How real are real numbers? (2004) • Hacker News Board

Re: How real are real numbers? (2004)

#271

Post by prmph » Wed, Apr 12, 2017, 9:35 PM UTC

Retra wrote: post #208

prmph wrote: post #153

Earlier quoted context omitted.

My philosophical take: the halting problem illustratres how free will can exist in a deterministic universe

Free will is just what it feels like to have a mind that can construct models of realities that are not fact. And you can model nondeterminism in a deterministic system just fine. So I would argue that it illustrates nothing of significance under either of those issues.

How do you model true non-determinism in a deterministic system? And if that can indeed be done, would that not support my original point?

Re: How real are real numbers? (2004)

#272

Post by mikhailfranco » Thu, Apr 13, 2017, 8:21 AM UTC

scythe wrote: post #44

>In addition to this mathematical soul-searching regarding real numbers, some physicists are beginning to suspect that the physical universe is actually discrete [Smolin, 2000] and perhaps even a giant computer [Fredkin, 2004, Wolfram, 2002]. It will be interesting to see how far this so-called “digital philosophy,” “digital physics” viewpoint can be taken. Here is how far: Everything written in words about the physi…

One of the few comments here that shows great insight. I think you should push the argument down a few layers, to talk about the discreteness of quantum observables, the underlying unobservable continuous wavefunction, and the randomness of the Born rule that maps between the two (Copenhagen collapse or forking of Many Worlds).

Tegmark takes a similar, but rather extreme, MW approach in his book 'Our Mathematical Universe'.

Personally, I struggle with Tegmark's use of Measure Theory, and proponents of various Anthropic Principles, because they seem to have a completely broken frequentist view of probability and inference.

Re: How real are real numbers? (2004)

#273

Post by mannykannot » Thu, Apr 13, 2017, 6:49 PM UTC

whatshisface wrote: post #3

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely."

So I guess that probability has to be a rational number?

Re: How real are real numbers? (2004)

#274

Post by mikhailfranco » Sun, Apr 16, 2017, 6:45 AM UTC

klodolph wrote: post #35

maverick_iceman wrote: post #16

Earlier quoted context omitted.

Our best theories, General Relativity and the Standard Model, say that the world is a continuum.

I'm a little uncomfortable with the language that the theories "say that the world is" X. General Relativity and the Standard Model both model the world using real numbers, but they're both known to be wrong, and the fact that they are continuous is not a great reason to claim that the universe is continuous. On the other hand, observations about Lorentz symmetry holding at distances on the order of the Planck scale…

I think it is easy to get around the problems of Lorentz invariance in discrete models, as long as it is not the spacetime that is explicitly discretized on the lattice. The lattice must be some other combinatorial graph structured algebra, with non-local 'propagation' of fields. Quantum Mechanics says that space is only defined relationally on the intervals between field interactions: a 'particle' is only localized as a particle when it interacts (position or momentum observable). So discrete space and time appear as values on some subset of lattice nodes in response to some propagating fields ('particles' only at the interaction event). The slogan for this is 'spooky distance at an action', because it is the (inter)action that defines the space(time).

QM (and QFT) assume a background time, it is not an observable, even though it appears to commute with energy (e.g. energy is momentum in the time direction). So it's more tricky to understand how time emerges in a discrete Quantum Gravity, but I suspect there is an intrinsic proper time, defined by interactions with the 'vacuum' (minimal field states on the underlying lattice), which bootstraps a relational time defined over intervals between interactions.

Re: How real are real numbers? (2004)

#275

Post by E6300 » Mon, Apr 17, 2017, 3:38 PM UTC

deepsun wrote: post #268

E6300 wrote: post #246

Earlier quoted context omitted.

> "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 w…

> 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

√ is a function. √2 is the result of applying √ to 2, which is a number. Since "how Giants scored last season" is a function, it can be applied to a value such as "2017-04-17", therefore "how Giants scored last season, and today is 2017-04-17" would be a number.

I don't follow your P=NP argument. Either P=NP or P!=NP. They can't both be simultaneously true. I also don't follow how the continuum hypothesis is relevant. = is not comparing the cardinalities of P and NP, it's asking whether they're the same set (i.e. all elements of P are in NP and vice versa). Again, either they're the same set or they're not. Even if there was a cardinality between that of N and that of R, I don't see how that would change how we compare sets for identity.