> one can specify algorithms which require no thermodynamic work to implement
Can you give an example? To my limited understanding, performing work without expending energy sounds like perpetual motion.
Can you give an example? To my limited understanding, performing work without expending energy sounds like perpetual motion.
Consider an algorithm which says:
while(true) state *= +1, state *= -1, state *= +1, ...
This is "perpetual motion", but the technical name in physics for this is inertial motion, and its common. Motion itself doesn't require work, using that motion for work, requires work.
The problem for the Damon is, that it needs to change the state of the trap according to the state of the incoming particle. And if you just hand wave "such a decision making thing exists", then you have your contradiction. But we tried, for several days, to come up with _any_ implementation (including fantasy materials) that could conceivably exist _and_ produce that effect. And for each and every attempt to build one, we came up _immediately_ with diffusive parts, where energy _must_ be lost. We concluded, that while non of us would feel confident to _rule out_ a possible existence of Maxwell's Damon, we wouldn't _at all_ be surprised if it could be ruled out.
So while you are entirely correct, with enough hand-waviness, you can build reversible computations. But I have yet to see an argument, where a _potential_ implementation of one is argued to the end.
Reality, i am sure, has many systems which are settable and measurable and changeable at some minimum energy... and which can interface with devices of interest. For any given problem, the limit case energy requirement is defined by the needs of the algorithm. If we require setting a highly complex input state, and if we require interactions with certain devices, then we've immediately ruled out a great deal.
These systems would provide you with a certain kind of "limit-case correspondence" between "logical bits" and "ideal physical bits" --- but we dont know what this system is. You dont get it from just playing around with units, nor these kinds of thought experiments. You need to know what algorithm you're talking about, and what it's requirements are.
If the algorithm is understood just to be "the whole of reality" and if we suppose that it is fundamentally just aggregates of discrete states being flipped (to me, highly unlikely).... then the energy requirements are Everything... which sum, i imagine (via energy conservation), to zero.
[1] https://www.goodreads.com/book/show/17274.Feynman_Lectures_O...
Also, I've said that before, but we already know that brains operate under an irreversible computation model.
We don't know that brains operate under any kind of computational model at all. It's often postulated, but it's not proved. Every attempt I've seen at a proof reduces to begging the question.
Edit: To be clear I’m not saying a computational model of the brain can’t be a useful tool. Newtonian physics works quite well quite often even though reality isn’t Newtonian.
If there is any model of how the brain works it will be a computational model. Perhaps a new one, and perhaps a radically different one, but it's still going to find the definition of a computational model.
Personally I lean towards latter view. I’m the first to admit that I have no proof. It’s just my belief, because I find the metaphysical evidence compelling. I don’t object to investigating the former possibility, but I also don’t care for it just being baldly asserted.
If the answer is affirmative, that means that science will probably never be solved and we’ll just have to content ourselves with incremental improvements in our understanding. That’s observably been the case up until now. Granted even if all processes are in fact described entirely by computable functions we might still never discover what they are.
I hope it’s clear how all that relates to the concrete problem of understanding human cognition and the brain.
I'm nowhere near smart enough to even begin to conceive of a mathematical framework for taming noncomputable functions in a pragmatic way, but I earnestly hope some genius comes along who is, supposing that noncomputable functions are needed to completely describe our reality.
So if the human brain was able to compute functions that Turing machines can't, it doesn't mean those functions are uncomputable, it just means the Church-Turing hypothesis is false.
That's why it's a hypothesis and not a theorem - it's pretty easy to prove that lambda calculus and Turing machines are equivalent if you have the mindset of a programmer.
Put in other words, the Church Turing hypothesis is that there is no computing model higher in hierarchy than the TM and lambda-calculus.
> It doesn’t actually tell us anything about reality
Well, of course it doesn't tell is anything about reality. That's because no one managed to prove it, and I suspect no one ever will.
Also, the notion of highest computing model is itself begging the question in the sense that the framing assumes a computational model.
So much for undergraduate philosophy. What’s your opinion? Are uncomputable functions just fanciful irrelevancies or are there processes in reality that can’t be described by computable functions? If so or if not can you propose how we might know?
I'm actually relatively confident the two are the same thing. I don't think any special name was given to the equivalency between lambda calculus and TMs.
> This comes back to the begging the question I initially objected to.
I don't think it does! Of course if someone assumes the thesis is true then they're begging the question, but I don't think that's what I am doing here, indeed :
> Also, the notion of highest computing model is itself begging the question in the sense that the framing assumes a computational model.
The definition of computation I'm using is that there exists a process somehow that can go from the input to the output. If that process can't be replicated by a TM then so be it, it just means Church-Turing is false.
Actually, the original wording that Church used was "any function that can be computed by a mathematician.." (could be computed by a TM).
So I think that it's reasonable to frame it as computation - it would be begging the question if I assumed church Turing was true.
It doesn't help that the uncomputable function definition assumes it! But in reality that a function is uncomputable doesn't actually mean there is no framework of computation that can compute it, just that a TM can't.
> So much for undergraduate philosophy. What’s your opinion? Are uncomputable functions just fanciful irrelevancies or are there processes in reality that can’t be described by computable functions? If so or if not can you propose how we might know?
I've honestly spent so much time on the question I don't even know anymore. I'm leaning towards the side that uncomputable functions can actually be computed in reality. There's an interesting paper here : https://www.sciencedirect.com/science/article/pii/S221137971... which also links to an equally interesting 2002 paper that provide theories as to how you could compute uncomputable functions in the real world (with or without blackhole evaporation), but this is still speculative because our best theories in physics are still iffy at those scales, and obviously the physics is beyond my understanding here.
Anyways, I hope this answers the question of how we might know!
> Actually, the original wording that Church used was "any function that can be computed by a mathematician.." (could be computed by a TM).
That explains the disconnect. My working definition of computation is essentially the one Church gives there, with the understanding that he was talking about specifically computing the exact value of some function that could also be computed with a slide rule or some other effective procedure.
> So I think that it's reasonable to frame it as computation - it would be begging the question if I assumed church Turing was true.
Even assuming C-T, it's an enthymeme and not a syllogism. The unstated leg is something roughly like "the universe is a computer." Assuming that is basically assuming what is to be proved with respect to whether or not the brain is a computer or some higher category of reckoner that happens to be able to do everything a TM can, at least with enough time, pencil, and paper.
> I'm leaning towards the side that uncomputable functions can actually be computed in reality
For clarity it would probably be sensible to use a term other than computation for determining an exact value in this case. Anyhow, I believe I follow and I lean that way too.
There's a related metaphysical question too. I'm not sure how to state it exactly, but it's implied by questions like "When I throw a ball is reality computing a parabola and applying necessary modifications?" The alternative is that whatever way reality determines how that ball moves, it's not by what we'd call an effective procedure.
> Anyways, I hope this answers the question of how we might know!
Thanks for your insight and the link!
I don't think I'm making that assumption. I'm making an assumption that the process by which the brain determines outputs and inputs has some degree of repeatability - which I think is more than reasonable. Whether the brain is or isn't a computer in the classical sense of the word, there is clearly computation happening that relates inputs to outputs, right?
That is to say, if I took the same person and made them live exactly the same life (that is, everything outside how they react reacts the same way), I'd expect the outcomes as we repeat it more and more to approach some kind of distribution after a fixed amount of time t (that diverges as t gets larger exponentially). Note that this doesn't assume the presence or absence of free will, by the law of large numbers, it works either way.
From then on there is some kind of effective, repeatable procedure, so the brain does do some kind of computation. And of course that's not the only thing the brain does. But we don't need to have the universe as a computer, just the brain.
> There's a related metaphysical question too. I'm not sure how to state it exactly, but it's implied by questions like "When I throw a ball is reality computing a parabola and applying necessary modifications?" The alternative is that whatever way reality determines how that ball moves, it's not by what we'd call an effective procedure.
That's an interesting question. My attempt at an answer is that, if we were to understand the universe as a continuous process which can be defined by some kind of (recursive) semi-random, differential rules - but with maximum frequency - then yes you could say that reality is an ongoing computation. Or you could take another crack at it and think that reality is just as set of probabilities and joint probabilities that merely get (partially) sampled - in which case there isn't really any computation happening, we're just observing a tiny sliver of reality as the ball travels the probability space. There are certainly even other ways to see it. I think we it's a matter of perspective as to how you see it.
Thanks for the conversation! It is great and I've been able to develop my understanding of these concepts :)
They are only connected when empirical-logical models are provided. p
A "computer" is a formal pure-mathematics notion, it is just a certain sort of "discrete mathematical model". One can define a computational model of any physical system, and hence, find computers in which energy is conserved.
You mention elsewhere that the whole universe might be construed as a computation in which energy is conserved. Why not take the absurd reduction for what it is though, and conclude that it means there's something wrong with calling anything-at-all a computer?
This is exactly my point.
If you want to use the formal machinery of computer science to say something empirical, you cant just hijack terminology and speak in this pseudoscientific way, ie., "the brain as a computational process" (etc.).
What is the empirical content of such a claim?
There might be some if you defined what algorithm the brain was computing, and what the empirical correspondance between the algorithm and the brain way (etc. etc.) -- but no one is doing this.
We are speaking as-if somehow "universal turing machines" had some empirical content, that there's something insightful about labelling the brain this way (vs. anything at all). There isnt.
Some of these theories work, others don't. Nobody had a truly solid reason to believe them before it was clear they worked (in the case of those that did). Scientists all believe things they have no strict reason to -- when they're wrong they're wrong, and when they're right it's a scientific discovery.
Planck had no reason to postulate the energy packets when he came up with them. He described it as a move of pure desperation. The explanation (such as it is) for why this worked wasn't developed until decades later.
I don't personally believe Turing machines reveal much about the human brain, but you're verging on a more general claim about how science should be performed. That to hypothesize, work within a model, or even get something out of it you need to already have a precise description of the phenomenon being described and "why" the model might work. None of that is required though.
To be clear this all applies much less to "bread and butter" science. What we're faced with here is a process that has no convincing description.
To convince people to give up on computational models of the brain you need to convince them that it doesn't work. Nothing has been revealed by it in 60+ years. It's never predicted anything. Neurobiology and pharmacology at least have some results to show about a real brain. What you're basically engaging in instead is philosophy -- "pure" math is distinct from the empirical world, what if the whole universe, science requires this specific method, ... But it's better if we can dismiss a scientific idea on scientific grounds, rather than philosophical considerations.