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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 pr

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
    Computers work with perfect representations of pi all the time. The TI-89 will do it routinely. All of your objections continue to apply just as strongly to human mathematicians as they do to computers. But you apparently believe there is a difference between what the mathematicians can do and what the computers can do. This is false. Any problem that occurs in computers' representations of values will also occur in human representations of values. Using your example, a system that is sensitive to differences so fine that they cannot be held in any realistic amount of memory is quite possible. But humans will have just as much trouble using it as computers do. If it is easy to show that x=pi in particular causes trouble, computers will find that easy too, using the same tools -- symbolic computation on pi -- that humans do. The fact that computers have discrete internal representations is not relevant to anything. All human mathematics is also performed using exclusively discrete representations.

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
  Computers work with perfect representations of pi all the time. The TI-89 will do it routinely. All of your objections continue to apply just as strongly to human mathematicians as they do to computers. But you apparently believe there is a difference between what the mathematicians can do and what the computers can do. This is false. Any problem that occurs in computers' representations of values will also occur in human representations of values. Using your example, a system that is sensitive to differences so fine that they cannot be held in any realistic amount of memory is quite possible. But humans will have just as much trouble using it as computers do. If it is easy to show that x=pi in particular causes trouble, computers will find that easy too, using the same tools -- symbolic computation on pi -- that humans do. The fact that computers have discrete internal representations is not relevant to anything. All human mathematics is also performed using exclusively discrete representations.