I’m back at the University of Dayton for another school year, and I’m gradually getting used to the mass of incongruities that is college. I think a recent bookstore purchase summed it up well for me: On top of a $130 textbook (with no used copies available), some reseller had taken the care to place a small gold sticker: Free CD-ROM inside!
If this isn’t irony, I don’t know what is.
Reader’s Note: This is what comes of thinking too hard about Zen koans, memes, and complex numbers. I assure you that no mind-altering substances were used in the creation of this post.
This post begins with the famous Zen koan: “What is the sound of one hand clapping?” The koan poses an interesting puzzle, as two hands are required to clap. However, the riddle specifies that only one hand is doing the clapping - in other words, the question is self-contradictory. Although the question is a perfect sentence in English (or in any other language), the rules of that same language prevent that question from ever being answered.
This familiar puzzle is of interest because of the existence of an almost parallel problem in mathematics: What number times itself is equal to -1? Alternately, and more commonly, what is the square root of -1? Again, the problem makes perfect sense in mathematical terms, but that same system of mathematics prevents an answer — any number multiplied by itself must be positive. Period.
Those readers who have studied more advanced algebra already know that the mathematical question has an answer: i, the imaginary number, which is defined as the square root of -1. With the introduction of imaginary numbers, the familiar number line becomes a plane: Some numbers are real, some numbers are complex, and some numbers are a little bit of both.
This is where the interesting pontificating begins: Mathematics is a sort of language, in a sense. It is a language designed to very accurately describe quantities, and to allow for easy manipulation and examination of those quantities. English is also a language, designed to accurately and conveniently describe the interaction of humanity with the rest of the world. However, because English (or whatever your native language is) is so pervasive in our daily lives, it’s easy to forget that the language is only a description of a system. Just as a tire does not have 35 psi, but rather a certain amount of molecular motion which we like to describe as 35 psi, so are physical objects, such as “hands,” by no means tied down by the language we use. A “hand” is simply a arbitrarily defined portion of a larger discrete organism.
The point of all this circumlocution is thus: If there are numbers not described in mathematics, such as imaginary numbers, that do in fact exist, what is there to say that there are not concepts which are not defined in language that do in fact exist? What if the sound of one hand clapping is an imaginary concept in the mathematical sense?
Certainly there is precedent for this sort of thing. Consider descriptions of higher spatial dimensions: Although they may very well exist, we don’t deal with them every day. Our language — and more to the point, our minds — are not adapted to deal with their existence. Like the sound of one hand clapping, they are imaginary concepts as far as our language is concerned. We can name them, certainly, just as we can name i. But try to visualize them — and likewise, tell me if i apples are more or less than two apples.
What are the possible applications of imaginary concepts (or imaginary memes, although it stretches the definition of meme a bit)? Aside from simply describing the divorce between our conceptions and the limits of what can be conceived, imaginary concepts such as these can serve a test for the limits of logic and reason. Consider the age-old philosophical debate about whether God (typically a Judeo-Christian God) exists. According to most Christian theologies, God existed before time, exists throughout all time, exists in all locations simultaneously, knows all, and is all-powerful. The idea of “existing before time” alone is clearly self-contradictory, not to mention the endless possibilities for self-contradiction arising from “all-powerful” (how many fifth-graders have asked if God could make a rock He couldn’t move?). Clearly, a Judeo-Christian God is an imaginary concept.
Unfortunately for those atheists reading this, I haven’t just sealed God’s fate. Rather, I am simply stating that the idea of God is beyond the comprehension of language and reason, just as imaginary numbers lie beyond the real numbers. This is why every philosophical attempt to prove the existence or nonexistence of God has always been blown to pieces — the task faced by these philosophers is similar to the task of a mathematician who tries to find two real numbers which add to i. Although God may or may not exist (I have my own views, or course), neither philosophy or science nor any rational expression of religion has the ability to prove or deny His existence.
Those who are familiar with Zen and the Art of Motorcycle Maintainence are likely to notice a striking similarity between the “imaginary concept” and Robert Pirsig’s definition of Quality. Pirsig likened Quality to the Tao: Although it exists, it exists before any intellectual comprehension. In fact, Pirsig stated that no definition of Quality was possible, as any definition would be bounded by the limits of intellectual thought and unable of capturing the true essence of Quality.
The only catch is that Pirsig thought his undefinable Quality was unique. Now, we are faced with a whole bevy of imaginary ideas. I will let this concept rest for now, but may come back and play with it later. And, as always, feel free to leave a comment or send an e-mail if you’ve got anything on your mind.
Herbal, I’m no stranger to stoner logic, but… — “Sketchy”, Dark Angel
Once again, DARPA’s Information Awareness Office has come up with a wonderful idea to fight terrorism. Unfortunately (and also, once again) the same office has demonstrated such an incredible lack of public relations skills that the same brilliant idea has been canned.
By now, I’m sure that you’ve heard of the proposed Policy Analysis Market, which was designed to create a sort of futures market for terrorist attacks and other such events. To quote from the website’s homepage (which is now only available through the Google cache).
Analysts often use prices from various markets as indicators of potential events. The use of petroleum futures contract prices by analysts of the Middle East is a classic example. The Policy Analysis Market (PAM) refines this approach by trading futures contracts that deal with underlying fundamentals of relevance to the Middle East. Initially, PAM will focus on the economic, civil, and military futures of Egypt, Jordan, Iran, Iraq, Israel, Saudi Arabia, Syria, and Turkey and the impact of U.S. involvement with each.
The contracts traded on PAM will be based on objective data and observable events. These contracts will be valuable because traders who are registered with PAM will use their money to acquire contracts. A PAM trader who believes that the price of a specific futures contract under-predicts the future status of the issue on which it is based can attempt to profit from his belief by buying the contract. The converse holds for a trader who believes the price is an over-prediction Ð she can be a seller of the contract. This price discovery process, with the prospect of profit and at pain of loss [sic], is at the core of a marketÕs predictive power.
At its core, the idea being proposed here is that of using a genetic algorithm to attempt to predict regional instability. (Genetic algorithms have actually been used to simulate stock markets before.) The basic idea behind a genetic algorithm is simple: You take a shotgun approach to predicting the answer to a question. Those rules (or genes) that do a better job of predicting the answer are weighted more heavily than those which don’t, and the process is repeated with the weighting in place. In a market, the feedback is equivalent to how much money each trader has to invest in futures. If you’re right, the amount of influence you have over the market goes up. If you’re wrong, the amount of influence you have goes down. Thus, most of the control over the market is given to those individuals who have the best track record in predicting events.
Unfortunately, DARPA decided to put this plan out on the internet, and to use real money. Both of these ideas have merit in an abstract, academic sense — using the internet opens the market to a wider range of perspectives, and using real money ensures that traders take the market seriously. Unfortunately, those ideas combine to form a public relations nightmare, which we saw explode today. Nothing kills a project like the phrase “Betting on Terror” splashed across every newspaper in the country.
Sadly, this is an almost exact repeat of what happened to the Terrorism (formerly Total) Information Awareness program. What started out as a very logical approach to identifying suspicious behavior using statistical patterns got demonized into visions of a 1984-esque police state.
Much of the problem is very, very bad PR management on the part of DARPA and the Information Awareness Office. The IAO in particular seems not to have realized that any proposal placed on the Internet is instantly subject to public scrutiny, and that using language that may be appropriate in an academic environment may cause a firestorm in a non-academic environment. The result of this political shortcoming is that potentially useful terrorism-fighting systems are getting the axe, and it is doubtful that any of the axed systems will be viable in the near future. Hopefully, DARPA can get its act together before another good idea hits the fan.
The New York Times has exhibited a wonderful degree of schizophrenia on this issue. Yesterday they presented an editorial Poindexter’s Follies, an editorial calling for the immediate firing of John Poindexter and the closure of the “wacky intelligence operation” he runs at DARPA. Today, a separate piece in the business section argues that the futures market was actually a good idea killed by bad press. Determining which section of the paper might know more about how a futures market works is left as an exercise for the reader.
Markets do not always operate perfectly in the larger world of stocks and bonds. The idea that they can reliably forecast the behavior of isolated terrorists is ridiculous. — Editorial staff, New York Times, July 30
Similar markets have been organized to predict shifts in Federal Reserve monetary policy, the outcome of political conventions and sales of consumer products. The results are that markets typically perform at least as well, and generally better, than feasible alternatives, and they are much cheaper to organize. — Hal R. Varian, New York Times, July 31