I think it is just so that once you hit a level of “systematicity” you are not missing out anything. It is not that the entities referenced when defining a turing machine are especially powerful. If there were “magic symbols” one could imagine that if a mathematician just scribbled the right sigils then a leaps better outcome could result. If you disregards what the mathematics is about it is black marks on a white background. What makes it “systematic” or “mathematical” is that the black marks give sufficient guidance on how to make more marks. One could try to relax this a bit. It is physically possible to write “2+2=5” (and it possible to make systems where “2+2=5″ is a valid move, but in disntinguishing valid from invalid moves we are assuming away things). But if what you write next doesn’t have to do what you have written before it becomes doubtful to call that a “result”.
One important idea is mututal emulation. If the mathematician can write a story how a turing machine would throw around symbols and a turing machine can throw around symbols how a mathematcian might move a pen, you don’t need to ask which capability is more “fundamental”. If you have a result in one language, you can either get it by native operation or by the other method emulating the other.
There are shadows of there migth be interesting things that might go a little beyond computation or efficient operation. To put is provocatively if you have advice that has a “asspull” in it then that is not a valid algorithm. One example could be “1. Try a thing. 2. If it fails try another thing”. One can turn this into a good algorithm with the flavor of “1. Enumerate all the possible answers 2. check each”. For some mathematical tasks it might be that you just do something and something ends up working, there might not be a method to come up with mathematical discoveries. But with everything that has a method, it is a technology and can be followed by a “dumb” instruction follower, ie the homework is “fair”. But some intelligent system can benefit from advice that is not a (complete) method, ie when following the instruction they need to do genuine creative/executive decicions such as “how to try a bunch of different things”.
One might be tempted to think that good science results is a human encoutnering the worlds sense data and executing correct inductive inference on it. But then you have stuff like Schrödingers equation being just guessed (rather than being constructed from some hints in a understanble construction). Another story would be that people hallucinate all kinds of hypothesis and natural selection just leaves those that lucked out to be in correspondence with reality to be alive. But if there is a method to the madness the story is trackable and we can say about its properties. So any kind of system that can deal with system descriptions will be able to do all describable work (or repharasing emulate/employ any kind of madness that would be neccesary for particular outcomes).
There are shadows of there migth be interesting things that might go a little beyond computation or efficient operation. To put is provocatively if you have advice that has a “asspull” in it then that is not a valid algorithm. One example could be “1. Try a thing. 2. If it fails try another thing”. One can turn this into a good algorithm with the flavor of “1. Enumerate all the possible answers 2. check each”. For some mathematical tasks it might be that you just do something and something ends up working, there might not be a method to come up with mathematical discoveries.
I am not sure what you mean by that. Are you actually suggesting that brains sometimes might do things that could not be done by any Turing machine? (which I don’t find very plausible (though on reflection if there is something in the universe that we don’t understand yet it’s probably brains, so if we were searching for something that couldn’t be modeled by a Turing machine it would be the right place to look?))
Or that there’s no algorithm that can discover all of “math”?
In that case I’d like to know what you mean by that and if you can give a specific example of a theorem or something for which a proof would exist (in some system? Provability was the subject that got skipped in our class due to a shorter corona semester. Regretting not knowing enough about this now), but it could not be found by a Turing machine or maybe to the right place/resource to learn more about this.
Its more about getting at it via negative definition by pointing to things that are not the thing in question.
For a positive definition we could go something like “everything that can be made into a step-by-step list”. Some of the negative “not steps:”
Neural networks when they get trained contain their training in weights. These are not the datatype “procedure”. Yeah you use the in summation procedures. But it is hard to tell a story of it being a thought process or anything like that
In movie Tenet the protagonist is adviced to “don’t try to understand it, feel it”. One way this could be taken to the limit is “even if you had the most sophisticated and adaptive understanding possible you would fail at the task”. Then the alternative object “intuition” might have different limits and possibilities than technological acruence.
Bugs are etymologically connected to the biological creastures because they made electrical connections that neither the bug or the circuit designers intended. Yet these machines had actually existed and operated. They could be desribed to be working “outside of procedure” even if doesn’t need laws of physics to break down its in a certain sense specificationless. So a worry would be that some sort of computer could be impossible to design but could come about accidentally. Or that “designing” processes are bootstrapped with accidental proccess, you can think of a cell as a machine but the fist cell might involve dice rolls to get assembled which could not be thought as machines.
I think in turings paper when they specify that in principle a “dump clerk” could carry out a computation kind of take the stance that “dumbness” is a atypical state for a human to be in. But if all of computation is already within “dumbness” what is there beyond that? This might be something like “insight”, a worker works according to external orders and designs but autonomous agent defines his own creations/solutions. If we have 100 house builders and 0 architects but the house gets built anyway this might seem suspicious (an ordinary engineer wil have no trouble doodling a blueprint in the absence of one, but truly mechanistic automatons might be forbidden to do so). Thus one could wonder whether “insightful clerk” would have different properties to “dumb clerk”.
Selection of sex over asexual reproduction relies on faster trait spreading and accumulation. But novel dna combinations are gotten throught mutations which are caused thing like cosmic rays and other disturbances. Such heavy reliance to “true RNG” is somewhat outside of algorithmic bounds. If a brain tumour causes brain circuits to fire differently then that is hard to account for as the brain following its own specifications as producing the behaviour. Biology is messy and algorithms are “clean”. So it might be that is one set out a “proof” that some animal is fit by modelling the evolution process by algorith this would be systematically frustrating as there is no canon or “correct” way to enumerate dna options. Thus “proof of elephants” might not a be thing of mathematical nature (just the ordinary natural selection of biological nature)
I think it is just so that once you hit a level of “systematicity” you are not missing out anything. It is not that the entities referenced when defining a turing machine are especially powerful. If there were “magic symbols” one could imagine that if a mathematician just scribbled the right sigils then a leaps better outcome could result. If you disregards what the mathematics is about it is black marks on a white background. What makes it “systematic” or “mathematical” is that the black marks give sufficient guidance on how to make more marks. One could try to relax this a bit. It is physically possible to write “2+2=5” (and it possible to make systems where “2+2=5″ is a valid move, but in disntinguishing valid from invalid moves we are assuming away things). But if what you write next doesn’t have to do what you have written before it becomes doubtful to call that a “result”.
One important idea is mututal emulation. If the mathematician can write a story how a turing machine would throw around symbols and a turing machine can throw around symbols how a mathematcian might move a pen, you don’t need to ask which capability is more “fundamental”. If you have a result in one language, you can either get it by native operation or by the other method emulating the other.
There are shadows of there migth be interesting things that might go a little beyond computation or efficient operation. To put is provocatively if you have advice that has a “asspull” in it then that is not a valid algorithm. One example could be “1. Try a thing. 2. If it fails try another thing”. One can turn this into a good algorithm with the flavor of “1. Enumerate all the possible answers 2. check each”. For some mathematical tasks it might be that you just do something and something ends up working, there might not be a method to come up with mathematical discoveries. But with everything that has a method, it is a technology and can be followed by a “dumb” instruction follower, ie the homework is “fair”. But some intelligent system can benefit from advice that is not a (complete) method, ie when following the instruction they need to do genuine creative/executive decicions such as “how to try a bunch of different things”.
One might be tempted to think that good science results is a human encoutnering the worlds sense data and executing correct inductive inference on it. But then you have stuff like Schrödingers equation being just guessed (rather than being constructed from some hints in a understanble construction). Another story would be that people hallucinate all kinds of hypothesis and natural selection just leaves those that lucked out to be in correspondence with reality to be alive. But if there is a method to the madness the story is trackable and we can say about its properties. So any kind of system that can deal with system descriptions will be able to do all describable work (or repharasing emulate/employ any kind of madness that would be neccesary for particular outcomes).
I agree with most of this except for:
I am not sure what you mean by that. Are you actually suggesting that brains sometimes might do things that could not be done by any Turing machine? (which I don’t find very plausible (though on reflection if there is something in the universe that we don’t understand yet it’s probably brains, so if we were searching for something that couldn’t be modeled by a Turing machine it would be the right place to look?))
Or that there’s no algorithm that can discover all of “math”?
In that case I’d like to know what you mean by that and if you can give a specific example of a theorem or something for which a proof would exist (in some system? Provability was the subject that got skipped in our class due to a shorter corona semester. Regretting not knowing enough about this now), but it could not be found by a Turing machine or maybe to the right place/resource to learn more about this.
Its more about getting at it via negative definition by pointing to things that are not the thing in question.
For a positive definition we could go something like “everything that can be made into a step-by-step list”. Some of the negative “not steps:”
Neural networks when they get trained contain their training in weights. These are not the datatype “procedure”. Yeah you use the in summation procedures. But it is hard to tell a story of it being a thought process or anything like that
In movie Tenet the protagonist is adviced to “don’t try to understand it, feel it”. One way this could be taken to the limit is “even if you had the most sophisticated and adaptive understanding possible you would fail at the task”. Then the alternative object “intuition” might have different limits and possibilities than technological acruence.
Bugs are etymologically connected to the biological creastures because they made electrical connections that neither the bug or the circuit designers intended. Yet these machines had actually existed and operated. They could be desribed to be working “outside of procedure” even if doesn’t need laws of physics to break down its in a certain sense specificationless. So a worry would be that some sort of computer could be impossible to design but could come about accidentally. Or that “designing” processes are bootstrapped with accidental proccess, you can think of a cell as a machine but the fist cell might involve dice rolls to get assembled which could not be thought as machines.
I think in turings paper when they specify that in principle a “dump clerk” could carry out a computation kind of take the stance that “dumbness” is a atypical state for a human to be in. But if all of computation is already within “dumbness” what is there beyond that? This might be something like “insight”, a worker works according to external orders and designs but autonomous agent defines his own creations/solutions. If we have 100 house builders and 0 architects but the house gets built anyway this might seem suspicious (an ordinary engineer wil have no trouble doodling a blueprint in the absence of one, but truly mechanistic automatons might be forbidden to do so). Thus one could wonder whether “insightful clerk” would have different properties to “dumb clerk”.
Selection of sex over asexual reproduction relies on faster trait spreading and accumulation. But novel dna combinations are gotten throught mutations which are caused thing like cosmic rays and other disturbances. Such heavy reliance to “true RNG” is somewhat outside of algorithmic bounds. If a brain tumour causes brain circuits to fire differently then that is hard to account for as the brain following its own specifications as producing the behaviour. Biology is messy and algorithms are “clean”. So it might be that is one set out a “proof” that some animal is fit by modelling the evolution process by algorith this would be systematically frustrating as there is no canon or “correct” way to enumerate dna options. Thus “proof of elephants” might not a be thing of mathematical nature (just the ordinary natural selection of biological nature)