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Anonymous at Sun, 7 Jul 2024 15:20:41 UTC No. 16271549
I was thinking of p-adic integers and realized something. Both are uncountably infinite, both are completions of a countable thing allowing for arbirtrary sequences in some sense, both have algebraic structure. But p-adics are less schizo than real numbers because, if you define p-adics by an algorithm that computes its n'th digit, you have the following computable properties
- You can add p-adics
- You can subtract p-adics
- You can multiply p-adics
- You can compare p-adics (one-sided computability, i.e. RE)
This is really nice in comparison with real numbers, wherein whatever representation you have, I don't think you have such nice properties. I think comparison being one-sided computable, or RE, forces something like having a real number be represented by its digits. But then addition is not computable.
So my questions are twofold
1. Is there a computable representation of real numbers that has nice properties like p-adics above?
2. If not, why? The construction seems very similar, just with a different metric. So why are p-adics algebraically nicer in this way?
Anonymous at Sun, 7 Jul 2024 17:00:10 UTC No. 16271635
>>16271549
Primes generate the reals. That is why you get rationals and most reals aren't rational because the index of each space is infinitely irrational compared to the prime generating function. In either case, numbers aren't real outside of the fact that you need a way to generate a number line lol
Anonymous at Sun, 7 Jul 2024 17:07:14 UTC No. 16271639
>>16271549
>I was thinking of p-adic integers and realized something.
You are getting Norman. J. Wildberger pilled.
>1. Is there a computable representation of real numbers that has nice properties like p-adics above?
No.
>2. If not, why? The construction seems very similar, just with a different metric. So why are p-adics algebraically nicer in this way?
There are many ways to define a real number:
> 1. Infinite fractions
> 2. Equivalence of Cauchy sequences
> 3. Dedekind cuts
> 4. Continued fractions
> 5. Axiomatic
All of them have their pros and cons but neither of them will tell you how to compute reals. There is no theory of reals that will tell you how to compute reals without already assuming that you can do infinite arithmetic. For example if you choose to define real numbers as the equivalence classes of Cauchy sequences [math]\{(a_n)_{n\in\mathbb{N}}\in\m
The problem with this assumption is that any structure that the continuum may have at infinity is disregarded. It's like assuming galaxies don't exist because every light in the sky looks like a star. Saying "oh, [math]e + \pi + \sqrt{2}[/math] is a *number* that's about 7.274088044421933..." is like saying "oh, Andromeda is a *star* with apparent magnitude 3.44", when in fact it's not just a star but an entire galaxy with an order of mangitude richer structure than a star.
Just like before Hooker telescope was built (which Edwin Hubble used to conclude Andromeda is not a star but a galaxy) there is currently no theory built that properly explains what a real number "is".
https://www.youtube.com/watch?v=XXR
Anonymous at Sun, 7 Jul 2024 17:33:24 UTC No. 16271667
$p$-Adic Mathematical Physics: The First 30 Years
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, E. I. Zelenov
$p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of $p$-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.
https://arxiv.org/abs/1705.04758
Anonymous at Sun, 7 Jul 2024 17:41:19 UTC No. 16271680
>>16271667
>Dragovich
>Volovich
careful with these guys anon
I used to be interested in their work about a decade ago, and they publish a lot in journals where one of them (I think Volovich) is the editor and stuff never seems to go through the usual process. They seem to be very much their own club.
Not to say their ideas aren't interesting (Volovich wrote an interesting book), but you have to check what these guys write due to this kind of behaviour.
Anonymous at Sun, 7 Jul 2024 19:20:28 UTC No. 16271808
>>16271549
Is comparison really computable? It’s not on the computable reals.
Anonymous at Sun, 7 Jul 2024 19:24:14 UTC No. 16271817
>>16271808
As I said, comparison is R.E.
Anonymous at Sun, 7 Jul 2024 19:48:19 UTC No. 16271838
>>16271817
Oh yeah. Then computable real non-equality is semidecidable (RE), but equality isn’t.
Anonymous at Sun, 7 Jul 2024 23:47:34 UTC No. 16272186
>>16271549
>1. Is there a computable representation of real numbers that has nice properties like p-adics above?
https://www.cs.bham.ac.uk/~mhe/.tal
Usual one is binary signed-digit, where the place values are powers of two but the digits are 1, 0, and -1.
Anonymous at Sun, 7 Jul 2024 23:53:35 UTC No. 16272192
>>16272186
in PDF
Anonymous at Tue, 9 Jul 2024 11:05:15 UTC No. 16273988
2023 - Jérôme Poineau, Nombres complexes, nombres p-adiques : à la croisée des chemins (1/2)
https://youtu.be/4PRACepKvjk
2023 - Jérôme Poineau, Nombres complexes, nombres p-adiques : à la croisée des chemins (2/2)
https://youtu.be/iGtBXkNECfg
Anonymous at Tue, 9 Jul 2024 12:14:26 UTC No. 16274044
>>16271639
>Hooker telescope
Kek. So math despite being the oldest field of study is severely underdeveloped. Maybe mathfags need to stop circlejerking over pedantry and start considering applications again
Anonymous at Wed, 10 Jul 2024 00:18:33 UTC No. 16275337
>>16274044
>Kek telescope
Anonymous at Wed, 10 Jul 2024 12:17:50 UTC No. 16275907
>>16271639
It's worth pointing out that Norman J. Wildberger knows damn well how to do computations with real numbers, to the extent that such computations are possible. I know he knows because he's explained it himself.
https://www.youtube.com/watch?v=I0X
tl;dw: Nested sequences of closed rational intervals with widths converging to zero, setoids rather than equivalence classes, sequences as algorithms rather than infinite choices.
It's the same thing as alternative notation #3 on page 20 of >>16272192. There are limitations, such as the fact that equality is only decidable if the two numbers are not equal, and the provably unsolvable problem of determining whether a computer program will actually output a sequence of intervals or go into an infinite loop. But it's good enough for some purposes.
Anonymous at Wed, 10 Jul 2024 13:07:57 UTC No. 16275964
>tranime
Anonymous at Wed, 10 Jul 2024 13:32:56 UTC No. 16275998
>>16271639
holy shit you're retarded
Anonymous at Wed, 10 Jul 2024 18:27:46 UTC No. 16276295
>>16275998
Projection and filtered
Barkon Approved Post at Wed, 10 Jul 2024 18:38:17 UTC No. 16276306
>>16271549
>uncountably infinite
In English please? WTF is academia doin? Making up words and sailing off in control of intellect in front of everyone. FUCK OFF
🗑️ Barkon Approved Post at Wed, 10 Jul 2024 18:43:21 UTC No. 16276309
>>16271549
I'd love to take a nice shit down your throat. Fag. Stop posting nonsense.
Anonymous at Thu, 11 Jul 2024 22:01:16 UTC No. 16277911
>>16272192
That's really neat
Anonymous at Fri, 12 Jul 2024 20:40:36 UTC No. 16279307
>>16272186
What's the catch though?
Anonymous at Fri, 12 Jul 2024 21:19:58 UTC No. 16279368
>>16275907
wilberger got too wittgensteinpilled. Having undecidable relations makes things more clean and doesn't obstruct the main goal which is having results with computational meaning. Wanting everything to be decidable is just nitpicking and turboautism