Duality and Reality

After Harshini's post, this is probably going to seem very pale. I've never written anything of this nature before. Please do comment, honest criticism will be well accepted and I will be more than happy to fill in the gaps. I am supposed to be giving a talk tomorrow on "Symmetry in nature and mathematics"in college and that has made me think again about something that I started thinking when I first began working on the General theory of relativity. This post is a result of those rambling thoughts.
It was in the book by Simon Singh, " Fermat's last theorem", where I first read Pythagoras' famous statement - " Everything is a number". This is indeed true. Theoretical physicists and experimental physicists are both trying to get meaningful numbers out of nature. Numbers are one of the most important things that define our reality. Theoretical Physicists have a way of getting to these numbers. Its called duality. Duals are paths to real numbers.
The first time I encountered duality was when I started out on general relativity and had to study tensor calculus [or according to G. Ramachandra, I was just raising and lowering indices ;-) ] There are vectors that are a part of dual vector spaces, the contravariant and covariant vectors. For those who are more into QM , its the Bras and the Kets. They do not make any physical sense individually. You need both to describe a physical quantity. They are the yin and yang of everything. For a long time I wondered why this was true. Why does a description of reality require two different things? Why not three or four? Can I not know the "one" thing knowing which I can know everything else? While working on Quantum Information, I was introduced to a new concept known as distinguishability.

Imagine a universe where you could not distinguish between any two things or events. Imagine not being able to differentiate between the correct answer and the wrong answer. The universe simply could not exist without any distinguishability. And now, while preparing for tomorrow's thought, it seems to me that duality is a consequence of symmetry. Can there be a perfectly asymmetric object?
I have come to realize so far that as long as a mathematical structure defines reality, there is an inherent duality. I may be wrong, but I haven't seen anything that doesn't satisfy my argument. For example, if you want to get rid of the notion of distance between two points, you need to get rid of the metric, which means ur getting rid of the dual vectors and the inner product between them. And the space thus becomes a simple topological space in which there in only a notion of connectedness but no distance.

I am yet to find a reason for this duality in nature to exist. Is it actually there? Or is the duality in our heads? Are our brains wired to think that way? At this point I silence a horrible thought occurring to my head about the struggle towards Quantum Gravity. I have in this post spoken about duality from a theoretical point of view. I will let you answer the question about the duality in experimental physics :) .

More on this and self-duality yet to come.


P.S: It may have no relevance with what I have said, but for some reason I am inclined to end by quoting Niels Bohr - " The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth."