Scientific Clearing House has a new web address. It is now at sciencehouse.wordpress.com. The reason I’m moving is because WordPress supports latex commands. I’ve been wanting to have the option of putting equations and mathematical symbols into my blog posts but it is impossible to do so in blogger. I tried using images of equations in my last post but it was extremely painful and didn’t work all that well. I have started to migrate my previous posts to the new address and I’ll eventually move all of my posts over.
As pointed out several times by Steve Hsu recently, a major instigator of our current financial crisis is the Credit Default Swap (CDS). As far as financial instruments go, this one is almost understandable. Basically, it is an insurance policy that is exchanged between two firms. So, say you just loaned a bunch of money and you want to insure it. Well, you can buy a CDS for some fee from someone who will pay you some agreed upon amount if the loan goes bad. It is a zero sum game, which is why the amounts insured (notional amount) can be larger than the GDP of the entire world. Although the notional amounts of these CDS’s could be large, the big banks and hedge funds that traded them are often hedged so that the net gain or loss are manageable. The problem was that when Lehman Brothers went down, the whole network became unbalanced and some parties were exposed to huge losses. However, given that there is no market for these things, no one knows who is holding what. Steve drew a complex graph and an example to demonstrate how difficult it would be to unwind everything.
This month’s Notices of the American Mathematical Society is highlighting formal proof, which is a proof where every step is justified with respect to the axioms of mathematics so there is no chance of error. Generally, this would be extremely tedious to do by hand so people have been developing computerized proof assistants to help do the job. My good friend and former colleague at the University of Pittsburgh, Tom Hales penned one of the articles. I actually lured Tom from Michigan to Pittsburgh several years ago. At that time, Tom had recently finished his proof of the Kepler conjecture, which asserts that the densest arrangement of balls in three dimensional space is face-centered cubic (FCC) packing. He had been battling with the editors of the Annals of Mathematics to get his proof published. The problem was that Tom’s proof was quite long (300 pages) and used a computer. Tom’s strategy for the proof was to show that all possible packings in infinite space could be reduced to a finite number of arrangements, which needed to be tested individually. Tom and a student then wrote a program with about forty thousand lines that used interval arithmetic to show that none of these other arrangements was ever denser than FCC. A panel of experts worked on reviewing the proof and essentially gave up. They were pretty sure it was right but were unwilling to certify it. Eventually, the Annals published the proof with an asterisk that it was not fully checked by referees. I may have played a small part in helping Tom finally get the proof published. I remember going down to Tom’s office one day. He was slumped in his chair and declared that he was giving up on the Annals and writing a book instead. I convinced him not to give up and gave him some tips on the strategy to deal with the editors and referees. I’m not sure exactly what happened but the next time I saw Tom he was beaming that the work would finally be published.
However, Tom was so frustrated with the whole episode that he decided that he needed to prove the Kepler conjecture formally, so that the asterisk could be removed. He began what he called the Flyspeck project, which is a stylized acronym for Formal Proof of Kepler. He’s already recruited a number of mathematicians around the world to work on Flyspeck and nearly half of the computer code used to prove the Kepler conjecture is now certified. He estimates that it may take as long as twenty work years to complete the project so if he gets enough people interested it can be done quite quickly in real time.
A formal proof essentially involves translating mathematics into symbol manipulation that is encapsulated within a foundational system. Tom uses a system called HOL Light (an acronym for lightweight implementation of Higher Order Logic), which I’ll summarize here. The details are fairly technical; they involve using a new axiomatic or logical system that involves types (similar to what is used in computer languages like C). HOL Light differs slightly from the Zermelo-Fraenkel-Choice axioms of set theory used in traditional mathematics. The use of types means that certain incongruous operations that any mathematician would deem nonsensical (like taking the union of a real number and a function), would automatically be disallowed. The system then involves mathematical statements or objects called terms involving symbols and logical operations or inference rules. Theorems are expressed as a set of terms called the sequent that imply the truth (or more accurately the provability) of another term called the conclusion. Proofs are demonstrations that using the allowed inference rules and axioms, it is possible to arrive at the conclusion.
What I like about formal proofs is that it reduces mathematics to dynamical systems. Each term is a point in the theorem space of all terms (whatever that means). A proof is then a trajectory between the initial condition (the sequent) and the conclusion. Technically, a formal proof is not a true dynamical system because the next step is not uniquely specified by the current state. The fact that there are multiple choices at each step is why theorem proving is hard. Interestingly, this is connected to the famous computer science problem of whether or not P=NP. Theorem proving is in the complexity class NP because any proof can be verified in polynomial time. The question is whether or not a proof can be found polynomial time. If it can be shown that this is possible then you would have a proof of P=NP and get a million dollars from the Clay foundation. It would also mean that you could prove the Riemann Hypothesis and all the other Clay Millennium problems and collect for those as well. In fact, if P=NP you could prove all theorems in polynomial time. This is one of the reasons why most people (including myself) don’t think that P=NP.
I think it would be very interesting to analyze the dynamics of formal proof. So many questions immediately come to my mind. For example, what are the properties of theorem space? We know that the set of all theorems is countable but the set of possible terms is uncountable. A formal system consists of the space of all points reachable from the axioms. What does this look like? Can we define a topology on the space of all terms? I suppose a metric could be the fewest number of steps to get from one term to another term, which might be undecidable. Do people even think about these questions?
Noted Harvard economist Greg Mankiw wrote an op-ed last week for the New York Times and posted to his blog, a letter to the president-elect. One of his recommendations was to listen to economists. Following up on Steve Hsu’s post on intellectual honesty, I think this exhibits an element of hubris since the majority of economists did not foresee this current financial meltdown. There were certainly those that were warning about the collapse of the housing bubble, like Robert Shiller, but other than Nouriel Roubini, I didn’t hear too much about it causing the worst crisis since the Great Depression. Even Paul Krugman admits that this took him by surprise. I could see that there was a housing bubble back in 2004, which is why I haven’t bought a house yet, but I had no idea that the bursting of a bubble could cause so much damage. The bursting of the internet bubble caused a lot of pain to some people but did not destroy the financial system.
The current crisis first became public knowledge when Bear Stearns went under in March of this year. Federal Reserve Chairman Ben Bernanke quickly engineered a buy out of Bear by JPMorgan and the market calmed for a while. Then in quick succession starting in September came the bailout of Fannie Mae and Freddie Mac, the sale of Merrill Lynch to Bank of America, the bankruptcy of Lehman Brothers, and the bailout of A.I.G. Shortly afterwards, Treasury Secretary Hank Paulson went to Congress to announce that the entire financial system is in jeopardy and requested 700 billion dollars for a bail out. The thinking was that banks and financial institutions had stopped lending to each other because they weren’t sure which banks were sound and which were on the verge of collapse. The money was originally intended to purchase suspect financial instruments in an attempt to restore confidence. The plan has changed since then and you can read Steve Hsu’s blog for the details.
What I want to point out here is that a narrative of what happened is not the same as understanding the system. There were certainly a lot of key events and circumstances starting in the 1980’s that may have contributed to this collapse. There was the gradual deregulation of the financial industry including the Gramm-Leach-Bliley Act in 1999 that allowed investment banks and commercial banks to coalesce and the Commodity-Futures-Modernization Act in 2000 that ensured that financial derivatives remained unregulated. There was the rise of hedge funds and the use of massive amounts of leveraging by financial institutions. There were low interest rates following the internet bubble that fueled the housing bubble. There was the immense trade deficit with China (and China’s interest in keeping the US dollar high) that allowed low interest rates to persist. There was the general world savings glut that allowed so much capital to flow to the US. There was the flood of physicists and mathematicians to Wall Street and so on.
Anyone can create a nice story about what happened and depending on their prior beliefs they can be dramatically different – compare George Soros to Phil Gramm. We really would like to understand in general how the economy and financial markets operate but we only have one data point. We can never rerun history and obtain a distribution of outcomes. Thus, although we may be able to construct a plausible and consistent story for why an event happened we can never know if it were correct and even more importantly we don’t know if that can tell us how to prevent it from happening again. It could be that no matter what we had done, a crisis would still have ensued. My father warned of a collapse of the capitalistic system his entire life. I’m not sure how he would have felt had he lived to see the current crisis but he probably would have said it was inevitable. Or perhaps, if interest rates were a few points higher nothing would have happened. The truth is probably somewhere in between.
Another way of saying this is that we have a very large complex dynamical system and we have one trajectory. What we want to know about are the attractors, the basins of attraction, and the structural stabilty of the system. These are things that are difficult to determine even if we had full knowledge of the underlying dynamical system. We are trying to construct the dynamical system and infer all these properties from the observation of a single trajectory. I’m not sure if this task is impossible (i.e. undecidable) but it is certainly intractable. I don’t know how we should proceed but I do know that conventional economic dogma about efficient markets needs to be updated. Theorems are only as good as their axioms and we definitely don’t know what the axioms are for sure. I think the sooner economists own up to the fact that they really don’t know and can’t know what is going on, the better we will be.
Marine dead zones are regions of the ocean, usually near the mouths of rivers or waterways, which receive a large amount of nutrient (phosphorous and nitrogen) run off mostly from fertilizer. This causes a great phytoplankton and algae bloom that takes up carbon dioxide. However, when they die they sink to the ocean floor where other aerobic bacteria break them down with such vigor that they deplete the oxygen supply leaving an anoxic zone that cannot support marine life. The environmental movement is striving to curb fertilizer use in an attempt to mitigate these dead zones. There are also theories that the increase in the number of these dead zones are related to global warming.
I have a heretical thought on this regard and I haven’t been able to find any information on it so if anyone knows please enlighten me. The earth’s oxygen was originally created by cyanobacteria, which make up the algae that are causing the dead zones. So, could these dead zones actually be removing and sequestering carbon dioxide? Once the ocean bottom becomes anoxic, would the phytoplankton and algae fecal matter and remains pile upon the ocean floor and turn into fossil fuels in a few hundred million years? I don’t think we should necessarily encourage dead zones but is there any data out there that they could be mitigating global warming? I don’t want to be another one of those staunchly leftist youths going conservative in their old age so please set me straight if I’m wrong.
I’ve been trying to reconcile the current political environment in terms of a consistent framework. In particular, I’ve been interested in dissecting how issues have been divided between the so-called left and right in the United States. My premise is based on ideas set down in my previous post on the genetic basis of political orientation. In that post I proposed that the political thesis of the right is that the wealth should be distributed according to a person’s direct contribution while the left’s premise is that wealth should be distributed equitably regardless of standing in the community. I think these are fair definitions based on historical notions of the right and left. What I want to do now is to see how current issues should be divided between these positions in a perfectly rational world.
Let me first summarize some positions currently held by the US right: 1) low taxes, 2) small government, 3) deregulation of industries, 3) free trade, 4) gun rights, 5) strong military, 6) anti-abortion, 7) anti-gay rights, and 8) anti-immigration. I would say positions 1) through 4) seem consistent with the historical notion of the right (although regulation can be consistent with the right if it makes markets more transparent), position 5) is debatable, while positions 6) through 8) seem dissonant. The left generally but not always take the opposite positions except possibly on point 5), which is mixed. The strong military position was understood as a right wing position during the Cold War because of the opposition to communism. The rationale for a strong military waned after the fall of the Soviet Union but 9/11 changed the game again and now the military is justified as a bulwark against terrorism.
The question is how to reconcile positions 6) through 8) and we could add pro-death penalty and anti-evolution into the mix as well. These positions are aligned on the right because of several historical events. The first was that many of the early settlers to the United States came to escape religious persecution at home and this is why there is a significant US Christian fundamentalist population. The second is slavery and the Civil Rights movement. The third is that middle class whites fled the cities for the suburbs in the 50’s and 60’s. These people were probably religious but a genetic mix between left and right.
In the early 20th century, the Republicans were an economic right wing party while the Democrats under FDR veered to the left although they were mostly Keynesian and not socialist. The South had been Democratic because Lincoln was a Republican. The Civil Rights movement in the 60’s angered and scared many suburban and southern whites and this was exploited by Nixon’s “Southern Strategy”, which flipped the South to the Republicans. This was also a time of economic prosperity for the middle class so they were more influenced by issues regarding crime, safety, religion, and keeping their communities “intact”. Hence, as long as economic growth continued, there could be a coalition between the economic right and the religious right since the beliefs of both sides didn’t really infringe on each other. Hence, culturally liberal New York bankers could coexist with culturally conservative southern factory workers.
Let me now go through each point and see if we can parse them rationally. I will not try to ascribe any moral or normative value to the positions, only on whether or not it would be consistent in a right or left worldview. I think low taxes, small government, deregulation and free trade certainly belong on the right without much argument. Gun rights seem to be consistent with the right since it is an anti-regulatory sentiment. Strong military is not so clear cut to me. It certainly helps to ensure that foreign markets remain open so that would help the right. However, it could also enforce rules on other people, which is more left. It is also a big government program, which is not so right. So my sense is that a strong military is neither right nor left. Abortion is quite difficult. From the point of view of the woman, I think being pro-choice is consistent with being on the right. Even if you believe that life begins at conception and I’ve argued before that defining when life begins is problematic, the fetus is also a part of the woman’s body. From the point of view of the fetus, I think it’s actually a left wing position to be pro-life. Gay rights seems to be clearly a right wing position in that there should not be any regulation on personal choice between consenting adults. However, if you view gay behavior as being very detrimental to society then as a left winger you could possibly justify disallowing it. So interestingly, I think being against gay rights is only viable from a left wing point of view. Anti-immigration is probably more consistent with the left since immigrants could be a competitive threat to one’s job. A right winger should encourage immigration and more competition. Interestingly, anti-evolution used to be a left wing position. William Jennings Bryan, who was against evolution in the Scopes Monkey trial, was a populist Democrat. He was worried that evolution theory would justify why some people had more than others. Survival of the fittest is a very right wing concept.
I doubt that political parties will ever be completely self-consistent in their positions given accidents of history. However, the current economic crisis is forcing people to make economic considerations more of a priority. I think what will happen is that there will be a growth in socially conservative economic populism, which as I argued is probably more self-consistent. Republican presidential candidate Mike Huckabee is an example of someone in that category. The backlash against Johnson’s Great Society and anti-poverty measures was largely racially motivated. However, as the generation that lived through the Cold War and the Civil Rights era shrinks in influence, I think a slow rationalizing realignment in the issues among the political parties may take place.
I think it is appropriate to add on Election Day that I don’t think either of the two major American political parties fall into the right or left camp as I’ve defined it. There are elements of both right and left (as well as a royalist bent) in both party’s platforms.