29 December 2009

Political note

I generally avoid political commentary when I'm sober, but I found the quote below (written by a former managing director from UBS turned mathematician and lecturer) one of the most highly illuminating descriptions of the mechanics of political psychology that I've read. In particular, it brings into view the self-contradictory nature that generally pervades states that grant people the so-called right of "self-rule."

" If you want to see what I mean by the arbitrariness of categories, check the situation of polarized politics. The next time a Martian visits earth, try to explain to him why those who favor allowing the elimination of a fetus in the mother's womb also oppose capital punishment. Or try to explain to him why those who accept abortion are supposed to be favorable to high taxation but against a strong military. Why do those who prefer sexual freedom need to be against individual economic liberty?

I noticed the absurdity of clustering when I was quite young. By some farcical turn of events, in that civil war of Lebanon, Christians became pro-free market and the capitalistic system—i.e., what a journalist would call "the Right"—and the Islamists became socialists, getting support
from Communist regimes (Pravda, the organ of the Communist regime, called them "oppression fighters," though subsequently when the Russians invaded Afghanistan, it was the Americans who sought association with bin Laden and his Moslem peers).

The best way to prove the arbitrary character of these categories, and the contagion effect they produce, is to remember how frequently these clusters reverse in history. Today's alliance between Christian fundamentalists and the Israeli lobby would certainly seem puzzling to a nineteenth century intellectual—Christians used to be anti-Semites and Moslems were the protectors of the Jews, whom they preferred to Christians. Libertarians used to be left-wing. What is interesting to me as a probabilist is that some random event makes one group that initially supports an issue thus causing the ally itself with another group that supports another issue, thus causing the two items to fuse and unify . . . until the surprise of the separation. "

The Black Swan by Nassim Nicholas Taleb, page 16

I would recommend anyone who takes an active interest in politics to contemplate this passage and see to what extent their own political passions are controlled by the powerful force being described by Mr. Taleb. It may help one realize more precisely the extent to which politics is radically unreal, a realization that has a tremendously freeing effect on the soul.

28 December 2009

Postscript to the preceding discussion

To those interested in the financial markets we may point out a partially related meditation resulting from our preceding discussion. Many (not all) financial models, having been developed by mathematically inclined persons used to thinking of continuous quantities, model asset prices as function taking place in continuous time. Hopefully we are all beginning to question now, given that we have shown that space is discrete, whether time is discrete as well (or whether these mathematical categories are even meaningful in relation to the essence of time and space). However, ignoring that point for the moment, we must point out as fact that time in the financial markets is not continuous, since the markets open and close at definite points, placing discrete cuts in the apparently continuous whole of market activity. This would be irrelevant if asset prices could not change after market close. However, we can observe, albeit rarely, enormous changes in value that occur (practically speaking) instantaneously at the market’s opening that result from accumulated orders placed overnight, usually as a result of news that came after the previous day’s market close.

This is especially important if we consider portfolios that attempt to remain neutral to market fluctuations through the use of derivatives. If asset prices are modeled using Brownian motion (the usual method), which was originally designed to model the behavior of particles of pollen suspended in water, then the model will not be able to appropriately price the risk of a discontinuous overnight movement, since the objects Brownian motion models cannot move discontinuously. This poses no problem to the original intended uses of Brownian motion, but it does pose a significant problem to attempts to hedge portfolio value, since it will result in the failure of such methods at precisely the point when they are most necessary. The financial engineers who designed such portfolios, instead of questioning the validity of their underlying theoretical assumptions, will all-too-often simply accept that the infinitely small probability of hedging failure predicted by their financial models had in fact transpired, effectively blaming fate instead of their own mathematically brilliant models, models that have the one drawback of not entirely corresponding to that which they seek to describe.

There are other issues with the usual formulas used to construct hedged positions, but these are beyond the scope of our present inquiry into the essence of space.

That space is composed of discrete units, and that continuous quantities cannot accurately describe space in its actuality

The previous argument we entered into, wherein we proved the existence of a finite and indivisible lower bound to space, had the helpful side effect of proving that space is composed of discrete units and is therefore not a continuous whole. However, this argument was not accepted by all, particularly those with some knowledge of calculus. Indeed, very often in discussions on this issue one encounters resistance precisely along these lines. Since the mathematical conception of space underlies the notion of space that many of us have come to accept, we naturally resist the notion that space does not completely correspond to its mathematical map, which has no finite lower bound. However, actual space cannot actually correspond to the mathematical representation in light of which we define the concept of space. Let us quickly revisit our initial argument to show why.

Our argument consisted (in brief) of pointing out that any real partition of space must contain some amount of space (this is clear from the definition). Since this is the case, there must necessarily be a smallest unit of space. We proved this by examining the contrary, namely, the belief that there is no smallest unit of space and that space is infinitely partitioned. We showed that, if this was so, there must be an infinite number of partitions in any given finite section of space, each containing some actual amount of space (else they would not be divisions of space at all), and therefore leading to the conclusion that there was an infinite amount of space in a finite amount of space. Other than being inherently contradictory, this argument would also make motion impossible, which is absurd, since things do move. Therefore, the contrary—that space is composed of finitely small units—must be true.

Distinguished interlocutors pointed out that any of these finite segments, even the smallest possible unit that was proved to exist, could be further divided into another segment, thereby proving that the smallest possible unit was not the smallest possible unit—showing an apparent contradiction in our position. The appearance of contradiction arises because we insist on our ability to further subdivide the smallest possible unit that the preceding argument firmly established. But by what right can we actually do this? It is certainly true that, given any real number, one can always find a smaller one through simple division. This also shows that the 3-dimensional real “linear” space that we intuitively think of (constructed by extending the real number line in all three dimensions) can always be divided further, since it is composed of real numbers. However, at this point the mathematical model breaks down in relation to actual space, since we have already demonstrated that there must be a smallest actually existing unit of space. If one takes this smallest unit and mathematically divides it into halves, one has not created a smaller unit of space, but has created a unit of a unit. This construct, while mathematically reasonable, and physically existent insofar as it is a division of a unit, is not in itself an actual unit of space.

To explain this (perhaps abstract) point by reference to a concrete example—suppose one takes a glass of water and sets to dividing it in halves. One will first have two half-glasses, then four quarter-glasses, and so on. Each new division we create will be a division of the water, since there is still water in each new division. However, when we finally have divided our glass of water into millions of separate glasses, each containing only one molecule each, what now can we do? Our mathematical method of procedure has let us divide the water by two, each time producing new divisions. Now, however, if we were to slice up the molecule of water even further, we would find that we no longer had any water at all, but simply a scattering of oxygen and hydrogen atoms. This is because water cannot be actually divided into any smaller piece than one water molecule. One could draw a water molecule on a piece of paper and proceed to slice it up in a million mathematically distinct segments—in fact, given infinite time and infinite paper, one could even infinitely divide it. But these divisions, while being no doubt impressive, would in no way exist.

The point of what we have gone through is to establish firmly that space is composed of a finitely-bounded quantity of discrete units. This conclusion comes analytically from our usual concept of space. Additionally, if space is itself composed of discrete units, then nothing in space can itself be continuous. Therefore, we must assert, despite their usefulness, that continuous quantities do not exist in any actual sense. That is, assuming that what we usually refer to as “space” is itself actual and not simply different than what we think it to be but essentially different than what we think it to be…

18 December 2009

That space, in its actuality, must be finitely subdivided (with a nod to Zeno)

There either is or is not a smallest actual unit of space. If there is not a smallest actual unit of space, then space is infinitely subdivided. If there is a smallest actual unit of space, then space is not infinitely subdivided, nor is it infinitely subdivisible. This matter cannot be settled by a direct appeal to empirical evidence, since it has already been established that there is a lower bound to the observable universe. The existence of this bound does not, however, prevent the potential existence of smaller unobservable particles—perhaps an infinitely large number of smaller particles (implying smaller subdivisions).

If there is not a smallest actual unit of space, then it follows that there are an infinite amount of subdivisions of space actually existing in any given finite region of space. If this is so, then it also follows that there is an infinite amount of space in any finite region of space. Now, it would seem that it would take an infinite amount of time to move through an infinite amount of space, unless one has the capacity to move at an infinite velocity. Since human beings do not seem to possess the means to means to travel at an infinite velocity, it follows that no one would be able to cross even a finite distance, since to do so they would have to move through the infinite subdivisions of the finite amount of space. In fact, nothing unable to move at an infinite velocity would be able to move at all, since it would have to cross infinite space to reach a point even infinitesimally farther away from where it started.

However, since clearly things do move, it follows that space cannot be actually infinitely subdivided. If space is not actually infinitely subdivided, it must be finitely subdivided; therefore, it is necessary that there be a smallest unit of space. If there is a smallest unit of space, then it follows that there must also be a smallest possible unit of matter or energy, though it is entirely possible that this unit may be too small to be detected.