191–200 of 205 posts
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That's what I was thinking too; If I do 1 / 3, then of course it will have to truncate, and integration errors would still be inevitable.
Actually it's possible to represent 1/3 perfectly accurately, but what about all the numbers that it's theoretically impossible to compute? (Almost all real numbers have this property)
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But for every integer there are also an infinite number of rationals. And since we can put the rationals into 1-1 correspondence with integers, that means that for every rational there are an infinite number of rationals. Infinity is funny like that. The conclusion that there are somehow more irrationals than rationals depends on subtle philosophical points that have no possible proof or disproof and usually get glos…
> The conclusion that there are somehow more irrationals than rationals depends on subtle philosophical points that have no possible proof or disproof and usually get glossed over Can you elaborate? The proof that there are more irrationals than rationals is very straightforward, from my perspective.
In Constructivism, all statements have 3 possible values, not 2. They are true, false, and not proven. All possible objects must have a construction. So instead of talking about a vague "Cauchy sequence", we might instead have a computer program that given n will return a rational within 1/n of the answer.
The first thing to notice is that all possible things that could exist is contained within a countable set of all possible constructions. There can't be "more" irrationals than rationals.
But what about diagonalization? That proof still works. You still can't enumerate the reals. But why not? The answer is because determining whether a given program represents a real is a decision problem that cannot in general be solved by any algorithm. You are running into the same category of challenges that lie behind Gödel's theorem and The Halting Problem. It is not that there are "more" irrationals than rationals. It is that there are specific programs which you can't decide whether they represent reals.
From a constructivist's eyes, the traditional proof that there are more irrationals falls apart because you're reasoning about unprovable statements about arbitrary sequences whose logic could have been constructed with the same sort of logic that you're applying to them. Is it any wonder that you wind up concluding the "existence" of things that clearly don't actually exist?
Now stepping back from BOTH philosophies, the differences between them lie in different attitudes about existence and truth. Attitudes that underly the axioms which we use, and cannot possibly be proven one way or another. (Gödel actually proved that. Any contradiction in Constructivism is immediately a contradiction in classical mathematics. But conversely there is a purely mechanical transformation of any proof in classical mathematics which resulted in a contradiction, into a constructive proof that also results in a contradiction.)
And immediately patents it... so no one else can use it. EDIT: and for some other methods: https://en.wikipedia.org/wiki/Unum_%28number_format%29 , particularly the latest one being the Posit method: http://superfri.org/superfri/article/download/137/232 EDIT2: of course other people can license it, but the other way to bring a new floating point to the scene would be through the same process that happened with IEEE 7…
> There are plenty of people who wouldn't touch anything patented at all, sometimes even with a patent clause. Then that's their loss. This seems like the ideal scenario for patent protections: small inventor developing a genuinely novel and useful invention that big, rich companies would otherwise shamelessly copy.
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The rational numbers are clearly countable. The irrational numbers are uncountable, which means that for every rational number there are infinitely many irrational numbers.
I wasn't talking about irrational numbers.
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> The conclusion that there are somehow more irrationals than rationals depends on subtle philosophical points that have no possible proof or disproof and usually get glossed over Can you elaborate? The proof that there are more irrationals than rationals is very straightforward, from my perspective.
The easiest way to understand it is to look at the question from a philosophical framework where it makes no sense to claim that there are "more" irrationals than rationals. And then untangle why it came to a different answer. In Constructivism, all statements have 3 possible values, not 2. They are true, false, and not proven. All possible objects must have a construction. So instead of talking about a vague "Cauchy…
Earlier quoted context omitted.
Actually it's possible to represent 1/3 perfectly accurately, but what about all the numbers that it's theoretically impossible to compute? (Almost all real numbers have this property)
>Actually it's possible to represent 1/3 perfectly accurately I'm curious, how?
Is possible to convert to rationals in calculations?
So if I do:
1/3 + 4 + sum / total it record values properly?
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>Actually it's possible to represent 1/3 perfectly accurately I'm curious, how?
And another dumb questions: Is possible to convert to rationals in calculations? So if I do: 1/3 + 4 + sum / total it record values properly?
Here’s the issued patent: https://www.google.com/patents/US9817662 Note that it’s a claim on the processing unit implementation (e.g. the FPU), not the method. Nonetheless, I’d be very surprised if this stands the test of interval arithmetic prior art.
The beauty of the patent is that it will never be tested, being of no practical value, for numerous reasons already mentioned in other comments, plus a few more not worth the bother of going into. So the inventor gets a patent number for his LinkedIn profile, and USPTO get their fee, and that's the end of it. A win-win for all involved. Earlier HN discussion of the phenomenon: https://news.ycombinator.com/item?id=160…
Patents like this have "threat value", which is often happily exploited by "IP monetization" companies, contingent law firms, etc.
This is the kind of stuff that turns into 100x $50k settlement demands.