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> This is the philosophical justification for why a (Turing) machine can calculate anything a human can (in fact, the same argument shows that a Turing machine

> This is the philosophical justification for why a (Turing) machine can calculate anything a human can (in fact, the same argument shows that a Turing machine can reproduce the behaviour of any physical system). I don't think this assertion follows. I don't think an argument like this can work without delving further into reasonable description of a "physical system". If you just just use the mathematics we employ to describe physical systems unreservedly it is possible to construct "physical systems" that exhibit non-computable behavior. For instance you can have computable and continuous initial conditions to the wave equation that produces a solution that is non-computable. See : https://en.wikipedia.org/wiki/Computability_in_Analysis_and_Physics https://en.wikipedia.org/wiki/Computability_in_Analysis_and_... I think it's important to emphasize that Turing stated his arguments in regards to "effective procedure" (which I see you mention in a different post). I don't think the substitution of "effective procedure" with "physical system" is justified.

chriswarbo 5y ago
> For instance you can have computable and continuous initial conditions to the wave equation that produces a solution that is non-computable. Thanks, that's a really nice example which I hadn't come across before (or at least not spent too much time studying). I may have to refine the language I use in future; although a cursory look seems to be compatible with my own understanding (my mental model is roughly: "if we found a halting oracle, we would have no way to tell for sure") > I think it's important to emphasize that Turing stated his arguments in regards to "effective procedure" (which I see you mention in a different post). I don't think the substitution of "effective procedure" with "physical system" is justified. Yes, Turing did not say as much (at least in his 1936 paper). He was essentially abstracting over 'whatever it is that a person might be doing', in an incredibly general way. Others have since taken this idea and applied it more broadly. Another useful caveat is that Turing machines are framed as (partial) functions over the Natural numbers. It's quite a leap from there to a "physical system". An obvious example is that no matter how cleverly we program a Turing machine, it cannot wash the dishes; although can simulate the washing of dishes to arbitrary precision, and it could wash dishes by controlling actuators if we decided to attach some (but even that would run into problems of time constraints; e.g. if calculating the first instruction took so long that the water had evaporated).
- wizzwizz4 5y ago
  The problem with your assumption is that you're assuming there are a finite number of states. There might be an uncountably infinite number of states, for instance if the states were the reals between 0 and 1.
  - chriswarbo 5y ago
    Which assumption are you referring to? If it's about the states in a finite region, note that I specifically limited this to distinguishable states. Whether or not a finite region can have an infinite number of states (countable or otherwise) is irrelevant; we can only distinguish finitely many in finite time. Two states being indistinguishable means they'll give rise to the same output behaviour (e.g. from a mathematician in a room, carrying out some procedure).

chriswarbo 5y ago

> For instance you can have computable and continuous initial conditions to the wave equation that produces a solution that is non-computable. Thanks, that's a really nice example which I hadn't come across before (or at least not spent too much time studying). I may have to refine the language I use in future; although a cursory look seems to be compatible with my own understanding (my mental model is roughly: "if we found a halting oracle, we would have no way to tell for sure") > I think it's important to emphasize that Turing stated his arguments in regards to "effective procedure" (which I see you mention in a different post). I don't think the substitution of "effective procedure" with "physical system" is justified. Yes, Turing did not say as much (at least in his 1936 paper). He was essentially abstracting over 'whatever it is that a person might be doing', in an incredibly general way. Others have since taken this idea and applied it more broadly. Another useful caveat is that Turing machines are framed as (partial) functions over the Natural numbers. It's quite a leap from there to a "physical system". An obvious example is that no matter how cleverly we program a Turing machine, it cannot wash the dishes; although can simulate the washing of dishes to arbitrary precision, and it could wash dishes by controlling actuators if we decided to attach some (but even that would run into problems of time constraints; e.g. if calculating the first instruction took so long that the water had evaporated).

wizzwizz4 5y ago
The problem with your assumption is that you're assuming there are a finite number of states. There might be an uncountably infinite number of states, for instance if the states were the reals between 0 and 1.
- chriswarbo 5y ago
  Which assumption are you referring to? If it's about the states in a finite region, note that I specifically limited this to distinguishable states. Whether or not a finite region can have an infinite number of states (countable or otherwise) is irrelevant; we can only distinguish finitely many in finite time. Two states being indistinguishable means they'll give rise to the same output behaviour (e.g. from a mathematician in a room, carrying out some procedure).