped me a lot during my PhD when I was getting up to speed on this subject, and some of the more nuanced ways that computer scientists have worked with these broad algorithmic problems.
Those were good days.
Our world appeared computable, but it isn't, even if P=NP.
- The physics of the universe can be completely modeled as computation, and
- It's possible to pose undecidable problems about the way the universe unfolds
This is intrinsic to the idea of undecidability even for Turing machines, e.g. "we equate computation with the functioning of Turing machines, but there are real processes executable in Turing machines that are undecidable".
For example, if aliens claim their machine solves the halting problem, we could test it on millions of inputs whose halting/not-halting behaviour we already know; but even if it works for all of them, there's no way to know that it works for all inputs. For all we know, it might be a huge lookup table which happens to cover all of those inputs we tried.
> if our universe is undecidable
My point is, there would be no way to empirically test this; and therefore, it would make no observable difference, there would be no way to exploit/utilise such effects, etc.
In essence: there's no way to tell the difference between a real halting oracle (which would imply an undecidable universe), versus a computable approximation which just-so-happens to be more powerful/sophisticated than the approximations we compare it against.
Sure, we can prove that some abstract systems are undecidable and that others aren't. Yet that distinction is inherently unfalsifiable, and hence physically "useless".
I want to push back a bit on this claim along two dimensions.
Imagine a physical Turing machine built out of atoms, gears, levers, and an electron parked on the read/write head and ask whether that electron ever crosses some fixed plane in space, which it does only when the machine enters its halt configuration. That's now a purely physical question about a trajectory (does this electron ever reach a certain target), yet answering it for the whole family of such machines is literally the halting problem, so there's a physical process that's undecidable.
Your examples about physical processes being undecidable are all basically just this... there examples of using reflections of light, or the flow of liquid, etc... and demonstrating that these physical processes in principle are sufficient to model a universal Turing machine.
And while it's fascinating that certain things you may not have expected can be used to model computation, it's misleading, or rather it's too strong of a claim to believe that there exist actual/real physical processes whose outcomes are undecidable. That's a subtle but very common misinterpretation of what undecidability is.
Undecidability, whether in physics or computer science, only applies to the infinitely broad class of a problem as a whole, it never applies to a specific instance of a problem. So it can never be the case that there's a certain configuration of reflections for which it's undecidable whether a ray of light reaches a target. Nor can it be the case that for a specific lattice of atoms, it's undecidable whether it has a spectral gap or not. It can only be the case that for the problem as a whole where the parameter space is entirely unbounded, there is no single algorithm that can decide if a ray of light reaches a specific target for all possible arbitrary (and infinitely many) configurations. Once you fix a specific system, then the undecidability goes away.
Not claiming that you are necessarily making this misconception, but I often see people misinterpret undecidability to mean that there exists a specific problem, like with specific inputs, where it's somehow impossible to know what the answer will be. Undecidability always requires an infinite family of instances, and it's a statement about the nonexistence of a single algorithm that correctly answers every instance in that family. It says nothing about any particular instance being unknowable/undecidable.
Feel free to flag this comment if I get an answer. I do want to know.
My apologies, and I do appreciate your reply.
It is depressing though, writing feels like it's in part becoming a game of outpacing the latest LLM's idiosyncrasies so we can signal authenticity, which perversely, is achieved through using an LLM enough so that you can become familiar with its flavor of communication.
I actually laughed quite a lot to begin with, GPT models saying things like "...might look like P, but is NP wearing a hat and a lab coat..." and "...is a haunted house disguised as a git repository..."; but alas when you've heard them a million times everywhere it really starts to bite.
What was the book in 1997? That's about the time of my first UAP sighting.
This is very helpful though, thank you.
For example, you can ask whether a Java program, run with infinite memory, will eventually halt. For any particular Java program, there's obviously an algorithm that says whether it halts or not. The algorithm is a single statement, which says either "yes" or "no". Might be hard to figure out which is the correct algorithm, but the Java program is fixed so the algorithm is definitely one of the two.
However, there is no algorithm which can take an arbitrary Java program as input and determine whether it will halt. It's about the class of all possible programs.
[1] https://en.wikipedia.org/wiki/Undecidable_problem
[2] https://en.wikipedia.org/wiki/Independence_(mathematical_log...
This isn't true.
In general, if a program hasn't halted yet you don't know if it will.
In particular, consider the Collatz conjecture. You can't even tell if your Java implementation of it will halt for a particular input, until it does.
We don't _know_ which algorithm it is, but that's not relevant to the definition of undecidability, which only requires that the algorithm exist.
This is slightly bizarre now I think about it: the definition of decidability allows the algorithm selection to be undecidable!
For your Turing machine example: even if we built such a machine, it would never truly be giving an answer to the halting problem, because any stray cosmic particle could excite the electron and cause it to cross whatever plane.
For a more realistic example: the ground state of an molecule is a physically relevant quantity, and in theory any molecule alone should lose energy and attain it's ground state, even if finding the ground state electronic configuration is undecidable. But in reality, no molecule is ever truly isolated and so would never actually be guaranteed to enter it's ground state (or if it were truly isolated, it would not be observed at all rendering the question moot)
No, since you cannot physically build a Turing Machine. A Turing machine requires infinite tape. Any physically realizable machine doesn’t have that, so has finite states, so is decidable: enumerate the states in finite time - it halts or repeats, so all programs on a finite state machine are decidable.
Your example is not an undecudable physical process.
Godel things also don’t apply: Godel theorems are about proof of this or that from within the same system. In logic one can prove such things from an outside system, then construct towers, avoiding Godel theorems. Godel theorems also require a model of integers including multiplication (without multiplication, such systems were proven complete and decidable). However the universe does not contain a model of integers, as the physical universe is not unbounded: relativity places a finite limit in spacetime on what can interact.
Mixing math as reality fails at these requirements.
sqrt(1-exp(-t/T))|1> + sqrt(exp(-t/T))|0>
If you want to claim that we could predict which specific atom decays next, I'd really like to see a lot more explaintion and exposition as it would upend current understanding.
But at least we agree there is nothing we can predict.
sqrt(1-exp(-t/T))|1> + sqrt(exp(-t/T))|0>
Do you think that's a kind of tunnel vision? If the only thing you focus on is computation, you'll probably end up seeing computation everywhere - it became a way of seeing the world.
"It's interesting to look back through history on this one. Each age has its pinnacle of technology, and each age uses that technology as a metaphor for nature, for the universe. In ancient Greece, the technological marvels were musical instruments and the ruler and compass. The Greek philosophers tried to build an entire cosmology from number, harmony, proportion, form, and so on — from mathematics, basically. Remember the music of the spheres? The Pythagoreans believed that nature was a manifestation of rational mathematics. Later on the pinnacle of technology was the clockwork. Newton wanted a clockwork universe, the entire universe as a gigantic clockwork mechanism, with all the parts interlocking and ticking over with infinite precision. Then in the 19th century along came steam power, and the universe was then depicted as an enormous heat engine, or thermodynamic machine, running down toward its heat death. Today the computer is the pinnacle of technology, so it's now fashionable to talk about nature as a computational process."
Which seems to source from https://www.edge.org/conversation/paul_davies-time-loops .
While "computer" may give us impressions of something with "a CPU" and "RAM" and "a disk drive", it does at least seem plausible that the universe as computation is a plausible base level, though. Unlike "the music of the spheres", which to the extent that it made predictions of the world, it got them wrong in the most basic way, viewing it through a lens of computation allows us to put some quite subtle and interesting limits on things. "Computation" is a pretty flexible substrate; it is difficult to imagine how the proposition "the universe is a computation and subject to the limitations thereto" could be falsified, and if it could, it is difficult to imagine how we would be able to know it was so falsified. Nevertheless the math of computation allows us to say non-trivial things about the universe as a result; it is not a vacuous generalization, though it is certainly a loose one... being able to say yet more concrete things about the nature of the computation, such as "this is exactly how gravity works", has quite a bit more utility.
"If Mathematics is the 'what', Computer Science is the 'how'".
This applies to each and everything.
The imo much more foundational relationship not everybody is aware of is https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
According to the currently known laws of physics. Which we know are incomplete/incorrect in several places.
"Computable" can mean probabilistic, and classical computers can function over probability distributions just fine.
Those quantum processes are interesting. Take the random numbers generated from radioactive decay. They are (after some cleanup) truly random. That is what we think. But how could we tell the difference from pseudorandom numbers, generated by a sufficiently advanced algorithm? We couldnt. So particles could simply be Turing Machines running sufficiently advanced algorithms that we cant reverse engineer. If so, quantum mechanics is computable even if we cant compute it.
(Particles being TMs doesnt mean they are FAs with an infinite tape, but that they are computationally equivalent to TMs.)
When I wrote Turing Machine, I was thinking about the classical determinstic Turing Machine.
Im not super knowledgable about Quantum Turing Machines, but as far as I know, they dont do better than the classical deterministic Turing Machines when we are talking about computability.
Not saying this is wrong or that I've watched all of the lectures above or anything, but it's just funny to imagine that aliens might look at us the same way we could look at a monkey society saying that the universe is like a big one of those rocks they use to smash nuts open.
Computation and information really does seem universal though, so this is just a funny thought and not serious commentary.
On I suppose a slightly more serious note it also strikes me that there will eventually, probably, maybe be _something_ that fits best; though we are not promised to find it. And it seems wild to say that about a model because all models are fundamentally wrong, but they are usually group-able by less-wrongness, so probably fundamentally feasible.
But decidabilty, Godel's theorm, busy beaver numbers, etc... those were unexpected and worth the price of admission.
Thanks Prof Hadas, you made it fun to have my mind blown.
When we constrain a formalism to reduce complexity, it feels like necessity emerges from within those constraints. For example, when we say 'CRUD app,' we immediately think of a specific pattern. In the same way, once you adopt a 'form,' the constraints that come with that form progressively expand the state space. In that sense, it feels like both discovery and invention.
Famous mathematicians and scientists often distinguish between model and reality, yet we tend to mistake the model's shape for reality itself. People like John Wheeler and Stephen Wolfram argue that computation is a fundamental property of the universe. But can we really say that when we downcast reality to fit human cognition, losing information in the process, and then upcast it back, the information is fully restored? I always find this point difficult.
Landauer's principle says that abstract logical operations, information erasure, necessarily increase physical entropy. That shows there's a thermodynamic cost to physically implemented information processing. But I don't think that proves computation is fundamental.
Whether it's computation or geometry, they're all abstract formalisms created by humans. But when we actually measure things, they're subject to physical laws. Still, whether that makes them fundamental is a difficult question. I think these are just results of the process where humans name phenomena and constrain them. I don't think they're the cause.
You can define computation broadly enough, as 'a process where a state changes to another state according to rules,' to make almost everything look like computation. But being able to explain something with computation and claiming that computation is fundamental are different things, aren't they?
Meaning exists within the structures and constraints of human-made formalisms. We artificially lower cognitive complexity and translate things into human language. Whether that's fundamental, I'm not sure.
Maybe I'm a reductionist. Plenty of intellectually brilliant scholars make those claims, but people like me, with slower minds, end up thinking these kinds of stupid thoughts. I wish I could organize my own thoughts bette
Regarding the downcast/upcast; I think it _can_ be possible to do this successfully;
> I have a glass, I throw it at a hard surface. What will happen? Well (duh) the glass will (most likely) break.
This hypothesis completely ignores nearly 100% of all relevant physics and the laws surrounding the problem; the arrangement of air molecules, the arrangement of the molecules in the glass, the physical forces governing me, it reduces the entire equation down to some really basic napkin physics.
But; does the outcome work? Has my interpretation of the universe and its physics actually predicted what will happen?
Probably a stupid example, but I think that a lossy picture of the universe can still yield a correct answer.
I can't physically run a simulation of the entire universe in my brain, as my brain is part of that same universe. Lossy representations/models are a necessity in the thinking-ham bound world in which we exist.
I think the key is that different phenomena require different approaches, and even if you interpret a single phenomenon through multiple mental models, none of them necessarily captures its essence. Ultimately, it's about which mental model is shared by a 'group'—not about what's fundamentally true.
We currently share the model of computation, but whether that makes it fundamental is a different story.
As a programmer, I'd put it this way: no matter how well an API describes the backend, the backend itself is not the API.
Here, the API is human knowledge, and the backend is the world
We know that universal Turing machines can emulate other Turing machines. Weirdos like Wolfram believe that a universal Turing machine can emulate reality. In a quick skim of this lecture series, the presenter doesn't talk about that, rather he just calls computation a scientific principal (universal and fundamental in the sense of physical laws, not fundamental in the sense of emulating reality on a computer).
A simple example for the computation: It’s like placing boxes next to each other. Yes I could say 1 box + 3 boxes = 4 boxes, via explicit calculation. I could also simply place 1 box next to three boxes, and without having to explicitly calculate, by nature of the interaction, the result (4 boxes) has been produced
There is no background computation or storage. The universe IS computation and storage. Each spacetime quanta IS storage, and each interaction IS computation.
There are even some philosophers/mathematicians who attest to finitism (they’re not mainstream, though), but if correct would undermine our idea of fields and continuous structures more broadly
An algorithm, at its root, is a procedure rooted in human understanding that human beings can follow.
When Turing and others first introduced this formal notion, not all mathematicians were even fully convinced it was an adequate representation of the informal intuitive notion. For example, some argued it was too broad because traditionally knowing that an algorithm eventually terminates was one of the requirements (to some) for something to be a legitimate algorithm. Why? Because the idea is rooted in human practical concerns and human understanding. Depending on what one cares about, one could actually reject the Turing formalization of algorithm on grounds of the class containing non-convergent (partial) functions.
All this is to say that "computation" is very much a human invention and little more than a formal model of human behaviors (we want to manipulate things algebraically). Elevating it to some objective substance of the universe is just doing 17th/18th century philosophical Idealism wrapped in new clothes.
> "Ergo is a nonprofit that publishes long-form philosophical lecture courses with leading scholars. Everything we publish is freely available, without ads or paywalls."
I know what to do this weekend if it rains!
Ergo: Long Form Philosophy Lectures
However, these rest on category errors. Consider two characterizations of computation:
1. A mental process constituted by logical and intentional acts.
2. A mathematical model or formalism (or a set of formally equivalent formalisms).
In the case of (1), intentionality rules out computation as an extra-mental phenomenon. Things in the world aren't about something else; they just are what they are. But computation as a mental act is about something else. To claim otherwise would be like claiming deduction is a broad feature of reality, which is effectively some kind of panpsychism.
In the case of (2), if it's a mathematical model, then either by definition it doesn't exist outside the mind as such, or it must be instantiated in some objective manner. The trouble with finding instantiations is that it's not clear what constitutes an instantiation. Can you find correspondences? Sure. In fact, our physical computing machines correspond to these models in some way. But instantiation is more than mere correspondence, and this becomes even more the case when you consider that the lambda calculus corresponds to the Turing machine.
Another problem is that even mathematical models of computation cannot be said to encode mathematical operations as such. Is a Turing machine moving symbols around on an abstract tape actually adding two numbers? I would say that it is merely simulating the addition by producing results that afford that kind of interpretation.
- given any system of quantum particles, a sufficiently large Turing machine can solve the matching system of Schrodinger equations
- the human body, including the brain, is certainly a system of quantum particles
- even if you think quantum behavior is directly relevant to consciousness (I don't) certainly there is nothing "beyond" quantum mechanics that would be necessary to explain it, so our system of Schrodinger equations should be able to model human consciousness
- so a Turing machine can simulate a human, including consciousness
The paper argues that this simulation of consciousness is not the same thing as actual consciousness.
Erm... this can't possibly work, can it?
- You have a piece of software
- That software does in memory compute only
- The software does not touch any peripherals, networking, or any other external source which introduce unpredictability (x)
I'm convinced that somehow this can be solved/proven whether the execution will halt or not.
(x) The second you touch any external peripherals or networking, you're effectively asking the question of "If I phone my friend, will they pick up the phone?" -> to which the only answer is, "They'll pick it up, only if they pick it up/are there". You can't answer that question without trying it.
Am I missing the point? I'm sure you can introduce other edges even in the limited model above, e.g. where a memory stick stops responding or something; but all in if you have reliable kit and don't touch anything external, why can't this be solved?
Linear bounded automata (LBA) the halting problem is decidable. But many properties of LBA are undecidable:
Emptiness: Does an LBA reject all possible inputs? Universality: Does an LBA accept all possible inputs over its alphabet? Equivalent: Do two LBA accept the same language? Finiteness: Does an LBA accept a finite number of strings.
I love the idea of this. So the BB problems are individual iterations of the halting problem right? To truly solve the problem one would have to come up with a program which would operate on all possible BB numbers?
- How long does it take to get from A to B? => Easy if you know where A and B are, and what mode of transport you're taking to get there.
- How long does it take to get from A to _somewhere_ => As long as it takes!!
Bear in mind checking whether or not the algorithm ever loops means taking the full state of the system and checking against a database of all previous states of the system. Bear in mind that the Atari 2600, and its whopping 128 bytes of RAM, has with that amount of RAM more states than there are planck volumes * planck time intervals in the known universe... by over sixty orders of magnitude. And every three additional bits you add to the RAM of the system your are looking at adds an order of magnitude (minus a bit) to that, so, nearly 3 orders of magnitude more states per byte... not per megabyte or gigabyte, per byte. Call it 2 orders of magnitude per byte if you want to be conservative.
It can be solved, if by nothing else simply by running it, in the mathematical sense. In the practical sense it's not even close. That's why we use the Turing machine analysis... technically it's an approximation because we don't actually have real Turing machines. However the size of the finite state machines we have is such that it is far more productive to simply say "the halting problem is unsolvable" than to argue about how many orders of magnitude of orders of magnitude of resources it takes to solve the question of whether or a given program terminates.
The approach you describe though is brute force. I don't think (if there even is an answer to this problem) that it can be brute forced; that's where you run into the limits of hardware/computation/energy and start talking about timeframes which exceed the life of the universe.
I think brute force might be a useful tool in places to validate results, but if there _is_ an answer to this problem it's purely mathematical.
Apologies for sounding both excited and naive; these sorts of challenges make me happy in strange ways that no other thing does!
For the same reason the halting problem doesn't even have a good heuristic, neither does this. Unpredictable chaos is not an exceptional case, it is the exponentially-normal case. You have to go the other way, and construct programs deliberately designed to have the ability to tell if they halt. The term for that if you want to learn more about it is "non-Turing complete programming language", sometimes called a "sub-Turing" programming language: https://increment.com/programming-languages/turing-incomplet...
You can read that as "this is how hard it is to construct code that we can make execution guarantees about". That focuses on code that is deliberately constructed to be finite in scope and may be something that can be strictly bounded in memory use or time or both. You'll note if you spend any time working with them how hard they are to work with. That's a reflection of the limits of generalizing any such proofs of time or space of a given program.
If there is a general algorithm that does what you think, we don't even have a clue what it would look like. And we have a lot of clues there can't be any such thing.
The second you wield a language which has constructs like Haskell, where in theory you can iterate over an infinite list of items (thinking about it even any language where for i in input_var is possible); the halting problem hits you in the face like a brick.
Its almost a chicken and the egg problem, where you can't know how long it will run for/whether it will halt without already knowing the answer, but if you knew the answer, you wouldn't need the program to find it.
My head is spinning.
Bonne lecture !
Assume H(P,i) to be a program that tells you whether the program P will halt on input i. Returns true if it halts and false if it doesn't.
Define a new program G(x) that halts if x does not halt on itself as input and loops forever if x halts on itself as input:
def G(x):
if H(x,x):
while True:
pass
else:
return
If G(G) halts then H(G,G) is true so we end up in the infinite loop, a contradiction. If G(G) does not halt then it returns without looping, also a contradiction.
So our halting oracle H does not exist, so there can not be a function that tells you whether another function will halt on itself as input, so there can not be a function that tells you in the general case whether some other computation will halt, QED.