- 14 Jan '10 18:16 / 1 edit

Let me try a form of an answer.*Originally posted by Palynka***Why care whether it violates the postulates of Euclidean geometry?**

Everybody knows that no mathematical entity exists in the real world exactly as we define them to be. The real point of discussion is if they do exist in a world outside our own and who put them there (if we think that the answer is that do they exist in an independent way).

But if you've read the original article you would realize why this article why this article is so gold.

This has to do with a a very profound theorem of quantum-mechanics (Kochen-Specker). And the author of the original article said that some experiments could nullify that theorem.

Well Peres picked his argument and "showed" that under the same logic the diameter of a circumference doesn't intercept it. That's just ridiculous. For a diameter to be a diameter it has to intercept the circumference!

"Why bother with Euclidean?" geometry you might ask. Because that's precisely the geometry we use in*normal*quantum-mechanics and that's the kind of geometry the Kochen-Specker's theorem assumes to hold.

So either the article by Meyer refers to another version of Quantum Mechanics which is not euclidean, and in that case it doesn't make sense to speak about the Kochen-Specker theorem (you do see why this is so, right?), or the argument given by Meyer is totally ridiculous.

The valid option is the second one. - 14 Jan '10 18:42

I still don't see it.*Originally posted by adam warlock***Let me try a form of an answer.**

Everybody knows that no mathematical entity exists in the real world exactly as we define them to be. The real point of discussion is if they do exist in a world outside our own and who put them there (if we think that the answer is that do they exist in an independent way).

But if you've read the original artic ...[text shortened]... argument given by Meyer is totally ridiculous.

The valid option is the second one.

A circle intercepting a line is something that doesn't really exist in the physical world. Does it? I don't even know what a line is in the physical world! Something 1-dimensional... Does that exist?

If not, then what's the big deal that Euclidean geometry isn't the best possible construct to describe quantum-mechanics (especially in point-like implications like intersections)? In another thread we were talking how Minkowski spaces are used to represent spacetime. Well, these spaces are also non-Euclidean!

I'm not saying Meyer is correct, but it seems to me that the counter-argument is silly. - 14 Jan '10 19:36

1 - Normal Quantum Mechanics is constructed in an Euclidean space*Originally posted by Palynka***I still don't see it.**

A circle intercepting a line is something that doesn't really exist in the physical world. Does it? I don't even know what a line is in the physical world! Something 1-dimensional... Does that exist?

If not, then what's the big deal that Euclidean geometry isn't the best possible construct to describe quantum-mechanics (especiall ...[text shortened]...

I'm not saying Meyer is correct, but it seems to me that the counter-argument is silly.

2 - A theorem is proven in Normal Quantum Mechanics

3 - Someone gives an argument that contradicts euclidean properties

Conclusion: That argument isn't to be applied in normal Quantum Mechanics. As simple as that. And Meyer applied that argument in normal Quantum Mechanics.

What I don't get is why keep saying that mathematical objects don't exist in the real world. That isn't relevant at all. What is relevant is that QM is constructed on a set of assumptions and if you don't respect those assumptions you aren't talking about quantum mechanics.

No physical theory represents the world as it is. There is always some discrepancies and the real deal is to make those discrepancies smaller and smaller in the given domain that we want theories to be applied.

Once again everybody knows that there isn't perfect triangles out there. The real discussion are:

Do*real*triangles exist somewhere else?

Where is that somewhere else?

And who put the triangles there?

Physics is all about realizing what can be neglected in some circumstances and still get meaningful results.

The first to realize that was Galileo. He talked about sphere that went down planes without rolling and without friction. Of course there's no such thing in the real world but one has to simplify in order to tackle these kind of problems. If one wants to incorporate all complexities in the model that's pretty what one will do: incorporate complexities in the model because there's an infinite number of them. One has to stop at a given point, then write down the equations, and then solve the equations.

After this is done we should check our predictions against what comes out of real experiments and see if our simple models of a very complex reality are applicable or not. Talking about right theories or wrong theories is just showing that one doesn't know the first things about science. Of course I use those expressions sometimes but it is just a way for me to simplify the discourse (just imagine this: "yes, under those set of assumptions, axioms and principles we have derived a series of results that when confronted with sets of data from experiments agree on a given probabilistic interval that is good enough for us to use that theory!" Scrap that! I'm just saying: "yes the theory is right" and get done with it) - 14 Jan '10 21:37 / 1 edit

I really, really, really, really doubt that Meyer didn't know his suggestion wasn't contradicting Euclidean properties. If he didn't, then he shouldn't be talking about topological properties.*Originally posted by adam warlock***1 - Normal Quantum Mechanics is constructed in an Euclidean space**

2 - A theorem is proven in Normal Quantum Mechanics

3 - Someone gives an argument that contradicts euclidean properties

Conclusion: That argument isn't to be applied in normal Quantum Mechanics. As simple as that. And Meyer applied that argument in normal Quantum Mechanics.

What I y!" Scrap that! I'm just saying: "yes the theory is right" and get done with it)

My points about these constructs not existing in reality, is just to illustrate that it to me it doesn't seem shocking*at all*that Euclidean properties are not respected. All your discussion about physics being an approximate description about what is, not an exact one, only reinforces the idea that we shouldn't view Euclidean properties as some "necessary" axiom for quantum mechanics. So that means the argument against his suggestion is a misguided one, to say the least.

I can imagine how hard it must have been for people to accept that time was relative to the observer and that we had to use a non-Euclidean space to represent this new concept of spacetime. All physics had been Euclidean up until then! (correct me if I'm wrong). Did it have to be wrong? Of course not.

So again, the argument by itself seems...silly to me. - 14 Jan '10 23:49 / 1 edit

Man you really are not getting this. It has nothing to do with physics it has all to do with logic.*Originally posted by Palynka***I really, really, really, really doubt that Meyer didn't know his suggestion wasn't contradicting Euclidean properties. If he didn't, then he shouldn't be talking about topological properties.**

My points about these constructs not existing in reality, is just to illustrate that it to me it doesn't seem shocking*at all*that Euclidean properties are n ve to be wrong? Of course not.

So again, the argument by itself seems...silly to me.

I'll try with a different example.

Imagine that I start to work on the context of spheric geometry and get some really nice theorem A.

Then some guy, let us call him Tom, comes along with an argument that disproves A. And says that my "theorem" isn't a theorem at all.

Some other guy, let us call him Dick, analyzes his argument and realizes that by a consistent analysis of his argument one can deduce a property that is a contradiction in the context of spheric geometry.

What this tell us is that what Tom actually did wasn't disproving my theorem since my theorem holds in the case of spheric geometry. What Tom did was to prove a theorem in a different kind of geometry.

Or in another way: imagine that I came to you and said "Palynka you got it all wrong! The sum of the inner angles of a triangle are always less than pi (obviously that having the high pedigree that we have we measure angles in radians, not degrees)!" I think your answer would be something like: "No, no it depends on your set of assumptions. Ypu can't say that I'm wrong since we are working based on different assumptions. What you can argue is about the applicability of my results based on my assumptions and the applications of your results on the set of your assumptions."

Now let us get back to our case. What Meyers did was exactly that. He thought that he had disprove a very deep theorem on quantum mechanics. But he didn't do that because that theorem is proved in the context of normal QM and normal QM is euclidean.

Since Meyer's argument is valid in the context of a non Euclidean geometry is point is mute.

I'm not saying, neither is Peres, that the right form of QM is euclidean what Peres showed is that Meyer's argument isn't in the domain of application of QM. And so can't really disprove nothing that is in the context of normal QM.

**I really, really, really, really doubt that Meyer didn't know his suggestion wasn't contradicting Euclidean properties.**

It's quite possible that he did. After all, his article was published so that means that it passed the peer review process. And that the means that the reviewer didn't get it. Once the paper got out some people started quoting it as being a great paper, so obviously they didn't get it too.

The point that Peres makes is really a subtle one. And that's why I find this article to be the best I've read.

These kind of things happens all the time in science. I'm not talking about frauds, I'm talking about real mistakes. I've done my share, and I've seen other people's mistakes too.

As a side note: von Neumann also got it wrong on this area. He also published a paper that was later found to be wrong.

So if von Neumann made a mistake I think it is highly possible that Meyer made a mistake.

*I can imagine how hard it must have been for people to accept that time was relative to the observer and that we had to use a non-Euclidean space to represent this new concept of spacetime.*

Yes it was hard! It was so hard that Einstein disliked it with a passion at first and only got to like it when he found it impossible to construct GR without the concept of space-time continuum whose geometry isn't euclidean.

Contrary to what most people think the idea of space-time doesn't come from Einstein. It comes from Minkowskii and at first Einstein was very much against it.

**All physics had been Euclidean up until then! (correct me if I'm wrong)**

This is mostly right so i won't even correct you I'll just add a little bit. People had used non-euclidean geometry in physics and math (and separating these two sciences before the 19th-20th century doesn't make much sense really) but they used it with a different mindset. They pretty much used in the context of specific problems but never with the notion that the geometry of the Universe might not be Euclidean (the being basically Gauss and Riemann that said that only experience could say what was the geometry of the Universe). For instance the geometry used in astronomy is spheric and it has been for a long time, but people saw that as being just a tool for solving a problem not as being a real contender for the geometry of the Universe.

A good book on this is this one: http://www.amazon.com/Poetry-Universe-Mathematical-Exploration-Cosmos/dp/0385474296

If you are in Portugal you can get in the subway stations bookstores by a very nice price (The portuguese title is A poesia do Universo).

Edit: http://plato.stanford.edu/entries/kochen-specker/ - 15 Jan '10 10:10

In short, the implication is that Meyer believes Euclidean geometry is not the best geometry to describe quantum mechanics. So?*Originally posted by adam warlock***Imagine that I start to work on the context of spheric geometry and get some really nice theorem A.**

Then some guy, let us call him Tom, comes along with an argument that disproves A. And says that my "theorem" isn't a theorem at all.

Some other guy, let us call him Dick, analyzes his argument and realizes that by a consistent analysis of his argu ...[text shortened]... se of spheric geometry. What Tom did was to prove a theorem in a different kind of geometry.

Contemporary quantum mechanics is an approximate description of reality at a quantum level. Today, you say Euclidean geometry is used. That's cool. But I still haven't seen a single argument why in the future we cannot say that it's not the best one.

And if you agree with that, then there is no guarantee that a mathematical deduction about the impossibility of hidden states will still hold. So IF Meyer (or anyone else) ever shows that finite measurement implies that our best description of quantum mechanics is using a rational-based geometry so be it.

In short, the paper seems to draw a simple corollary of Meyer's proposal. Since I don't see any particular reason why that corollary is an impossibility, I don't see how he disproved anything. - 15 Jan '10 12:23
*Originally posted by Palynka***In short, the implication is that Meyer believes Euclidean geometry is not the best geometry to describe quantum mechanics. So?**

Contemporary quantum mechanics is an approximate description of reality at a quantum level. Today, you say Euclidean geometry is used. That's cool. But I still haven't seen a single argument why in the future we cannot say that ...[text shortened]... ular reason why that corollary is an impossibility, I don't see how he disproved anything.**In short, the implication is that Meyer believes Euclidean geometry is not the best geometry to describe quantum mechanics. So?**

You haven't understood. Nor Meyer's article, nor Peres one... By the way have you read Meyer's one?

Read this again:**imagine that I came to you and said "Palynka you got it all wrong! The sum of the inner angles of a triangle are always less than pi (obviously that having the high pedigree that we have we measure angles in radians, not degrees)!" I think your answer would be something like: "No, no it depends on your set of assumptions. Ypu can't say that I'm wrong since we are working based on different assumptions. What you can argue is about the applicability of my results based on my assumptions and the applications of your results on the set of your assumptions."**

The problem is that Meyer is criticizing a theorem in QM based on a property that isn't inside the model of QM. And his critic is made in the context of normal QM. Do you got it now? Nobody's saying euclidean QM is the way to go. What Peres said is that since you are criticizing a formal theorem of normal QM with methods that are outside normal QM you aren't really criticizing it.

Again: If i came to you saying that you solved a problem assuming that the sum of the inner angles of a triangle are pi and that's wrong because they're sum is always less than pi. You'd say. Yes adam but I'm working in the context of Euclides so your reasoning doesn't really apply. You're not saying that Euclides is the one and only way. You're saying that you are in that context and I shouldn't mix things up.

And if you don't get like this I'll just give up.

**But I still haven't seen a single argument why in the future we cannot say that it's not the best one.**

Neither do I. And I've said so in the previous posts. Actually a lot of people work in non-euclidean QM these days. But the thing is that they don't mix things up like Meyer did. And that was his mistake: Mixing things up.

**So IF Meyer (or anyone else) ever shows that finite measurement implies that our best description of quantum mechanics is using a rational-based geometry so be it.**

Agreed. But once again the secret is to not mix things up, like Meyer did. You can't come with a*dis-proof*of a theorem when your assumptions negate the assumptions of a theorem. That's just illogical. Imagine that I proved that if a function is continuous than it is integrable. If you came along saying "Yeah but some discontinuous functions aren't integrable." Yes that is right, but it has nothing to do with the theorem I proved since I was only talking about continuous functions. Since QM is only talking about euclidean space, you can't come with something that works in non euclidean spaces to criticize something that was proved with the assumption of a euclidean space. Simple, straight logic. - 15 Jan '10 14:24 / 2 edits

The way I see it, he first disproved the theorem based on finite precision measurement and ONLY THEN suggested an framework where the theorem also doesn't apply.*Originally posted by adam warlock***[b]In short, the implication is that Meyer believes Euclidean geometry is not the best geometry to describe quantum mechanics. So?**

You haven't understood. Nor Meyer's article, nor Peres one... By the way have you read Meyer's one?

Read this again:**imagine that I came to you and said "Palynka you got it all wrong! The sum of the inner ang ith the assumption of a euclidean space. Simple, straight logic.**[/b]

So let's just agree to disagree. You are repeating what you said before, I still don't see it and we're not making any progress either way. Sorry.