Friday, November 12, 2010

Thought Experiments in Linguistics

One fundamental idea of linguistics is the continuity of language and dialect; that if we were to start in some remote village of Africa and walk up all the way to Europe at a slow enough speed, along the way it would never seem like the language was changing very quickly. This model of continuity, however flawed it may be, lends to us the natural extension to linguistic calculus. That, considering linguistic differences as a function, we can safely find the rate of change of that function, and make accurate predictions about linguistic differences elsewhere.

Imagine if you took samples and found a function to model how the way an “r” is pronounced across some relatively long line (eg, all of Europe). We would get some kind of curve; perhaps a linear looking one, or perhaps an exponential one. Maybe it would be sinusoidal even, or piece-wise. Now let’s say we took samples along the same line and instead tallied, say, the words that were chosen to represent a particular symbol. That sounds very abstract, so let me give a concrete example that should make sense to most American readers. In different parts of the U.S., we use different words for a “soda”. In some places we say “soda”, others “pop”, others “soda pop”, and in many places “coke” (among other words). We could find out how the tendency to use one of these words over the other changes as we move along some curve.

Imagine the questions we could hypothesize about with just these two data sets! How does the curve of the pronunciation of “r” compare with the curve on the vocabulary? What does this say about our brain, how it communicates with and controls our voice box, how it perceives words as symbols, or how it represents a single sound compared to a whole word? How do the curves compare along the same distance, but over a different distribution of principality? For example, compare these two curves over all of Europe to over all of the US—what does this say about the difference between “languages” and “dialects”, or the sociological relationships between language and the idea of nationality? Or even the formation of nations in the first place? Is the high rate of change of language over national borders a natural consequence of how our brain processes language, or is it a result of the territory divisions? In other words, if states and countries never developed, would our language still be as different?

Sunday, October 17, 2010

Brain Range

To get started, I wanted to give a solid example of the type of ideas I'm talking about. One area that I find particuarly interesting (because it is so relevant to us, and yet so mysterious) is neurology. In an ironic twist of fate (or perhaps "fate" is not the right word), we do not understand how our own minds work. I have a lot to say about neurology and the kinds of questions we can ask therein, but I wanted to start with just a simple question:

Let us define a function f:(B,S)->B that takes the current state of the brain and any stimuli (eg, sight or sound) to a new state of the brain. Given any state of the brain, is there a set of stimuli that could lead to any other desired brain state? Put more mathematically, for all b and b' in B, does there exist an s in S such that f(b,s) = b'? It is hard to wrap your head around what this means, exactly, but you can see the weirdness of what I'm talking about.

For example, is there always some set of stimuli that will make me think of chocolate chips and hamburgers, no matter what mood or mindstate I'm in? You might say, yes of course, if you see chocolate chips and hamburgers, or taste them, you will probably be thinking about them. But what if we ask a trickier question (that is still a specific case of the above generalization)? One state of the brain could include "understanding the Hodge conjecture". I will be the first to admit that my brain has never entered this state. But given the right stimuli (no matter how strange, or alien to our human ideas or perceptions), could I suddenly understand it?

You will notice that this post was not very rigorous, and while I could have been much more specific about a lot of these things, I just wanted to give any readers a flavor of the kind of things I think about.

Looking forward to hearing your ideas!

On the shores of a Deep New Sea

Welcome, and thanks for taking the time to look at my blog. My name is Alex King, a 4th-year Mathematics student at the University of Virginia. I have come to many realizations this year-- some mathematical, some far from it-- but my appreciation for the beauty of mathematics has bloomed to a point where I can no longer contain my ideas. While traditional mathematics are certainly intriguing, I am most fascinated by the power of axiomatic thinking in fields that are NOT seen as mathematical in nature.

I truly believe that the mathematicization of academia is humanity's next epistemic venture. Axioms and abstraction are powerful tools that allow us to draw significant conclusions from very lenient assumptions. Many say that our "last frontier" is space, and yet others contend that this title belongs to Earth's oceans, the deep blue sea. I instead urge you to reconsider the way you think and explore a deep new sea, one that is filled with possibilities, exotic species of problems, and elusive answers.

With this blog I aim to:

-Share ideas that I find fascinating in the hopes that someone else will as well

-Inspire and encourage dialog, especially among young people, about the aforementioned topics

-Learn and grow from the ideas of others as I further my study of mathematics

Thanks so much, and don't hesitate to contact me,