December 2007 Archives

Question: Is it better to be a pig satisfied or Socrates dissatisfied?

Answer: If you are asking the question it is already too late.

A while ago, someone told me of an interesting paradox of Banach and Tarski which is worth relating.

Consider a ball in space of dimension 3 or greater. One can partition the ball into a finite set of subsets that can be reassembled to form two new balls identical to the original ball. In other words, one can cut up a solid ball and reassemble the pieces into two duplicates, with volumes equal to the original. The volume is effectively doubled with no operation other than cutting--no stretching, twisting or deforming of any kind. Moreover, one can move the pieces in such a way that they do not run into each other.

Intuitively (because I don't understand it mathematically), the key to this process is performing cuts of an infinitely convoluted nature. Clearly this is physically impossible, so we will not be multiplying any balls in reality, but it is entirely possible geometerically.

See Banach-Tarski Paradox for a slightly more rigorous explanation.

When one reads Hobbes, the typical reaction is to be impressed by his innovative technique (at least in the context of 17th-century political theory) and but discard his conclusion that an all-powerful sovereign is the best form of governance. Given the political background of an American in the early 21st century, this attitude is not surprising, but is it an accurate one? While I suspect that it is generally correct, Singapore provides one extremely interesting counterexample.

By most measures, Singapore is not a very free state. Though nominally a democracy, free speech is actively discouraged, there is only one major political party, and law enforcement is repressive. Petty theft is dealt with harshly, and the government sentences an astonishing number of people to death. In fact, Singapore executed 13.57 people per million to death between 1994 and 1999. The next highest country was Saudi Arabia, with 4.65 executions per million people. I am no expert on the status of Singaporean civil liberties, but numerous sources appear to have a low opinion of them.

On the other hand, Singapore is an incredibly successful nation. Its GDP per capita and human development index is the highest in Asia for a non-oil-producing nation. They have an excellent education system and remarkably low crime. It is even rated as one of the least corrupt governments in Asia, despite its oppressive policies. Apparently transparency and oppression are not entirely inconsistent. And apparently the Leviathan might exist somewhere.

The bulk of my thinking about political theory and practice for the last year and a half has focused on liberal democracies with capitalist economies. When selecting my reading list for the winter break, it only seemed appropriate to break with this relatively well-accepted theme and to delve into the theory of power--unlimited power. I noticed on my bookshelf an unread copy of Machiavelli's The Prince that I had acquired for my aborted class on the history of European civilization and a copy of Hobbes' Leviathan which I had received for free from a student who was trying to get rid of old coursebooks (somehow this seemed so un-Hobbesian). Needless to say, the political theory of power is fascinating stuff, much more so than the democratic theory that we digest over and over from the time of our births in this country.

The most remarkable fact about Machiavelli is that, despite his reputation, he was actually heavily in favor of the republican model of government. In the book, he repeatedly states this, and even gives a brief analysis of why it is so. The point then is, that if there exists a nondemocratic form of government such as a dictatorship or monarchy, then the sole incentive that leaders possess is to maintain their power, not strictly to maximize the welfare of society. After all, if they cared about maximizing the welfare of society, then they would immediately instate a democracy. This does not happen empirically. So, with the objective of power in mind, Machiavelli gives "advice" to a power-seeking "prince". In so doing, he (if inadvertently) constructs a remarkably complete deductive model of how a rational dictator ought to operate. This alone is innovative enough, but he also tests the facets of the model (i.e. his particular pieces of advice) through extensive historical analysis. Remarkably, I can't think of any significant claim in the work with which I strongly disagree.

Hobbes, on the other hand, is very disagreeable, but even more interesting than Machiavelli. Hobbes attempts to use mathematical argumentation to construct a purely deductive model of political behavior and thereby demonstrate the need for a single supreme sovereign with unlimited power. This, he claims, is the optimal form of government. While I am only about one-third of the way finished, the main problems with his argument appear to be in his initial propositions.

It is a very long and comprehensive work, so I will not go into the precise details of my counterarguments, but I believe one can successfully apply the model to explain why democracy tends to fail more often in third-world countries. To see this, we first must look to some of the general shortcomings of Hobbes' argument. Most generally, Hobbes fails to consider forces of social cohesion that cause the individual's optimal behavior to not be "warre-like". One good example is the existence of a stable economic institutions. Supposing these institutions exist, as long as there some semblance political order, democratic, aristocratic, or dictatorial, it is very likely that individuals will have an incentive to remain peaceful rather than return to the state of nature. Even with a small amount of order, the benefit that can be reaped from a vibrant economic environment will be greater than that that comes from destroying it and trying to gain more political power in a system devoid of economic activity.

In third-world nations, however, these baseline institutions frequently do not exist, so Hobbes's argument about the state of nature holds because the cost incurred when the nation reverts to the chaos is much smaller than in an economically vibrant nation. In such a state, there will therefore be more people vying to seize power. Moreover, in such a climate Hobbes's argument that all-powerful sovereigns are more stable than democracies is probably true. Democracies allow competition for power, so all parties will simultaneously be attempting to defeat each other with no strong leader in place. Thus, in third-world nations, the democratic outcome is highly unstable, while the dictatorial outcome is very stable: the dictator can just systematically eliminate his opposition, given enough force.

That being said, I still cannot agree with Hobbes's conclusion that the all-powerful sovereign is the optimal outcome even in the third-world case, because of the incredible economic and human capital benefits that come from having a free and open society. Such a society would be highly improbable under an all powerful sovereign.

"A simple criterion for science to qualify as postmodern is that it be free from any dependence on the concept of objective truth. By this criterion, for example, the complementarity interpretation of quantum physics due to Niels Bohr and the Copenhagen school is seen as postmodernist."

--Madsen and Madsen

So, if quantum mechanics has no objective truth, then why can I lose points on my examinations? Maybe my handwriting is just not convincing enough...?

Due to my mathematical ineptitude, I'm not very fast at solving problems in group theory. This can have disastrous consequences on examinations. So, in order to correct this deficiency, I asked my algebra professor where I could obtain problems similar to those that will be on our final. My plan was to practice speed by doing heaps of problems in rapid succession. Her most useful suggestion was to look on the web for preliminary and qualifying examinations given to graduate students. Many reputable schools publish their old exams or lists of practice problems to assist graduate students in studying.

The preliminary exams were not particularly useful due to their very broad content. On the other hand, the qualifying exams for Ph.D. students are in more specific areas like algebra or even just group theory. I downloaded a few exams, and started tackling the group theory questions. As I worked through more and more problems, I realized to my surprise that these problems were often easier than the ones on our exams. This was not uniformly true: Berkeley and Harvard had a few tricky questions on which I was clueless. Nevertheless, I think it's fair to say that the problems and exams in my class are at least as hard as the qualifying exams given to graduate students at reputable institutions (maybe not the top-10 for mathematics, but still good places): Dartmouth was mostly trivial, Rochester was mostly doable, and others didn't seem too bad.

Damn algebra.