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Ultimately, the open question here is this: are uncomputable functions just a mathematical fancy, or do there really exist processes that can only be fully corr

by User23 4y ago

Ultimately, the open question here is this: are uncomputable functions just a mathematical fancy, or do there really exist processes that can only be fully correctly described by uncomputable functions? 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.

sudosysgen 4y ago
Well, those functions are called uncomputable because, as far as we know, there is no repeatable way of computing them, right? 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.
- User23 4y ago
  The Church-Turing hypothesis just means that the lambda calculus and Turing machines are isomorphic. It doesn’t actually tell us anything about reality, no matter what enthusiastic misunderstandings might say.
  - sudosysgen 4y ago
    No, it doesn't. The Church Turing hypothesis is much stronger - it states that all real-world computation can be done by lambda calculus or Turing machines. 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.
    - User23 4y ago
      Ah yes. Fair enough. I confused the thesis with the hypothesis. This comes back to the begging the question I initially objected to. 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?
      - sudosysgen 4y ago
        > I confused the thesis with the hypothesis. 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/S2211379719304036 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!

by User23 4y ago

sudosysgen 4y ago

Well, those functions are called uncomputable because, as far as we know, there is no repeatable way of computing them, right? 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.

User23 4y ago
The Church-Turing hypothesis just means that the lambda calculus and Turing machines are isomorphic. It doesn’t actually tell us anything about reality, no matter what enthusiastic misunderstandings might say.
- sudosysgen 4y ago
  No, it doesn't. The Church Turing hypothesis is much stronger - it states that all real-world computation can be done by lambda calculus or Turing machines. 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.
  - User23 4y ago
    Ah yes. Fair enough. I confused the thesis with the hypothesis. This comes back to the begging the question I initially objected to. 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?
    - sudosysgen 4y ago
      > I confused the thesis with the hypothesis. 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/S2211379719304036 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!