Things I know are true, but I don't really believe
abstrusegoose.com
abstrusegoose.com
There is no intuitive understanding of a subject like that because neither 100K years of evolution of our brain now decades of real world experience expose us to it.
Entanglement is slightly mysterious though as what is the physical mechanism behind it?
Imagine you have two particles in 1 space dimension. The first particle has a position x (1 real number), the second particle has a position y (1 real number). "Physical space" would be the set of real numbers (R^1), while configuration space is the set of pairs of real numbers (x,y) (this set is R^2). The first number in the pair represents the position of particle 1, the second the position of particle 2.
An entangled state is simply a probability distribution on R^2 which is not a product state (i.e. P(x,y) != P_1(x) P_2(y)). This means that knowledge of the position of particle 1 gives you (possibly incomplete) information about the position of particle 2.
One example of this is "probability 1/2 that x=0, y=0, 1/2 that x=1, y=1". The minute you know x, you know y.
That's what entanglement is. What makes it weird is the idea that "knowing x" (measuring it) changes things. An explanation of this is trickier. Still understandable in principle (not just 'follow the math'), but too long for this post.
How about another post, then? :-)
There isn't even a standard measure of entanglement for three or more particles. It's a pretty big mystery, still.
I doubt the writer meant it the same way you do. He was talking about the known things that hes not wrapped his mind around. In this case, explaining why the theories don't necessarily conflict should do.
On a more serious note, it was understanding the Axiom of Choice, or that our "intuitive" notions of reality aren't really consistent with themselves:
Imagine my distress: the same problem occurs, Banach-Tarski still holds, unless you go into the rather more weird, Smooth Infinitesimal Analysis or Synthetic Differential Geometry; basically, treating geometry as made up of little discrete infinitesimally small points.
To prove Banach-Tarski, you need to assume Zermelo-Frankel set theory (just ordinary set theory) plus the uncountable axiom of choice.
In my view, this means the uncountable axiom of choice is false. This actually breaks far less than you would expect. For most practical purposes, you only need the countable axiom of choice, which is the intuitively reasonable version.
The main thing the loss of the uncountable version breaks is the Hahn-Banach theorem on spaces of dimension higher than countable infinity. This in turn breaks quantum field theory on free space (not in a box), but that's broken anyway.
- our "intuitive" mental model of the world which says one thing can't become two of the same thing without using any more material, is wrong
- the mathematical model that says you can do that, is wrong
- the mathematical framework on which the model is built, is limited.
In other words, while I can't follow the maths, I would have no problem believing the mathematical weirdness is self consistent because I don't believe it would have any rational link to the real world anyway.
That such a human model can be as accurate and precise as quantum mechanics is, now that's hard to believe!
Everybody knows the theorem isn't real. Now, I'm not just being pedantic here, because the fundamental and obvious failure of the theorem to have any connection to the real world is at the heart of one of the philosophical questions about the nature of math it engenders, along with its good friend the Axiom of Choice. While there's nothing wrong with either thing mathematically, does it really matter when anything that invokes the Axiom of Choice basically just opted out of concrete reality entirely?
Basically, I'm saying that the contents of your post are true, but trivial; there's no interesting argument about correspondence to reality to be made. The fun lies at the next level. I have no easy answers for you there.
(Personally, I have no problem breaking math into "Things that Might Be Practically Useful" and "Things to Entertain Mathematicians", especially since the latter tends to migrate to the former at quite surprising times, but many people seem to have trouble with giving such a rich meaning to the single word "math".)
It doesn't matter a jot when something "just opted out of concrete reality entirely", but it's also not especially interesting. Talking about pink elephants and how they can double in size may be internally consistent in a story, but it's not something that's "true but really difficult to believe in" anymore than "dividing by zero equals infinity" is difficult to believe in.
If the fun is at the next level, perhaps that's where I ought to go...
Funnily enough, it was actually while I was sat in church with my family when I was younger (I've since lapsed profoundly and comprehensively, of course) that my mind first began to wander to and wrestle with such concepts.
Reading Clarke and imagining being in a spaceship in orbit around Jupiter and seeing it dominate the exterior view.
Trying to imagine a ball of fire as big as the Sun.
Seeing the solar giants in images like these: http://www.rense.com/general72/size.htm or videos like this: http://sciencehack.com/videos/view/_WEL117xXpw
Antares is huge, lol. Well, not lol, more ... scared.
Hubble Deep Field.
It's awesome in the literal sense of awesome.
From the [axioms of math] one can show that there is a well-order of the reals; it is also possible to show that [those axioms] alone are not sufficient to prove the existence of a definable (by a formula) well-order of the reals.
http://en.wikipedia.org/wiki/Well-order
So you can prove that set of real numbers can be well-ordered, but you can also prove that an order cannot be defined.
Kind of how God exists, but cannot be perceived or comprehended ;)
(or maybe not)