4 ms·

Almost all functions aren't computable, as they aren't discrete. A "computer" is just a function from {0,1}^N -> {0,1}^M

jojobas 4y ago
Symbolic engines are way above and beyond number crunching in many respects.
- mjburgess 4y ago
  Mathematics is constrained by properties of abstract objects that the symbols are about. Here, for example, the mathematician has to imagine a scenario to describe with mathematics (two couter-flow fluids, etc.). The notation gains its meaning from this imagined scenario. Rules for manipulating symbols are therefore insufficient. The proof has to follow from the scenario, which the machine is unable to represent.
  - chmod775 4y ago
    > which the machine is unable to represent. If you believe that a computer will eventually be able to accurately simulate a human brain, you might as well give up right now. Since if a computer with all its constraints is able to simulate a human (brain), but cannot do this, then a human can't do it either. Conversely if a human can do this but a computer can't, then a computer can never simulate a human. Don't tell this to a software developer working on AI. They might quit their job and become a baker instead. I don't think you would have any effect on a mathematician, since they would already be acutely aware some things provably cannot be done.
    - mjburgess 4y ago
      Yes, I don't believe a concrete computer can "simulate" anything in the relevant sense, let alone a human brain. Computer, as defined abstractly, is just any abstract function from {0,1}^N->{0,1}^M. Any realisation of that, eg., by providing each {0,1} as an electrical switch, realises physical properties associated with electrical switches only. The reason that electrical computers are useful has vastly more to do with the electrical part than the computer part. The "computational properties" of the electrical devices we call computers are relatively trivial. But in any case, no system in virute of being an implementation of a discete function thereby acquires physical properties. A woodern "comptuer" is useless precisely because you can't play video games on it. Likewise, even if the brain can be described by a discrete function -- (which is so implausible as to be a bit mad and certainly purely an act of faith) -- then it still requires the relevant physical properties to implement. These properties are extremely unlikely to be those of electrical switches. The "computational work" done by biochemical signalling alone should probably be regarded as "infinite", saying much about the limitations of discrete conceptions of information.
MarcelOlsz 4y ago
What level of math do I need to be at to "compute" this comment?
- mjburgess 4y ago
  Very little. A computer is an abstract mathematical description (eg., like "prime") of a certain mathematical object, a function. A computer is a way of specifying a discrete function (ie., one which maps a finite number of bits to a finite number of bits), in terms of a sequence of mathematical transitions. It's an "algorithmic" way of specifying the domain and codomain of a discrete function. Electrical digital computers aren't actually computers in this sense, and are extremely aproximately described by them. Inasmuch as the shape of the earth is aproximately "spherical". In any case, pretty much all of physics does not use discrete functions (indeed, I can't think of a single case). In every way physics describes reality, ie., parameterised on space and time, functions are continuous. They map an infinite amount of spatio-temporal information to an infinite amount of spatio-temporal information. And there is yet no reason whatsoever, other than the AI PR machine, to suppose that all of physics is wrong in this regard, and the universe is describable by anything else. This is relevant here, since the problem that cannot be represented to the machine uses ordinary equations of physics, none of which are computable.
- ben_w 4y ago
  > function from {0,1}^N -> {0,1}^M "{0,1}": the set containing the values 0 and 1 "{0,1}^N": a discrete n-dimensional space, where the possible values in each dimension are 0 or 1 So they're saying a computer takes a length N binary sequence input and produces a length M binary output. (As for "what level is this", I didn't cover any of this in my double A-level[0] in maths/further maths, but I am covering it in brilliant.org and some popular maths books, so my best guess is it's first or second year degree level?) [0] https://en.wikipedia.org/wiki/A-Level https://en.wikipedia.org/wiki/A-Level
Maursault 4y ago
A function is merely a rule, and rules do not do anything other than define relationships, but this is not quite right either, because who is defining? Taking into account Newton's Second Law of Motion, which the practical application of cause and effect, a computer is always always a person: one who computes. Consider that no matter how complicated they become, pencils do not calculate, cars do not drive, and guns do not shoot. You tell everybody. Listen to me! You've gotta tell 'em! Computers are people! We gotta stop them! Somehow!

jojobas 4y ago

Symbolic engines are way above and beyond number crunching in many respects.

mjburgess 4y ago
Mathematics is constrained by properties of abstract objects that the symbols are about. Here, for example, the mathematician has to imagine a scenario to describe with mathematics (two couter-flow fluids, etc.). The notation gains its meaning from this imagined scenario. Rules for manipulating symbols are therefore insufficient. The proof has to follow from the scenario, which the machine is unable to represent.
- chmod775 4y ago
  > which the machine is unable to represent. If you believe that a computer will eventually be able to accurately simulate a human brain, you might as well give up right now. Since if a computer with all its constraints is able to simulate a human (brain), but cannot do this, then a human can't do it either. Conversely if a human can do this but a computer can't, then a computer can never simulate a human. Don't tell this to a software developer working on AI. They might quit their job and become a baker instead. I don't think you would have any effect on a mathematician, since they would already be acutely aware some things provably cannot be done.
  - mjburgess 4y ago
    Yes, I don't believe a concrete computer can "simulate" anything in the relevant sense, let alone a human brain. Computer, as defined abstractly, is just any abstract function from {0,1}^N->{0,1}^M. Any realisation of that, eg., by providing each {0,1} as an electrical switch, realises physical properties associated with electrical switches only. The reason that electrical computers are useful has vastly more to do with the electrical part than the computer part. The "computational properties" of the electrical devices we call computers are relatively trivial. But in any case, no system in virute of being an implementation of a discete function thereby acquires physical properties. A woodern "comptuer" is useless precisely because you can't play video games on it. Likewise, even if the brain can be described by a discrete function -- (which is so implausible as to be a bit mad and certainly purely an act of faith) -- then it still requires the relevant physical properties to implement. These properties are extremely unlikely to be those of electrical switches. The "computational work" done by biochemical signalling alone should probably be regarded as "infinite", saying much about the limitations of discrete conceptions of information.

MarcelOlsz 4y ago

What level of math do I need to be at to "compute" this comment?

mjburgess 4y ago
Very little. A computer is an abstract mathematical description (eg., like "prime") of a certain mathematical object, a function. A computer is a way of specifying a discrete function (ie., one which maps a finite number of bits to a finite number of bits), in terms of a sequence of mathematical transitions. It's an "algorithmic" way of specifying the domain and codomain of a discrete function. Electrical digital computers aren't actually computers in this sense, and are extremely aproximately described by them. Inasmuch as the shape of the earth is aproximately "spherical". In any case, pretty much all of physics does not use discrete functions (indeed, I can't think of a single case). In every way physics describes reality, ie., parameterised on space and time, functions are continuous. They map an infinite amount of spatio-temporal information to an infinite amount of spatio-temporal information. And there is yet no reason whatsoever, other than the AI PR machine, to suppose that all of physics is wrong in this regard, and the universe is describable by anything else. This is relevant here, since the problem that cannot be represented to the machine uses ordinary equations of physics, none of which are computable.
ben_w 4y ago
> function from {0,1}^N -> {0,1}^M "{0,1}": the set containing the values 0 and 1 "{0,1}^N": a discrete n-dimensional space, where the possible values in each dimension are 0 or 1 So they're saying a computer takes a length N binary sequence input and produces a length M binary output. (As for "what level is this", I didn't cover any of this in my double A-level[0] in maths/further maths, but I am covering it in brilliant.org and some popular maths books, so my best guess is it's first or second year degree level?) [0] https://en.wikipedia.org/wiki/A-Level https://en.wikipedia.org/wiki/A-Level

Maursault 4y ago

A function is merely a rule, and rules do not do anything other than define relationships, but this is not quite right either, because who is defining? Taking into account Newton's Second Law of Motion, which the practical application of cause and effect, a computer is always always a person: one who computes. Consider that no matter how complicated they become, pencils do not calculate, cars do not drive, and guns do not shoot. You tell everybody. Listen to me! You've gotta tell 'em! Computers are people! We gotta stop them! Somehow!