Math and Physics, Again....

My Philosophy of Quantum Mechanics class (24.111) is making me think of things again, which I learned to give up thinking about. It's this whole issue of taking as postulates that which we observe and then figuring out the mathematics from that.

I mean, this really bothers me and in a lot of ways, instead of sympathizing with the physicist, I've become more hard-lined as a mathematician. I often find myself asking the question "Why are the eigenvectors and eigenvalues of a certain operator the only possibly observed states?" Often the response from my physicist friends is "That's just a question you don't ask in Quantum Mechanics." So that response, really bothers me. The second someone says "You can't ask that question," when apparently the question is well-formed, that is a quick line to raising my ire.

The other response I often get is "That's the way things have to be so that we can accurately model the observed data." This I suppose is what gets at the heart of the difference of the mathematician and the physicist. One views the sense-data as the starting point for inferential reasoning, the other only starts with well-defined things and checks periodically (or doesn't check at all) if it happens to match sense-data. It really is the ultimate clash between the bottom-up versus top-down approach, and which direction you prefer to go ultimately determines which camp you belong to. For me, sense-data stresses me out. It's confusing. You have to average things, because systems are often so complicated and have so many confounding variables, that you have to ignore the non-linearities of pretty much everything before you can actually say something about the "real world." By then, you're not talking about the real world, you're talking about some idealization of the real world which exists in platonic space and by then you should be doing mathematics and just burn the bridge that supposedly connects you to the real world.

Additionally, I'm a philosopher. I was fed skepticism for breakfast ever since I was a child. I'll never forget auditing my Dad's Philosophy 101 class at James Madison University when I was in middle school and first learning about Descartes and his meditations. The rational skepticism and inability to tell sleeping states from waking states fundamentally shook my faith in sense-data. Sure the electrons always split into "spin-up" and "spin-down" states when past through an inhomogeneous magnetic field, but I could just be dreaming and then what? I think always having that uncertainty that I might devote my life to understanding and explaining sense-data and then wake up into a parallel universe, where I find out my entire life was just some strange dream in some strange universe with strange physics, really undermines the feeling of value and worth in taking that path as a profession.

This brings me to the path which I have taken: the top-down approach. With this approach, at the end of the day, you really don't care too much if the structures you've built match what is "out there" because they're your structures and they are intrinsically beautiful,
independent of their relevance to the "outside world."

However you should do what you enjoy the most. If you are actively squirting dopamine and serotonin by thinking about the path of a charged, spinning particle, passing through an inhomogeneous magnetic field, great. More power to you. When you die or wake up into that alternate universe, and find out that everything is different from what you thought it won't matter, because you enjoyed your life, and that is all that matters. So find out what you like and if its not math or physics, its bound to be something else, and if it is washing dishes so that you can spend your evenings partying like a rockstar, that's cool too. If its raising funds to feed hungry babies in Africa, that's also great. As long as your dopamine levels are high and this very moment you are glad to be experience something rather than nothing, you're living the good life. I realize the logic is backwards, but if I could've convinced myself that physics is worth doing, then I would enjoy doing it and would be happy devoting my life to it even it turned out to be a farce. Instead, my skepticism about the meaning of my activities, has led me to take a different path, but I'm enjoying the path and the sights it has to show me are truly breathtaking.

However the things which really bug me is when our beautiful mathematical structures do find their place in the "real world." It makes you wonder if this is the case because our mathematics shapes the way we can think about things, and thus the bias lies in the observer and not the observed, or it could be there is some weird underlying fabric of reality that is mathematical and does reside in some Platonic world of ideas, and our eyes, ears, nose, mouth, and tongue are just the biological interface for this digital graphing of mathematics into our minds.

So I'm left with a goal: To master all the mathematics in Roger Penrose's The Road to Reality and see if really does lead me there. The journey will certainly take a while, but I'm up for the ride.

Fall 2007 Course Selection

Alright! Enough! Here it is: My Fall Course Selection.

For Credit:
18.101 - Analysis on Manifolds
18.701 - Algebra I
18.904 - Seminar in Topology
24.111 - Philosophy of Quantum Mechanics
21W.775 - Writing about Nature and Environmental Issues

For Listening Fun:
18.102 - Introduction to Functional Analysis
18.905 - Algebraic Topology

Classification Theorem for Compact Surfaces

So this was actually the "Theorem of the Day" about a month ago, but allow me my delayed public display of math wanking. Get use to it, because at Cambridge I was surrounded with mathematicians and these urges were often released in more...err...natural, consensual ways. Now back on the other side of the puddle, surrounded by engineers, I have to release my math juices all over this wonderful interweb. Taken from W.S. Massey's Algebraic Topology: An Introduction (GTM 56). Only a taste of things to come in 18.904 - Seminar in Topology.

Theorem 5.1 Any compact surface is either homeomorphic to a sphere, or to a connected sum of tori, or to a connected sum of projective planes.

Ahh Man... So pretty. Spend some time in projective space and you might never want to come back to your drab everyday world.

Quantum Mechanics and Experience

One of the classes I'll be taking this semester is 24.111 - Philosophy of Quantum Mechanics. Today I just bought the required text, Quantum Mechanics and Experience by David Z. Albert. The book looks pretty good, with a solid layman introduction to the foundational principles of quantum mechanics.

My dance with Quantum Mechanics (QM) has been a really interesting one. Prior to my coursework this past year at Cambridge, I objected to the conclusions of quantum mechanics advertised in popular science books because it ran against my mental model of the universe. My struggles with the conceptual foundations of QM were very much in the same vein as Einstein's critique of the theory. The idea that probability has a real existent status deeply troubled me. Always before, probability was a description of averages and distributions. A coin toss -- the idol of probability -- was still fundamentally governed by Newton's Laws, the forces in the room, the initial configuration of the coin in your hand, the torque, the impulse of your thumb on the coin, the chaotic turbulence of the air molecules in the room, the list goes on but all the initial configurations and momenta could theoretically be specified and then the entire motion could be simulated as the solution of a large system of nonlinear ordinary differential equations (ODEs) -- Laplace's Demon all over again. But QM is a whole different bag of nuts... Instead of solving specifically for things like position and momentum and energy, you have state vectors, and your eigenfunctions of these operators form the basis of an infinite-dimensional space known as Hilbert space. What's new about all of this is that our description of a particle is described by the time evolution of the coefficients of each of these eigenfunctions. Now we can't perceive this superposition, but rather -- and this is the really crazy bit of the whole theory -- only the eigenfunctions can be measured. So our particle is like a point wandering around in infinite dimensional Hilbert space when no one is looking. It's like Mr. Observer finally shines his flashlight in the car window and asks "Hey!?! What are you kids doing?!?" and our couple is always caught in any one of these (usually an infinite number of) compromising positions, but they're never caught scrambling to put clothes on or in a acrobatic transition or superposition of pages of the Kama Quantum Sutra.

So it will be interesting to see what insights 24.111 will bring, and complemented with 18.102 - Introduction to Functional Analysis, this should be a tantalizing semester spent exploring Hilbert space!