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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#61

Post by xelxebar »

testaccount28 wrote: post #28
xelxebar wrote: post #19

Earlier quoted context omitted.

> What do "real" numbers buy you? They're well-known and have a simpler implementation, and we are familiar with their quirks. There is a giant body of useful knowledge built up around standard real analysis. That doesn't really exist if you insist on using only computable numbers. The computables are also more fiddly in many ways. Because equality is undecidable, you can't have discontinuous functions, you need to c…

> equality is undecidable equality is always undecidable until you see the light of intuition. consider the rational number whose numerator is 0 if $theorem is true, and 1 if it is false, and whose denominator is 1.

Okay, theorem=generalized-continuum hypothesis. If you use exotic axioms to give that a definite result, the go eat a Gödel.

We define computable numbers to be Turing machines, lambda reduction processes, or whatever your favorite model of computation happens to be. If you don't like this kind of definition, then we need to talk philosophy of computation.

To decide equality, we let your machines clunk along until they both produce a result, which we then compare (using another machine). Hello Mr. Halting Problem. Specific programs are fine, but comparing against arbitrary classes of program is the bugger. This is why discontinuous functions cannot exist in a hardline computable analysis theory.

Re: How real are real numbers? (2004)

#62

Post by d4ng »

dhosek wrote: post #2

I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-phy…

What numbers in R would not be possible to express in an infinite universe?

Re: How real are real numbers? (2004)

#64

Post by jojomodding »

d4ng wrote: post #62
dhosek wrote: post #2

I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-phy…

What numbers in R would not be possible to express in an infinite universe?

Chaitin's constant. Or rather, you can never see that it has been laid out in your infinite universe.

Re: How real are real numbers? (2004)

#65

Post by exac »

ttctciyf wrote: post #29

I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs. I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adop…

> According to Pythagoras everything is number, and God is a mathemati- cian. This point of view has worked pretty well throughout the development of modern science. However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information. In other words, now everything is software, God is a computer programmer, not a mathematician, and the world is a giant information-processing system, a giant computer [Fredkin, 2004, Wolfram, 2002, Chaitin, 2005]

¯\_(ツ)_/¯ For some reason I get the same vibe from this as people referring to LLM inference using gendered pronouns instead of "it".

Re: How real are real numbers? (2004)

#66

Post by cgio »

twothreeone wrote: post #44

> To prove that Ω is computationally and therefore logically irreducible, requires a theory of program-size complexity that I call algorithmic infor- mation theory (AIT) [Chaitin, 2005] Interesting, I think everyone else calls this Kolmogorov complexity.

Actually it’s Solomonoff. But it’s called Solomonoff-Kolmogorov-Chaitin. So Chaitin is among the few people entitled to call it something and AIT is real and less pretentious than using his name.

Re: How real are real numbers? (2004)

#67

Post by cgio »

exac wrote: post #65
ttctciyf wrote: post #29

I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs. I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adop…

> According to Pythagoras everything is number, and God is a mathemati- cian. This point of view has worked pretty well throughout the development of modern science. However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information. In other words, now everything is software, God is a computer programmer, not a mathematician, and…

Arguably starts with Wheeler and “it from bit”. Zuse also, who predates Wolfram. For me it’s a little bit like calling the egg you hold in your hand the centre of the Universe while rolling on a skateboard.

Re: How real are real numbers? (2004)

#68

Post by tialaramex »

Dylan16807 wrote: post #38
Nevermark wrote: post #34

It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed numbers, consistent with the forms we actually use to represent quantities and perform numerical operations. I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.

> I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers. They're worse. Just having another dimension is significantly more relevant to reality. And the reals also ruin the word "normal".

I actually think both "real" and "normal" are helpful in building the correct intuition about how complicated the world we inhabit is rather than how simple we want it to be.

Re: How real are real numbers? (2004)

#69

Post by nyc111 »

exac wrote: post #65
ttctciyf wrote: post #29

I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs. I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adop…

> According to Pythagoras everything is number, and God is a mathemati- cian. This point of view has worked pretty well throughout the development of modern science. However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information. In other words, now everything is software, God is a computer programmer, not a mathematician, and…

> However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information

Can this be true? DNA is the information system of living creatures and as far as I know, it is not coded with 0s and 1s. So, how can we justify that "the world is built entirely out of digital information"?

Re: How real are real numbers? (2004)

#70

Post by atoav »

nyc111 wrote: post #69
exac wrote: post #65

Earlier quoted context omitted.

> According to Pythagoras everything is number, and God is a mathemati- cian. This point of view has worked pretty well throughout the development of modern science. However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information. In other words, now everything is software, God is a computer programmer, not a mathematician, and…

> However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information Can this be true? DNA is the information system of living creatures and as far as I know, it is not coded with 0s and 1s. So, how can we justify that "the world is built entirely out of digital information"?

For all I, a Victorian everyman, know the world is built from small pistons, gears and pulleys.

Every age has their technology which they will project onto the world. A century ago one may have started to talks about everything being electrical wires and switches.

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