Mathematical Interests

OK, so before this accidentally turns into a philosophy blog and I forget about the fact that I was going to go through my interests, next up math.

My main mathematical interests are in algebraic topology and category theory. More specifically in the stuff that Stefan Stolz and Peter Teichner are doing in trying to examine QFT correlations with elliptic cohomology (as a beginning grad student I am nowhere near this level though). I’m also interested in the theoretical side to differential geometry (none of this calculate the curvature stuff), i.e. the Whitney embedding theorem etc. Also putting differential forms on manifolds and looking at cohomology.

Category theory interests me, but not all that strongly. I have major philosophical viewpoints on how language affects mathematics, and I agree with David Corfield’s opinion that category theory will eventually take over since it is a more compact and powerful language. In this sense it interests me. In a pure sense, it doesn’t (sort of like set theory interests me philosophically, but not mathematically).

On a weaker level, I’ve always found ring theory, point-set topology, complex analysis, and algebraic geometry fascinating. They aren’t the subjects for me as a career or anything, but very interesting none-the-less. Probably my least favorite subjects would be pure finite group theory, the excessive terminology in point-set topology, any analysis that requires excessive approximation, and numerical analysis.


Uncertainty II

Well, I’ve been thinking about that first uncertainty post a little throughout the day and here is what I came up with. Why does this only have to hold for the Tao? There are really two schools of though on philosophy of language dealing with the meaning of words. The one says that words are defined in terms of the system in which they are used, and the other is that words are defined in terms of other words.

Either way, words are a superposition of other things. We can almost take quantum mechanics now as a special case of Wittgenstein and Kripke. They say that everything is language. Without language nothing would exist, including consciousness. It is how we think. So maybe the uncertainty in a wavefunction of a particle is really due to the fact that their is uncertainty in the superposition of terms describing it. When we pinpoint the terms and collapse the wavefuntion, it is no longer uncertain. This is the case for every physical object. They exist and are concrete precisely because we have named them and collapsed the wavefunction.

Probably lots of holes with this since I haven’t thought about it much, but I think there might be something there.

Tao and Uncertainty

Well, I’m pretty mad that I came up with this idea before reading about it, and then it turns out that other people have already thought about it. I haven’t quite found it in the formal way I would like to think about it, though. This assumes knowledge of Taoism and the Heisenberg Uncertainty Principle. It is also a good example of tying together philosophy of language, religion, and physics together.

Take the line from the Tao Te Ching: The Tao cannot be named. Why is this? Maby the uncertainty principle can shed some light on this. If you have a particle, then you cannot know both its position and momentum perfectly. Uncertainty is inherent in the system. We could say by definition. To know these perfectly, you must observe the system (make a measurement) and collapse the wavefunction.

Now, maybe the true definition is like the superposition of wavefunctions of the particle. Instead of wavefunctions, though we have terms. Tao is a superposition of all terms (or maybe just infinitely many or maybe just two). To precisely define what Tao is, you need “silent,” “unnameable,” “the way,” etc. When you give it a single name, such as Tao, you have collapsed the true Tao of superposition into a single state, which is a changed object. You no longer have the true Tao, but just one of the superimposed parts of the Tao. In order to keep all assets of the Tao, one must not name it and collapse its wavefunction. By definition it cannot be named without being changed.

Philosophical interests

So I guess I’ll just do a single post on each of the four topics that this blog is on to give an overview of my interests in that area. Here goes for philosophy.

I have lots of interests here, since I believe this is probably the main way in which each of the subjects cross-over. First off: Philosophy of Consciousness. This is a sort of minor philosophical interest with major intersection potential with physics/quantum mechanics. Is consciousness necessary to collapse a probabilistic wavefunction?

A major interest would be Philosophy of Language. How does language affect how we perceive things and think about things? Is mathematics a language? Do different languages make it easy to understand certain concepts in math? Does changing the language of math make it easier to solve certain problems? I think even the uninitiated would answer definitive “yes”s to these last two questions, but why and how? This leads to my next interest.

Philosophy of Aesthetics. Technically this is what my undergrad thesis was in. It is definitely the most elusive of my interests (we’ll see why in the next paragraph). This clearly is about how philosophy and art coincide. Can mathematics be considered an art form? Combining the phil consciousness, language, and aesthetics: how does changing the language of math affect how we think about problems and how we perceive the beauty of problems. Eventually I’ll post either all of or parts of my thesis which is a gigantic exploration of that question.

Finally, I’d like to point out that I started as an analytic philosopher. This is essentially philosophy done solely on clearly defined terms and logical manipulation/arguments using these terms. I have since become more of a continental philosopher . This is more of showing artistically why things must be true. Terms are usually considered to be more ambiguous, and arguments not as logical (hence a “lower” form of philosophy).

Here is why I not only think that continental philosophy is not “lower”, but is actually a much higher form of philosophy. Suppose you want to argue for Wittgenstein’s view of language. You could do this in a 300,000 page rigorous logical argument, but you probably wouldn’t convince very many people. In fact, even those that you do convince would probably still not know why it is true, just that it is true from a logical perspective. Compare this to reading David Foster Wallace’s The Broom of the System. Not only would you become convinced of the truth of Wittgenstein’s argument, you would see why it must be true in your life.

Ah…This is a rant for another day, though.