Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

How real are real numbers? (2004)

arxiv.org

Post reply on HN
Print view View original

261–270 of 275 posts

Re: How real are real numbers? (2004)

#261

Post by Dr_Segfault »

hzhou321 wrote: post #241
stymaar wrote: post #237

Earlier quoted context omitted.

> I'll try to physicalize countability. > Start counting the naturals: 1, 2, 3, ... > At future timelike infinity you'll reach infinity. > Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth. > 0 0.0........ > At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point. > In the first case, at any finite time before \breve{i}^+ you will h…

> but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this. It is only straightforward once you accepted infinity and all other definitions/description based on infinity. If you, like me, cannot accept infinity, then all the proofs/descriptions that contain infinity become apparent non-sensical. >> Start counting the naturals: 1, 2, 3, ... >> At future timelike infinity you'll…

The mathematical concept of infinity generally derives from the axiom of infinity in Zermelo-Fraenkel set theory, which asserts the existence of the inductive set (which is the basis for the construction of the natural numbers). Specifically, it states that there exists a set N such that the empty set is in N, and for all x in N, x union {x} is also in N. This is an axiom, so it cannot be proven to be right or wrong. You can either accept this axiom, or attempt to create your own system of mathematics that does not require the axiom of infinity.

Re: How real are real numbers? (2004)

#262

Post by Bigbestbooty »

socmag wrote: post #249
tel wrote: post #175

For those interested in constructive and intuitionistic approaches here Dummett's [0] Elements of Intuitionism is an extremely good read. Intuitionism is a form of a constructive foundation for mathematics which (a) notes that any attempt to deny the uncountability of reals leads to difficulties and (b) any attempt to internally define them violates constructivity. The resolution proposed is to posit the existence of…

All numbers are ultimately "probabilistic" in calculations. Sure you can state them as a "fact", but that undermines quantum theory itself and the Universe. If you want to do the math right you really need to consider terms as peak probabilities and work from there. (See Chaos theory) IMHO Of course doing math with hard constraints is perfectly adequate for most macroscopic calculations. That said, errors do add up g…

are u retrded loool

Re: How real are real numbers? (2004)

#263

Post by alexbecker »

stymaar wrote: post #244
CuriouslyC wrote: post #176

Earlier quoted context omitted.

Interesting. How do you describe the difference between two successive real numbers?

> How do you describe the difference between two successive real numbers? There is no such things as «two successive real numbers».

Yeah, there aren't even two successive dual numbers. What's after 1? If it's 1 + epsilon, then what about 1 + epsilon / 2?

Re: How real are real numbers? (2004)

#264

Post by stymaar »

stymaar wrote: post #253
gertef wrote: post #76

Earlier quoted context omitted.

The number you are paraphrasing cannot be precisely identified in finite time.

Oh, you're right. For that I would need to explicit the construction of the sequence n0, n1, … but that's impossible : if I can find such sequence, my proof holds but at the same time such sequence shows that the set is countable => contradiction, hence there is no such sequence. But well, now that we've proven that such sequence doesn't exist, we've proven that the given set is not countable ! Not the most elegant p…

Actually it doesn't: what if the given sequence exists but cannot be expressed with words ? (This is exactly the same thing than «a real number that can't ne expressed with words» which is what I'm trying to demonstrate not to exist. I'm going nowhere)

Re: How real are real numbers? (2004)

#265

Post by klodolph »

Smaug123 wrote: post #214
Smaug123 wrote: post #20

Earlier quoted context omitted.

Then my statement is vacuously true. (This was intentional, but perhaps a bit obscure.)

I'm a bit confused why this got downvoted. I can understand my parent post being downvoted, but the explanation for my parent post? Is it false?

I simply can't make heads nor tails of the parent comment. It was probably downvoted because people think it is nonsensical. Here is my explanation:

1. What does it mean to be "discrete on a macroscopic level"? (If a universe were discrete, would it not be discrete at any scale? When we use the term "discrete", are we talking about state space, or discrete spacetime, or what?)

2. What does it mean for a universe to "inherently [be] a natural number"? (Universes are not numbers, right? Is some sort of claim that the universe's state space is finite?)

3. What does have to do with whether a universe was "intelligently created"? (How can it possibly make sense that "intelligent creation" is an alternative to a discrete universe? The claims seem entirely unrelated.)

The "explanation" comment doesn't attempt to explain anything so I'm not sure why you call it an explanation. (Well, perhaps it explains that a conditional with a false antecedent is logically true, but most people here already know that, so pointing it out is not a great way to contribute to the discussion.)

Re: How real are real numbers? (2004)

#266

Post by Confusion »

mbid wrote: post #215
Confusion wrote: post #213

Earlier quoted context omitted.

A surjection seems silly: doesn't that imply that some reals would be indistinguishable? Or can you indeed not prove that more than the natural number of reals are distinguishable in constructive mathematics?

Honestly, the result hinted at in the wikipedia article caught me by surprise as well: There are some flavors of constructive mathematics (notably CZF), that are still consistent if you also add the statement "There is a subset A of the natural numbers such that there is a surjection from A onto the real numbers". Note that this does not imply that you can prove this in CZF, it only means that you cannot disprove it.…

OK, then it isn't as silly as I thought. I guess 'distinguishable' was my layman's approximation of the formal term 'decidable'.

Re: How real are real numbers? (2004)

#267

Post by mikhailfranco »

For the parallel historical development of 'the continuum' in physics, I recommend this readable survey:

Paper:

https://arxiv.org/abs/1609.01421

https://math.ucr.edu/home/baez/continuum.pdf

Blog summary with discussion:

https://johncarlosbaez.wordpress.com/2016/09/08/struggles-wi...

https://johncarlosbaez.wordpress.com/2016/09/09/struggles-wi...

Re: How real are real numbers? (2004)

#268

Post by deepsun »

E6300 wrote: post #246
deepsun wrote: post #219

Earlier quoted context omitted.

> Would you mind producing such a phrase? Sure: "Current quarter of the Moon". Or "How Giants scored last season". Or "One, if P=NP, zero otherwise". Well, I see your point, if we limit our phrases to only formal language, then you're right.

> "Current quarter of the Moon". Or "How Giants scored last season" These sentences describe functions, not numbers. > "One, if P=NP, zero otherwise". This value is a constant. It doesn't change through time. We simply don't know what it is.

> These sentences describe functions, not numbers.

Yes, and if we think deeper about it, we'll find out that in mathematics we very often use functions in place for numbers, like it was the same thing. Simple example: √2. Even it form suggests that we apply function "square root" to number 2. Less obvious example: π. It's also a relation between diameter and circumference.

If we go even deeper, trying to understand what, for example, number "2" means, we'll come up to their definition through functions (or classes) of equivalence of sets cardinalities (cardinal numbers), or order (ordinal numbers), where sets are defined using ZFC, for example.

I met this discrepancy while naively trying to program mathematical logic in college. In mathematics you just "take" a number, while in programming you "create" or "instantiate" a number, and it has a ton of consequences.

>> "One, if P=NP, zero otherwise".

> This value is a constant. It doesn't change through time. We simply don't know what it is.

Not necessarily a constant, as it's not proven to be a constant yet. As an example of how it may turn out to be non-constant, check out Continuum hypothesis proof (https://en.wikipedia.org/wiki/Continuum_hypothesis) : solution is independent of our axiom set. In other words -- we can state that ℵ1=c, or we can say ℵ1!=c, and both statements will not contradict anything else. We'll just get two different models, with one more axiom each.

Re: How real are real numbers? (2004)

#269

Post by tel »

pron wrote: post #259
tel wrote: post #256

Earlier quoted context omitted.

> it is as impossible to determine whether two lawful sequences are equal as it is to determine whether two lawless sequences are I also think that's enough. I think when you mention that it's easier to build a mechanistic theorem prover on an intuitionistic foundation that it's a bit more "foundational" a problem than it seems. Theorem provers need (some kind of partial) equality on propositions and on proofs in ord…

> but propositions are too rich to support finite equality Why do you say that? The finitary deduction rules of all logic systems provide both equivalence ( ) and partial order (=> or |-) relations. Classical logic gives rise to a boolean algebra, while intuirionistic logic forms a Heyting algebra, of which boolean algebra is a special case. The latter is more general, but both are bounded lattices, and both are perf…

Sorry, on the propositions themselves, yes, but it's hard to internalize equality: the big sticking point between ITT, OTT, HTT, etc.

Re: How real are real numbers? (2004)

#270

Post by pron »

tel wrote: post #269
pron wrote: post #259

Earlier quoted context omitted.

> but propositions are too rich to support finite equality Why do you say that? The finitary deduction rules of all logic systems provide both equivalence ( ) and partial order (=> or |-) relations. Classical logic gives rise to a boolean algebra, while intuirionistic logic forms a Heyting algebra, of which boolean algebra is a special case. The latter is more general, but both are bounded lattices, and both are perf…

Sorry, on the propositions themselves, yes, but it's hard to internalize equality: the big sticking point between ITT, OTT, HTT, etc.

It's only hard if you're constructive (and so all relations have to be computable, including equality).
Post reply on HN