The Economy, Prayer, and Idolatry

There has been a long-standing argument in halachah (Orthodox Jewish law) over whether Christianity is considered avodah zara (the technical term for idolatry in Jewish law). The problem of a human being declared to be divine is particularly troubling and Jews have been debating this issue for almost as long as Christianity has been around.

However, some Christians have recently been nice enough to make the Jews have an easier time figuring out how to treat Christianity. According to Wonkette, Christian groups have gathered at the statue of the bull at Wall Street to pray for a better economy. That's right: they are praying at a Golden Calf. Really. You can't make this sort of thing up. I have tried in the past to argue against claims that evangelical Christians are Biblically illiterate, and then they go and do something like this. Between this and Reverend Jeremiah Cumming's inability to quote basic scripture I should just give up.

Hat tip to Pharyngula for bringing this to our attention.

Censorship, guns, and Lone Star College Tomball

Lone Star College Tomball, a small college in Texas, is at the center of criticism by various right-wing groups for perceived censorship. The college's branch of Young Conservatives of Texas distributed a "satirical" "gun-safety flier" for which they are now in trouble. The branch may be put under probation at the school and the students may face disciplinary action. While this has received some attention in right-wing blogs, so far the go to source is from that bastion of fair and balanced news World Net Daily.

I am normally strongly against anything resembling censorship, but in this case the circumstances are not clear cut. The list of safety tips included "Always keep your gun pointed in a safe direction, such as at a Hippy or a Communist". Now, I'm not going to discuss the fact that this sounds like it was from 1965 other than to speculate that possibly Tomball is so isolated that the upheaval of the 60s is just starting to reach there. I'm not going to discuss that the Cold War is over. (And Emily, if you are reading this- as far as I'm concerned I still think that communism in most of its variations a threat to basic liberties and to humanity and all remarks about what to do when at war still apply. My views on this are absolutely unchanged. Don't think otherwise). I don't need to discuss the abject stupidity of the remark because it is far from what constitutes reasonable political discourse. You don't make jokes about how segments of society are ok to shoot. In this case, it amounts to a borderline threat. This line by itself is simply appalling.

Moreover, I'm disturbed that none of the right-wing groups commenting on this matter have had the good graces to say something like "well, we think this is within free-speech rights but we understand how you would be offended or threatened." These people are joking about how it is ok if students who they don't like get shot. If there is anything that pushes the bounds of acceptable speech, this is it. And I haven't even discussed how the list also included "Don't load your gun unless you are ready to shoot something or are just feeling generally angry." It doesn't take much effort to put these together.

Frankly, I hope that the college does not punish the Young Conservatives. The speech is arguably on the ok side of the border and we should be careful not to impinge on speech even when it is idiotic. But I also have a selfish reason for wanting the college to leave them alone. I really want to see what their next piece is that passes for "humor."

Don McLeroy Jenkins: Epic failure by the creationists. Again.

Some readers may be vaguely aware that Texas is going through their aperiodic burst of anti-evolution sentiments from their school board. Don McLeroy, the current Chairman of the Texas State Board of Education has managed to effectively lay the groundwork for any possible legal challenge by his opponents.

First a bit of background: The creationists used to try to teach creationism next to evolution. The federal courts said that was unconstitutional because of that whole First Amendment thing. It can be so pesky and inconvenient sometimes. Then suddenly a mutation showed up and they started to teach “creation science” which was completely different from “creationism.” The Supreme Court in Edwards v. Aguillard ruled that this was nothing more than thinly disguised creationism.[1] Then miraculously, a new mutation occurred. Now, there was “intelligent design” which had nothing to do with that unconstitutional “creation science” thing. Not all. (Actually it wasn't a single mutation but a series including a well-preserved transitional form). In Kitzmiller v. Dover, this new variation was ruled unconstitutional. The latest version is “teaching the controversy.”

However, McLeroy has gone and shot himself in the foot. In a recent editorial, he justified teaching the controversy by saying that, under the new proposed curriculum “claims about evolution … will be challenged by creationists.” Oops. He said the c-word. I can see his legal allies carefully planning a case in front of a federal judge to explain how what they want has nothing to do with creationism at all. And then he arrives, running into the courtroom screaming “McLerooooy Jennnnkins! Creationism!”

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[1] If one does have opportunity to read this decision, I strongly recommend reading Scalia’s dissent as well. It gives one real appreciation for Scalia’s intelligence and thoughtfulness and raises serious issues that are worth thinking about.

On the complete uselessness of the TSA

Jeffrey Goldberg has a piece in this month's The Atlantic on how easy it is to penetrate airport security in the United States. Goldberg relies primarily on the advice of security guru Bruce Schneier. To say that Goldberg demonstrates the uselessness of post-9/11 security procedures is an understatement. I strongly recommend that people read this article. The article does drive one point home which is worth repeating explicitly here: none of the security precautions will stop a smart terrorist. All they will do is possibly catch dumb terrorists.

Three youtube videos related to the Presidential campaign

Three people have endorsed candidates for President recently: Sarah Silverman, Jackie Mason and Marty Chalfie. The first two are well known comedians. The third is a Nobel Prize winning scientist.

Comedian Sarah Silverman recently posted a Youtube video calling for young Jews to persuade their retired relatives in Florida to vote for Obama. Jackie Mason (yes he is still alive) posted a rebuttal video. Frankly, they both seemed like close to a waste of time. Silverman's video wasn't that funny and Mason's video missed the entire point of Silverman's video. Mason seems to think that Silverman is using her video to persuade people to vote for Obama. That's clearly not what she intended. The video is to persaude people who are already voting for Obama to talk to their elderly relatives. Mason attacks Silverman's video for being contentless but his own is about as contentless. Also here's a minor hint for Mason if you are reading this: most old people in Florida probably aren't going to spend much time watching Youtube videos.

But enough about the comedians. The video out that should matter a lot more is that by Chalfie. Marty Chalfie, one of the three nobel prize winners in Chemistry for 2008, has endorsed Obama. That brings the total to 65 Nobel Prize winning scientists who have endorsed Obama. That total includes all the American winners of the 2008 prizes. When James Watson is endorsing a black man that says something. And Chalfie's video gives a detailed, contentful rationale for voting for Obama. An interesting note: As of writing this message Chalfie's video has been viewed 4,120 times. Silverman's video in contrast has been viewed 189,661 times and Mason's video 7,873. So even with the information contained in our little hint above, Mason's video has still been viewed over twice as many times as Chalfie's. Apparently people prefer listening to comedians over Nobel Prize winning scientists for advice.

One interesting argument I've heard in the context of the Nobel endorsements of Obama is that the scientists are just another special interest group. That argument is flawed. Most seriously, no endeavor is as influential to the long-term success of the United States or humanity as science. Scientific research impacts everyone from the discoveries that make many consumer goods possible to the drugs that stop cancer to the agricultural techniques that provide cheap food. Moreover, even aside from issues of funding (where one might be able to make something resembling a coherent argument that scientists function like a standard lobbying group) science provide guidance on what policies make sense and what do not. In that regard, science offers far more relevance than any other "interest" group.

(I'm having some trouble embedding videos for some reason for now here are three links to the videos: Silverman. Mason. Chalfie.)

Note: This post underwent substantial revisions after talking to a few people about it. Also, I acknowledge that the comparison between Chalfie and the comedians suffers from a variety of flaws. If I have time I'll write another post discussing this in more detail.

Bill Maher's Religulous

I saw Bill Maher's Religulous last night. I had mixed feelings about the movie. The movie was amusing but some of Maher's comments were cheap shots. At other points one felt that Maher was sloppy. Here are my thoughts in a not completely organized fashion:

The thesis through most of the movie was that religion is silly and that it would be harmless but for the fact that it also causes serious violence. Given Maher's stated views that isn't that surprising.

In an odd way, I could not help comparing the movie to Expelled(see my previous review of that movie here). And unfortunately, there was a lot to compare. As with Stein's Expelled, Maher frequently used clips intended for humorous effect rather than seriously considering the points or claims made by the people he was interviewing. Unlike with Stein these clips were actually funny and sometimes even moderately germane. However, over all, direct confrontation of the ideas and claims presented would have served better since most of the ideas so mocked could be easily dismantled. There was however, one example that did strike me as amusing and well-played. During an interview with Jeremiah Cummings (more on him later) Cummings talks about how he counseled a young man who wanted to kill himself over a lady that the young man should instead "have that sort of passion for God. Imagine what it would be like if people had that sort of passion for God?" Maher then switched immediately to a clip of an apparent suicide bombing. The connection is tasteless, amusing and certainly more directly relevant than any of Stein's clips.

Another similarity between the movies was that it was not clear how much material was being left on the cutting room floor. The interviews appeared to be heavily edited and it was difficult to tell whether comments were in sensible contexts. Unlike with Expelled this was a serious concern since the people being interviewed did frequently come across as idiots.

In a few cases there was no way to imagine a better context for the comments. For example, Cummings in his interview tries to defend his lavish lifestyle by attempting to assert that there is a Bible verse that supports being rich. Maher immediately makes clear that the verse that Cummings seems to be trying to stumble over is the statement found in the synoptic gospels that "it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God." How Cummings can manage to be taken at all seriously by his congregation given his ignorance of basic Christian scripture is beyond me.

Similarly, Senator Mark Pryor came across as about as much of an idiot as was expected by the previews. And yes, he really does try to defend his views at one point by saying that "You don't have to pass an IQ test to be in the Senate." The statement in fact comes in the middle of an extended clip with minimal editing anyways.

One interesting interview was with Aki Nawaz who is a radical Islamic rapper whose lyrics glorify suicide bombing and terrorism. Nawaz attempted to argue that he had a free speech right to his lyrics but that Salman Rushdie did not have a similar right. People like Nawaz make me want to believe in a vengeful God who will make sure that Nawaz burns for a very long time.

Maher focused almost exclusively on Christianity and Islam and never addressed the non-Abrahamic religions aside from a few minor cults. Judaism is addressed mainly as an opportunity to make self-deprecating jokes (I suspect that Maher is quick to discuss that his mother is Jewish is part to allow him to make such jokes).

Maher was in a variety of circumstances just sloppy. For example, he interviewed one of the members of Neturei Karta. While Maher noted that Neturei Karta was an extreme minority view in "Orthodox" Judaism, that's an understatement. Neturei Karta is an extreme minority view in Charedi Judaism.

Similarly, Maher at one point talks to Dean Hamer about both the "Gay Gene" and the "God Gene" and fails to note that there are serious reservations in the scientific community about much of Hamer's work.

Maher also excepted uncritically certain claims about parallelism the stories of Horus and Mithra to those of Jesus. Some of these claims are disputed and they weaken what would otherwise be a good case.

Maher also used tactics similar to those used in Expelled to get his interviews including lying to some interviewees about what the movie was going to be about. That is unfortunate and intellectually dishonest. Maher was however much more up front about this tactic than the makers of Expelled were and Maher seemed to make clear in the movie when people were unhappy with his interviews or worried about what they were focusing on.

Maher's concluding claims did not fit well with the rest of the movie and felt a bit tacked on. The end of the movie consisted of Maher engaging it what amounted to a call-to-arms to atheists and agnostics to speak up while at the same time calling for moderate religious individuals to stop giving aid and comfort to extremists. The last part seems particularly difficult; in general moderately religious individuals don't like extremists and rarely will the defend them or support them. The moderate religious individuals who do support such individuals will be likely extreme enough that they won't be listening to Maher anyways.

There has been some discussion on the blogosphere comparing this movie's box office ratings to those of Expelled. This film opened on about half as many screens and did about as well on its first weekend. Moreover, there was no similar campaign as there was in Expelled to pay churchs and schools to send students to the movie. Thus, it seems that by reasonable measures of success Religulous comes out ahead of Expelled.

I'm frankly a bit uncomfortable comparing this movie too closely to Expelled. Expelled claimed to be about science and was really about demagogery and deception. Religulous did not make any claim to be about science as a subject matter. Moreover, a focus on Expelled v. Religulous puts to much emphasis on a science v. religion view of things which is simplistic and unhelpful (if someone wants to talk about rationalism v. religion that would be a different situation).

While I disagree with a large part of Abbie's more positive assessment of the movie she is correct to point out that many of the responses by religious individuals to the movie have been kneejerk responses to criticism. Claiming that Maher somehow doesn't understand religion is obviously false. He may get details wrong but many of the interviewers and facts speak for themselves. A lot (and possibly the vast majority) of religions out there have some pretty silly ideas. And people kill over them.

Overall, this is not a movie brimming over with intellectualism and deep thought. Religulous does not substantially raise the level of dialogue. The movie is at times sloppy and inaccurate. However, this blog entry's focus onprimarily negative aspects should not be construed as a reason not to see the movie. Religulous was funny and brought up serious issues that society needs to discuss. I strongly recommend that readers go see it.

Mersenne Primes and Perfect Numbers

This is the second part of a series of three posts about why people care about large prime numbers. The first part can be found here. This post will focus on primes that are of the form 2ⁿ-1. Such primes are generally called Mersenne primes.

The historical reason for caring about Mersenne primes is that they are related to perfect numbers. A positive integer n is said to be perfect if n is the sum of all its proper divisors. For example, 6 is perfect because 1+2+3=6 but 8 is not perfect since 1+2+4=7 and 7 is not equal to 8. Perfect numbers were first studied by the ancient Greeks and have been connected to various numerological ideas. For example, St. Augustine noted that the world was created in 6 days and 6 is a perfect number.

The first few perfect numbers are 6, 28, 496 and 8128. These were the only perfect numbers known to the ancients with the next one 33550336 not discovered until much later. Now, how are these numbers related to Mersenne primes? Well, it turns out that every even perfect number is associated with a Mersenne prime. More particularly, an even number is prime if and only if it is of the form (2ⁿ-1)(2^n-1) where 2ⁿ-1 is a Mersenne prime. For example, 6= 2*3 = (2²-1)2¹ and 28=7*4 = (2³-1)2². That numbers of this form are perfect was proven by Euclid about 2300 years ago. About 2000 years later, Euler proved that these were the only even numbers that were perfect.[1] The main aim of this post will be to sketch out Euclid’s and Euler’s proofs in modern notation and then to discuss a few other properties of Mersenne primes.

First we need a small lemma:

Lemma 1: 1+2+4+8…+2ⁿ = 2ⁿ⁺¹-1.

Proof: Consider S=1+2+4 + 8…+ 2ⁿ. Then 2S=2+4+8…+2ⁿ⁺¹ = S - 1 + 2ⁿ⁺¹. So

2S=S - 1 + 2ⁿ⁺¹ and solving for S now gives us the desired result. Readers who recall their high school algebra will note that this is just the proof for the sum of a finite geometric series in the specific case with the first as 1 and with each succeeding term as twice the previous.

Proposition: A number of Euclid’s form is a perfect number. That is (2ⁿ-1)(2^n-1) with 2ⁿ-1 is a perfect number.

Proof: The proof isn’t anything that complicated in modern notation. Euclid had to do this using highly geometric notation but we can use modern notation to do this easily. All we are going to do is take the sum of the proper divisors of N=(2ⁿ-1)(2^n-1) and note that they are precisely N. Divisors of N take two forms, they either have a 2ⁿ-1 in them or they do not. The proper divisors that do not have such a factor are precisely the powers of 2. 1+2+4+8…+2^n-1 = 2ⁿ-1 by our previous lemma. Then there are the divisors that do contain a factor of 2ⁿ-1. Those divisors are all of the form (2ⁿ-1)2^k. For those divisors we get 1*(2ⁿ-1) +2(2ⁿ-1) 4*(2ⁿ-1) ... + 2^n-2 = (2ⁿ-1) (1+2+4+8…+2^n-2) =

(2ⁿ-1)(2^n-1-1) by Lemma 1. So the sum of all the proper divisors is (2ⁿ-1)(2^n-1-1) + (2ⁿ-1) = (2ⁿ-1)(2^n-1) = N. And so we are done.

The proof of Euler’s result (that every even perfect number is of Euclid’s form) is more involved than than Euler’s proof but is not difficult. First a small notational change: Let σ(n) be the sum of the positive divisors of n. For example, σ(4)=1+2+4=7. Note that unlike are previous sums this includes the number itself. Observe that N is perfect if and only if σ(N)=2N. It turns out that for most purposes σ(n) is nicer to work with than the sum of the proper divisors which is of course σ(n)-n.

Lemma 2: σ is multiplicative. That is, σ(ab)= σ(a)σ(b) whenever a and b are relatively prime.[2]

Proof: Observe that for any divisor d of ab we can write d=a₁b₁ where a₁ divides a and b₁ divides b. Moreover, since a and b are relatively prime this representation is unique. Thus, σ(ab)= (1+d₁ + d₂… + d_k) = (1+a₁+a₂..+a_m)(1+b₁+b₂…+b_n)= σ(a)σ(b) since every d_i can is the product of some a_j and b_h. Q.E.D.

We also need one other function. Define h(n)= σ(n)/n.

Lemma 3:

1) h(n) equal to the sum of the reciprocals of the positive divisors of n.

2) Furthermore, h(n) is multiplicactive.

3) If a divides b a is less than b then h(a) is less than h(b).

The proof of this lemma will be left as an exercise to the reader.

Also, it is helpful to note that N is perfect if and only if h(N)=2. It turns out that for constructing a general theory h(n) is the function in that some sense matters the most.

Now we are ready to prove Euler’s result:

Proposition: If N is an even perfect number then we may write N = (2ⁿ-1)(2^n-1) where 2ⁿ-1 is prime.

Proof: Assume N is an even perfect number. We may write N as m2^k where m is odd. By our earlier remarks this means σ(m2^k) = 2m2^k = m2^k+1. Now, since m is odd it is certainly relatively prime to 2^k. So σ(m2^k) = σ(m)σ (2^k)= σ(m)(2^k+1-1) (by Lemma 1). So σ(m)(2^k+1-1)= m2^k+1. Now, 2^k+1-1 divides m2^k+1. Since 2^k+1 is a power of 2, we must have 2^k+1-1 divides m. Now, if m is not prime we must have h(m) >= 1 + 1/(2^k+1-1) + 1/d >>= 1 + 1/(2^k+1-1) for some non-trivial divisor d (see part 1 of Lemma 3 above). But then an easy calculation establishes that h(m2^k) > 2m2^k which is a contradiction. So m must be prime and we are done.

Throughout this blog post I have referred to Mersenne primes as primes of the form 2ⁿ-1. In fact, in order for such an integer to be prime one also needs that n is prime. If n=ab for some a,b>1 then we can use the factorization trick x^m-y^m =(x-y)(x^m-1 +x^m-2y +x^m-3y²… xy^m-2+y^m-1) to get 2^ab-1 = 2^ab-1^a = (2^b-1)(2^b(a-1) … +1). Thus, mathematicians talk about Mersenne numbers being numbers of the form 2^p-1 prime. Not all such numbers are prime. For example, 2¹¹-1 = 23*89. But it turns out that there are very quick ways of testing whether such numbers are prime. Those methods are beyond the scope of this blog post which is already becoming quite lengthy. However, I will say that those fast methods have resulted in Mersenne primes being the largest known primes for most of the last 100 years.

There is a distributed computing project searching for more such primes. The project, called “GIMPS” for Great Internet Mersenne Prime Search, has found most of the Mersenne primes found recently. That project allows you to download software which runs in the background and uses your computer’s spare processing power to search for Mersenne primes. Recently they discovered the 45^th and 46^th known Mersenne primes. These two numbers 2^43,112,609-1 and 2^37,156,667-1 are each in the range of 10 million decimal digits long. These are big numbers. I strongly urge readers to go download the software for GIMPS. It doesn’t cost anything, doesn’t take up your time and you might even find something interesting.

In this post I’ve laid out the historical basis and some of the intellectual basis for caring about large primes. In my next post in this series I will discuss practical reasons why people care about large primes. I’ll discuss how you can share a secret with someone without having to worry about eavesdroppers and how that is relies on prime numbers.

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[1] The astute reader will note that we have switched from talking about perfect numbers in general to only talking about even perfect numbers. What about perfect numbers that are odd? The answer is that no one knows. There are many restrictions on what an odd perfect number must look like. Any odd perfect number must be very large (greater than 10³⁰⁰) and have at least 9 distinct prime factors. There are good reasons (especially those presented by Carl Pomerance) to believe that no such numbers exist. However, a proof is not in sight.

[2] If anyone suspects that I’m writing posts on this subject so I can talk about multiplicative functions which are some of my favorite functions, they may be right.

Large Primes and Mersenne Primes

In the last two months, two new Mersenne Primes have been discovered. Mersenne Primes are prime numbers that are 1 less than a power of 2. So for example, 7 is a Mersenne prime. These two new primes 2^43,112,609-1 and 2^37,156,667-1 are the 45^th and 46^th Mersenne primes known and are the two largest primes known. Their discovery has made it into the popular press and has prompted multiple people to ask me what the big deal is and why mathematicians care about big primes anyways. This blog entry will be the first of three entries in which I will attempt to discuss these issues.

First, note that there are infinitely many prime numbers. This has been known since the ancient Greeks. An elegant proof is given by Euclid which I sketch out here: Assume I have a finite list of primes, p₁, p₂, p₃….p_n .If I take their product and add 1 I get p₁p₂p₃….p_n +1 and this number is clearly not divisible by any of the primes on my list. If this number is prime then I have a new prime that was not on my list. If this number is not prime then its smallest divisor that is not 1 must be a prime not on this list. Thus, for any finite list of primes I can always construct a new prime not on the list. There is a common misconception that if one takes , p₁, p₂, p₃….p_n to be the first n primes then it p₁p₂p₃….p_n +1 must itself be a prime. The smallest one where it does not lead to a prime is 2*3*5*7*11*13+1=59*509.

Unlike with generic primes, no one knows if there are infinitely many Mersenne primes. It is strongly believed that there are infinitely many but there is no proof as of yet. Mersenne primes are independently interesting for a variety of reasons. First, there is the issue of whether or not there are infinitely many. Second, Mersenne primes correspond to even perfect numbers. A number is said to be perfect if you add up all the positive divisors less than the number and end up with the number. So for example 6=1+2+3 so 6 is perfect. Euclid showed that for every Mersenne prime you could construct an even perfect number from it. Euler showed that the only even perfect numbers are those from Euclid's construction. Thus, information about Mersenne primes is information about perfect numbers, and discovering a new Mersenne prime leads to a new perfect number. I will discuss this in more detail in my next blog entry on this subject.

Third, Mersenne primes are interesting because there are tricks to test whether a number of the form 2^n-1 is prime that make it much easier to check whether such numbers are prime than a generic number. Thus, for most of the last 100 years the largest primes known have been Mersenne primes.

Now, why do we care about finding large primes? One simple reason is for the same reason that scientists might try to construct larger and larger atoms or might look for new species; because they are there, learning is fun, and you don't know what you might find.

Also, in a more general setting primes (although generally primes much smaller than Mersenne primes) are useful for cryptography which are used in many modern software applications. Many readers have likely used those cryptographic protocols without even realizing it when you've logged onto to secure websites and such.

Modern browsers handle the procedures for you automatically. So the ability to find large primes and related procedures (such as theability to factor large numbers quickly) are very relevant to the functioning of the modern economy as well as to various national security issues. I will discuss this cryptographic use of large primes in the third blog entry in this series.