[ Epistemic status: way more tentative than it sounds. Talking out my hat. ]
As I made a point of in my other post, it is a simple bivalent truth that definability is beyond computability and as humans can define things that cannot be computed, they do not work solely computationally.
Wait. Definability is beyond computability. Gotcha. But I think there’s a reference problem when you say humans can do “it”. Humans can’t actually solve any of these problems. They can define them. I don’t know how much exploration have been done about whether definitions are computable, and whether a theoretical Turing machine can define problems that it can’t solve.
I suspect the answer is “yes”, definitions can be computable even if the thing defined is not. Which then squares my intuition that humans are actually computing devices, just very complicated ones.
I also object to the sloppy title—“choose one” has a pretty open meaning, and “one” is an absolutely valid choice. It was performed by my brain (or by the section of the universe that computes my thinking, if you prefer), so it was done computationally. Further, obXKCD: https://xkcd.com/221/
>But I think there’s a reference problem when you say humans can do “it”. Humans can’t actually solve any of these problems
You got me here. Problem can be a “trick” term in this regard, as this is a very simple definition. I am just calling it problem to lure people into a computational logic to see for themselves that it makes no sense and is not how they address the question or instruction.🙂
In fact definition is also not necessarily the right term.
Internally I did uncomputationally deduce that X=1, because it is the first natural number and X can be 1, and hence X is 1 in this particular context of me actually writing this text. Of course one could deduce something else, but it is what I deduced.
Only afterwards did I define X=1, as I recognized in principle I could have deduced something else if my natural numbers were sorted differently, but I deduced based on logical order not geometric order.
>I don’t know how much exploration have been done about whether definitions are computable, and whether a theoretical Turing machine can define problems that it can’t solve.
They are not always (in bivalent logic), as that is contradictory (for example I can self-referentially state: “This definition is not computable and X=1”, which is a valid definition, and true by definition, as you can define something to not be computable, even though it is already not computable as a definition is not a computation (for example I could define that the busy beaver function is “really really really not computable”, even though it is already proven it is not computable).
You can define the busy beaver function, but you cannot compute it. The abstract set of Turing machines and their computations define the busy beaver function, but it cannot be computed by a Turing machine. That was proven by contradiction, as you can verify on the internet.
I guess in polyvalent logic you can through all terms together as you have infinite options what the terms can mean. Also a bit of a pitfall, as I think in general polyvalent logic might be be more valid, as it applies in more situations, but in mathematical reasoning bivalent logic is used for good reasons (like if 0s might or might not be 1 your numbers system becomes pretty complex pretty quickly).
>I also object to the sloppy title—“choose one” has a pretty open meaning, and “one” is an absolutely valid choice. It was performed by my brain (or by the section of the universe that computes my thinking, if you prefer), so it was done computationally
Choose one has a well defined meaning. You take one number of the set of all natural numbers, and put it on the right side of the equation. It is open, as the set of natural numbers is infinite.
That is inherent to the process, not sloppy in any way. I would say it is a bit sloppy to suggest otherwise but whatever, I think my term “problem” was also sloppy, so maybe let’s say instruction😉.
You said “one” is an absolutely valid choice, I unfortunately have to say it is literally not a valid choice as it a string not a natural number. It seems you just converted what I said into a string and outputted it back to me. That is a valid computation, it is however not a valid definition for X.
It seems just internally restated your assumption that your brain performs definitions computationally. The question is if that is true?
I would say per definition it is not true as definition and computation have a different semantic meaning, and are also different terms syntactically as definition!=computation. It might be point out the obvious, but although it is obvious we are not always conscious of it. We are not always conscious of any definitions which is maybe because definitions are not that important. Even just in math there is undefinability like the undefinable real numbers which make up almost all of the real numbers. Let alone outside of math.
Of course syntactically you can mix up definition and computation somewhat arbitrarily as you can define “definition”=”computation”=”undefined”=”ojfg99″=”krümpeldefinatorisator”=”????!??!”. That is just a matter of syntax then, and that gets absurdly complex if you produce many complex definitions which might be different than earlier definitions. Hence maybe sometimes the long time it can take to wash all of that away to start with clearer definitions.
Unquestionably if you are not an absolute lover of math it might be a bit boring to talk about definitions, and I am only partially fond of math as bivalent logic and well-defined numbers are only of so much interest to me. That is emojis for the people that want hearts on the other hand: 💚💛🧡❤💜🖤🤎 Also computer are cool haha.🖱👾💻
[ Epistemic status: way more tentative than it sounds. Talking out my hat. ]
Wait. Definability is beyond computability. Gotcha. But I think there’s a reference problem when you say humans can do “it”. Humans can’t actually solve any of these problems. They can define them. I don’t know how much exploration have been done about whether definitions are computable, and whether a theoretical Turing machine can define problems that it can’t solve.
I suspect the answer is “yes”, definitions can be computable even if the thing defined is not. Which then squares my intuition that humans are actually computing devices, just very complicated ones.
I also object to the sloppy title—“choose one” has a pretty open meaning, and “one” is an absolutely valid choice. It was performed by my brain (or by the section of the universe that computes my thinking, if you prefer), so it was done computationally. Further, obXKCD: https://xkcd.com/221/
>But I think there’s a reference problem when you say humans can do “it”. Humans can’t actually solve any of these problems
You got me here. Problem can be a “trick” term in this regard, as this is a very simple definition. I am just calling it problem to lure people into a computational logic to see for themselves that it makes no sense and is not how they address the question or instruction.🙂
In fact definition is also not necessarily the right term.
Internally I did uncomputationally deduce that X=1, because it is the first natural number and X can be 1, and hence X is 1 in this particular context of me actually writing this text. Of course one could deduce something else, but it is what I deduced.
Only afterwards did I define X=1, as I recognized in principle I could have deduced something else if my natural numbers were sorted differently, but I deduced based on logical order not geometric order.
>I don’t know how much exploration have been done about whether definitions are computable, and whether a theoretical Turing machine can define problems that it can’t solve.
They are not always (in bivalent logic), as that is contradictory (for example I can self-referentially state: “This definition is not computable and X=1”, which is a valid definition, and true by definition, as you can define something to not be computable, even though it is already not computable as a definition is not a computation (for example I could define that the busy beaver function is “really really really not computable”, even though it is already proven it is not computable).
You can define the busy beaver function, but you cannot compute it. The abstract set of Turing machines and their computations define the busy beaver function, but it cannot be computed by a Turing machine. That was proven by contradiction, as you can verify on the internet.
I guess in polyvalent logic you can through all terms together as you have infinite options what the terms can mean. Also a bit of a pitfall, as I think in general polyvalent logic might be be more valid, as it applies in more situations, but in mathematical reasoning bivalent logic is used for good reasons (like if 0s might or might not be 1 your numbers system becomes pretty complex pretty quickly).
>I also object to the sloppy title—“choose one” has a pretty open meaning, and “one” is an absolutely valid choice. It was performed by my brain (or by the section of the universe that computes my thinking, if you prefer), so it was done computationally
Choose one has a well defined meaning. You take one number of the set of all natural numbers, and put it on the right side of the equation. It is open, as the set of natural numbers is infinite.
That is inherent to the process, not sloppy in any way. I would say it is a bit sloppy to suggest otherwise but whatever, I think my term “problem” was also sloppy, so maybe let’s say instruction😉.
You said “one” is an absolutely valid choice, I unfortunately have to say it is literally not a valid choice as it a string not a natural number. It seems you just converted what I said into a string and outputted it back to me. That is a valid computation, it is however not a valid definition for X.
It seems just internally restated your assumption that your brain performs definitions computationally. The question is if that is true?
I would say per definition it is not true as definition and computation have a different semantic meaning, and are also different terms syntactically as definition!=computation. It might be point out the obvious, but although it is obvious we are not always conscious of it. We are not always conscious of any definitions which is maybe because definitions are not that important. Even just in math there is undefinability like the undefinable real numbers which make up almost all of the real numbers. Let alone outside of math.
Of course syntactically you can mix up definition and computation somewhat arbitrarily as you can define “definition”=”computation”=”undefined”=”ojfg99″=”krümpeldefinatorisator”=”????!??!”. That is just a matter of syntax then, and that gets absurdly complex if you produce many complex definitions which might be different than earlier definitions. Hence maybe sometimes the long time it can take to wash all of that away to start with clearer definitions.
Unquestionably if you are not an absolute lover of math it might be a bit boring to talk about definitions, and I am only partially fond of math as bivalent logic and well-defined numbers are only of so much interest to me. That is emojis for the people that want hearts on the other hand: 💚💛🧡❤💜🖤🤎 Also computer are cool haha.🖱👾💻