Easter must be coming

Curious.

There's a new film out, claiming to tell of a shocking new discovery. It's one of those discoveries that, in the breathless words of the news-reporters, could change 2000 years of religious history.

There's a totally unrelated story from another part of the ancient world. The king of Persia sent a messenger to the leaders of a league of Greek cities. That messenger said, "Our king is coming with his army. You may fight us--and if our army wins the struggle, your cities will be ours. Your men will die, your wives and children will become our slaves, your riches will gild the capital city of our empire."

The reply was a single word: "If."

This story about a possible tomb of Jesus (claimed to have never been emptied by an Easter event)--if true, it is a serious challenge to Christianity.

If.

Such challenges to the central claims of the Christian faith have cropped up many times in history. If we restrict our view to last few decades, challenges to the historicity of orthodox Christianity crop up on a regular basis. Usually, they recur during the season of Lent. (For those who aren't familiar with the liturgical calendar, Lent is the 40 days preceding Easter.)

Each time, the claims are loudly reported in the popular press. Often, the claims aren't given the kind of rigorous analysis that would be expected of material published in scholarly journals of archeology and history.

In yet another development, there won't be any riots or threats against the creator of the film attempting to tell this story.

Math Geekery, pt. 2

In one of my rambles across the Internet, I ran across a math-oriented website called Project Euler.

Named after the most prolific mathematician in the 18th Century, Project Euler challenges people with programming skill to tackle mathematical problems. Among the ones that I have attempted are these:

• Find the 10001st prime number
• Add all natural numbers below 1000 that are nultiples of both 3 and 5
• What is the difference between the sum of the squares of the first 100 natural numbers and the square of the sum of these numbers?

These problems have obvious, brute-force solutions. Often, there are elegant shortcuts which significantly reduce the time spent producing the answer.

Looking at the third example on my list of problems, I see that if a formula could be found to add up the first n natural numbers, then things would be much simpler. It turns out that there is such a formula, and it is fairly easy to remember:

1+2+...+n = ½(n)(n+1)

So adding the first 100 natural numbers becomes much easier, and squaring the result isn't too hard.

Adding the first 100 squares of natural numbers looks equally hard at first, and the formula escaped me at first. Still, it was not impossible to find.

After finding these formulas, I find that a computer isn't strictly needed to solve these problems. If pressed, I could have found both formulas in textbooks. The mathematics isn't too challenging if done solely on paper.

However, working out an algorithm to produce the result on a computer isn't too hard. Still, the challenge is mentally stimulating, especially when compared to explaining lower-level algebra to University freshman.

Literature

Despite my distaste for entertainment-driven news media, I cannot quite escape its influence.

One side-effect of the influence of TV news is that I will occasionally see a person who becomes the subject of a news blitz, and suddenly realized that I've met the archetype for this person before--in some piece of literature that I've read.

The effect can produce strange parallels. A couple of summers ago, a certain protester loudly proclaimed that the Chief Executive of the United States needed to feel the pain she felt as a grieving mother of a dead soldier. Her claims resonated strangely with my memories of the central character in The Count of Monte-Cristo.

In an early scene in that novel, while discussing crime and punishment, the Count opines that execution is relatively quick and painless compared to the pain and suffering inflicted by the criminal on others. The Count expounds to one of his comrades his own theory of punishment, in which the criminal is made to feel some degree of the loss, deprivation, and pain that he has inflicted on others. This theory, and the effects it has on the Count and on his old enemies, dominate the rest of the story.

More recently, a different juxtaposition has occurred.

Within the last decade, John Grisham penned a legal thriller about a billionaire, a will, and lots of claimants on the will. The Testament contains all the usual elements from a legal-thriller; it also contains good characterization and a carefully-crafted plotline.

The elements of that story which resonate with the big news of the past few weeks is the gathering of vultures around the estate of a wealthy person who has recently died. The legal wrangling (and the lottery-like effects the will) was all described accurately by John Grisham. A variation of that story is being played out by people in the real world before an attentive audience of reporters.

It is a mild surprise to note that many of the events of life have already been told of by a story-teller. It is encouraging to realize that meeting these events in the world of literature gives an insulation from the breathless, spectacle-driven frenzy of the TV news world. People have dealt with such things before.

While the examples that I have chosen seem small, the principle can be extended. Emergencies, new developments, and shocking events have followed the nations, tribes, and families of humanity since before the dawn of history. They should not be ignored; but they need not be obsessed over.

Downturns

Two weeks ago, the governor of the State of Michigan gave the annual State of the State address.

While possibly not of interest to many outside the State, it does provide a snapshot of a rare situation among the States in the Union. That is, the economy of Michigan has been coasting downhill for some time. The State budget is facing serious shortfalls, and the Governor is caught between her desire to bring more money in (through raising tax rates) and a legislature that doesn't want to raise tax rates.

According to the U.S. Bureau of Labor Statistics, Michigan is one of the few States int the Union which had more than 7% unemployment as of December 2006.

But unemployment rates--even rates that are above the national average of 4.6%--don't say much. When two of the largest corporations in the Detroit area (and a third corporation that had been purchased by a foreign competitor, but could be up for sale again) are struggling to post profits, the entire business environment in the State's largest city suffers. While these large companies make the big headlines, there are a handful of other large firms in the area which depend upon the auto manufacturers for business. Not to mention the fact that when large numbers are laid off by large companies, the banks, restaurants, malls, office-supply stores, etc.

There are other industries in the State--but none of them can bring in money in the way that the Big Three can.

Whether the State budget troubles are directly related to these problems or not, the State government will have a hard time getting any increases in tax revenue from a business-base that is shrinking. Personal income depends mostly on those businesses, and that base is likely to be held stable (or to shrink) in the near future.

In dealing with the resulting funding shortfall, both State and local governments are going through exercises in belt-tightening.

It's not as if this state (or other states) have not seen such down-turns before. But it does give room for wonder--after the last time the Big Three were in trouble, wouldn't they (and the State government) have figured out how to avoid the worst effects of the next downturn?

Or did they only figure out how to fix the last downturn, after it had passed?

Sadness

Apropos of...being a week and a half late in blogging about this...to my own credit (or shame), I had very little idea who this woman was before she died.

Those who wish to find a fulmination against the shallow-mindedness of the modern age can probably find it easily--elsewhere. This death is a sad event.

Every death is sad event.

This one is being trumpeted across the airwaves and news channels as if it is unique. Her sad life, tangle of friendships and lovers, the doubtful parentage of her child, and her desperate need for the love of others has been turned into a made-for-TV melodrama with an improvised script.

It either magnifies the sadness of the death in the mind of the viewer--or encourages the viewer to treat the death like an act at a carnival sideshow.

One more word on climate

A little while ago, I did some blogging on a subject I know a lot about: math-driven computer modeling. I also did some blogging about something I know very little about: climate change.

One of my assertions was that the science of global climate isn't well-understood enough to know whether a computer model is producing valid results for the future.

Come to find out, someone else who's been blogging (heavily) about climate change came to the same conclusion. (It's stated in the UPDATE section at the end of that post.) Even more interesting, he came to this conclusion from another direction--discussing scientific observation, scientific theories, alternate explanations, and scientific methodology.

In the spirit of the season...

For the mathematically astute: Happy Cardioid Day!

For the rest of the readers, a small riddle is posed: how did the Roman Catholic holiday in honor of three different martyrs who shared a name become a secular celebration of romance?

Stories about the future

My recent reading has allowed me to compare two very different authors. These two authors each had (still have, to my knowledge) a distinct following.

Both Robert Heinlein and Philip K. Dick wrote in the genre of science-fiction. Both men were craftsmen; their ability to construct written art made their work stand out from shills writing in the same field.

One story idea which both men used was the idea of military technology created by humans becoming a threat to the future existence (or freedom) of the human race. Heinlein created the story The Long Watch in which one man had the opportunity to stop nuclear blackmail of all governments on Earth. This opportunity also involves a high risk of death for the hero; this is a risk that he gladly takes.

Dick also created a story about military technology which threatens the lives and freedoms of all humanity. In The Second Variety, a series of intelligent hunter-killer robots have been created to replace a decimated army. These robots have manage to create better robots, near-perfect imitations of human beings. These new robots have the same basic programming to seek out and kill humans, but no ability to distinguish friend from foe.

Heinlein tells about his hero saving the day for everyone involved. Dick tells about the hero's final realization that the forces set in motion cannot be stopped--the robots will be able to exterminate humanity, and will fight each other for dominance afterwards.

These two stories underline two fundamentally different ways of looking at human nature, technology, and the future. One vision of the future tells of strong characters who will be able to channel society into the right directions through the use of wits, intelligence, and good sense. The other vision of the future is wrapped in the foreboding that technology will eventually create something which will, in its application, irreversibly doom the human race.

Popular culture returns to these themes occasionally. In the realm of politics and nuclear power, Tom Clancy made a major novel in which one man has the ability to avert catastrophe. In the realm of robotic weapons turning against their creators, the movie Terminator and its sequels are very widely-known.

It does seem that neither vision is capable of explaining all of the future. Both the story of the individual who turns the course of history and the society that is unable to stop its slow march to destruction have been seen in human history.

The history of atomic weaponry in the West was changed significantly by a single letter from Albert Einstein to President Roosevelt.

However, the history of warfare and political intrigue in the West was on a march towards the dominance of a single great nation for quite some time. That development, combined with the development of culture inside that nation, has created a strange combination of insurmountable military capability and a seeming inability to rally political will to win most conflicts which arise.

It has been argued elsewhere that the democratization of technology has enabled the creation of "non-state actors", groups of men who wish to alter world politics while working outside of the strictures of the government of a nation-state. This enablement seems to have played out in a way that might have been predicted by Philip K. Dick: the inherent advantages of non-state actors to the less-powerful nations of the world mean that such groups will always have unofficial support from some nation. But non-state actors can be powerful de-stabilizers. Their long-term goals may be inimical to the long-term goals of the sponsor state. At the very least, the sponsor state must deal with any international opprobrium and threat of military action by aggrieved nations.

It is easy to envision a fictional world in which terrorist-style non-state actors have suborned all the governments on earth. However, these groups would then have to deal with becoming a de facto governmental force of some kind, and construct a new international order--which would strip them of most of the freedoms they now enjoy as non-state actors.

It is also good to remind ourselves that the future of the human race depends on the individual decisions of many billions of people. The individual effect of many of these decisions appears vanishingly small--but that does not separate any individual from living with the consequences of their decisions. Nor should it separate any individual from the knowledge that they are also affecting the decisions that their children and grand-children will make.

Perhaps a comparison with another author's treatment of evil, human nature, and the future is also in order.

"...And so, a great evil in this world will be removed.
Other evils there are that may come...Yet it is not our part to master all the tides of the world, but to do what is in us for the succour of those years wherein we are set, uprooting evil in the field that we know, so that those who live after may have clean earth to till. What weather they shall have is not ours to rule."

(Gandalf, speaking in Return of the King, written by J.R.R. Tolkien)

Tolkien did not share the vision of futility. His heroic vision assumes that not all evil will be cured by the sacrifice of the hero. He sometimes hints that the hero would be worthy of adulation whether or not the hero succeeded in his quest. Tolkien also insists on never letting go of hope, even in the darkest hours of the quest.

The heroic vision as promulgated by Tolkien is probably the best way to view the future. Men will never eradicate evil from the world; but men can eradicate much of the evil from the present world.

Down-time

Blogging will be interrupted for the rest of this week: outside events will keep me far away from blogging time (and internet connections).

Global Climate

Quick note: quite apart from any mathematical or computer models of global climate change, there could be problems with the assertion that the temperature rise is entirely caused by greenhouse gases.

That large nuclear fusion furnace which provides energy to the Earth might be burning a little hot.

This isn't reported as settled science or as new scientific consensus. It's broadening research in the field of Planetary Climate.

This further confirms my suspicion that the climatologists who try to predict the average temperature a century from now don't have a good, solid climate model to work with. They're still trying to figure out which parts of the model are primary, and which parts are secondary.

Computer Models and the Climate

In my previous post, I mentioned computer models and their uses and limitations.

Here, I would like to look at a well-known use of computer models.

In the wake of the recent report by the International Panel on Climate Change, I am reminded once again that mathematically-based models are powerful, but imperfect, tools.

Far too often, when a news reporter (or a publicity agent) says that they are reporting the results of computer modeling, they imply that the model is flawless. At the very least, they rarely (if ever) report the ways in which the model might fail to predict reality.

Using computers to analyze models of real-world problems will always involve some assumptions to simplify the problem. If the model is of a small system, the number of assumptions is small. If the model is of a well-understood system, the simplifying assumptions (and their limits) will be easy to quantify and explain.

If the model is of a large system--large in the number of interconnected variables and data points, not in sheer physical size--then the model becomes less trustworthy. If the behavior of the system is hard to test, or the system contains interconnected forces which aren't fully understood, the model is much less trustworthy.

One physically small system which is a regular subject of mathematical modeling is the spray of fuel being injected into a diesel engine. The turbulent flow of the fuel/air mixture is defined by a well-known set of differential equations. However, these equations are known to have no general solution; specific solutions can be approximated by the use of Computational Fluid Dynamics.

This system has an astonishingly huge data-set of velocities, temperatures, fluid vectors, and time-steps. The accuracy of the model depends on several assumptions. The most banal assumption probably has the largest cumulative effect--the assumption that the small amounts of rounding inherent in doing computer calculations will provide a good approximation of exact arithmetic. These roundings can happen millions of times during the long process of the calculations. Each error introduced by rounding will have an effect on each calculation that depends on the results of that particular calculation.

In considering the main subject for the post--global climate change--I know that the specific job of CFD is not exactly like the job of predicting the future average climate of Planet Earth.

For one thing, the temperature of the planet depends on a variety of factors. Almost all of the factors (except for the nuclear fusion furnace known as our Sun) are influenced by other factors in the system. The oceans absorb or release gases based on their surface temperature. These gases may help hold more heat energy in the atmosphere. But the clouds also affect the global climate--and cloud formation may change if the atmospheric temperature rises.

But how does each of these changes alter the central question of the direction of average temperature in the atmosphere? How does the respiration of 6,000,000,000 human beings alter the level of CO₂ in the atmosphere? What about the animals and plants we raise to feed ourselves? What about our machines?

How much do scientists know about the cycles of the climate? The farmers of humanity have known about the annual cycles of weather. If there are any century-long cycles, should they have been noticed? What about millennia-long cycles? What began and ended the Ice Ages which once held our planet in an icy grip?

These questions hover in my mind every time I hear about a study of the global climate. To my knowledge, the understanding of all these factors and connections is imperfect.

It is true that speak as an untutored amateur in the atmospheric science, meteorological science, and the other related sciences. However, I speak as an experienced (if still unpaid--hence, amateur) student and creator of mathematical and computer models.

The scientists who constructed these models ought to know something about the limitations of their work. If they don't tell us about the limitations in their predictions, I suspect that they either don't know--or don't care to ask--what might make the predictions wrong.

At the moment, all I have is this suspicion.

Modeling

Next weekend, groups of college students from around the world will take part in a large (but rarely publicized) competition.

These students typically are in mathematical, engineering, or hard-sciences curricula at their school. They attempt to work in teams to analyze, model, and solve complex problems drawn from the real world.

The problems range from efficient construction of stunt sets in Hollywood to using computers to help medical doctors use sophisticated cancer-fighting tools (at the same link, further down the page).

The students are challenged to do research, create a model of the problem, and use the model to produce a solution. Before the end of the competition, the team must produce a paper which presents their results.

Out of the hundreds of teams which take part each year, a large number are included in the Meritorious category, and a small number achieve the Outstanding category. Outstanding papers are published by the agency that oversees the competition.

Some years, the Outstanding teams get an opportunity to present their results before government or industry leaders who are interested in the problem. In 2001, for example, a large number of teams studied the evacuation of South Carolina coast under the threat of a hurricane landfall. The authors of the Outstanding papers were invited to present their results to the government agencies of South Carolina who oversee such things.

I mention this partly because I owe a great deal to this competition. Without it (and without the professor who encouraged me to become involved), I would have not thought of pursuing my interests in mathematics.

It also taught me the strengths and weaknesses of computer models. A good modeling team can create a powerful model, but if the model ignores (or assumes to be unimportant) an important real-world variable, then the model has only limited use.

One example may help this: in a problem dealing with computation of lawful capacity of a public building, the modeling team must take several variables into account. Among these variables are the floor space in the building, the pathways to the emergency exits in the building, the type of the exit (single door, double door, sliding door, etc.), and the distribution of the exits around the edge of the building. However, the modelers will also need good information on how to model crowds of people moving through a restricted passage. Each of these factors may need to be modified by the kind of emergency which occurs.

If the modelers ignore any of these factors, then the usefulness of the model decreases significantly. However, if they pay attention to all pertinent factors, the model can be used as a good resource. At this level of certainty, the model is often said to have proved a particular result.

Of course, there are also computer modeling attempts which are much more open-ended. When studying events like asteroid impacts, the number of possible variables scales upwards into the hundreds. There are wind patterns, dust clouds, the short-term (and long-term) effects of asteroid impact in the middle of a glacier. There is also the possible alteration in reflected and absorbed solar energy, due to the aforementioned dust clouds. Such alterations will affect different regions of the planet in different ways; they may also affect the severity and quantity of storm systems over the oceans. The number of variables involved in each of these calculations is large. The number of possible interactions between the variables is also large. Lastly, the interactions aren't all well-understood.

In these situations, the model may produce a large number of predictions--but it is much harder to say what exactly the model proves. That question is rather open-ended, especially if empirical data is lacking on the subject.

In the case of large asteroid impacts on Earth, I suspect that few people want to actually see such an event happen, even though it will generate great amounts of empirical data.

Computer models are powerful tools. But like most results of academic study, the results of the model are as dependable as the sources used to create the model; the results are also as rigorous as the analysis done by the model architect. Lastly, the usefulness of the modeling results are heavily dependent upon the assumptions incorporated into the model.