A Quick Note on a Silly, Pseudo-recursive Class of Functions

Let f(n) be a function from the natural numbers from the natural numbers. Suppose further that f(1)=1, and assume that f(n) is equal to the number of positive integers k that are less than equal to n and satisfy f(k) | k. Then it is not hard to show that in fact f(n)=n. Now suppose we instead look at the exact same thing, but insist that f(n) count the number of k that are at most n and satisfy f(k)|P(k) where P is some fixed polynomial with integer coefficients.

For some choices of P, we do not have any function f which would satisfy the desired recursion. For example, if P(x)=x+1 we don't have any function f satisfying the recursion. To see this, note that we have either f(2)=1 or f(2)=2. If f(2)=1, then f(1)|1, and f(2)|2, so in fact f(2)=2. Consider the other case where f(2)=2. If so, since 2 does not divide 3, we must have f(2)=1. Contradiction and nd of p.

Here's my question which I have not been able to make substantial progress on: Are there any valid choices of P and f that satisfy the recursion and don't have f(n)=n for all n?

Thoughts and links after the recent Massachusetts election

First, I have to say that a lot of liberals seem upset over Coakley's loss to Brown. Frankly, given Coakley's poor record on civil liberties (including her despicable behavior handling the Amirault situation) I can't find myself getting worked out over the matter. Two pieces I recommend on the matter: My twin has a piece at the Huffington Post talking about lessons the Democrats should take away from this election. Ed Brayton has a piece up talking about what this says about Coakley and the Democrats chances for their legislative agenda. Brayton makes the point that the Democrats still have strong majorities in both the House and Senate. So if they can't push through their agenda then it is hard to explain it as anything other than general incompetence.

A Different Take on Ireland's New Blasphemy Law

Readers are likely aware that Ireland has a new anti-blasphemy law. The response to the law has been understandably negative. It is hard to reconcile outlawing of blasphemy and modern notions of free speech. The law has been met with mockery and derision. See for example here and here. However, Whiskey Fire has an interesting piece up arguing that the law makes much more sense in context. In particular, an anti-blasphemy law is required by the Irish Constitution. The new blasphemy law helps actually minimize the chance that blasphemy prosecutions will occur since the large fines mandated by the law force any blasphemy prosecutions have to occur under the Irish High Court rather than the circuit courts. This effectively prevents local yahoos from filing blasphemy charges. Whiskey Fire's entire piece is very worth reading and is a good example of how political situations can often be more complicated than they first appear.

Hat tip to Almost Diamonds.

Jack Chick, Native Americans and Henotheism

Everyone's favorite Christian fundamentalist tract writer, Jack Chick, has a new tract, Crazy Wolf. This tract is apparently aimed at Native Americans and attempts to show how their traditional religious beliefs are really demon worship.

The tract starts with two stereotypical Native Americans talking about how one of their own, Mary, has accepted the "White God" and how they are unhappy with her. One of them, Margaret, is particularly unhappy because Mary tried to evangelize to Margaret's young daughter Sarah. I guess Jack Chick can't quite understand why someone might be justifiably upset if someone tried to interfere with one's kids’ religious upbringings.

Margaret then discusses how they tried to get a medicine man to put a curse on Mary, but "some strange power" prevented the medicine man's curse from working. So, they decided to ask the assistance of a powerful witch named Crazy Wolf.

Notice that every individual so far in this tract has an English name except for the old, evil witch. I guess it's just a sign of how baddass he is that he as stereotypic name, or something like that. At least his name isn't "Injun Joe."

Of course, Crazy Wolf tries to use his Devil-granted powers to shapechange into a massive wolf to eat Mary. He fails because of Mary and her pastor's prayers. An angel materializes which beats up Crazy Wolf. Mary then further prays that Crazy Wolf will accept Jesus as his personal lord and savior.

We all know this part of the routine: Injun Joe, sorry, Crazy Wolf, talks to Mary and accepts Jesus as his personal lord and savior. Crazy Wolf declares that "my real name is Billie Wolf." Apparently, he has a good name, but it only gets used once he's saved. Then, as happens in so many Jack Chick tracts, he dies a violent death, as Margaret shoots him with a shotgun in revenge for failing to kill Mary.

Crazy Wolf goes to heaven and is told that "You just made it by the skin of your teeth! You believed on(sic) Me and that saved you. Billie Wolf, enter thou into the joy of thy Lord." Margaret of course goes to hell to burn for eternity.

This tract raises some interesting issues about Chick's theology. For example, as with some prior tracts, demonic forces are not only real, but very powerful. There's an almost henotheistic aspect to the story. Henotheism is the belief that many deities exist while only worshipping one. Chick’s Jesus becomes relevant primarily after death or during the apocalypse. In Chick’s pantheon, there are many deities but Jesus, the deity of death and destruction, reigns supreme.

Implied in this narrative is the teaching that prayer for a soul can actively lead to salvation. This is confusing. The entire point of Chick’s theology is that all that matters is whether an individual has accepted Jesus or not. If God and prayer can alter that decision, then the even minimal theological explanation of why everyone is not saved breaks down. It becomes within God’s power to alter whether or not individuals are saved. This renders the primary evangelical apologetic of such a deity non-feasible. In particular, damnation is usually defended by arguing that God cannot force people to accept Jesus as their savior. Yet here we see God apparently doing exactly that.

The importance of names is also worth noting. Aside from the not so subtle racism associated with Chick’s name choices, this is part of a general pattern in Chick's theology. What one calls something matters. Thus, for example, in previous tracts aimed at Islam, Chick argues that Allah is not just another word for God. This brings up an issue: Consider the following hypothetical: Someone is explained the entire evangelical belief system but with the words "Satan" and "Jesus Christ" swapped throughout. Then that person accepts Satan as personal lord and savior, does Chick think that that person is saved or not? If names matter then presumably Chick would believe that such an individual is not saved.

The tract of course ends with the usual warning that only Jesus saves. But the wording is worth noting: "Trusting religion, idols, ceremonies, nature gods or the Virgin Mary to save you is only chasing the wind!" That last phrase is not normally in these tracts. I suspect that to Chick "chasing the wind" sounded like an Injun phrase. This fits Chick's Jesus using the phrase "skin of your teeth" which is much less formal than how Chick's faceless, glowing Jesus normally talks.

So what is the overall lesson of this tract? The take away message seems to be that Native Americans are primitive savages but they get cool magical powers. And as long as you accept Jesus eventually, you get to play with the powers for a long time.


Note: Between drafting and posting this review I ran across another review that is worth reading.

Unscientific America and Pluto: The problem isn't the scientists

I recently read Chris Mooney and Sheril Kirshenbaum's Unscientific America. Their thesis is that anti-science attitudes and the general lack of scientific knowledge in the United States are not the fault of the public or the media, but largely of scientists failing to communicate effectively. My response to this thesis is not positive. This book has been extensively discussed elsewhere. (Of these discussions, I recommend PZ Myers take, Joshua Rosenua's, this balanced analysis by Scicurious, this positive review by Chad Orzel, and this review by Mike at Real Climate)

A major theme of the book is that scientists are out of touch with the general public and spend too much time mocking the public or denouncing journalists rather than trying to engage the public and journalists to understand science. This argument may have some merit. However, the authors offer little evidence for their claim. Simply put, Mooney and Kirshenbaum are wrong: bad science journalism is far more the fault of bad science journalists than it is the fault of disengaged scientists.

Mooney and Kirshenbaum’s primary example of scientists being out of touch with the general public is the 2006 decision to recharacterize Pluto. After the discovery of Eris in 2005 and discovery of other similar objects, it became apparent to astronomers that consistent classification standards required that either Pluto be classified as a non-planet or that many more objects would need to be labeled as planets. Thus, in 2006, the International Astronomical Union constructed a new definition of planets which reclassified Pluto to the status of “dwarf planet.” Readers will likely recall the media outcry that erupted in response to this “downgrading” which included internet petitions and resolutions by various state governments supporting Pluto’s right to continue to be a planet. Mooney and Kirshenbaum point to the popular reaction and astronomers’ failure to anticipate or understand the reaction as an example of how scientists are out of touch with the general public. (There has been some discussion over the validity of Mooney and Kirshenbaum’s point and see for example these two remarks).

However, while reading Unscientific America, serendipity struck and I had occasion to read an article in the December 26, Puerto Rico Daily Sun, from Scripps Howard News Service. This article discussed extrasolar planets and how astronomers have recently discovered smaller extrasolar planets which are closer to what life-sustaining planets would look like. From the article:

"The first extrasolar planets were discovered 15 years ago, and now more than 400 have been found and at an accelerating pace. The early discoveries were gas giants on the order of Jupiter and Pluto and they have orbited far too close to their stars. But as techniques have improved, astronomers are able to identify smaller, occasionally rocky, planets, orbiting far enough from their stars to be close to what is considered a habitable zone."

The author is apparently trying to talk about astronomy while discussing "gas giants on the order of Jupiter and Pluto." Of course, Pluto is not a gas giant. Pluto is a tiny ball of rock, so small it isn't able to keep orbital debris out of its path. That's the entire reason it got downgraded from being a planet. It is small and rocky, not a gas giant. Anyone paying any attention to the faux controversy over Pluto should remember this. Indeed, anyone who remembers anything from grade school would know that Pluto is small and rocky.

This foolish statement about Pluto shows what is really wrong. While scientists may not be the best communicators, putting a majority of the blame for the status quo on scientists is wrong. The problem isn't the scientists. The problem is not that the scientists mistreat journalists and the media. The problem is that the media is full of people so ignorant that we get an article from a major news service claiming that Pluto is a gas giant. Not all journalists are this clueless. But cloning Carl Zimmer simply isn't a viable option. If we're going to deal with this problem, we need to focus on the actual causes. And poor journalism is far more the fault of ignorant journalists than it is of unengaged scientists.

Orthodox Judaism, Science, and Natan Slifkin

Both Orthodox and Ultra-Orthodox/Charedi Jews commonly reject many basic parts of the scientific understanding of the world. For example, many Orthodox Jews believe in a global Nocahian deluge some five thousand years ago and reject evolution. In the charedi world, this rejection of science is even broader. The charedi rejection of science is substantially different from that of other branches of Orthodoxy. Rather than simply reject specific theories based on their own theological predilections, charedim, including charedi leaders (called by their followers "Gedolim" which is Hebrew for "great ones") take an actively hostile view of science.

I was unaware of the depth of charedi fear of and disdain for science until I recently began examining the controversy surrounding Natan Slifkin. Slifkin is a charedi Rabbi who wrote a series of books looking at the interplay between Judaism and biology. Slifkin made three primary of arguments: First, he argued that the evidence for evolution was overwhelmimg. Second, he argued that belief in evolution was not incompatible with Judaism. Third, he argued that the Rabbis of the Talmud could be wrong about science. For the charedim, the second two points apparently caused far more concern than the first. In 2005, many of the Gedolim joined to issue a ruling in which Slifkin was labeled a heretic. Possession and reading of his books was banned.

The charedi rejection of science goes far beyond simple opposition to evolution. For example, I was recently disturbed to learn from a conversation with Slifkin that some major charedi Rabbis believe in spontaneous generation of small rodents. I had been aware that such beliefs had survived until the early 1900s, but I was shocked to find out that many prominent charedi rabbis still believe spontaneous generation of small creatures. The charedi attitude towards science is in many ways connected to a deep worry of persecution. Moshe Sternbuch, the current chief Rabbi of the Edah Charedis, a prominent organization of Israeli charedim, stated that scientists say the world is old because "they want to refute the words of our Sages and undermine the faith that exists amongst the Jewish people. Their main concern is to try to shake the faith in G-d — which has been accepted by us generation after generation."

For another example, see these videos of Rabbi Aharon Schechter in which Schechter gets actively angry at the thought of people trying to investigate evolution, the age of the earth and related questions:






This attitude, one of perceived persecution and anger, seems to stem from two sources: First the charedi worldview is very wrapped up in the history of persecution against Jews. Thus, the charedim see any modern event in that light. Second, the charedi worldview is profoundly self-centered. They assume that essentially everyone cares about what they are doing. Thus, if scientists come to a conclusion that clashes with standard charedi beliefs, the charedi infer that the scientists are trying to target them. In this regard, comparison between the charedi leaders and the leaders of fundamentalist Christianity today is not favorable to the charedim. While many evangelical Christians and fundamentalist Christians reject much science, it is rare for their leaders to claim that scientists are trying specifically to destroy their religion.

However, the anti-science beliefs discussed here are not held by just the charedim. The modern Orthodox also have serious problems with much of science. Alexander Nussbaum has examined modern Orthodox attitudes towards science (see his article in Skeptic as well as Nussbaum's article "Creationism and Geocentrism Among Orthodox Jewish Scientists." in the January-April set of Reports of the National Center for Science Education). Nussbaum found that even among orthodox Jews attending secular colleges, a large fraction reject much of biology, astronomy, geology and other branches of science. About three quarters of the respondents when asked about the age of the Earth, said that it was less than 7000 years old. The vast majority (around 90%) believed that all land animals descend from animals on Noah's ark. Possibly most disturbingly, around a quarter of the students believed that evolution was not only false, but that scientists were deliberately concealing this fact.

Nussbaum also found that undergraduates majoring in scientific areas were less likely to accept many aspects of basic science. Nussbaum proposed that:

It seems that the science majors and degree holders — precisely because they were more likely to be exposed to evolution — were subject to additional community influences not to be “taken in” by the “heresy” they would hear, and were even less accepting of evolution. And individuals with a science background from that community have the added responsibility to use their knowledge and standing to promote religious doctrine in scientific matters.

Presumably the science majors would respond that they are more involved in science and so are more able to see the terrible problems with vast swaths of modern science.

While I had seen Nussbaum's work many years ago, I had generally assumed that something was wrong with his work. While these anti-science viewpoints were not unknown among the Orthodox students when I was an undergraduate at Yale, these views were not as popular as they appeared to be in Nussbaum's study. However, I'm now a graduate student interacting with students at Boston University. Here it seems that the profile of the Orthodox beliefs fits Nussbaum's data much better. Indeed, I recently found myself in a situation with six Orthodox students in the room and five out of the six believed in a literal global flood. When a conversation ensued, one student was unwilling to say whether he believed or not some of the more interesting claims in the Talmud such as the aforementioned spontaneous generation or the existence of the phoenix. That may have been in part due to the student not wanting to discuss the matter, but the overall reaction still agrees strongly with Nussbaum.

Why are otherwise moderate theists so willing to disregard large sections of modern understanding of the world? There are a variety of factors at play. However, one factor that is worth considering is the Orthodox attitude towards Talmudic rabbis, Talmud, and associated midrashic texts. While in some respects Jewish willingness to look at associated commentary or to interpret verses using Oral Law allows for moderation and incorporation of new knowledge. That willingness can also backfire. In particular, for many Orthodox Jews, statements made by Talmudic rabbis are by nature intrinsically infallible. Thus, instead of using the Oral Law as a way of reconciling science and religion, it is used to add additional statements that must be taken as literally true. whether they are about mice arising from mud, or birds burning themselves to regenerate for another life.

So far, this anti-scientific attitude does have some limits. I'm not aware of any Orthodox Jews (regardless of type) who believe in a flat earth. But geocentrism does certainly exist among Orthodox Jews. Most disturbingly, however, is that these anti-science views seem to be becoming more common rather than less in the Orthodox world, especially in the charedi world. The charedi world is not disconnected from the rest of the Orthodox world. If the charedi world becomes more extreme, it will likely pull the rest of the Orthodox world in the same direction.

If Modern Orthodox Judaism is to be taken seriously as a reasonable religion, able to survive in the modern world, then these trends need to be countered by responsible Orthodox leadership.

Amazon, Internet Sales Tax, and Gratuitous Promotion of Family Members (again)

My father has a piece up at the Oxford University Press blog. The piece discusses a serious discrepancy in how sales tax is administered in the United States. Sales tax is broken down by state. However, companies are not required to add on sales tax themselves unless they have a physical presence in given state. If a company does not have a physical presence in a given state, then customers in that state are expected to pay the sales tax themselves. In practice, almost no customer pays this sales tax. Thus, in practice companies such as Amazon which engage in sales primarily over the internet don't pay any any sales tax on their sales. This gives them a competitive advantage over physical stores. The piece is worth reading. The only issue that I have is that he does not address one issue which is also worth discussing: It may be that such sales tax practices are not only economically unfair but also regressive. If more well-off people are more likely to buy on the internet then they will not pay a sales tax where poorer individuals buying from brick-and-mortar stores will. At this point, internet access and internet use is so common that this may not be an actual problem. However, it is an additional concern with the current system. In any event, his piece is worth reading. Go and read.

Minyanim and Mathematics: A protocol for anonymous counting.

In Judaism, many communal prayers require a minimum of ten people. This quorum is called a "minyan" which is often also used to mean a service in general. (In Orthodox Judaism, only Jewish adult males count for a minyam while in Conservative Judaism all Jewish adults count). However, certain liturgical elements on fast days (such as a Torah reading at the afternoon prayer) require not just ten present, but ten who are actually fasting.

However, people may not wish to advertise that they are not fasting even if they are willing to assist with the minyan. Moreover, even a very observant individual might have a reason not to fast and would not want people to know about it (such as a medical condition). This leads to the following problem: How do we determine how many people are fasting without requiring people to disclose whether or not they are fasting?

In a recent minyan I attended, this question was handled by the gabbai (the person in charge of running the minyan) asking everyone to close their eyes and hold up one hand if they were fasting. This is obviously unsatisfactory since the gabbai finds out who is fasting. One could suggest slips of paper or the like. but they could be easily connected to particular individuals. To make this problem interesting, it is helpful to assume that everyone knows the identity of the sender of any communication they receive. Is there a solution under which individuals' statuses remain private while the minyan is assured of 10 fasters?

Curiously the answer is yes albeit the solution is a bit cumbersome. Here's how: The gabbai picks a random integer (the distribution doesn't matter although in practice one might want a reasonable distribution that approximates a bell curve or an exponential distribution or something similar). The gabbai takes this integer and adds 1 to it if he is fasting. The gabbai whispers this number to another person. That person adds 1 to the total if he is fasting and similarly passes it on to another person. The last person gives his number to the gabbai. The gabbai now subtracts the random integer that they initially added. The total the gabbai has is the number of people fasting.

A few notes about this algorithm. First, it is important that, when the gabbai is picking a random integer, that every integer has some non-zero probability of being chosen. Consider, for example, what would happen if the gabbai was known to only pick positive integers. If the gabbai picks 1, the next person can tell if the gabbai is not fasting. Thus, the gabbai can never pick 1. But then, by the same logic, the gabbai won't be able to pick 2. Or 3. And so on. This problem is avoided if the gabbai can pick any integer whether or not it is positive.

There is also a degenerate case of if only the gabbia is fasting or only the gabbai is not fasting. In those cases, the exceptional individual can work out the status of everyone else.

The algorithm also assumes that people are not going out of their way to obtain information. If two people bracket a third and cooperate, they can figure out the middle person's status as a faster.

In fact, it turns out that it is possible to make a more sophisticated protocol that would prevent this and similar tricks. This would be a variation of an anonymous voting algorithm. However, these algorithms rely on clever cryptography with large prime numbers which is only believed to work rather than proven to work. Implementation also would require much more computation than can be done by hand.

Less intensive algorithms can be used to increase the number of people who must cooperate. It is, for example, an interesting exercise to construct a variation of the above such that finding out someone's status requires the cooperation of at least three people.

This algorithm in practice is cumbersome. Moreover, the goal it seeks to accomplish is minor since, if the fast day liturgy is used, those sections of the services actually need to be performed by people who are fasting. Thus, the gabbai will need to know the status of at least a few people. However, this algorithm has the slight advantage that, if there are not enough people to recite the fast day liturgy, then no one will know which people cause the deficiency.