201–210 of 275 posts
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> I think that the set of possible French texts is exactly as large as the set of reals. How? The set of French texts is countable and the set of reals is uncountable.
0,1,2 etc are French words. All reals are French words 8)
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Mathematician here. If you mean to say that you cannot constructively prove "the real numbers are not countable", then you're wrong. As a rule of thumb, you can usually prove negative statements constructively as you would prove them classically. A constructivist would probably state the result more positive (and stronger, constructively): To every countable set M of real numbers, there is a real number not contained…
I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since…
The computational techniques from Automatic Differentiation use these types of entities to calculate derivatives exactly without approximating infinite (limiting) processes.
Calculus can be done constructively without infinite limiting processes purely algebraically using these nilpotents. And from a geometric interpretation there is nothing nonsensical about a tangent line to a curve.
Also you don't differentiate numbers (real or rational), you can only differentiate functions.
Also the idea of a actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent.
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I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since…
Reals are not equivalently as unrealizable or dubious as infinitesimals. Infintesimals can be constructively specified as nilpotent/nilsquare entities, numerical entities which when squared equal 0 but where those entities themselves aren't reducible to 0. All of this can be done in a constructive manner avoiding any use of infinity or classical logic that depends upon indirect proofs (ie excluded middle). John Bell'…
I don't have problem with calculus -- or I wouldn't be able to do physics. I am having problem with calculus based on infinity.
> Also the idea of an actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent.
Well said.
> Also you don't differentiate numbers (real or rational), you can only differentiate functions.
Of course I meant distinction.
> "the halting probability Ω, which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason" I really enjoyed the beauty of this statement.
My philosophical take: the halting problem illustratres how free will can exist in a deterministic universe
This was a light and interesting read. Here is the direct link: https://arxiv.org/pdf/math/0411418.pdf "Indeed, the most important thing in understanding a complex system is to understand how it processes information. This viewpoint regards physical systems as information processors, as performing computations. This approach also sheds new light on microscopic quantum systems, as is demonstrated in the highly develop…
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only that in a single symbolic system we can't have expressions for all of them at once. I don't think that's true. There are only countably many different symbolic systems[1], and as we can only express countably many numbers in each, we don't leave the realm of countable. [1] - A "symbolic system" must at least come with a procedure to tell whether a sequence is a part of it, and, unless you disbelieve the Church-T…
Doesn't this (and by extension, Curch-Turing) rely on the assumption that the universe is discrete? Provided it were not, and one were able to harness infinite precision, one could presumably make a new symbolic system based on it. Not saying I believe it, just teasing out assumptions. If one is arguing whether the universe is continuous and using the Church-Turing thesis as justification for something, there's a dan…
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0,1,2 etc are French words. All reals are French words 8)
This is not true, no French word has an infinite length. Any set of finite strings is countable.
> "the halting probability Ω, which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason" I really enjoyed the beauty of this statement.
My philosophical take: the halting problem illustratres how free will can exist in a deterministic universe
So I would argue that it illustrates nothing of significance under either of those issues.
This entire subject is very academic and theoretical, and will never impact anything in the real world. Using Big Fractions instead of floating-point numbers is a far more concrete argument with definite real-world impact! https://news.ycombinator.com/item?id=13855198
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A set being "countably infinite" only means that you can write a function that maps each distinct entry in the set to exactly one natural number (0, 1, 2, etc.) without duplicates. That's it. So for example, the set of natural numbers is countably infinite and we know this because we can write a function that maps each natural number to exactly one natural number: the id function. We can extend this and say that the…
Why a pre-definable function matters here? We have allowed a real number (that cannot be exhaustively described) in, so why a real function (that you cannot exhaustively define) has to be excluded? Isn't it unfair that I am only allowed to use a finitely definable function while you can choose a non-pre-finitely-describable real number? Isn't that a loop proof -- that the set of real numbers is uncountable simply bec…