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"That’s because it’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity, but it can’t actually reach it" Why

by somat 4y ago

"That’s because it’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity, but it can’t actually reach it" Why not? Is it impossible to calculate infinite values in general? I suspect not, My understanding is that a lot of calculus is in fact on how to calculate infinite values. And a computer is a universal machine, this means that while it can not calculate everything, it can calculate anything that is calculable.

raverbashing 4y ago
Because infinity is a mathematical trick, to say where something is going but it is not calculable (it is not a number in itself)
- agumonkey 4y ago
  well you end up in bounded axiomatic ranges
mjburgess 4y ago
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!
marginalia_nu 4y ago
An ideal computer is an universal machine. A real computer has real limitations on what is calculable, even among the things that are theoretically calculable.
- moffkalast 4y ago
  Sounds like we need to make a virtual universal machine and account for the limitations when emulating it.
  - majewsky 4y ago
    The problem with this is whether the emulation ever terminates. Simply put, you cannot emulate an infinitely powerful machine in finite time.
e12e 4y ago
> Is it impossible to calculate infinite values in general? I would say that it's impossible in a finite universe to calculate infinite values. But it's quite possible to define and manipulate relationships that involve infinity. You can't calculate the sum of all integers - but you can relate it to the sum of all real numbers. Or compare the sum of all integers lager than one hundred, to the sum of all integers larger than ninety.
fay59 4y ago
In context, it sounds like they relied on simulations that don’t use exact numbers. I’m guessing that they saw an IEEE-754 floating-point infinity and then had to determine whether they got it because the accurate result was infinity or if the infinity they saw was the result of floating-point calculation artifacts.
credit_guy 4y ago
> Why not? They don't explain this very well, I think. Take the ordinary differential equation x'(t) = x^2(t), with initial condition x(0)=1. It has the solution 1/(1-t) which blows up to infinity when t tends to 1. If you try to solve it numerically, using, let's say Euler's method, then this is how you go about it. You pick a step size, let's say 0.1. And iterate this way: you know x(0) = 1, and you also know its derivative x'(0) = x^2(0) = 1. You assume x follows a straight line, so you get x(0.1) = 1 + 0.1 = 1.1. At the next step you add more because x'(0.1) is now 1.21, so x(0.2) = 1.1 + 0.121 = 1.221. You keep going like that. The numbers will go bigger and bigger, but they will never be infinite. Of course, floating numbers with double precision overflow around 10^308, but if you use a multiple precision library you'll be able to keep going forever and ever. If you make the time step smaller, the solution will be closer to the actual solution, but still, the algorithm will produce finite values at all times (until it hits overflow).
bheadmaster 4y ago
Computers can, in fact, calculate infinite values in style of calculus, but they must use symbolic methods. Computer Algebra Systems often implement such methods. I believe this article is talking about numerical methods, which are always bound to finite values, because of finite memory.
1letterunixname 4y ago
Infinite values are representable in IEEE floating point. I think the author didn't understand the concepts of precision, range, or of arbitrary precision calculations, and so mades bullshit proclamations without sufficient understanding of the fundamentals. PS: I notice many people I work with (in MAANG) lack an understanding of Computer Science, aren't formally educated, and lack curiosity. It's discouraging because they also lack concern for code craftsmanship and don't know how to use a debugger. This and they're paid $350-450k / year. If you had to recommend a computer science postgraduate program, which would you choose?

by somat 4y ago

raverbashing 4y ago

Because infinity is a mathematical trick, to say where something is going but it is not calculable (it is not a number in itself)

agumonkey 4y ago
well you end up in bounded axiomatic ranges

mjburgess 4y ago

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!

marginalia_nu 4y ago

An ideal computer is an universal machine. A real computer has real limitations on what is calculable, even among the things that are theoretically calculable.

moffkalast 4y ago
Sounds like we need to make a virtual universal machine and account for the limitations when emulating it.
- majewsky 4y ago
  The problem with this is whether the emulation ever terminates. Simply put, you cannot emulate an infinitely powerful machine in finite time.

e12e 4y ago

> Is it impossible to calculate infinite values in general? I would say that it's impossible in a finite universe to calculate infinite values. But it's quite possible to define and manipulate relationships that involve infinity. You can't calculate the sum of all integers - but you can relate it to the sum of all real numbers. Or compare the sum of all integers lager than one hundred, to the sum of all integers larger than ninety.

fay59 4y ago

In context, it sounds like they relied on simulations that don’t use exact numbers. I’m guessing that they saw an IEEE-754 floating-point infinity and then had to determine whether they got it because the accurate result was infinity or if the infinity they saw was the result of floating-point calculation artifacts.

credit_guy 4y ago

> Why not? They don't explain this very well, I think. Take the ordinary differential equation x'(t) = x^2(t), with initial condition x(0)=1. It has the solution 1/(1-t) which blows up to infinity when t tends to 1. If you try to solve it numerically, using, let's say Euler's method, then this is how you go about it. You pick a step size, let's say 0.1. And iterate this way: you know x(0) = 1, and you also know its derivative x'(0) = x^2(0) = 1. You assume x follows a straight line, so you get x(0.1) = 1 + 0.1 = 1.1. At the next step you add more because x'(0.1) is now 1.21, so x(0.2) = 1.1 + 0.121 = 1.221. You keep going like that. The numbers will go bigger and bigger, but they will never be infinite. Of course, floating numbers with double precision overflow around 10^308, but if you use a multiple precision library you'll be able to keep going forever and ever. If you make the time step smaller, the solution will be closer to the actual solution, but still, the algorithm will produce finite values at all times (until it hits overflow).

bheadmaster 4y ago

Computers can, in fact, calculate infinite values in style of calculus, but they must use symbolic methods. Computer Algebra Systems often implement such methods. I believe this article is talking about numerical methods, which are always bound to finite values, because of finite memory.

1letterunixname 4y ago

Infinite values are representable in IEEE floating point. I think the author didn't understand the concepts of precision, range, or of arbitrary precision calculations, and so mades bullshit proclamations without sufficient understanding of the fundamentals. PS: I notice many people I work with (in MAANG) lack an understanding of Computer Science, aren't formally educated, and lack curiosity. It's discouraging because they also lack concern for code craftsmanship and don't know how to use a debugger. This and they're paid $350-450k / year. If you had to recommend a computer science postgraduate program, which would you choose?