Sunday, August 23, 2009

Lockerbie and Guantanamo: YMGPFM

My twin has a piece up at the Huffington Post arguing that Abdel Basset Ali al-Megrahi return to Libya is an argument for keeping the Guantanamo prisoners in the United States. The argument in essence is that if we keep them in the US we will have better control over what happens to them in the long run. I am however, somewhat jealous: His piece has already gotten him called all sorts of nasty names. Nothing I blog about seems to ever do that.

Wednesday, August 19, 2009

The International Community Should Recognize Somaliland

Somalia is the perfect example of the failed state and has been so for almost twenty years. Since the fall of Siad Barre’s dictatorship in 1991, the country has been in a perpetual state of civil war. Most of the country is ruled by various warlords, and the country is in a state of perpetual war, with so many different clans and different Islamic extremist groups that it is hard to even keep count. In large parts of the country, it isn’t even clear who is in charge. All attempts by other countries to alleviate the situation have failed. Yet the international community continues to pour billions of dollars of aid into Somalia.

However, there is an exception. After Barre’s fall, the north of Somalia declared independence and named itself Somaliland. Like the rest of Somalia, Somaliland is composed of rival clans. Unlike the rest of Somalia, clan members have been willing to embrace the rule of law over the short-term dominance of their clans. Somaliland has not been perfect. It has had on and off conflicts with the neighboring Puntland, which is another autonomous entity created after the collapse of the Somali government. However, Somaliland has been relatively stable and, moreover, has attempted to comply with international law. Unlike Puntland and the various warlords controlling the remaining areas, Somaliland has moved to prevent piratical behavior by its residents. It has gone so far as to convict people simply for plotting piracy.

Despite the strides Somaliland has made as a nation, its situation is precarious. Its independence is not recognized by any country. The economic situation is much better than in the rest of Somalia, but Somaliland is by no means prosperous. Most importantly, Islamic extremists have recognized the success of Somaliland and have deliberately tried to destabilize it with terrorist attacks.

The Islamic extremists and the clan warlords recognize that, as Somaliland continues as an example of peace and stability, the example of a successful state threatens them and their respective domains. The United States and the rest of the international community need to recognize this as well. There are two simple steps to help this fledgling state: First, we must give formal recognition of Somaliland as a separate country. Second, some of the international aid directed to Somalia must instead go directly to Somaliland. Much of this aid is wasted as food supplies and other forms of aid are often seized by warlords and other groups. The people of Somaliland will actually benefit from this aid.

Somaliland has stood on its own feet for almost twenty years in a land of bloodshed and violence. During that time, the country has embraced the rule of law and helped the international community address piracy. No one else in the area is either willing or able to confront piracy. It’s time the international community had the wisdom and courage to recognize Somialand as an independent state in the community of nations.

Monday, August 17, 2009

Revisiting the Four Fours

In an earlier blog entry, I discussed a generalization of the four fours problem. In that entry, we discussed the function f(n) defined to be the least number of 1s needed to represent n as a product or sum of 1s using any number of parentheses. (Thus, for example, f(6)=5 since we may write 6=(1+1)(1+1+1)). That entry inspired some discussion both in the comments thread and later by email on the general behavior of f. Harry Altman and I made some progress on the general behavior of f, that may lead to a paper.

One open problem is whether f(2^a*3^b)=2a+3b in general (ignoring a=b=0). This conjecture essentially says that the most efficient way to represent such a number is in the obvious fashion of writing (1+1) a times and writing (1+1+1) b times. Harry has shown that for any value of b if a is at most 15, this holds. Harry's proof involves a large amount of case checking but may be able to be pushed up to larger values of a. He thinks he may be able to construct a computer program that can examine the relevant case types in a systematic fashion.

My own work (with some input from Harry) has focused primarily on the global behavior of f. In the blog entry, I commented that the best known bounds on f were 3log3 n ≤ f(n) 3 ≤ log2 n for n>1 . I have reduced the upper bound to 2.65 log2 n. The remainder of this blog entry will attempt to give the general idea of the proof by sketching out the simpler result that we may take
f(n) ≤ 2.95 log2 n.

The strategy of our proof is similar to the proof that f(n) ≤ 3log2 n but more involved. We will prove this inductively on. For general n ≥ 2, let S(n) be the statement "If for 2≤ k ≤n-1, we have f(k) ≤ 2.95 log2 k, then we have f(n) ≤ 2.95 log2 n." We will show that S(n) holds for all n ≥ 1, and thus the induction holds.

First, observe, that if 2|n then S(n) since we may write n = (1+1)(n/2) and so f(n) ≤ f(n/2) + 2 ≤ 2.95log2 (n/2) +2 ≤ 2.95log2 n + 2 -2.95log2 2 ≤ 2.95log2 n. We thus may assume that n is odd. A similar remark then applies if 3|n. Now, we have either n ≡ 1 or 2 (mod 3). If n ≡ 1 (mod 3), we may write
n = (1+1)(1+1+1)((n-1)/6)+1 and so we get f(n) ≤ f((n-1)/6) + 6 and similar logic applies.

By nearly identical logic we can through arduous case checking obtain that we have S(n) unless n ≡ 7 (mod 8), n ≡ 8 (mod 9), and n ≡ 4 (mod 5). Note that the difficult cases for any modulus always occurs at n ≡ m-1 (mod m). This is not a coincidence, but discussing why that occurs would take us farther afield.

Before proceeding further, let us introduce a helpful notation. We will write [x,y](n) to mean (n-x)/y. Thus, our above results can be phrased in terms of this notation. For example, n the case that n ≡ 1 (mod 6) above, we could write f(n) ≤ f([1,6](n))+ 6. We will also use the notation [x,y]^i(n) to mean repeating the [x,y] function i times. Thus for example, [1,2]^2(11)=2. This notation makes things nicely compact. Now, let a be the largest integer such that 2^a|n+1 and let b be the largest integer such that 3^b|n+1. By the above remarks, we may assume that a ≥ 3 and b ≥ 2 and c ≥1. It does not take much work to see that we then have f(n) ≤ f([1,3]^b[0,2][1,2]^a (n)) + 3a + 4b + 2.

Now, assume that S(n) is false. We thus have 2.95 log2 n ≤ 2.95 log2 (f([1,3]^b[0,2][1,2]^a (n)) + 3a + 4b + 2 ≤ 2.95 log2 n/(2^(a+1)3^b) + 3a + 4b + 2. Canceling the 2.95log2 n on both sides and bringing the log terms over to the right hand side we obtain 2.95(a+1) + (2.95log2 3)b ≤ 3a + 4b + 2 which implies 13.5b + 19 ≤ a. Note that we did implicitly use that 5|n+1 since otherwise we would not be assured [1,3]^b[0,2][1,2]^a (n) is not negative or 0 since otherwise we would be unable to take its logarithm.

Using similar logic but reducing first by 3s before reducing by 2s we can get a similar lower bound for b in terms of a: 1.387a +2.391 ≤ b. The pair of equations has no solutions with a ≤ 3. Thus, our assumption that S(n) fails for some n is false.

The essential idea of this proof is that just as the proof for 3log2 n used the base 2 expansion, of n, we can be assured that we can in some sense reach a number with a good expansion in either 2 or 3 without expending much.

This, is of course, a toy example. With not much more effort, we can reduce the constant to 2.8, using slightly stronger inequalities and using bases 2,3 and 5. The full proof for 2.65 is more detailed but uses the same basic idea with 2,3,5,7,13 and 17. It also turns out that it helps to not think of the bases so much as the p-adic expansions, something which I hope to discuss in a later blog entry.

Sunday, August 9, 2009

Skepticism Is Not an Excuse for Sloppiness

While browsing a local bookstore a few days ago, I ran across a copy of James Randi's "The Supernatural A-Z: The Truth and The Lies." Randi is a professional magician and has been at the forefront of the skeptical movement for some time. The book is an encyclopedia of supernatural and fringe claims described from a skeptical perspective. Randi is a witty and clever writer. I therefore bought the book and looked forward to an entertaining learning experience.

Unfortunately, this was not to be. Browsing through the book, I found that it contains many errors and misleading statements. And these were only those detected by me from the (small) set of entries of which I had some prior knowledge.

One of the most glaring series of errors occurs in the entry Tetragrammatron. The entry reads (with internal formatting suppressed):

In the kabalah, this is the term for the four-letter name of God. In effect, it is the name of a Name. It varies from text to text. Some versions are JHVH, IHVH, JHWH, YHVH and YHWH. Since these are too sacred to be spoken outloud, the word `Adoni' is used when the name is spoken. This has led to a serious misunderstanding, since in Hebrew texts only the vowels of Adoni (or of `Elohim' - this makes it more confusing) are printed. Thus are produced the reconstructions Yahweh, Jehova, etc.

There's so much wrong with this entry that I'm not sure where to start. I'm going to refrain from pointing out the many minor errors, such as that the term "tetragrammatron" isn't actually connected to kabalah. There's no circumstance where only the vowels are printed. I'm not completely sure where Randi got this idea or what statement that this was based on. The most obvious is that in Hebrew generally only consonants are printed. It is possible that somewhere Randi got vowels confused with consonants and then thought it was something which applied only to the four letter name. The other likely possibility is that Randi was confused by the practice that, on the occasions when something is printed with vowels (such as prayer books and certain religious texts), sometimes the four letter name is printed with the correct consonants but using the vowels from Adoni. However, this practice is not the root of the vowelization in either "Yahweh" or "Jehovah."

This is not the only severe error. The entry for the Necronomicon reads:

Several additions of this grimoire have appeared. Said to have been first published in about AD 730, in Arabic, as Al Azif, by Abdul Alhazred, an English translation is attributed to John Dee. It relates powerful formulas for calling up dangerous demigods and demons who are dedicated to destroying mankind.
It is a bit surprising that a nominally skeptical work would discuss the Necronomicon without mentioning that it is a completely fictional work. The Necronomicon was originally written about by H.P. Lovecraft in his horror writing in the 1920s and 30s. It is in his explicitly fictional universe that all the details above are correct. Since Lovecraft, various hoax Necronomicons have been written, but those are all very much modern creations. While this is an error primarily of omission rather than commission it is a massive mistake which makes one wonder how much attention Randi has paid to the subject.

These are not the only entries with errors. There are misleading statements about the doctrines of Christian Science, and there are claims that are so wrong that two-minutes of fact checking would find them. For example, Randi claims that Cotton Mather presided over the Salem Witch Trials.

All these errors I found from browsing through the book for about an hour. There are many entries about which I know little or nothing and I have made no effort to check the accuracy of these entries.

There are serious pragmatic and ethical concerns with this sort of sloppiness. Pragmatically, there are three major issues: First, a moderately credulous individual might pick up this book, read through it and react against skepticism as a result of seeing such a major spokesperson of skepticism engaging in such intellectual laziness. Second, a skeptic might read the book, and rely on the incorrect information for later use and thus be caught out in a debate or discussion. Third, it is common for members of fringe groups to accuse skeptics of not taking the time to understand what they are analyzing. This gives unfortunate weight to that charge.

There are three ethical problems: Most seriously, readers expect when they buy a book by James Randi to buy a book that is accurate and has been subject to minimal fact-checking. It does a disservice to readers to sell them such poorly researched material. Second, Randi and the skeptical movement as a whole have repeatedly and correctly criticized various fringe groups for engaging in poor research and outright sloppiness. It is thus the height of hypocrisy to engage in the same behavior. Third, it is in general unethical to promote falsehoods and misunderstandings.

I'm also disturbed that I can find little discussion on the internet about the flaws in this book. The skeptical movement cannot be skeptical of others and then turn a blind eye to the flaws of their own. That's not skepticism. That's tribalism.