That’s a solid counter argument. I assume you’re saying we don’t know of a single physical process that’s not a Turing machine because all of the models we have built to simulate them are Turing computable? That’s a strong point. Maybe I should readjust my prior on this. It does sound like you’re better versed in this topic. I like to learn by asking questions, so if you’re amenable to answering please do. If you don’t, then please don’t. It can be overwhelming for some and I’m not trying to prove you wrong. I admit I rushed into a position in haste without actually being well prepared on the topic and had way too much confidence in my own opinion.
I have some devil’s arguments but maybe they’re all bad. Genuinely, my technical grounding and knowledge here has atrophied to (at least I feel like) extremely laughable point and didn’t start high to begin with as undergrad engineering teaches a very different kind of math and I really limited my extracurricular need to seriously study beyond the core engineering topics. Even there, doing the bare minimum to just cram through exams rather than actually learning and understanding the topics throughout the year.
If we have only very primitive models of how the world works. After all we can simulate how the smallest atoms work to maybe some more complicated chemical interactions. Still, as it gets to biology our ability to simulate things breaks down rapidly. I don’t mean at a performance level, but at a “we’re waaaaaaay off in the applied aspects”. Drug discovery is one I’m thinking of. Or predicting someone’s facial features from their DNA (that last one always feels extremely dubious but news friendly). Or general AI. Those have seem to hit a wall in results. That’s close to a god of the gaps argument so let me know if that’s a bad faith one because that’s just too small a gap for non computability to live vs we just aren’t smart enough yet as a species? I could see that.
Or what about that we don’t know of any actually infinite Turing machines at a physical level (thermodynamics and heat death off the universe puts an upper bound there I am thinking now randomly to justify my position rather than considering that before-hand). So is anything really a Turing machine in reality or are Turing machines themselves just a useful tool/model to model the universe but not 100% accurate and the world works slightly differently and the error comes from non computability and not randomness? Let me know if that’s just a kooky supposition on my part in case the math of Turing machines already proves that non-infinite things are always Turing computable and is very basic results I have forgotten/never learned.
What if the Turing machine model is an easy tool to simulate parts of it but not possible to simulate the thing itself? Like the universe itself is not computable on a Turing machine. As far as I know something like that’s been proven in the past few years - there was a proof that if we are living in a simulation, then the physical rules of the thing running the simulation would have to be very different and not look like our universe. Doesn’t that indicate that there’s a limit to the size of any Turing machine we could build to model the universe itself, and thus maybe the infinite model required to build a theoretical Turing machine itself doesn’t map to how reality works. I’ll admit freely this one may have an improper grounding in Turing machines even at a popular level as I only read the abstract to that paper and maybe I misread it or I’m misremembering something that has so much technical nuance that I’m missing what that actually means and misremembering.
Has it been investigated on the theoretical side whether the interaction of networks of Turing machines (whether intelligent or not) all interacting is itself Turing computable? Would that simulation itself be an instance of hypercomputation or some other non computable area of research? I don’t understand the basics well enough to interpret that research so I’m hoping you do. I started reading the Wikipedia article and my eyes just glazed over.
I would think these would all be very significant problems but I’ve admittedly not kept up on my understanding of how this stuff works at a deep mathematical level (and have let those skills atrophy over the years) and maybe that’s all been answered or I have significant flaws in my intuition. I really and sincerely appreciate you taking the time to respond and explain and talk about this stuff. I don’t like studying but I love learning what the higher level intuitions are of people who do study this stuff more deeply are.
Oh, and I’m going to read [1] now. I thought the heading was timely and lovely since we were discussing this and I look forward to reading it (I haven’t yet so I do t know what the article says yet)
> No method of computing carried out by a mechanical process can be more powerful than a Turing machine.
Although widely adopted, as there is no clear way to prove or disprove its validity the proposition still remains a conjecture.
I think that’s what we’re basically discussing, right? Still, the way that’s phrased puts it into the P!=NP camp for me so I think you may be right.
I have some devil’s arguments but maybe they’re all bad. Genuinely, my technical grounding and knowledge here has atrophied to (at least I feel like) extremely laughable point and didn’t start high to begin with as undergrad engineering teaches a very different kind of math and I really limited my extracurricular need to seriously study beyond the core engineering topics. Even there, doing the bare minimum to just cram through exams rather than actually learning and understanding the topics throughout the year.
If we have only very primitive models of how the world works. After all we can simulate how the smallest atoms work to maybe some more complicated chemical interactions. Still, as it gets to biology our ability to simulate things breaks down rapidly. I don’t mean at a performance level, but at a “we’re waaaaaaay off in the applied aspects”. Drug discovery is one I’m thinking of. Or predicting someone’s facial features from their DNA (that last one always feels extremely dubious but news friendly). Or general AI. Those have seem to hit a wall in results. That’s close to a god of the gaps argument so let me know if that’s a bad faith one because that’s just too small a gap for non computability to live vs we just aren’t smart enough yet as a species? I could see that.
Or what about that we don’t know of any actually infinite Turing machines at a physical level (thermodynamics and heat death off the universe puts an upper bound there I am thinking now randomly to justify my position rather than considering that before-hand). So is anything really a Turing machine in reality or are Turing machines themselves just a useful tool/model to model the universe but not 100% accurate and the world works slightly differently and the error comes from non computability and not randomness? Let me know if that’s just a kooky supposition on my part in case the math of Turing machines already proves that non-infinite things are always Turing computable and is very basic results I have forgotten/never learned.
What if the Turing machine model is an easy tool to simulate parts of it but not possible to simulate the thing itself? Like the universe itself is not computable on a Turing machine. As far as I know something like that’s been proven in the past few years - there was a proof that if we are living in a simulation, then the physical rules of the thing running the simulation would have to be very different and not look like our universe. Doesn’t that indicate that there’s a limit to the size of any Turing machine we could build to model the universe itself, and thus maybe the infinite model required to build a theoretical Turing machine itself doesn’t map to how reality works. I’ll admit freely this one may have an improper grounding in Turing machines even at a popular level as I only read the abstract to that paper and maybe I misread it or I’m misremembering something that has so much technical nuance that I’m missing what that actually means and misremembering.
Has it been investigated on the theoretical side whether the interaction of networks of Turing machines (whether intelligent or not) all interacting is itself Turing computable? Would that simulation itself be an instance of hypercomputation or some other non computable area of research? I don’t understand the basics well enough to interpret that research so I’m hoping you do. I started reading the Wikipedia article and my eyes just glazed over.
I would think these would all be very significant problems but I’ve admittedly not kept up on my understanding of how this stuff works at a deep mathematical level (and have let those skills atrophy over the years) and maybe that’s all been answered or I have significant flaws in my intuition. I really and sincerely appreciate you taking the time to respond and explain and talk about this stuff. I don’t like studying but I love learning what the higher level intuitions are of people who do study this stuff more deeply are.