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

Pi Is Wrong

math.utah.edu

Post reply on HN
Print view View original

61–69 of 69 posts

Re: Pi Is Wrong

#61

Post by Dn_Ab »

lutorm wrote: post #12
pwpwp wrote: post #2

Instead of whether we should use Pi or 2*Pi, I find much more interesting R. Buckminster Fuller's statements that "Pi is operationally irrelevant" and that "Nature is not using Pi". "Inasmuch as the kind of mathematics I had learned of in school required the use of the XYZ coordinate system and the necessity of placing π in calculating the spheres, I wondered, "to how many decimal places does nature carry out π befor…

Hogwash. Mathematics is a way of describing nature, not the other way around. And doesn't Gödel's incompleteness theorem make it clear that any such attempt must necessarily be flawed?

The quote is not hogwash, in fact it looks to me like you do not oppose the thesis of the quote.

Godel's incompleteness theorems are only applicable to a very specific type of mathematical system: formal axiomatic systems of a specific power (allow (+), (-), succ, ( * ), = , and a few more axioms relating these). Gödel says:

a) Within your formal system some statements simply cannot be proven nor disproven. Kinda like junk dna of your formal system. Very, very roughly, imagine a bunch of islands connected by bridges. Some bridges lead nowhere. Getting from A -> B, is a proof of something's truth. The set of bridges and islands is your deductive system. Godel 1st ICT says there are some islands of legend where you cannot show that no bridges lead to them nor can you find a path to get to them. They are effectively unreachable, "independent". You need a boat or plane to prove that the islands are even real instead of just a trick of fog.

b) GIT2 follows from 1 and says you cannot prove the consistency of your full formal system within your formal system.

Notice that our scientific theories thus far have been neither formal, consistent or complete. But what about nature? For Gödel to apply to nature the question basically is, does a sufficiently powerful formal system underlie nature? That is, is the universe a Turing machine? Buckminster Fuller looks to say yes or less. Buckminster Fuller is basically saying that nature does not compute with arbitrary reals (well what he is saying is actually stronger since he disallows computable reals). That is, hypercomputers do not exist in reality. A very reasonable stance I agree with. http://en.wikipedia.org/wiki/Hypercomputation

-----------

The next lines of musing leave the realms of fact and edge into philosophy.

If nature is Turing equivalent and hence axiomatically encodable by a formal system or as a computable function then is it complete and consistent? Is a mind + universe a subset or superset of the universe? A system can be complete and consistent without us having access to a stronger system to prove it. So it is possible that the universe is complete. It could also be incomplete, we may never know. Something interesting is that Heisenberg's original terminology translates to indeterminacy not uncertainty. Work's attempting to bridge to Godel typically do so via leveraging Kolmogorov randomness.

Re: Pi Is Wrong

#65

Post by loup-vaillant »

pbhjpbhj wrote: post #57
loup-vaillant wrote: post #49

Earlier quoted context omitted.

> We can't be sure that the models' inner workings mimic nature's inner workings William of Ockham says we have reasonable hope of coming damn close. http://en.wikipedia.org/wiki/Occam%27s_razor Anyway, infinite certainty doesn't exist. Not even for 2+2=4. http://wiki.lesswrong.com/wiki/Absolute_certainty

Occam's Razor itself actually says that you mustn't multiply entities beyond necessity. It gives no clues as to the necessity and so is, IM[not very popular it seems]O, absolutely of no worth. Of course if you can prove whether an entity is necessary to describe an outcome then you've no use of Occam's Razor, so it seems rather to excise itself from being useful.

Ockham basically meant that the "simplest" explanation that fits the facts is the most likely. And as a matter of fact, we do have a precise definition for "simplicity": http://en.wikipedia.org/wiki/Kolmogorov_complexity

You will note that this is a quantitative reasoning, not a qualitative one. An equivalent way to come up with the same results is http://en.wikipedia.org/wiki/Inductive_inference

Anyway, it all boils down to http://en.wikipedia.org/wiki/Bayesian_probability , with what we commonly call "Occam's prior". Probability theory is wonderful, but to use it, you have to start from a set of prior probabilities. When you have zero knowledge, starting with probabilities "inversely proportional" to Kolmogorov complexity seems the most reasonable thing to do.

Re: Pi Is Wrong

#66

Post by loup-vaillant »

mattheww wrote: post #54
loup-vaillant wrote: post #44

Earlier quoted context omitted.

> In physics, it does not make sense to discuss position (nor boundaries) of object with infinite or absolute precision. No, no, no, you are over interpreting. The fact that you cannot measure with infinite precision doesn't mean it doesn't even makes sense to talk infinitely precisely about positions. You can for instance make thoughts experiments, by imagining an initial state infinitely precisely, then conclude th…

Actually, it's you who is doing the over interpreting. The Heisenberg uncertainty principle is a physical constraint that wave functions obey. If you define a state with an infinitely precise location, then the range of momentums is infinite. This result is fairly easy to reproduce, so it would probably be instructive to try it. The measurement interpretation of the Heisenberg uncertainty principle is both a useful w…

Just to be sure: the wave function is still a distribution of complex amplitude over a configuration space, right ? If I understand, you are saying is that if we have a Dirac peak at some point, then the momentum spread everywhere (the infinite momentum). I don't dispute that.

I only meant that we could describe (in principle) a hypothetical wave function with infinite precision. Meaning, at each exact point in our hypothetical configuration space, we would specify an exact amplitude. (With the usual caveats due to the fact that we're talking about a distribution, not a function.) From then, we could predict the experimental results, which are bound to yield finite precision.

Now, current physics say we will never infinitely accurately measure our wave function. But it also says that this wave function behaves lawfully, with infinite precision, from some (I think?) unknown initial state. Is there a problem left ?

Edit: it just hit me that there is a problem if you don't believe in Many Worlds. Just know that I think the Copenhagen interpretation is crazy (I don't believe in the collapse of the wave function), and that currently, I find some form of Many World Interpretation most likely.

Re: Pi Is Wrong

#67

Post by batista »

m_for_monkey wrote: post #53
batista wrote: post #36

Earlier quoted context omitted.

This is not mathematics though, nor science, it's just the meta-mathematical philosophical opinion of a guy (mostly self-taught, by the way, and know for his cool inventions and ideas, not for his deep mathematical insights). He also use Pi in computers, e.g to draw circles, and computers (and displays) are even more discrete that nature. Has he come up with something better to use in Pi's place? That we don't use "a…

Yes, it's "only" philosophy, but that doesn't mean it's utter BS. Look we either talk about the exact pi, or not pi at all. In my computer there's no pi. As for the alternative, at least he tried something: "Fuller also claimed that the natural analytic geometry of the universe was based on arrays of tetrahedra. He developed this in several ways, from the close-packing of spheres and the number of compressive or tens…

Look we either talk about the exact pi, or not pi at all. In my computer there's no pi.

You'd be surprised to hear about the wonderful concept of approximation.

As for the "alternative", sounds like bogus science to me...

Re: Pi Is Wrong

#68

Post by pbhjpbhj »

loup-vaillant wrote: post #65
pbhjpbhj wrote: post #57

Earlier quoted context omitted.

Occam's Razor itself actually says that you mustn't multiply entities beyond necessity. It gives no clues as to the necessity and so is, IM[not very popular it seems]O, absolutely of no worth. Of course if you can prove whether an entity is necessary to describe an outcome then you've no use of Occam's Razor, so it seems rather to excise itself from being useful.

Ockham basically meant that the "simplest" explanation that fits the facts is the most likely. And as a matter of fact, we do have a precise definition for "simplicity": http://en.wikipedia.org/wiki/Kolmogorov_complexity You will note that this is a quantitative reasoning, not a qualitative one. An equivalent way to come up with the same results is http://en.wikipedia.org/wiki/Inductive_inference Anyway, it all boils…

Occam tells you nothing of truth. It simply says that your knowledge of a situation may be limited. Which seems as close to truism as any aphorism could get.

I'll say it again: Occam's Razor (as told by Ponce at least) has nothing to say on whether one knows the truth. Neither whether one has simplified sufficiently nor if one has failed to add a necessary entity.

You appear to say here that the ability to calculate the Kolmogorov complexity, K, is necessary to establish the simplicity of a given form/function/algorithm/state and so is an entity essential to applying Occam's razor. However, we know that we can't calculate K in all situations and so, it seems, Occam's razor as modified by your requirement to determine the simplest explanation is itself insufficient.

>When you have zero knowledge, starting with probabilities "inversely proportional" to Kolmogorov complexity seems the most reasonable thing to do.

For example, take the current situation with particle physics. It looks like particle soup, very complex, varied interactions. But more knowledge - perhaps entities which currently appear unnecessary to create a working theory - could well precipitate a far simpler theoretical model that revolutionised the analysis of particles and their interactions (a fully working unifying string theory maybe).

To recapitulate, Kolmogrov complexity appears to assume that you know everything and therefore are certain that you're providing the best simplification. You don't and you're not. Occam's Razor has no truth generating/revealing ability.

As I'm sure is clear I've not studied Kolmogorov or BLC before. WRT Occam's Prior how do you judge the K of different entity types (like are more spatial dimensions somehow less complex than more axiomatic constants).

Re: Pi Is Wrong

#69

Post by brlewis »

Confusion wrote: post #40
brlewis wrote: post #38

Earlier quoted context omitted.

Why does he assume nature requires decimal representation for her calculations?

He doesn't; the entire argument is base independent.

He assumes nature needs to calculate Pi out to a certain number of decimal places. The flaw in that reasoning is base independent.
Post reply on HN