huh

You'd think that since F_n can be written in closed form, it wouldn't be that hard to do, but this is, indeed, rather tricky. Certainly beyond my level of math understanding/interest.

And thus beyond mine. But heck, its worth a shot. I spent a month last year trying to mathematically contradict string theory. I got no where but it was a valuable experience nonetheless despite the fact I was in over my head.

Hmm, what would that mean in this case, Diplomat?

Which case specifically? The F_n one or my string theory one?

String theory.

Oh, now I get your question. It means, if I really was bored, I could try and prove or disprove something high over my head mathematically. But with a new semester and some workload increases I think I would not have that time. I cannot cruse through Philosophy as easily as Biology of course.

No, I just meant, what would it mean to mathematically contradict string theory? Given that it is a physical theory, people usually think of contradicting it physically, by experiment. So I was curious what kind of contradiction you were looking for. Not that one can't imagine one -- find that it gave contradictory predictions for some situation or something. Just curious what lines you were exploring.

Anyway, keep it up with philosophy. It's an excellent thing to have some background in philosophy.

Ah just trying to see if it could be mathematically contradicted. I spent the better part of the attempt trying to learn the math which failed. And then there is M-Theory anyway.

@ Semck Yeah. Now I can troll with intelligence!!!!!!!!!!!!

Nomenclature was your problem. String theory only yields up its secrets to those who call it Superstring theory.

And yes, or at least with sophistry.

@semck

"No, I just meant, what would it mean to mathematically contradict string theory? "

String Theory is heavily based upon mathematics. Just like Quantum Mechanics, which was almost entirely developed before being verified experimentally.

Exactly. Proponents of string theory (read: all the cool physicists) will tell you that internal MATHEMATICAL consistency of string theory is one of its greatest strengths. The fact that millions of experiments confirmed it is of course another one ;)

@abge, I'm well aware (of course) that string theory is heavily based on math, like every branch of physics. But just like quantum mechanics, it consists ultimately of certain assumptions about the physical world (in mathematical terms), and then figuring out the mathematical consequences of those.

I'd ask the same thing about QM: what would it mean to _mathematically_ contradict quantum mechanics? Well, I guess you could find some kind of logical flaw in the foundations of functional analysis or something, but that wouldn't really be contradicting QM so much as find problems with its mathematical foundations. The closest I can think is that maybe you'd find that some physical observable couldn't be realized as a Hermitian operator or something; even that, though, seems as much like a physical refutation as a mathematical one.

Incidentally I wouldn't agree that QM was "almost entirely developed before being verified experimentally." The Stern-Gerlach experiment (1921), for example, which was a foundational QM experiment, preceded the development of the theory by several years. Of course, as with any theory, experiments were carried out afterward to test its novel predictions.

@ulytau, Indeed, which is why I asked if he was referring perhaps to a desire to find contradictory predictions.

@semck

QM is internally mathematically consistent. It was built off of axioms, which did not have to be correct. Should a contradiction have arose following those axioms, QM would have been disproven, based solely off of mathematics.

" but that wouldn't really be contradicting QM so much as find problems with its mathematical foundations."

This sentence makes no sense. QM *is* a math used to describe parts of the universe.

QM is good stuff. Even though I doubt I will have a career in it I still want to take a class or 2 in it just to know some of the mathematical basis for it.

"Incidentally I wouldn't agree that QM was "almost entirely developed before being verified experimentally.""

Compared to all other branches of physics, pretty much all of which start with experiments and then math is formed to explain what was observed.

@D33

The math needed to get a cursory understanding of QM isn't that bad. Any good engineering/hard science curriculum should give you the proper math to take an Intro QM course.

Example those Newtonian laws and such that while maybe contradictory to QM and GR still have use and validity.

@abge, well, kind of. As I said, QM's axioms are not exactly mathematical axioms -- they're axioms about how math relates to physics. Let's review a standard presentation of them (I'm quoting Shankar):

I. The state of a particle is represented by a vector in [projective complex] Hilbert space.

II. The classical variables X and P are represented by Hermitian operators that act on the position basis by

<x|X|x'> = xd(x - x') (where d = Dirac delta).
<x|P|x'> = -ih d'(x - x')

III. If a particle is in a given state |A>, then measurement of some observable (corresponding to a Hermitian operator Z) will yield an eigenvalue of Z, and it will yield the eigenvalue z with probability [density] |<z|A>|^2.

IV. The state vector evolves according to Schroedinger's equation.

So, there's a lot of physical content to these, but yes, if one culd find a contradiction between them, I guess you'd call that a mathematical disproof. The problem is, that's almost inconceivable. The one obvious opportunity would have been if the Schroedinger equation were not norm-preserving, so that probability was not globally conserved at 1. Another possibility perhaps would have been if the spectral theorem weren't true, so that the probability of the eigenstates did not add up to 1. But what else would a contradiction here look like?

Right, Diplomat, but the Newtonian laws you're referring to are not _mathematically_ inconsistent. They just don't match the real world under extreme conditions.

@abg Yeah I am aware I have looked at some collage courses and such. I plan to do a science career of some sort. Maybe minor in math and do QM and stuff. Maybe even do med school and still squeeze them in.

@abge,"'Incidentally I wouldn't agree that QM was "almost entirely developed before being verified experimentally.'

"Compared to all other branches of physics, pretty much all of which start with experiments and then math is formed to explain what was observed."

Right, I'm saying I just don't agree with this. Quantum Mechanics was the same way, or at least, as much as Newtonian mechanics was. They didn't pull it out of their hat. They were trying to explain what had been observed over the last 30-50 years.

Yeah I know. I never said they were "wrong" just contradictory to some of QM/GRs points. I also love how off topic this thread has gotten.

Yeah. If you know anything of the history of QM (which since I lack the knowledge to do the math I became interested in) semeck is making a valid point there.

@Diplomat, QM is best understood in terms of infinite-dimensional vector spaces. If you learn calculus through Calculus III and some linear algebra, plus basic classical mechanics, then you should be ready for a good course in QM (though it will be some work). Shankar is a superb book. Shankurai gets good reviews as well. I would avoid Griffiths -- although more accessible, it does not give a very good picture of what is actually going on.

@D33, gotcha. That is certainly true, and quite interesting. (Especially interesting is that classical mechanics is the limit of both theories under "normal" conditions, but that they don't agree with each other).

In contrast, string theory was kind of the reverse in a sense in how some of the math was there and worked with before the theory was developed. If you know of Susskind and Oiler's equations (which I read about in Susskind's The Black Hole War, a great read) then you might get what I am pointing at.