Wednesday, December 30, 2009

Integer Complexity: Why Blogging is Fun

A while back I wrote a blog entry about a generalization of the four fours problem. That entry focused on the function f(n) defined as the minimum number of 1s needed to represent n using just addition and multiplication. So for example, f(6)=5 since 6 = (1+1)(1+1+1) and there's no way to represent 6 with four or fewer 1s. Surprisingly little is known about the general behavior of f.

After that blog entry, two commentators, Etienne and Harry, took up examining f in more detail. Harry and I together produced new work on the behavior of f that will likely turn into a paper.

I will attempt here to briefly summarize what was known and how we've improved it. (Reading my original blog entry and my previous follow-up will probably help matters).

It was known that f(n) satisfied the inequalities 3log3 n ≤ f(n) ≤ 3log2 n for n greater than 1. We've improved the upper bound somewhat, replacing 3log2 n with 2.64log2 n. We've also shown that f(n) - 3log3 n is unbounded (that is one can make it as large as one wants if one chooses suitable n). We did this by defining what we call the "defect" of d(n)= f(n) - 3log3 n. We defined Ak to be the set of numbers with defect at most k. Then, we were able to show that each set Ak was way too small to contain every natural number. More technically, we showed that if one defines Ak(x) to be the number of elements in Ak that are at most x, then Ak(x) = O( (log x)^c) with c a function of k.

Also, it was conjectured that if one had a number of the form n=2^a3^b that f(2^a3^b)=2a+3b (aside from the trivial case when a=b=0). Essentially this conjecture stated that the most efficient representation for n arose from the obvious one, that is writing n = (1+1)(1+1)...(1+1) * (1+1+1)(1+1+1)...(1+1+1). Harry, was able to prove that this conjecture held for all such n as long as a was at most 10. Since then, we've been able to improve that bound to a at most 19.

It turns out that for much of the behavior of f, including both of the two problems discussed above, the sets of Ak and some closely related sets are natural objects to consider. Thus, we tried to work on making c as small as we could in the equation Ak(x) = O( (log x)^c). The initial proof gave a very poor bound on c (being able to take c=1024^k). After a series of improvements, primarily by Harry, he was able to take c=4^k and then using a combination
of ideas from the two of us, we got a linear bound. In particular, one can take c=3k/(5- 3log_3 5). Note how this bound grows much slower than the previous bounds. However, we do know that A1(x) is in fact of order log(x)^2 so there's some realm for improvement in that this proof gives an upper bound for c of about 4.96 for A1(x). One thing to note: While these results were proved using induction arguments (as one would expect), one cannot induct on Ak with k a positive integer. One actually needs to induct on Aαk for fixed α less than 1. A good choice of what alpha to use then becomes very important.

Harry has on his blog a long post that summarizes what we have accomplished and discusses them in more detail. See his posts here and here and the other posts he links to.

The moral to this story is that you don't know what interesting things will happen when you blog, especially when you have readers that are as smart or smarter than you.

Sunday, December 27, 2009

Three Religions Meme

James McGrath has recently tagged me (well. really a very large set of people) with the meme of naming three religions you find fascinating of which you are not or have not been a member.

I'm going to place a few additional restrictions on my choices:

First, I am not going to pick any purely fictitious religions. Thus, for example, I'm not going to pick any religions from Brandon Sanderson's Mistborn series (although there is a very large temptation to try to pick some of those). Similarly, I'm not going to pick any religion
that started as a fictional religion and then became real. Thus, Melkorism and Cthulhu worship are both out. Finally, I'm not going to pick any parody religions. Thus, Invisible Pink Unicornism and Flying Spaghetti Monsterism are both out although I will note that IPUism was around long before FSMism which is a sad little upstart in comparison (I mean, c’mon. It doesn't even have an oxymoron in its title.).

So of the real, non-parody religions, what will I choose are Cargo Cults, Roman Catholicism and Bahai in no particular order.

Cargo cultism is a religion with which some may not be familiar. It is also as far as I am aware the only example where nearly identical religions arose independently which already makes it very interesting. However, where and how cargo cults arose explains this feature. At the same time, it makes them all the more fascinating. During World War II, the United States used islands whose populations previously had had little or no contact with the outside world. Many people in Western and other civilizations believe in forms of sympathetic magic or the like, but small tribal groups often take this to an extreme. In these cases, the tribes saw the US soldiers doing apparently ritualistic activity that included marching in formation, talking to themselves on pieces of wire, and clearing out large fields. The tribes further saw the immediate responses to this activity: Airplanes came and dropped cargo to the soldiers, including food, medicine and clothing. The tribal members concluded that if they could engage in the same ritualistic behavior, they might get the same or even better results.
Thus, the cargo cults were born.

When the soldiers left, the tribes assigned priests who talked to spirits on the radio which would generally consist of a few pieces of wire. They cleared out or kept clear airplane landing fields. And they wore uniforms and conducted drills. These religions were derisively labeled "cargo cults." These remained strong through the mid 1970s. Members retained their beliefs and practices even when confronted with more features of modern civilization and explanations of what US soldiers had been actually doing. Cargo cults still exist in limited numbers today, seventy years after World War II.

The second religion is Roman Catholicism. The Church has been and remains a fascinating set of contradictions. On the one hand, it is hard to reconcile the gilded architecture and massive hierarchy with the ascetic messages preached by Jesus. The Church has a history of encouraging violence from the Crusades to the Inquisition. When the Church has not encouraged war and killing for its own ends, it has been silent when speaking up could save lives. The Church has also actively persecuted individuals such as Galileo who disagreed with the Church. On the other hand, the Church helped preserved learning that would otherwise be lost. The Church was a bastion of knowledge against the tides of ignorance (tides possibly encouraged by aspects of the Church. but still) The Church has provided shelter and relief to the poor for centuries. In the last fifty years, the Church has strived to modernize and reconcile its beliefs with science while many other religions have gone out of their way to attack science.

The Roman Catholic Church is the Church of Torquemada But it is also the Church of George Coyne and J. R. R. Tolkien. Whether the Church survives this next century and whether it embraces modernity or not will be some of the major questions impacting this century.

The third is the Bahai. The Bahai are the youngest of the major Abrahamic religions. They have consistently embraced reason over dogma and thus far have a history of being the least persecutory of the Abrahamic religions (although how much this is due to their still small numbers remains to be seen). If I had to name a single religion that I might be comfortable with drastically increasing in size over the next few centuries, I would likely name the Bahai.

Edit: It was pointed out to me by multiple people that I neglected to tag this meme for anyone. So I'll tag Kurt because he hasn't blogged much since he got married, Apikores because I'm just curious what he has to say, and Kat because she commented on this post already so hey, why not? And I suppose anyone else who has enough free time who reads this blog regularly and has a blog can consider themselves tagged.

Sunday, December 13, 2009

Off-Topic Lovecraftian Horrors: Adventures of the Squamous and Rugose

Here are two marginally related videos that I could not resist sharing:

First, we have the adventures of Lil' Cthulhu. It's a new day and the stars are right. It's time to play. Wake up Lil' Cthulhu!

This is an excellent video for those who want their children to be inducted into the ceremonies and blasphemous, mind-shattering secrets of the Great Old Ones from a young age. Starting early is important!

And following this, we have Heartache Over Innsmouth. This is a moving romantic song, about a man who has fallen in love with a girl who joins the cult of Dagon.

As far as I am aware, Dagon is the only deity in Lovecraft which actually appears in the Bible. Dagon was a common Semitic agricultural deity. For many centuries, commentators(such as Rashi) noted the similarity between Dagon and the Hebrew word for fish, dag. This lead them to believe that Dagon was some sort of oceanic or fish deity. Laboring under this belief, Lovecraft chose Dagon as the name of the horrific creature under the ocean which rules over the Deep Ones. Some later writers have attempted to retcon this discrepancy by suggesting that Dagon was not that horrific being's name but rather was a name given to it by the cultists as they needed a name. They thus chose that name due to their own mistaken belief that Dagon in the Bible was a fish deity.

Monday, December 7, 2009

Virgin Galactic, Star Trek and Retconning

Virgin Galactic has unveiled their newest model of spaceship, SpaceShipTwo. Virgin Galactic, part of Richard Branson's Virgin Group, is trying to make affordable space tourism. Where by affordable, we mean affordable if you have $200,000 to burn. And all you get is a quick sub-orbital hop. We are at the dawn of private space travel, and one could easily spend much time discussing the long-term political, social and religious implications. I'm not going to do that.

Instead, I'm going to focus on the names of the ships. The first ship will be named the VSS Enterpise. VSS stands for "Virgin Space Ship." The second ship according to some reports will be the VSS Voyager. Yes. Enterprise and Voyager. As in from Star Trek.

Officially, the VSS Enterprise is named not just after the fictional ships but also the real ships that have been given that name, including the space shuttle Enterprise.

Now here's where things get complicated. The space shuttle Enterprise was named after the USS Enterprise (NCC-1701), commanded by James Tiberius Kirk; the real shuttle Enterprise is named after the fictional ship. But it gets better. The writers of Star Trek retconned in the existence of the Enterprise space shuttle into their universe. That is, in the Star Trek universe, the Enterprise shuttle is now one of the long line of ships that existed before the Federation ships. But the Enterprise shuttle in the Star Trek universe is not named after the starship USS Enterprise because in the Star Trek universe, Star Trek never occurred as a 20th century television show. There has even been a Star Trek novel which had a plot focusing on the space shuttle Enterprise finally getting a chance to fly (in both the Star Trek universe and in our own universe the Enterprise was used a test shuttle but never flew into space). I've been unable to track down this novel although I did read it many years ago. If anyone can track down the novel I'd be very grateful.

I suspect that in a few years we will see the same thing for Branson's ship. That is, there will be official Star Trek material which includes Branson's ship as a ship that existed in the Star Trek universe.

I'm hoping that the ships three and four are named Serenity and the Millennium Falcon. Star Wars at least has the advantage of having occurred a long, long time ago, in a galaxy, far far away. So they won't have to deal with the oddity of incorporating into the universe real ships that were named after ships from their universe.

Sunday, December 6, 2009

Excessive Executive Compensation, Health Care and More Gratuitous Promotion of Family Members

My twin has a piece up at the Huffington Post in which he examines the sections of the proposed health care reform that concern compensation for insurance executives. Under current regulations, corporations can easily deduct executive compensation. This legislation will substantially restrict such deduction by health insurance companies. Aaron argues that the proposed caps on such compensation do not go far enough. In particular, there is no good reason to have such restrictions for insurance companies and not other companies. He makes a good case and also explains in detail the relevant regulations that govern the status quo. My one nitpick is that in many locations a game of Ms. Pacman costs more than a quarter. Go and read.

Tuesday, December 1, 2009

Westboro Baptist Church at Boston University

Readers are likely familiar with the Westboro Baptist Church as the strange cult run by Fred Phelps which is devoted to telling the world that God hates them. A lot. They are most known for the slogans "God Hates America" and "God Hates Fags." Pretty much there is no group they don't hate. They even have a song about called God Hates the World, to the tune of "We Are the World." They have a fascinating eschatology in which in the end time God will ask them what should be done with the rest of humanity and they'll gleefully tell God to send humanity to hell. Some people who don't know much about the Church have speculated that the Church exists really just to make the right-wing look bad.

Why am I thinking of the Westboro Baptist Church? Today they came to Boston University to protest. I think this time they were nominally protesting Jews. Or something like that. Apparently they had signs attacking a lot of different groups.

Unfortunately, I had office hours when they were protesting. This isn't fun. When Ray Comfort was distributing his version of Origin of Species to a hundred universities across the nation, he neglected to include Boston University and I was unable to get to Harvard or MIT where there was distribution.

I've concluded that there is a God. God is taunting me by dangling interesting groups just close enough that I'm aware of them but not so close as to actually get to talk to them. God probably sees me as sort of like a cat and sees the various crazies as akin to a laser pointer.

Just my luck, Neturei Karta will probably come to campus and I'll miss them also.

Friday, November 27, 2009

C.S. Lewis, the Trilemma, and Cultural Norms

A common Christian apologetic argument is the Trilemma. First introduced by C.S. Lewis, this argument since Lewis has undergone modification. However, the basic argument has not changed. As the argument goes, Jesus, was either telling the truth when he said that he was God, Jesus was lying, or Jesus was insane. This is generally abbreviated as “Lord, Liar or Lunatic.”

Lewis used this argument primarily as a response to people who thought that Jesus was a good person. but not the Son of God. Lewis argued that this was not a possibility since, if Jesus was not telling the truth, then one of the other two possibilities must hold. In its more modern form, the argument is identical, but evidence is presented as well that the last two possibilities don’t hold.

The argument in either the original form of Lewis or in other variants, suffers from flaws. The most serious flaw is the reliability of the Gospels as record of what Jesus said. It is not at all implausible that Jesus didn’t claim to be the Son of God, but such claims were later asserted by followers. Or Jesus could have in fact said exactly what he is quoted as but have been genuinely mistaken. These are two are two good detailed discussions of these and other flaws. Rather than discuss the flaws, I’d like to examine why this argument is so effective apologetics.

The argument, especially in its post-Lewis form (such as that advanced by Joshua McDowell in his “Evidence that Demands A Verdict”) does not explicitly invoke the presupposition that the audience thinks that Jesus was a great man . However, in an unstated form, this approach is far more effective. We live in a society with few taboos stronger than saying negative statements about Jesus. Indeed, even if one asks most Jews who live in the United States what they think of Jesus, they will feel compelled to say something like “I think he was a great teacher” or something similar. Thus, to most people, the notion that Jesus was either a lunatic or a liar is so repulsive (or politically incorrect) that when faced with those alternatives, they have no choice but to rush to the third possibility. The Trilemma thus rests on implicit cultural norms that Lewis was willing to make explicit. His successors have been less forthcoming.

Tuesday, November 24, 2009

A Quick Note about Bill Sparkman

Some readers may remember Bill Sparkman. Sparkman was a federal census worker who was found dead, hanging from a tree with the word "fed" scrawled across his body. At the time, the general consensus among progressive bloggers was that Sparkman's murder was evidence for the deep problems being created by right-wing rhetoric that stoked anger and paranoia. (See for example this note). The reply by the right-wing was interesting with all sorts of explanatory hypotheses proposed. Some of the responses on the right-wing were simply put, insane, such as incredibly baseless claim that Sparkman had been killed because he was pedophile. All of this looked like the standard behavior for the blogosphere and pundits but for one detail: It nows turns out that Sparkman wasn't murdered. According to police, he committed suicide and tried to make it look like murder to help get insurance money. I hope that all the bloggers who used this as evidence of the problems of the current right-wing rhetoric will post follow-ups but I'm not optimistic.

Note that this isn't intended to say that there aren't good reasons to be concerned with the increasing radicalization and paranoia of the American right. But Bill Sparkman's death should not be part of those concerns.

Sunday, November 22, 2009

Spam and Commenting II

Given the discussion in the comments for the thread below I am turning on CAPTCHA for commenting. I haven't been able to figure out how to ger reCAPTCHA working for blogger. If someone knows how to use that, I'll do that instead.

Wednesday, November 18, 2009

Spam and Commenting

Over the last two weeks there has been a problem with spam bots. The problem has become annoying enough that it needs fixing. I am either going to turn on comment moderation or turn on a CAPTCHA system. Which do readers prefer?

Monday, November 16, 2009

Barack Obama, the Turing-Lovecraft Theorem and Horcruxes

One nice thing about a blog is that you can see how people arrived at your blog. Many hits come from Google searches. Some of those Google searches are phrased explicitly as questions while in other cases what the individual is searching for is obvious. This blog entry will examine some of the more common search strings and questions which blog entries here do not already answer.

Search string: "Is Barack Obama a clone?" Number of searches: At least 15 counting all variations.

People searching for this question or some variation thereof normally find my entry on Barack Obama's position on cloning. So is Barack Obama a clone? No. Of course not. That's even stupider than thinking that Obama's parents deliberately faked a birth certificate so he could run for President 40 years later. That's even stupider than thinking that 9/11 was an inside job. That's even dumber than thinking that scientists invented evolution to undermine belief in God. Cloning is a really difficult technology. We've had trouble until recently even cloning mammals. The idea we could clone people 40 years ago is absurd. And there's no coherent aim to cloning Obama.

On second thought, maybe it isn't so unreasonable. It would explain a great deal such as why he's been so uncooperative about giving a long-form birth certificate for the birthers to examine. And it explains why he is so charismatic. Someone tell Orly Taitz! Obama wasn't born in Hawaii or Kenya or anywhere else! Maybe he was never born but cloned in a vat by the Illuminati! Sadly, this hypothesis contradicts the preexisting conspiracy theory that Obama is a reptilian infiltrator. Maybe he is a cloned reptilian hybrid?

Search strings: "Turing-Lovecraft Theorem" and "proof of Turing-Lovecraft theorem" and others. Number of searches: Around 10.

According to Charlie Stross's "The Atrocity Archives," (discussed in this blog entry) Alan Turing did not commit suicide but rather was killed by the British government because he discovered a very dangerous theorem. This theorem that disproves the Church-Turing thesis and if thought about the wrong way could summon Lovecraftian horrors. This theorem's exact statement and proof are not included in the book. The book is fiction. As in, not real. As in, no such theorem exists. Just as H.P. Lovecraft wrote fiction. As in, not real. As in, his monstrosities came from his imagination, not from horrific realms beyond the understanding of mortals.

Or maybe that's just what they want you to think. Don't you find it interesting that not only has Barack Obama never denied being a reptilian clone but he's also never denied that the Turing-Lovecraft theorem is real?

Search string: "How do I make a Horcrux in real life." Number of searches: Too many to count. I get this search or some variation almost every single day. The total number of searches is easily in the hundreds.

This search and very similar search turn people to the blog entry arguing that it is acceptable under halacha (Orthodox Jewish law) to make a Horcrux if one had the ability to do so. Fortunately, horcruxes are not real. They are fictional. They are from the Harry Potter books. The Harry Potter books are fictional. Again, fictional means not real. You cannot split your soul into pieces using a magic wand. Sorry. But no.

Again. This may just be what Barack Obama wants you to think. That way, if the public does ever find out about his reptilian clone heritage and tries to kill him, they won't realize that he'll be unstoppable unless his Horcrux is destroyed. What would his Horcrux be? Remember, that if he is a reptilian clone, he obviously enjoys laughing at us while he and his reptilian compatriots slowly take over the world. So he might leave clues about his intentions. Isn't it a bit suspicious that Harry Potter, who turns out to be an accidental Horcrux for Voldemort, sees the shape of an acorn in his tea leaves in "Harry Potter and the Prisoner of Azkaban" and yet Obama was elected with the help of an organization called... ACORN?! All the pieces are coming together!

Sadly, there were two common hits from Google that don't seem to fit this. Indeed, they seem to be questions from people who don't have a problem distinguishing fact from fiction. I'll try to put together serious blog entries about those topics when I have time.

And apologies to Glenn Beck.

Tuesday, November 10, 2009

Malcolm Gladwell, Memes and Intellectual Honesty

I recently read Malcolm Gladwell’s “The Tipping Point.” As with Gladwell’s previous books, I found the book to be full of interesting tidbits, but lacking a coherent thesis. The book was sufficiently mediocre that I decided against reviewing the book. This is not a review. This is a discussion of Malcolm Gladwell’s lack of intellectual rigor.

Gladwell’s primary thesis is that ideas or behavior can, under the right circumstances, spread like an epidemic. What precisely this means is not clear. However, there is another, serious problem with this book, a glaring omission. People sometimes talk about something being “blindingly obvious.” In this case, the omission is so obvious that it is more akin to staring at the sun at high noon. This omission can be summarized by a single word that appears nowhere in the text: Meme.

In his 1976 book, “The Selfish Gene,” Richard Dawkins, coined the term “meme” to mean, in essence, a unit of cultural transmission that reproduces and is selected for, akin to how genes are selected for by natural selection. Since 1976, “meme” has been used more loosely to mean an idea or behavior that spreads in a viral fashion. The idea of a meme is similar, if not identical, to what Gladwell discusses. Despite that, the word “meme” never appears once in Gladwell’s book. Nor is Dawkins mentioned or referenced once in the text.

Gladwell’s book was first published in 2000, many years after Dawkins’s coinage. This is not a case of two minds independently and simultaneously arriving at the same idea, like Leibniz and Newton. This is one mind, presenting an idea and then another mind publishing a similar idea while failing to acknowledge prior work. By academic standards, this is completely unacceptable.

One might think that, possibly, Gladwell’s research was sloppy, and so he never learned about Dawkins’s term “meme.” However, I read Gladwell’s book on Kindle, which includes many updates since the original book was published. It is implausible that, in the last nine years, Gladwell has never heard the term “meme.” And yet, he felt no need to add even a footnote or a sentence about Dawkins’s important prior work.
Most damningly: Gladwell does mention the word “meme” in passing on his website in the FAQ about the book:
5. Are you talking about the idea of memes, that has become so popular in academic circles recently?
It's very similar. A meme is a idea that behaves like a virus--that moves through a population, taking hold in each person it infects. I must say, though, that I don't much like that term. The thing that bothers me about the discussion of memes is that no one ever tries to define exactly what they are, and what makes a meme so contagious. I mean, you can put a virus under a microscope and point to all the genes on its surface that are responsible for making it so dangerous. So what happens when you look at an infectious idea under a microscope? I have a chapter where I try to do that. I use the example of children's television shows like Sesame Street and the new Nickelodeon program called Blues Clues. Both those are examples of shows that started learning epidemics in preschoolers, that turned kids onto reading and "infected" them with literacy. We sometimes think of Sesame Street as purely the result of the creative genius of people like Jim Henson and Frank Oz. But the truth is that it is carefully and painstaking engineered, down to the smallest details.

So an idea “very similar” to his doesn’t deserve a single mention in the entire book. Furthermore, the claim that the idea of a meme isn’t well-defined is simply false, as one can see from actually reading “The Selfish Gene.” Gladwell’s claim that individual memes have not been examined in detail is also false. Indeed, there was a Journal of Memetics for almost a decade that examined memes in detail. And if one accepts Dawkins notion that religion is inherently memetic, then all of religious studies is essentially an examination of memes. Some modern scholars have specifically examined religion in that context. In fact, Gladwell himself examines the rise of Methodism in the United States as an example.

Gladwell comes across as a child trying to explain why his hand was in the cookie jar. He advances a series of unconvincing, somewhat contradictory explanations, hoping that we will ignore the larger problem. So far as I can tell from Google searching, this strategy has worked; people have noted that Gladwell is talking about memes but no one has called him out for his failure to acknowledge this prior work. This isn’t acceptable. Gladwell’s behavior is intellectually dishonest. His failure to credit Dawkins or others who have thought about these ideas before him does a disservice to those individuals and to honest intellectual discourse. I don’t think Gladwell’s behavior constitutes plagiarism, but it certainly would be punished if it occurred in an academic setting. Failure to cite prior work results in a paper being rejected from any legitimate journal. If a student hands in an assignment that fails to cite prior work, the student receives a bad grade, if not outright failure. Gladwell owes his readers and Richard Dawkins an apology for his failure to acknowledge that Gladwell’s idea recycles Dawkins’s earlier work.

Saturday, October 31, 2009

Dan DeLong, Animal Sexuality and High School English

Mike Dunford has drawn my attention to a situation in Piasa, Illinois where high school English teacher Dan DeLong assigned extra-credit reading about gay animals. The situation seems to be a bit complicated. The assigned article discusses how homosexuality is common in the animal kingdom. DeLong has been suspended from his job and faces a disciplinary hearing.

If a teacher simply assigned reading from areas of study outside the teacher's area that would be a legitimate concern. However, it appears that the assignment was to read the essay and examine the essay's structure and argumentative form. Therefore, this is simply a normal high school assignment. Teachers assign all sorts of different reading about different subjects. That's part of the normal school curriculum. One might be able to describe a problem if one had evidence that DeLong had assigned the essay to make some sort of political, moral, theological or other point. However, I haven't seen any claim that that was the case. The objection simply seems to be that some parents were uncomfortable with the reading.

Unless the school had some sort of blanket policy about the teaching sexual material requiring parental permission I don't see anything that DeLong has done. While news reports are still sparse with details it seems like this is more about fear of gays and homosexuality extended to such a point that even discussing gay sex in animals triggers a backlash. That's stupid. There's simply no other way to describe it. DeLong should be restored to his position as soon as possible.

Update: DeLong has been reinstated.

Thursday, October 29, 2009

Benford’s Law: Human Intuition, Randomness and Fraud

Suppose we look at some set of fairly natural data, say the populations of various countries. Let’s look at the leading digits of their populations. For example, the United States has a population slightly over 300 million. So, for the United States, the leading digit would be 3. What fraction of countries would you expect to have a leading digit of 1? Most people would guess 1/9th since there are 9 possibilities (1,2,3,4,5,6,7,8,9) and no obvious reasons to prefer any digit over any other digit. However, you’d be wrong:The actual percentage is slightly over 25%. In fact, it turns out that we should expect about 30% of countries to have a leading digit of 1. This strangely large number of 1s shows up frequently in natural data and is known as Benford’s law. It turns out that for many natural data sets we expect about 30% to have a leading digit of 1, about 18% to have a leading digit of 2 and so on. Benford’s Law is an important statistical rule that has a variety of generalizations and has practical applications (such as in detecting fraudulent or manipulated data).

Where is this strange pattern coming from? I didn’t have a good understanding of this until it was explained to me by Steven Miller. Consider a finite list of numbers that is reasonably well-behaved. For each member of our list, we can write it in scientific notation. So, for example. if we had 236 on our list, we would write it as 2.36 * 10^2. Note that the lead digit is then determined by what the lead digit is in the part of the scientific notation that isn’t the exponent. (This part is sometimes called the mantissa when one wants to be fancy) Now, for each number on our list, instead of looking at the number x, we can look at log10 x. What does this do to the scientific notation? Well, scientific notation then corresponds to the log of the mantissa + an integer that is the power of the exponent. So, for example, log10 (2.36 * 10^2)= .3729...+ 2.

Let’s examine the non-integer part of log10 x (call this f(x)) What distribution do we expect for f(x)?, It is a fixed value between 0 and 1 with no clear cut offs or biases in any direction so the most obvious thing to do is to make it a uniform distribution. That means that there’s about a 5% chance that f(x) falls below .05, about a 20% chance that f(x) falls below .2, about a 50% chance that f(x) falls below .5 and so on.

So what does this tell us about the leading digit? If the mantissa is below 2, then we have lead digit 1. If the mantissa is between 2 and 3, then we have lead digit 2 and so on. The mantissa is below 2 if f(x) is less than log10 2 = .301. Accordingly, we should expect that the mantissa is below 2 about 30.1% of the time. Thus, a number should have a lead digit of 1 about 30.1% of the time. Similar logic works for how frequently we should expect the lead digit to be 2, or 3 or so on.

What is wrong with the intuition that every lead digit is just as common? When we calculate probabilities, we are used to using the simplest probability distribution we can imagine, something like picking a positive integer from 1 to 10^n for some fixed n. We are used to this approach primarily because it is easy to calculate. Consequently, most probability problems in high school and college assume that we have such a uniform distribution since that assumption makes the math much easier. But actual distributions in real life don’t often look like this. For example, we might have a Bell curve or some other distribution. For almost any distribution that arises in nature, Benford’s law will apply due to the logic we used earlier.

So what does this do for us? Perhaps most importantly, we can use this insight to detect fraud. When humans try to make up data, it often fails to fit Benford’s law. In general, humans are bad at constructing data that passes any minimal test for randomness. Failure to obey a generalized version of Benford’s law was one of the major pieces of evidence for election fraud in the last Iranian election. The recent questions regarding whether Strategic Visions Polling was falsifying poll data arose when Nate Silver noticed that its results diverged substantially from Benford’s law.

For more information on Benford’s Law and related patterns in data, as well as more mathematical discussions of that data, see Terry Tao’s blog post from which I shamelessly stole the hard data about populations of nations.

Sunday, October 18, 2009

HPV, Cancer and Friends

Stephanie Zvan who blogs at Almost Diamonds has cervical cancer. A recent Pap smear came back with abnormal results and follow-up work found the cancer. Stephanie has talked about her experience and given follow-up information about her prognosis. Pieces responding to Stephanie's situation are at ERV and PalMD that are both worth reading.

I'm going to use this as an opportunity to go over some very basic issues:

First, for many cancers, getting them caught early matters. This is the case for many cancers. If you are a woman make sure you get regular Pap smears. If you are a man make sure to get regular prostate exams. No matter who you are, if you are older, make sure to get a regular colonoscopy. Etc. Etc. These exams save lives. When cancer is found doesn't just impact what treatment options there are but can be the difference between life and death.

Second, get the HPV vaccine. Unfortunately, this vaccine is not yet available for men but is available for women. HPV is one of the leading causes of cervical cancer. HPV is also associated with penile cancer in males. Once one has a given strain of HPV, getting a vaccine later isn't going to help matters. Moreover, HPV transmission is not prevented by condom use. HPV is a nasty little bugger and is much more easily transmitted than most STDs.

Stephanie grew up before there was a vaccine. However, in another respect, she is lucky. Stephanie and her husband don't intend to have children. However, for others they aren't so lucky. If you are a woman, get vaccinated. Men, when the vaccine is finally approved for males, please get vaccinated. Not only will this protect you, it will prevent you from getting a nasty disease which could then be given to those you love.

Stay informed. Stay protected. Stay safe.

Edit: Apparently the vaccine has now been approved for males. Good timing.

Thursday, October 15, 2009

Where Are All the Smart Apologists?

Recently I read The Screw Tape Letters, a series of letters written by C.S. Lewis. The letters purport to be from an old demon giving advice to a young demon. Lewis is a smart, funny and talented writer. What the demon Screwtape says is as revealing and clever as what he does not say. This is not the only great work of apologetics by Lewis. Why are there no great Christian apologists like Lewis today?

Lewis is far from perfect. Lewis gave birth to one of the most annoying apologetic arguments, the Trilemma. However, even there his intelligence and originality shine through. I have not seen any contemporary apologist produce any argument that isn’t a tired repackaging of pre-existing arguments.

Who are the major apologists today? There really aren’t any in the influential way that Lewis was. But if one had to identify those who today continue the tradition of Christian apologetics, one would probably list Ray Comfort, William Demsbki, and Alister McGrath.

Do any of these people measure up to C. S. Lewis? No. Consider these writers individually:
Does Ray Comfort stack up to Lewis? No way. Ray Comfort is an idiot and an ignoramus. He’s the man who most famously tried to claim that the modern shape of the banana was evidence for a divine creator. Yes, the banana, a fruit that has been heavily modified by extensive breeding by humans, a fruit whose wild form is a nasty hard thing full of seeds.

Does William Dembski stack up to Lewis? Wililam Dembski isn’t an idiot like Comfort. He has a real PhD in mathematics. But this also is a man who, after intelligent design failed in the courts, was reduced to teaching apologetics at a second rate seminary while giving course credit to students for trolling pro-evolution websites. I can’t see C.S. Lewis doing that. Moreover, Dembski’s writing ability resembles that of a 7th grader trying to sound like he’s really bright and well read. I should know. I used to write like Dembski when I was in 7th grade. Demsbki also seems to spend most of his time fighting with other Christians. (He really, really doesn’t like theistic evolution.)

Does Alister McGrath stack up to C. S. Lewis? Now we are getting closer. McGrath is a respected theologian who also has a degree in biophysics. He’s bright. He’s willing to accept both science and religion. He has on occasion made cogent arguments. But there are two problems: First, he’s a dreadfully boring writer. I have trouble staying awake when I read anything he writes. Someone needs to get Ben Stein to do a book on tape of one of McGrath’s books. It would be the ultimate sleep aid. Or maybe it would be a weapon of mass destruction as just playing it nearby would cause individuals within a hundred mile radius to fall into irreversible comas. This brings us to the other issue with McGrath: The subjects and titles of his books are equally dreadful. His two most well known books are "The Dawkins Delusion?" and "Dawkins' God." Ok, Alister. We get the point. You don’t like Richard Dawkins.

So why are there no great apologists for Christianity today? Here are four possible explanations:
First, perhaps great apologists are simply rare and C.S. Lewis is a great outlier. This isn’t a satisfactory explanation. I could compare the modern stock of apologists with G. K. Chesterton and they would still not match up.

A second argument is that Christianity is not the common belief among intellectuals that it was fifty or sixty years ago. Since a smaller fraction of intellectuals today are deeply Christian and since apologetics is valued less today as it has been in the past, intellectuals are much less likely to go into apologetics.

Third, the state of the evidence has changed over time to make belief in Christianity less probable. This argument is almost certainly wrong. The major modern controversies implicating Christianity and Judeo-Christian religions in general have existed for a very long time. The Documentary Hypothesis and similar theories about other Biblical texts have been around for more than a century. So has evolution. Thus, the need to address these issues (either by reconciling Christianity with them, or by refuting them) has existed for a long time.

Fourth, the modern focus of apologetics has been the watchmaker analogy and variations thereof. The watchmaker analogy is an argument for the existence of God based on an analogy to a watch found in the desert which one would immediately realize had a designer. It is no coincidence that the three apologists listed above, all have arguments that revolve around the watchmaker. Ray Comfort uses a particularly stupid form of the watchmaker argument. William Dembski uses a particularly obfuscated form of the watchmaker argument. And Alister McGrath doesn’t really use the argument itself but rather spends most of his time arguing that Richard Dawkins hasn’t sufficiently refuted the watchmaker argument and that if Dawkins is fallible God must then exist.

This focus is understandable: The watchmaker argument and other teleological arguments for the existence of God are some of the hardest to refute. However, the focus of all contemporary apologetics on a single argument has left the industry stagnant and uncreative. In such circumstances, it isn’t surprising that apologetics fails to attract many intellectuals. Moreover, the focus on the watchmaker argument has caused much of modern apologetics (and thus many of modern apologists) to go head to head with much well-established science. C. S. Lewis in contrast was open to the possibility that evolution was correct. If the entire apologetic system revolves around attacking basic science, one shouldn't be surprised that not many bright, educated people are willing to lead it.

I’m not completely satisfied with any of these explanations. However, the decline of contemporary Christian apologetics needs explanation.

New Largest Prime Number Found

The largest known prime has now been bumped up. The recently discovered prime is 2^43112609 − 1 which in base 10 has around twelve million digits. As with all the largest primes discovered for most of the last hundred years, the prime is a Mersenne prime, that is it is one less than a power of 2. As with the last few Mersenne primes discovered, this was discovered by the Great Internet Mersenne Prime Search which uses distributed computing to search for Mersenne primes. I've blogged before about Mersenne primes. For more details on why we care about Mersenne primes and their history dating back to the ancient Greeks see this post and this post.

There are reasons to suspect that if 2^n-1 is prime that n-1 should be likely to have a lot of small prime factors. In this case, 43112609 -1= 2^5 * 7 * 11 * 17497. So while some of the prime divisors look very small, it has at least one prime very large prime divisor and so doesn't seem to fit this pattern well. This is in contrast to the last discovered Mersenne prime 2^42643801 - 1. In that case one has 42643800 = 2^3 * 3^3 * 5^2 * 53 * 149. It remains to be seen if this new example is simply an outlier or if we need to reevaluate what we expect the exponents of Mersenne primes to look like.

Monday, October 5, 2009

Glenn Beck, International Law, and Hypocrisy

The last time I discussed Glenn Beck on this blog, we discussed Beck's general ignorance and contempt regarding basic legal history. Most astounding of Beck's claims was the claim that the notion that laws change over time was an idea that arose in the 1920s as a response to "Darwinian evolution."

After remarks like that one might think that Beck isn't a bad person but just incredibly ignorant. However, Ed Brayton has drawn attention to Beck's latest interaction with international law demonstrates his true hypocrisy.

Beck frequently use the "I'm just asking questions" gambit to insinuate hurtful and borderline libelous claims. In response, One enterprising individual started a satirical website "Did Glenn Beck Rape and Murder a Young Girl in 1990?"Frankly, this seems to me to be over the line of reasonable civil discourse (even if it is amusing).

How has Beck responded to this website? Beck has petitioned World Intellectual Property Organization to remove control of the domain from the satirist. Beck's argument is that the domain creates confusion with his brand name and therefore is a violation of international conventions on trademarks. That claim is so profoundly stupid that I'm not going to bother addressing it.

The lawyers for the website responded by explaining in detail what was wrong with Beck's claim and noting that his appeal to WIPO seemed to be an attempted run-around of the First Amendment of the Constitution of the United States. What makes Beck's behavior noteworthy is that, as one of the commentators to Brayton's blog observed, Beck has a history not just of complaining about transnationalism but of any intervention of international law into U.S. affairs. Indeed, Beck wrote in a March 30, 2009 column:

"Once we sign our rights over to international law, the Constitution is officially dead. When you say things like, 'We are not going to put the Constitution behind international law,' you say that in the international court, if you say that on the floor of the United Nations, you are a freak show."
In the same column Beck also wrote this gem:

"Let me tell you something. When you can't win with the people, you bump it up to the courts. When you can't win with the courts, you bump it up to the international level."
I'll let those comments speak for themselves.

Wednesday, September 30, 2009

Lockhart’s Lament: A Cogent Criticism of Math Education

I’ve received multiple requests to blog about Paul Lockhart’s “A Mathematician’s Lament.” Lockhart is a math teacher who is fed up with elementary and high school math education. I haven’t blogged about Lockhart's piece primarily because I agree with most of what he has to say and also a lot of people have already talked about it. (See for example, Scott Aaronson’s insightful commentary.)

Lockhart’s thesis is that much of mathematics education is simply wrong. According to Lockhart, the vast majority of our math education before college is rote learning that does not convey what mathematics is about. Lockhart argues that much of what children do in high school would be the equivalent of painting by numbers if we translated it into art. Mathematics is far more about exploration and understanding than it is about rote memorization. Lockhart argues that, by failing to let children understand and explore, we are not even teaching them mathematics. Lockhart further argues against rote math education based on practicality i.e. that these are techniques children will need when they are older.

Lockhart makes many good points and I recommend that people read his piece. As someone who has worked for many summers with the PROMYS program which uses a method similar to that outlined by Lockhart, I have much sympathy for his viewpoint. However, there are three problems with his thesis.

First, Lockhart overemphasizes the willingness of students to do exploratory mathematics. Exploration is intrinsically difficult. Moreover, it is difficult to get people to do math exploration if they don’t want to. If one tries to get youngsters to explore and they can’t do it effectively , the result is that parents will do the “exploration” for them. I’m sure there are readers of this blog who remember their parents “helping” with art projects back in elementary school.
Second, Lockhart underemphasizes the actual importance of rote learning and drills in picking up basic mathematics. Students need to be able to add, subtract, divide and multiply. They need to be able to do these things quickly in real life. Moreover, they need to do them enough times that they develop an intuition for orders of magnitude and when answers look right or wrong. That requires drilling in arithmetic from a young age. Lockhart addresses this issue briefly, but his response is unsatisfactory.

Third, Lockhart’s choice of focus on specific aspects of the high school and elementary curriculum is poor. He is correct in his criticism of the large amounts of trig memorization that occur. But he is incorrect in his example of the quadratic formula. In order to have an intuitive understanding of parabolas and other curves of degree 2, you need to know the quadratic formula. Moreover, the formula comes up frequently enough in later math classes that not knowing it would be a serious barrier. Finally, there are some items that educated people just need to know. Understanding the quadratic formula is one of those things that educated people just need to know in the same way that you can’t be an educated citizen of the United States and not know who Abraham Lincoln was.

Despite these criticisms, Lockhart is essentially correct. There are many serious problems with how we teach math and Lockhart correctly identifies many of them. While the massive overhaul that he outlines may not be necessary, it would substantially help matters if children were exposed at a much earlier age to what mathematics actually is, a subtle and beautiful art.

Thursday, September 10, 2009

Alan Turing, Apologies, and Cthulhu

The British Government has finally apologized for its treatment of Alan Turing. Turing was one of the greatest mathematicians of the twentieth century. He was responsible for founding computer science and he lead the effort to crack the Enigma encryption used by the Germans during World War II. This work saved many Allied lives and according to some historians proved crucial to the victory over the Axis forces. Without Turing's work, our world would look very different. However, Turing was gay. In 1952, Turing was convicted for engaging in homosexual acts. He was forced to undergo hormone therapy which lead to weight gain and other problems. Turing's security clearance was revoked. At the time, homosexuals were considered a security risk because of the potential of blackmail. The fact that the entire risk of blackmail was because they were considered a security risk apparently did not matter. Nor did it matter that since Turing was publicly gay, there was no possible risk of blackmail. Turing's ongoing consulting work with the government was terminated. Turing's life took a steady downhill side. In 1954, he committed suicide.

I am ambivalent about this apology. On the one hand, it is good to acknowledge how horribly Britain treated one of the saviors of civilization. On the other hand, apologies to the long dead always strike me as hollow. The living always face more than enough issues that are of far more practical importance than assuaging the feelings of the long-deceased.

Rather than discuss the pros and cons of such apologies, I am instead going to suggest three pieces of further reading.

First, Wikipedia has an excellent biography of Turing which explains his accomplishments and his mistreatment in far more detail than one can easily do in a short blog entry.

Second, Greg Egan, an excellent science fiction writer, has written a short story imagining a world in which Turing's life went slightly differently. In this case, "slightly differently" means had the assistance of a time-traveling robot. The story is more serious than one might think from that summary. The story looks at Turing's interactions with C.S. Lewis. I'm not sure the story is completely fair to Lewis overall, but it is very well-written and is an amusing what-if. Like most of Egan's writing, there's just enough plausibly correct mathematics to make it interesting.

Third, Charles Stross has written an amusing novel The Atrocity Archives in which Turing figures in the background. The essential premise is that Turing did not commit suicide but was assassinated by the British government to cover up far scarier discoveries he made (so presumably the Brits still owe Turing an apology in that universe). In that novel, mathematics is deeply connected to magic and thinking about certain theorems can accidentally lead to summonings of Cthulhu and other eldritch horrors. Turing was killed for discovering a series of powerful theorems including a proof that P=NP which if invoked improperly could destroy our universe. Unlike the Egan story, this is not a story I can claim has much in the way of serious merit. But it is very fun. By most accounts, Turing was a man with a sense of humor about things. I'd like to think that he'd smile to know that fifty years after he was dead, Great Britain would be apologizing to him at the same time that people were reading novels which linked him to Lovecraftian horrors.

Tuesday, September 8, 2009

Health Care, Death Panels and GPFM

Recently, Sarah Palin and other prominent Republicans have repeated claims that the proposed health care legislation would result in the creation of so called "death panels."This claim was quickly determined to be an egregious falsehood. However, the damage was already done and all elements of the proposed legislation that dealt with end-of-life issues were removed. My father has a piece up at the Oxford University Press Blog arguing that in fact death panels aren't such a bad idea. Essentially, the point is that one of the major reasons our health care costs are so high in the United States is because we go through tremendous effort to extend life during the last few months of life for many elderly. It is an issue very much worth discussing.

However, there are two related matters I'd like to mention. First, we should not lose track of how incredibly mendacious Palin and her compatriots have been about this matter. The fact that Bush's earlier proposed health care legislation had nearly identical end-of-life provisions is just one of the many ways in which Palin has simply become divorced from anything resembling reality. At this point, one must wonder if reality has taken out a restraining order on Palin.

Second, as someone who grew up with a very Jewish ethic about end-of-life issues it is emotionally very hard to agree to actively deciding that we will no longer provide health care to certain people. In classical Jewish thought, every effort must be made to continue to sustain life, regardless of cost. However, this view was constructed when medical technology was very different. Two-thousand years ago we didn't have the option to add a few painful months of life to someone at the cost of millions of dollars. We do now. This different situation may require serious reexamining of this sort of belief. This probably isn't an issue for most people in the US or at least the form it will take will be very different. Catholicism for example has for a long time tried to make a distinction between ordinary and extraordinary standards of medical care. In event, as medical technology becomes even more advanced, more and more value systems are going to need to serious think about these issues. Go read his piece here.

Thursday, September 3, 2009

Jack Chick’s New Tract

Jack Chick has a new tract, “Some Like it Hot.” I’ve discussed Chick’s tracts before. This one, however, takes us to a new perspective: The entire theological system as explained by the Devil. Like all good Jack Chick tracts, even the title of the tract provides amusement. The title is an apparent attempt at something resembling wordplay referring to the 1959 movie of that title starring Marylin Monroe and Jack Lemmon. Apparently, Chick’s idea of a clever pop cultural reference is to a film that came out fifty years ago. It seems that Chick’s understanding of pop culture stopped sometime in the 1960s. This is consistent with other data, such as his continuing bashing of rock music as satanic while not addressing more modern forms of devil music such as rap.

The premise of this new tract is that a grandfather and his grandson find themselves both in Hell. While wandering around the landscape, they wonder why Henry, the intervening parent, is not there and then realize that he accepted Jesus as his personal lord and savior. The Devil then decides that he wants to talk to the two of them. The Devil gives a long rambling speech, outlining the basic theology. Jack Chick’s Devil seems to be not only incompetent, but doesn’t understand basic time management. As with any Jack Chick tract, we need some bashing of other religions. Thus, the Devil declares:

I alone control every major religion in the world! Isn’t that a pleasant surprise. We’re overrun with religious leaders down here… and all their followers! We’re “blessed” with popes galore and “holy men” like Buddha and Muhammad… all of them got here by trusting their “good works.”

This paragraph is vintage Chick. We’ve got the required Catholic bashing and we’ve got the deep misunderstanding of other religions. Memo to Jack: There’s nothing in Buddhism remotely resembling a notion of “good works.” We understand that you can only think in terms of your religion and its own theological disputes, but that doesn’t mean that everyone else thinks in the same way as you.

The Devil’s rant is accompanied by pictures of various people burning in Hell. One of them appears to be dressed as a Pope, another has peyos and a black hat, while another is wearing a turban. Apparently good Christians never wear turbans.

The Devil continues his rant, explaining that he really hates Henry because Henry saved so many souls. Since the Devil cannot take out his wrath on Henry who is in Heaven, he decides to take it out on these two and thus throws them into a bottomless pit.

This raises a number of issues. First, even Jack Chick seems to think that there’s some element of justice regarding people who go to Hell. So what theological justification allows the Devil to torture people who happen to be related to people he doesn’t like? Second, is this tract really a great argument for accepting Jesus if the most likely result is that one’s family is stuck in a burning hot wasteland where one can actually find and talk to people you know? It may be that Chick is just getting soft in his old age and so we don’t have the default setting of eternal all consuming flame while surrounded in darkness. However, if I had accepted Jesus as my personal lord and savior and had relatives who had not, I’d be tempted not to witness to people. Maybe, I’m just a horrible human being, but I’d have a lot of trouble trying to save strangers if it meant my friends and loved ones would suffer more in Hell. I don’t think that’s what Jack Chick is trying to accomplish.

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.

Monday, July 13, 2009

Are We in the Middle of a Technological Revolution?

It is a common meme that we are in the midst of a technological revolution. When I made an off-hand remark in an earlier post that I did not think we were in such a state, Etienne disagreed. This post explains why it is inaccurate to state that we are in the middle of a technological revolution.

What do we mean by a technological revolution? The term is difficult to define. Here are four possible definitions: First, a technological revolution can be defined as a period during which new technologies result in rapid alterations to societal and cultural behavior. This is a vague concept but it has more precision than that of a generic technological revolution. We shall call this the social definition of a technological revolution. This definition has the advantage that it places little emphasis on when a given technology was actually invented and instead emphasizes the impact of technology.

Second, a technological revolution can be defined as the development and improvement of technologies at a pace more rapid than has been the historic norm. We shall call this definition the historical definition of a technological revolution.

Third, a technological revolution can be defined as the rapid development of new technologies faster than has been the historic norm. We shall call this the inventor's definition of a technological revolution. This inventor’s definition differs from the historical definition in that it focuses only on technologies which are new, not improvements upon existing technologies. Unfortunately, the distinction between significant improvement of an existing technology and fundamentally new technology is not that clear.

Fourth, a technological revolution can be defined as the accelerated construction of new technologies such that much of the population are unable to cope with the rapid changes in daily life. We shall call this the old-geezer definition of a technological revolution. Given that, whenever new technologies are developed older individuals have trouble adapting to them, the fourth definition seems to be less useful than the other definitions. We shall therefore focus on the first three definitions.

We are arguably in the middle of a technological revolution by the social definition, but the argument is not as strong as one might think at first glance. When most people point to a technology today which has altered social behavior, they point to the internet. The internet has allowed people to communicate conveniently when they are continents apart. It has also produced a series of media where the barrier to entry for the common citizen is almost non-existent. This has led to the free flow of all sorts of content, whether blogs, videos or collaborative projects such as Wikipedia.

TIME Magazine went so far as to declare the Time Man of The Year last year to be "You." However, this should raise alarm bells. Any bandwagon on which TIME has jumped is almost certainly overhyped. This is no exception.

The actual social impact of the internet has been small. Even the major social-networking sites such as Myspace and Facebook have relatively few people on them. Myspace has about 100 million registered accounts, but many are simple spam or are bands and small corporations using Myspace as webhosts. This is not fundamentally different from what people were doing fifteen or twenty years ago with personal webpages on sites like geocities. The total number is simply larger.

The barrier to entry to the internet may be small, but the barrier to being substantially noticed on the internet is higher, arguable as high as it was in the age of conventional publication. Moreover, much of the content (such as silly videos on Youtube) is content that would have likely been created without the internet and the only change is that the content is publically available.

Thus, the argument that we are in the midst of a technological revolution is weak under the social definition of such a revolution.

What about by the historical definition? Here again the argument for a contemporary technological revolution is weak. Moore's Law (that computer power doubles approximately every 2 years ) is valid , but it has been valid for almost sixty years. Similar observations apply to other technologies.

Most of the time when people speak of a current technological revolution they speak of technologies related to computers. However, for this particular definition, others might point to advances in the biological sciences. However, here also they would be wrong to assert that something unique is occurring now. We have in the last few years made major strides in our understanding of genetics and related areas of biology. PCR in particular has opened new frontiers. Yet even here, there are clear limits to what has been accomplished. Our improved understanding of genetics has produced not much in the way of direct medical breakthroughs but, rather, has increased our understanding of how very complicated and non-Mendelian many diseases are (see for example this discussion by Carl Zimmer). Meanwhile, genetic tests for some simple Mendelian diseases such as Tay-Sachs have existed for about forty years. Thus, the pace we are advancing in biology is not so rapid as to justify the label “revolutionary”.

What about the inventor's definition? This is by far the one with the clearest answer. And the answer is “no”. All of the major modern technologies are old. The electronic computer itself is almost 70 years old. The internet (even if one doesn't count ARPAnet but rather the internet as a large scale network) is 30 years old. The World Wide Web is 20 years old. The cellphone is about 30 years old. The primary improvements in the contemporary world have made existing technologies faster, smaller and cheaper. But little has changed with the underlying technologies.

By the most reasonable definitions of a technological revolution, we are not living in one today. Technologies continue to improve at a rapid, but not revolutionary, pace.

Monday, July 6, 2009

Sonia Sotomayor and More Gratuitous Promotion of Family Members

My father has a piece up at the Oxford University Press blog arguing for the confirmation of Sonia Sotomayor. He was earlier in favor of the confirmation of Samuel Alito. He argues that the most important condition is once again satisfied: Sotomayor is a qualified legal professional.

I'm not sure I agree with his argument. Frankly. there seem to be many qualified candidates who have a great deal of experience. I personally am also deeply concerned about Sotomayor's attitude towards both free speech and civil liberties issues ( Doninger v. Niehoff being the most serious example). I worry that with her on the Court many close decisions that would otherwise support civil liberties will otherwise go in the other direction.

In any event, the piece is worth reading and raises a number of interesting points. Go read it.

Sunday, June 21, 2009

Perez Hilton is a Sexist Bigot: A Rant on Misogyny and Math

While researching for my last post on Fox News and informed blog readers, I read too much Perez Hilton. Perez Hilton traffics in negative stereotypes damaging to young girls. He thinks stereotyping himself gives him license to stereotype others.

In a recent post on Jessica Alba playing a mathematician in an upcoming movie , Hilton wrote:

They want us to buy into her as a math genius?

Ha ha ha ha!
To help things along, we bet she's going to do the 'ugly' thing, a la Charlize Theron in Monster, right?

What a joke.
The formatting is the original with internal links suppressed. Perez Hilton thinks that hot chicks can't do math. I have taught math to children of a variety of ages. It is difficult to convince young women in our society that they can do math. Young women often lack self-confidence about mathematics. Many of them think that "only the ugly, unpopular girls" do math. One high school student once told me that she didn't want to go to a summer math program because if she did "everyone will think I'm uncool."

Perez Hilton reinforces this perception. Hilton is gay, an oppressed, frequently stereotyped minority- and he fits the stereotype. He does after all work as a "reporter" on celebrity gossip. He thinks stereotyping himself allows him to do the same to others.

One might say: "well, maybe Perez is correct." He's not. Danica McKellar, who played Winnie on The Wonder Years, is an accomplished actress, a published mathematician, and is smoking hot. There are many examples of female accomplished scientists and engineers who are very good looking. Julie Payette is Canada's most accomplished astronaut and was described by one teenage acquaintance as an "AILF," an acronym I will not spell out so I can keep this blog's PG-13 rating.

Perez, sir, you are an ass. You are actively harming young children's education. You are perpetuating damaging prejudices. Think before you speak. Or even better, just do the mathematicians and math teachers of the world a favor and shut up.

Saturday, June 20, 2009

Bloggers, Fox News, and an Informed Audience

In April of 2007, the Pew Research Center released a study about the knowledge levels of individuals in the United States. Many self-identifying liberals and liberal bloggers have pointed out that the study showed that regular viewers of Fox News have one of the lowest levels of political knowledge compared to many other groups. Only 35% of Fox News viewers were classified as being in the high knowledge group. This was low compared to many other samples. For example, 41% of CNN viewers, 54% of Daily Show watchers, and 54% of users of major newspaper websites were classified as being in the high knowledge group.

However, these bloggers have ignored other data in the study. In particular, the study found that people who use blogs as a news source are almost as ignorant as Fox News viewers. Only 37% of people who use blogs as a news source are in the high knowledge group. The difference between the Fox News viewers and the blog readers is statistically indistinguishable. Moreover, although Fox News viewers performed poorly, by some metrics conservatives performed better than liberals.

How were people determined to be in the high knowledge group? A series of 26 questions was asked of which 23 (decided in advance) were used to judge the examinees' knowledge level. An example of the questions asked is what party is currently in control of the House. People who correctly answered 15 questions or more were placed in the high knowledge group; people who correctly answered 10 to 14 questions were classified in the middle knowledge group; people who answered fewer than 10 were classified in the low knowledge group. The study had a total sample size of 1,502 and a percent error of 3.5%.

One interesting question to ask is if a breakdown of the mid -and low level knowledge groups allows us to distinguish the knowledge levels of the blog readers and the Fox News viewers. Here again, Fox News viewers perform poorly. 35% of Fox News viewers fall into the low knowledge group. Fox News has the second highest percentage of people in the low knowledge group (excepting people who do not get news regularly). However, the real poor performers are the blog readers of whom 37% fall into the low knowledge category. What is happening with these blog readers? I propose five possible explanations.

First, it is likely that part of this result stems from the aggregation of all blogs compared to just a single news network. Thus, the group of blog readers includes those who are reading Perez Hilton or TMZ and not much else. Thus, they only know about what celebrity is cheating on whom or what celebrity threw a tantrum. These celebrity gossip mongers could be reducing the apparent knowledge base of blog readers. One way to test this hypothesis is to examine the breakdown of what blogs people are reading. However, the study does not do so. It is likely that the inclusion of Perez Hilton readers and the like is bringing down the knowledge numbers for the blog readers. However, I doubt that this explains entirely the extremely poor performance of blog readers.

Second, a liberal can point to the low knowledge level of Fox News viewers and suggest that conservatives as a whole have poorer knowledge levels. From this premise, the argument would continue, what is bringing down the knowledge level of the blog readers is the presence of conservative blog readers who have abysmal levels of knowledge. However, this argument doesn’t fit with the other data from the Pew survey: viewers of the O’Reilly Show and listeners to Rush Limbaugh both score relatively high for news awareness (51% and 50% in the high knowledge group respectively). Furthermore, Democrats were substantially more likely to fall into the low knowledge group than Republicans, with 26% of Republicans in the low knowledge group and 31% of Democrats. This may be due in part to the slightly lower average income of Democrats .The study confirmed a strong correlation between income level and knowledge level, but did not investigate whether Democrats and Republicans have closer knowledge levels when income is a fixed variable.

In short, the Pew data refute the standard liberal perception that viewers and listeners of conservative shows are uninformed. Presumably, O’Reilly and Limbaugh’s audiences are at least as conservative as the generic Fox News viewer. Consequently, there is no reason to think (given this data) that readers of conservative blogs are less informed than readers of moderate or liberal blogs or that conservatives are in general less informed than liberals.

Third, the apparently low knowledge numbers for blog readers could be due to the highly specialized nature of many blogs. If a person only cares about a small number of issues, he may only read blogs focusing on those issues. If those issues have little to do with general political concerns, he may not have reason to learn or recall data such as who controls Congress. This is again a hypothesis that could be tested by looking in more detail at what blogs people are reading.

Fourth, many of the most successful news blogs are hosted on major news websites. A variety of New York Times reporters have blogs at where they report on stories, add their own commentary or add follow up notes to earlier articles they have published. This leads to the question of what one means by blog. The Pew study made no attempt to provide a coherent definition. Thus, people who read blogs by reporters may be classifying those blogs as part of major newspaper websites which could be leading to an artificially uninformed collection of people who identify themselves as reading blogs.

Fifth, blog readers could be genuinely unknowledgeable. This is, to me, the most distasteful explanation. However, it is the simplest explanation for the data and does bear serious consideration.

Note that the percentage of Fox News viewers who are in the high knowledge group is not low compared to the overall percentage of people in the United States who are in that group. That number is also 35%, and is brought down primarily by the large number of viewers of local TV news and viewers of network morning TV shows. So under this metric, blog readers and Fox News viewers look very much the same, with political knowledge levels close to those of the general population.

One implication of the Pew data is clear: Liberal bloggers need to stop using this study to attack Fox News and must stop attacking the knowledge base of conservatives as a whole. More studies need to be done. If this data continues to hold under further scrutiny and when Hilton-type blogs are removed from the picture, then bloggers as a whole need to ask why their readership is so dismally ignorant and what bloggers can do to alleviate the situation.

Friday, June 19, 2009

Iran and Gratuitous Promotion of the Other Brother

Normally when I point to something a sibling has on the Huffington Post it is something by my twin. However, this time, there's a piece by my little brother. Nathaniel talks about the history of the V for Victory symbol and how it is now making its way into Middle-East politics. It is very worth reading. The history of the symbol and how it has evolved makes for fascinating reading. I think that he may be underestimating how diverse its symbolism is in the Middle-East and how fast the symbol's meaning is changing. Anyways, check it out.