> 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?
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.
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.
So this is just not true, and I'm not exactly sure where these premises are coming from. Is it a misunderstanding of theoretical computer science, mathematics, engineering, or what?
But I can at least now see why you're attached to extremely strange notions about, eg., floats being sufficient representation for mathematical reasoning. Ie., some article of faith that "computers" must be capable of everything.
There is no "symbolic computation on pi" that arent rules of inference created by people. We arent born with these rules, we create them. So if we havent yet created them, there's no sense in saying any actual computer is capable of anything. Actual computers are merely implementations of rules we'd have to create.
The process of conceptualising the world is, in my view, continuous and non-cognitive. One example of it is in the generative capacities of the imagination, which presents situations as wholes and it's latent space imv is continuous -- having to do with the structure of the sensory-motor system.
In any case, regardless of whether you believe animals have access to a continuous reality which cannot be formalised in discrete mathematics, we arent talking about whether there are possible computers which can reason this way -- we're talking about actual computers. (Though we have no reason to suppose there are such possible computers, and proofs against such things, ie., the non-computability of the reals).
It's relatively trivial to show that all existing computers are woefully incapable of a vast amount of things. Consider, only, the exponential space complexity of storing the parameters of a chaotic system. In any existing computer, we'd need an electronic system the size of a planet merely to track what's going on inside an atom.
It requires vast arrays of machines to track surface properties of particles interacting in the LHC, for example.
Yet, of course, we can formulate QFT. There are a near infinite number of such "existence proofs" of the power of animal mental capacities: AND NOT A SINGLE ONE! Of machine capacities.
No existing actual computer has ever created a system of concepts to formalise a hitherto unformalised domain. No one has even solved the problem of how it would be possible for a machine to do so (ie., the framing problem).
This makes actual computers, and all possible ones we can presently even imagine useless for open problems with unformalised domains.
The only role a computer can play here is providing an implementation of a discrete aproximation we have created, and this aproximation is woefully inadequate to the task. Even using a computer here is just a means of improving the power of human speculation.
In any case, this article of faith in the power of discrete mathematics and the electrical systems which we use to implement it, blinds you to the overwhelming and woeful inadequacy of all existing systems.
To the point you're even defending floating pt representations of infinity. If you really wish to cling to that religion, you're going to have to get better at choosing which hills to die on. Saying floats here are a sensible means of representing problems in continuous mathematics is absurd, and discredits your views greatly.
The only computers you should be defending here are "presumably possible" ones, yet to do be defined, yet even to be specified.
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?