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A computer is always just a function from {0,1}^N -> {0,1}^M The "Inf" interpretation, of, eg., 11111111111111111 isnt infinity. And, in general, almost ever

A computer is always just a function from {0,1}^N -> {0,1}^M The "Inf" interpretation, of, eg., 11111111111111111 isnt infinity. And, in general, almost every function isnt representable on the discrete domain above. floats are a hacky interpretation of discrete bit patterns

thaumasiotes 4y ago
> A computer is always just a function from {0,1}^N -> {0,1}^M > The "Inf" interpretation, of, eg., 11111111111111111 isnt infinity. This is incoherent nonsense. If you want to say that the floating point value "infinity" isn't really infinity, you must also say that nothing else is really infinity either. That is true in a completely useless and uninformative sense, but it's false in every sense a person would ever use. And it fails to distinguish the correct mathematical proof demonstrating an infinite limit from the simulation suggesting, but not demonstrating, an infinite limit. Neither of them is really infinity. Each of them can represent infinity just as accurately as the other, though. A paper is just a function from a bounded countable subset of ℝ² to another bounded countable subset of ℝ². What would you conclude, from that, about the limitations of what you can represent on paper?
- mjburgess 4y ago
  I'm not sure where your misunderstanding comes from, but at least, you might consider you're disagreeing with an article on quanta magazine which writes up a project by experts in their field. In any case, no. The idea that a finite number of bits in a particular state "must just be infinity!!!! because the IEEE ref docs say so" is strange to say the least. The issue is to demonstrate that a given function, say f, for given real-valued inputs, say x, has an output y which is not real-valued and goes to infinity. A computer cannot demonstrate such a thing, because real-valued functions aren't computable. No sequence of bit patterns on a computer can ever show the above, because the above is an issue of the limits of a continuous function. Relevant inputs to the function have an infinite precision, and relevant outputs have an infinte precision. Eg., x = pi, y = pi^100 A computer may very easily mistakenly conclude x = pi is an input which becomes infinite in y, because (1) a computer cannot represent pi; and (2) cannot represent pi^100 either. infinity isn't a bit pattern; and isn't here in any relevant sense even a number; the IEEE standard may as well have said "Overflow"
  - thaumasiotes 4y ago
    > The issue is to demonstrate that a given function, say f, for given real-valued inputs, say x, has an output y which is not real-valued and goes to infinity. Why would the output need to be not real? There's no difficulty with saying a real-valued function has a singularity. The issue is to demonstrate that this function has a singularity at some point, yes. Simulation is a bad way to do that, though conceivably you could get lucky. > A computer cannot demonstrate such a thing, because real-valued functions aren't computable. Obviously false; computers are fully capable of providing proofs that some function has an infinite limit somewhere. > The idea that a finite number of bits in a particular state "must just be infinity!!!! because the IEEE ref docs say so" is strange to say the least. That is the only way anything is ever infinity - by designation. As I pointed out elsewhere, IEEE infinity has all the correct mathematical properties of positive infinity in the extended reals, so it's difficult to see what you think you're saying. > I'm not sure where your misunderstanding comes from, but at least, you might consider you're disagreeing with an article on quanta magazine which writes up a project by experts in their field. Writing about an expert doesn't make you any smarter. The reason proffered by Quanta is nonsense. They are correct that the experiment they describe cannot achieve the goal sought; they are quite obviously wrong about why. > infinity isn't a bit pattern; and isn't here in any relevant sense even a number; the IEEE standard may as well have said "Overflow" That's what infinity is. In every sense. Overflowing is defined by exceeding a boundary; infinity is defined by exceeding all boundaries. I'm morbidly intrigued by your fetish for the idea of "bit patterns". Infinity is also not an image on paper. How do you expect a correct mathematical proof to represent infinity?
    - mjburgess 4y ago
      The issue with bit-patterns are, at least, they're discrete. And so cannot, eg., represent pi. This project is about real-valued functions which are taken to describe physical reality. Almost all of physical reality has no closed-form analytical description that "traditional mathematics" can operate on. So there arent any relevant symbolic rules of inference yet invented to resolve this problem. If you want to program a computer to perform these rules on these functions, there arent any -- hence the millenium problem. And if there were some, we wouldnt bother using a computer. What "using a computer" here means is finding a discrete approximation to this system, searching through that discrete input space until something which looks "infinity-like" occurs in the output space. Now, a priori, this is never going to constitute a proof of anything. Since the discrete approximation needs, independently, to be analytically shown to be reliable. And, a priori, it's likely to be highly highly unreliable. It would be trivial to show, for example, an iterated chaotic system is sensitive to an x=pi initial state at "decimal places" that no possible physical computer could provide a discrete approximation of; and hence inferences made via this approximation would be, routinely, false. (This is, for example, why most "climate" models only predict a global mean temperature, and very little else). So this all comes down to the need to formalise a non-discrete system in discrete terms, and worse in terms that are physically possible implement using electrical switches. In this case, every output of the system including special designation of bit patterns is, a priori, profoundly suspect.

thaumasiotes 4y ago

> A computer is always just a function from {0,1}^N -> {0,1}^M > The "Inf" interpretation, of, eg., 11111111111111111 isnt infinity. This is incoherent nonsense. If you want to say that the floating point value "infinity" isn't really infinity, you must also say that nothing else is really infinity either. That is true in a completely useless and uninformative sense, but it's false in every sense a person would ever use. And it fails to distinguish the correct mathematical proof demonstrating an infinite limit from the simulation suggesting, but not demonstrating, an infinite limit. Neither of them is really infinity. Each of them can represent infinity just as accurately as the other, though. A paper is just a function from a bounded countable subset of ℝ² to another bounded countable subset of ℝ². What would you conclude, from that, about the limitations of what you can represent on paper?

mjburgess 4y ago
I'm not sure where your misunderstanding comes from, but at least, you might consider you're disagreeing with an article on quanta magazine which writes up a project by experts in their field. In any case, no. The idea that a finite number of bits in a particular state "must just be infinity!!!! because the IEEE ref docs say so" is strange to say the least. The issue is to demonstrate that a given function, say f, for given real-valued inputs, say x, has an output y which is not real-valued and goes to infinity. A computer cannot demonstrate such a thing, because real-valued functions aren't computable. No sequence of bit patterns on a computer can ever show the above, because the above is an issue of the limits of a continuous function. Relevant inputs to the function have an infinite precision, and relevant outputs have an infinte precision. Eg., x = pi, y = pi^100 A computer may very easily mistakenly conclude x = pi is an input which becomes infinite in y, because (1) a computer cannot represent pi; and (2) cannot represent pi^100 either. infinity isn't a bit pattern; and isn't here in any relevant sense even a number; the IEEE standard may as well have said "Overflow"
- thaumasiotes 4y ago
  > The issue is to demonstrate that a given function, say f, for given real-valued inputs, say x, has an output y which is not real-valued and goes to infinity. Why would the output need to be not real? There's no difficulty with saying a real-valued function has a singularity. The issue is to demonstrate that this function has a singularity at some point, yes. Simulation is a bad way to do that, though conceivably you could get lucky. > A computer cannot demonstrate such a thing, because real-valued functions aren't computable. Obviously false; computers are fully capable of providing proofs that some function has an infinite limit somewhere. > The idea that a finite number of bits in a particular state "must just be infinity!!!! because the IEEE ref docs say so" is strange to say the least. That is the only way anything is ever infinity - by designation. As I pointed out elsewhere, IEEE infinity has all the correct mathematical properties of positive infinity in the extended reals, so it's difficult to see what you think you're saying. > I'm not sure where your misunderstanding comes from, but at least, you might consider you're disagreeing with an article on quanta magazine which writes up a project by experts in their field. Writing about an expert doesn't make you any smarter. The reason proffered by Quanta is nonsense. They are correct that the experiment they describe cannot achieve the goal sought; they are quite obviously wrong about why. > infinity isn't a bit pattern; and isn't here in any relevant sense even a number; the IEEE standard may as well have said "Overflow" That's what infinity is. In every sense. Overflowing is defined by exceeding a boundary; infinity is defined by exceeding all boundaries. I'm morbidly intrigued by your fetish for the idea of "bit patterns". Infinity is also not an image on paper. How do you expect a correct mathematical proof to represent infinity?
  - mjburgess 4y ago
    The issue with bit-patterns are, at least, they're discrete. And so cannot, eg., represent pi. This project is about real-valued functions which are taken to describe physical reality. Almost all of physical reality has no closed-form analytical description that "traditional mathematics" can operate on. So there arent any relevant symbolic rules of inference yet invented to resolve this problem. If you want to program a computer to perform these rules on these functions, there arent any -- hence the millenium problem. And if there were some, we wouldnt bother using a computer. What "using a computer" here means is finding a discrete approximation to this system, searching through that discrete input space until something which looks "infinity-like" occurs in the output space. Now, a priori, this is never going to constitute a proof of anything. Since the discrete approximation needs, independently, to be analytically shown to be reliable. And, a priori, it's likely to be highly highly unreliable. It would be trivial to show, for example, an iterated chaotic system is sensitive to an x=pi initial state at "decimal places" that no possible physical computer could provide a discrete approximation of; and hence inferences made via this approximation would be, routinely, false. (This is, for example, why most "climate" models only predict a global mean temperature, and very little else). So this all comes down to the need to formalise a non-discrete system in discrete terms, and worse in terms that are physically possible implement using electrical switches. In this case, every output of the system including special designation of bit patterns is, a priori, profoundly suspect.