Friday, March 27, 2009

Failure of Unique Prime Factorization

In an earlier entry I discussed Unique Prime Factorization and the ancient Greeks. Some systems very similar to the integers have unique prime factorization while others do not. Let's look at some of those systems.

Consider the Gaussian integers, normally denoted Z[i]. Z[i] is the set of all complex numbers of the form a+bi where a and b are integers and is the square root of negative one. It is not that hard to see that Z[i] is closed under both addition and multiplication. That is, if I take two elements in Z[i] and take their product or sum then the result is also in Z[i]. Furthermore, every element in Z[i] has an additive inverse, that is a number to which I can add to it to get 0. For any a+bi this is just -a-bi. Multiplication and addition are both associative and commutative, and multiplication distributes over addition. Thus, Z[i] is very similar to Z, the set of integers. Sets which satisfy these properties are called rings and are generally symbolized with bold letters. It turns out that rings are the natural setting to look for unique prime factorization.

At first glance, it might seem that the Gaussian integers do not have unique prime factorization. For example, (1+2i)(1-2i)=5=(2-i)(2+i). However, this can be easily corrected. What we have done is something sneaky called multiplying by a unit. In the integers we can do this also. (-3)(-5)=3*5. This isn't because unique prime factorization breaks down. We can always multiply terms by some product of numbers that equal 1. In the case in Z, we used -1 and -1. In the case with (1+2i)(1-2i) we used i and -i. Numbers in a ring which divide 1 are called units. Thus, for example if we were in R, the set of real numbers, every non-zero number is a unit since for any x, we can multiply it by 1/x to get 1. In the integers, the only units are 1 and -1. It turns out that the Gaussian integers have a total of 4 units, -1,1, i and -i. Factorization in the Gaussian integers is unique up to multiplication by units just as factorization in the integers is unique up to multiplication by units.

In the early part of the 19th century it was taken for granted that all well-behaved rings had unique prime factorization. However, this turns out to be false. The standard counterexample is A, the set of numbers of the form a+bk where k is the square root of -5. Like the Gaussian integers, A forms a ring. Unlike the Gaussian integers, A does not have unique prime factorization. We have 2*3=6 and (1+k)(1-k)=1-k+k+k^2=1+5=6. 2,3, 1+k and 1-k cannot be broken down further in any useful way. Thus unique prime factorization fails. The failure of unique prime factorization turns out to be deeply linked with a number of serious issues. In particular, the lack of unique prime factorization in general rings was the major reason that mathematicians in the 19th century were unable to prove Fermat's Last Theorem.

Although A is the standard example of unique prime factorization failing, it isn't a very satisfying example. It isn't clear why unique prime factorization is failing and it isn't clear that if one were looking for an example ring where unique prime factorization fails that one would construct A. There is a slightly more abstract ring that does the job a bit better. Consider the ring B = R[a,b,c,d]/(ab-cd). What we mean by this ring is to take all the products and sums of four formal symbols, a,b,c,d and all real numbers. Thus we would have for example 2a+b as an element. By /(ab-cd) we mean to treat ab-cd as zero. Thus, we do arithmetic formally on this set of four symbols, but we may whenever we see ab or cd replace it by the other. Now, this set is a ring. And it clearly fails at unique prime factorization. Just look at ab and cd. They are different prime factorizations for ab. Unlike with A, there's no difficulty involved in showing that the numbers in question are somehow genuinely behaving like primes. It is immediately obvious from the construction that there is no way of representing a,b,c, or d as an interesting product of terms.

It isn't clear to me why this example isn't used more often. Although it is more abstract than A, the failure of unique factorization is much clearer.

Monday, March 23, 2009

AIG Bonus Tax and Married Couples

It's time for more gratuitous promotion of family members.

My sister has a piece up at the TaxProf Blog arguing that the tax on the AIG bonuses is sexist. The essential argument is that as written the tax will trigger additional taxation for married couples even when only one spouse received a bonus. Since most of the people receiving bonuses are men, this tax effectively punishes women who are married and are not stay-at-home wives. I'm not sure I would call this sexist since the result is not intentional. However, the result is unacceptable.

Sunday, March 22, 2009

Conservapedia, The Telegraph and the BNP

Conservapedia founder Andrew Schlafly is at it again. Shane Richmond, a technology writer for The Telegraph, wrote about a blog entry about PopModal, a right-wing alternative to Youtube (incidentally, this is not the first attempt at a conservative alternative to Youtube). In mocking PopModal, Richmond mentioned Conservapedia in passing:
Like Conservapedia, the encyclopedia that has no time for reality's liberal bias, PopModal offers a "conservative alternative" to YouTube.
Conservapedia responded by declaring on their mainpage that "Conservapedia is the benchmark for the liberal Telegraph in England to use in describing other new media alternatives." By itself, this would be mildly humorous and not worth a separate blog entry. However, what happened next is Conservapedia at its finest. People attempted to explain to Schlafly the myriad problems with the declaration. One user wrote:

I don't think that Telegraph article you're posting on the Main Page is appropriate. First, the Telegraph is anything but liberal and has always been considered the most conservative of all UK daily broadsheets, with an almost exclusively an older, suburban Tory Party readership...
Secondly, the article itself is clearly mocking Conservapedia.

Andrew responded in his typical fashion by declaring that the writer clearly has a liberal bias and that Britain was so hopelessly liberal that "Saying something is the most conservative newspaper in the UK is like saying someone is ... the smartest student in a remedial class. "And now the real fun starts. The conversation gets more meandering and comes to the topic of the British National Party where Schlafly states that his main problem with the BNP is that they are in favor of universal health-care:

The BNP might get my vote for its position alone on education: "We will end the practice of politically correct indoctrination." Of course, I expect knee-jerk liberals to claim there is something racist about that or about the BNP in general, but overuse of the racist label as a political tactic has gotten pretty tiresome for everyone on this side of the pond.

I'm not impressed by the lack of social positions (like abortion, marriage, prayer in the classroom, etc.) on the BNP website you reference. This position by the BNP is not conservative at all: "We are wholly committed to a free, fully funded National Health Service for all British citizens."
This is particularly interesting since Conservapedia's own entry on the BNP describes them as neo-Nazis. There's an old rule on the internet known as Godwin's Law which in its simplest form states that as any conversation on the internet continues, the probability that a reference to Nazi's will occur approaches one. Generally, it is taken for granted that if one side compares their opponents to Nazis then that side has by default lost the argument. I don't know how to apply such a rule when someone on their own accord compares their opinions to those of neo-Nazis. I'm also not aware of any other occasion in which an individual has stated that their biggest problem with neo-Nazis is that the Nazis support universal health care.

Wednesday, March 18, 2009

Kashrut Supervision

Orthodox Jews and many Conservative Jews keep kosher. This is a series of dietary laws as laid out in the Torah and explicated in the Talmud and later works. Major rules include prohibitions on the consumption of certain types of animals, a prohibition on mixing meat and milk, and requirements for how animals must be slaughtered in order for the meat to be kosher. Many Orthodox Jews will only consume food that has been specifically supervised to ensure that no non-Kosher contaminants have entered the food. Such supervision is marked on food packaging with small labels indicating what organization supervised the food. This symbol is called a hechsher (plural hechsherim). For example, in the United States if you look on many foods in the United States you will see a small U inside a circle on the packaging. This symbol indicates that the food is under the kashrut supervision of an organizaton called the Orthodox Union.

Why am I thinking about kashrut supervision today?

Wolfish Musings talks about a new milk being sold in New York and New Jersey that is aimed at the ultra-Orthodox population has no less than four hechserim. The vast majority of Orthodox Jews have no issue with food that has a single hechsher. Indeed, it is generally difficult to find a food with multiple hechsherim on it. Moreover, while arguments do exist in the Orthodox community over which hechsherim are really trustworthy enough, most of those arguments are about the hechsherim for meat products. The laws concerning milk products are much simpler than the laws for meat. Ultra-Orthodoxy seems to be about trying to find new ways to be strict that no one previously has ever worried about. Indeed the motto of the milk company is "Machmirim bnei Machmirim" which translates as "The strict who are children of the strict." If anyone can give an explanation for why someone would think that four different hechsherim would be at all useful or halachically better I'd like to hear it.

Tuesday, March 17, 2009

Wikipedia, Australia and Censorship

I've previously discussed censorship of Wikipedia by Great Britain. It appears that Australia may be the next democracy to attempt to censor Wikipedia. Australia currently has a secret blacklist of websites which are censored. Nominally the list is to deal primarily with child pornography but no one can tell what precisely is one the list since it is secret. As with the websites censored by Great Britian, there is no clear system of oversight of what gets added to the list. Activists have attempted to show that additions to the list are arbitrary and poorly thought out. Recently, they succeeded in getting an anti-abortion website with graphic pictures of aborted fetuses added to the list. Moving on, the activists then added links to the website in various online fora, using it as an example of a how the list was running rampshod over peoples free speech rights. The Australian Communications and Media Authority responded by sending one of the online fora a nasty note telling them to take out the link or be subject to large fines and risk having their forum added to the list. Wikipedia in reporting on this ongoing controversy has linked to the blacklisted abortion website. The activists understand what this means:

"If ACMA blacklists their own Wikipedia page, well that says it all doesn't it? If they don't, that is a very, very strong reason to call them hypocrites for making vastly different responses to two sites linking to the very same material."
It should be interesting to see over the next few days how ACMA responds.

Thursday, March 12, 2009

107th Skeptics is Now Up

The 107th Skeptics Circle is now up at The Skeptics' Field Guide. It includes a lot of interesting posts including my own post here on electronics and the supernatural. I strongly recommend interested readers go and check it out.

Wednesday, March 11, 2009

Barack Obama and Human Cloning

Obama took a step forward in ending the restrictions on federal funding of stem cell research. This decision is correct. However, in announcing this decision, Obama declared that under no circumstances would he allow human cloning which he called “dangerous, profoundly wrong.” Obama went on to declare that cloning “has no place in our society, or any society.” On the cloning issue, Obama is both scientifically and ethically wrong.

Cloning falls into two broad categories: “therapeutic cloning” and “reproductive cloning.” Therapeutic cloning involves making a clone embryo and then extracting its stem cells for use in therapy. Reproductive cloning involves making a cloned embryo and bringing it to term.

The chief ethical issue for both cloning and stem cell research is the moral weight to assign to embryos and fetuses. Generally, people who are strongly “pro-life” assign a high moral status to embryos , at or close to the level they would assign a baby. Others assign less ethical value to embryos and fetuses.

Reproductive cloning should not be bothersome from a strong pro-life perspective. Indeed, reproductive cloning is functionally identical to the process which creates identical twins. No one can claim that reproductive cloning involves the destruction of a human life or of the potential for a human life whereas such claims can be made about certain types of stem cell research. At minimum, if one believes that embryos have the same moral rights as adult humans, then embryonic stem cell research benefits from murders. Even if one assigns some but not much value to embryos, this concern still exists. Similar concerns should apply to therapeutic cloning since such clones might potentially be able to be brought to term. (Indeed, I’m continually puzzled by studies showing far more people approving of therapeutic cloning than reproductive cloning). The serious ethical moral issues raised by embryonic stem cell research are not raised by reproductive cloning.

Reproductive cloning does raise legitimate worries. There are concerns about egotistical people trying to make copies of themselves. And we must worry that cloning on a large scale could reduce genetic diversity in the human population. Genetic diversity is necessary to protect against disease and sudden environmental shifts. And as long as mammalian cloning continues to have its high complication rate, we cannot ethically attempt it with humans. However, these are all concerns that can be addressed by further research and informed debate. None of these concerns justify President Obama’s claim that there is something fundamentally wrong with cloning.

Sunday, March 8, 2009

Euclid, Unique Prime Factorization and Sapir-Whorf

Unique prime factorization is a basic fact about the positive integers that many of us learn from a young age. Start with any positive integer and express it is as a product of prime numbers, say 105=3*5*7. This prime factorization is unique. That is to say, except for rearranging the order of these primes, there is no other way to express 105 as a product of prime numbers. Thus, for example, I don’t need to do any calculations to know that 105 is not equal to 11 *13. We can do this for any positive integer and regardless of what integer we start with we will get a unique result. (Another way of stating this is that if two people are both given the same number and told to factor it into primes, they will always get the same result as each other).

This idea is often called the "Fundamental Theorem of Arithmetic." Unique factorization is taught to children as young as 6th grade. When I have helped to teach number theory to high school students at PROMYS, many students take unique prime factorization for granted. Indeed, it often takes effort to convince them that the statement needs to be proved and is not just an obvious fact.

Despite the “obviousness” of unique prime factorization to the modern mind, the ancient Greeks did not know about unique prime factorization. This apparent failure of the ancient Greeks is well known to historians

of mathematics, but not known by many mathematicians. The only math text I am aware of which mentions this is Hardy and Wright's “Introduction to the Theory of Numbers. Euclid in his seminal work “The Elements” has many results that are close to unique prime factorization. For example, Euclid was able to show that, if a is relatively prime to b and relatively prime to c, then a is relatively prime the produc of b and c. This result is almost as powerful as unique prime factorization, but it is not all the way there.

Why did the ancient Greeks not have this result? Simply put, they lacked the words to express the statement. To a mathematician of ancient Greece, there was no way to say "take a bunch of numbers and take their product." A product of two numbers x and y was easy; it was the area of a rectangle with sides of length x and y. Similarly, a product of three numbers -- x, y and z -- was the volume of a box with side lengths x,y and z. Fixed in this geometric mindset and with no concept of higher dimensions, the ancient Greeks could not talk about general products. Being unable to talk about general products, they apparently did not think about them.

To be sure, they groped towards this result. For example, Euclid proved a result that in modern language would be stated as follows: if n is the least positive integer divisible by some list of distinct primes p1,p2.p3… pk, then n is not divisible by any prime not on the list. This result is trivial if one knows that prime factorization is unique this is obvious since one can simply has n=p1 * p2... *pk.

Apparently, the ancient Greeks, including Archimedes, Euclid and Diophantus, along with many lesser luminaries could not conceive of this idea because they lacked the language to express it. This failure is closely related to a hypothesis in psychology and linguistics known as the Sapir-Whorf hypothesis. Sapir-Whorf states that our language constrains our thought processes. There are different degrees of Sapir-Whorf. Very strong versions of the hypothesis are obviously false (if they were to be believed, translation between languages would be impossible) while very weak versions border on the trivial. This failure of the ancient Greeks is evidence for some strong form of Sapir-Whorf.

There's a lesson here for people other than psychologists. We cannot know how much we are missing simply because we lack the language to express an idea. Mathematicians over the last three centuries have taken this idea very seriously and spent much time trying to find optimal notation to express ideas. Yet, we cannot tell if there is some fundamental idea that we simply do not notice or appreciate because we lack the necessary language.

Thursday, March 5, 2009

Science appointments on hold.

Members of the US Senate are holding up the appointments of Jane Lubchenco as NOAA administrator and John Holdren as Science Advisor. Moreover, while the specific Senators causing the delay are still anonymous they are apparently Democrats using these positions as pawns in a bargaining game for unrelated political gains. This situation is unacceptable. I and many others voted for the Democrats in the last election because we were sick of poor treatment of science by the Republicans. Unfortunately, I am cynical enough that holding up critical appointments of qualified appointments for necessary science positions is change I can believe in; it is no change at all.

You can help deal with this. Mike Dunford has multiple pieces discussing the situation and what you can do about it. Mike has the details of which Senators to contact. This situation is unacceptable. The Democrats in the Senate need to understand that people who care about science care about empirical results. We won't just listen to your rhetoric if you don't follow through. Having these appointments confirmed by the Senate is vitally important. Please look at Mike's pieces and contact your Senators.

Monday, March 2, 2009

Creationism and Intelligent Design in Great Britain

In the comments thread to my last post Lautreamont pointed to a recent study about belief in creationism and intelligent design in Great Britain. The study showed that over 80% of the people in Great Britain reject intelligent and creationism. A larger percentage (89%) rejected intelligent design than those which rejected creationism(83%). Given the big-tent nature of ID this is surprising. This rejection may be explained by definitional issues since some theists will accept "creationism" as the general idea that God created the world but reject "intelligent design" in the sense that they consider this belief to be exclusively theological rather than scientific in nature. There are also some young earth creationists who reject intelligent design as not having enough explicit emphasis on the Bible. Without a more thorough examination of the study and the phrasing of the questions it is hard to say. Note also that the percentage who are rejecting creationism is much higher than in the United States where approximately half the population are young earth creationists.

Razib over at Gene Expression has also blogged on this story and has some analysis on the geographic breakdown of belief. He raises two points that are worth noting: He notes an unusally high percentage of creationists in London and suggests that this may be due to Muslims living in the city. The study's authors speculated in contrast that this may be due to the relatively high Pentecostalist presence in London. Razib also discusses the high percentage of creationists in Northern Ireland and speculates that this may be due to the internecine religious fighting creating a general push towards more extremist views.

There are two paragraphs specific paragraphs from the article that are also worth noting:

The poll also revealed some extraordinary views on more recent writings, with 5% of adults thinking Darwin wrote A Brief History of Time, a bestseller on the science of spacetime, which was written by the Cambridge physicist Stephen Hawking and is widely regarded as the most popular science book never to be completed by its readers.

A further 3% of those surveyed thought Darwin wrote The God Delusion, by the arch-atheist and Oxford biologist Richard Dawkins, while 1% thought Darwin was the author of The Naked Chef by Jamie Oliver.
It could be worse. We could have people who thought that Darwin, Dawkins and Hawking wrote the Bible.

Sunday, March 1, 2009

Electronics and the Supernatural

Many modern day conveniences are electronic in nature. These devices, such as cell phones, computers, stereos and televisions, are not only useful; they are also testaments to the triumph of science and technology. Ingenuity and reason have given us capabilities far beyond anything that we have naturally. Yet the popular imagination connects the fruits of that labor with the supernatural.

There are two prominent ways in which the supernatural is connected to electronics. First, there is a claimed connection between electromagnetism and ghosts. Ghost hunters are fond of claiming that ghosts create electromagnetic radiation or consist of electromagnetic energy. See, for example the repeated claims made on Ghost Hunters.

Second, a direct connection between electronic devices and the supernatural is often asserted. The most common form of this claim is the so-called EVP or electronic voice phenomena. These occur when individuals claim that supernatural entities, especially spirits of the departed, communicate through the static in electronic devices.

EVP, despite their obvious pareidolic nature, are taken seriously by many people. See for example the American Association of Electronic Voice Phenomena . EVP also is common in movies and other elements of popular culture. White Noise, for example, is about using EVP to communicate with the departed.

Many similar ideas exist in popular culture that are not precisely EVP but are similar. One Missed Call is about a vengeful ghost that travels from cell phone to cell phone, killing each owner and then moving onto another number listed on that phone. The Ring is about a cursed videotape which kills anyone who watches it.

In many other movies, the presence of the supernatural manifests itself with interference with electronics, causing static or causing lights to flicker. In many films, ghosts appear on videotapes even when they cannot be seen with the naked eye. This is related to the idea that photographs can reveal what the naked eye cannot. Serious ghost hunters take this idea seriously for both photographs and videos.

The general idea that electronics are somehow connected to the supernatural is not new. There is an old story about Thomas Edison trying to make an electric device to contact the dead; this story is roughly contemporaneous with Edison. In fact, the Edison story was used as the main premise of the truly wretched movie The Brink in which an engineer discovers Edison’s old plans and uses them to contact the dead.

Why is this connection so prevalent in the popular culture? One cause is that there is a self-reinforcing element. The more common the idea ,the more it will get used. This is especially true given how much derivative junk is written by half-baked writers with as much chance of having an original thought as I have for being the next cover model for the Sports Illustrated Swimsuit edition. There are two other reasons for the connection between electronics and the supernatural. First, poor understanding of science makes electronics seem similar to magic. Second, there is an element of particular terror when our technological marvels betray us.

The general public has little to no understanding how electronics function. If one asks individuals on the street what charge an electron has , they are as likely to say positive as to say negative. As far as they are concerned electricity is supernatural. When they turn on a light the electrons coursing through the filament may as, well be faeries. Thus, it is not at all unreasonable to the common mind that one supernatural phenomenon may be connected to another. Why shouldn't ghosts use the supernatural television set to manifest themselves? And if both spirits and electromagnetic waves are invisible things of which one has only a very vague notion, why shouldn't they be connected or be the same thing? This is made all the more plausible by the capabilities of our electronic devices. They can repeat the words of the dead to us. We can with the click of a mouse hear the speeches of saints and sinners who died before we were born. It is not a far leap for the imagination to wonder if such beings still exist in the same devices which record what they have to say.

When this phenomenon manifests itself in movies ,there is another element at play. The notion that the modern devices on which we rely for our every day luxuries could betray us, or open portals to horrors, is a potentially scary premise.

There's a distinct way that electricity and science also interact with the supernatural, more common in books than in movies. Rather than see electronics as connected to the supernatural, they see electronics as an enemy of the supernatural. When this occurs, science as a whole is set up as a system that opposes magic Thus, for example, in Harry Potter electronics do not function at Hogwarts or at other locations of heavy magic use. Similarly, in John DeChancie's Castle Perilous series, every universe has some amount of "science" that works and some amount of "magic". The more science a universe has, the less magic. . Similarly, in Randall Garret's Lord Darcy stories, people in the late middle ages discovered the laws of magic instead of science. In an old Spiderman comic, when Spiderman fights a certain villain who uses alchemy, Spiderman zaps him with electricity since the opposite of alchemy is science and electricity is scientific.

Again, this is an attitude that does not just exist in fiction. I've met at least one New Ager who said that (slight paraphrase) "science only works if you believe in it. I believe in magic and that works for me." In this view, mystical reasoning is not opposed to rationality just as a conflict of world views. It is not just that either rationalism explains the universe or mysticism explains the universe. Rather mystical thinking and rational thinking are actual opposing forces at work in the universe. Given the triumphs that we have through electricity, it is not surprising that it is considered to be the champion of science.

Both these views of science and electricity-- electricity as something supernatural; electricity as something anti-supernatural -- reflect profound misunderstanding of how science works and what rationality and science represent. The first view is one of classical bad reasoning, relying on ignorance and sympathetic magic.

The second view has problems, but it takes slightly more work to articulate them. In a nutshell, if there were special people who could use sticks of wood and point them and say "Crucio" or "Wingardium Leviosa" or "Latinus Mangledus Maximus" and get specific results on a consistent basis, this wouldn't cause the scientific method to vanish, nor would it negate anything that had been learned with the scientific method. Indeed, we could apply the scientific method just as well to these people and their powers. We could perform controlled experiments and derive laws and theories to explain their powers. We could test hypotheses about how magic worked. I'm reminded of the psychics who claim that their powers don't work around skeptics because the skeptics' skepticism creates a psychic dampening field. Such attitudes reflect the worldview that how we think about the world alters it. This is the essence of magical thinking. If someone believes in a dichotomy between the forces of reason and the forces of magic, they don't really believe in a dichotomy; they believe in a world which is just supernatural.

Both of these views are symptoms of irrational and ignorant populations. To the general public, there are two options. Either electricity has little or nothing to do with science, but is connected to things that go bump in the night or electricity is connected to science where science means some natural force that humans exploit, a force that is opposed to forces of magic.

Edit:See also this followup piece discussing why one can be reasonably certain that there is no genuine connection between ghosts and electronics.