Philadelphia-area artist Paul Santoleri draws/paints amazing images that are evocative of fractals. Not the Mandelbrot-class fractals that are the staple of most fractalologists and fractal software, butreminiscent of the fiendishly delicate and organic-looking fractals created by Clifford Pickover.

Only Pickover does his with a computer, while Santoleri paints his. The image at the top of this post is titled Red Spider, and is a 2’ by 2’ painting using acrylic on canvas. (Santoleri also paints on a very large scale: he is also a muralist, and has done 70+ large murals around the world, with a few in Philadelphia.

Here’s a fairly well-known Pickover creation titled From the X-Files. See more of Picover by clicking here.

While Pickover has done an enormous amount of spreading the fractal word, and has been a great spokesman for the melding of art and mathematics, I find Santoleri’s work much more interesting because its organic-like nature is truly evident, paintedwithout any dependence on computers. In Santoleri’s own words:

In my works I erase the borders between the visible and invisible matter and create a new medium generating object and beings whose meaning, gender, and whereabouts is unclear and not important. My works should be viewed just as a part of the moving beyond their bounds whole that knows no spatial or temporal limitations.

Pickover certainly captures an artistic view of infinity with his images, but they are more of space alone and can’t compete with Santoleri’s temporal infinity.

The tag line for this blog, indeed the overarching theme of the Chaos and Fractals course is the fuzzy three-legged monster of Modeling, Understanding, and Prediction. Fuzzy because the boundaries are never clear; they are themselves fractal-like. Suffice it to say that the non-linear dynamic modeling of most systems is used for both the understanding the models provide about why something happens the way that it does as well as the prediction of future states.

Which brings me to quantum mechanics - a field where models routinely predict experimental results with extraordinary accuracy, yet there is still debate on what it all means. Taking this to an extreme, if there is disagreement among physicists on the meaning of quantum mechanics (specifically, the meaning of the quantum mechanical wave function and the nature of observation) then there is a lack of understanding. Whether one considers this "good prediction, no understanding" scenario unsatisfying or not comes down to one’s proclivity for philosophizing.

Actually, for me I developed a "proclivity for philosophizing" because of Quantum Mechanics.

I remember as a physics student being totally mystified at and angry with quantum mechanics. Sure I could do the mathematics, but I really had no clue as to what a stationary state was, or what it meant for a wavefunction to collapse. Even after getting very good grades in both semesters of quantum, I really couldn’t articulate the connection between the mathematics and reality in a reasonable way, or at least not in the tangible way that I could describe the (apparent) reality of Newtonian physics.

Click to enlarge the Quantum CatGrad school and a post-doc in solid state physics finally did bring some aha! quantum moments; I could finally talk-the-talk of quantum mechanics interpretation as well as theory. Like most students, I was taught the Copenhagen Interpretation of Quantum Mechanics as promulgated by Bohr and Heisenberg in the 1920’s. I adopted it whole heartedly, and soon Schrödinger Cats and Wave-Particle Duality were common topics of late-night sessions, often fueled by single-malt Scotch. But there was something very strange about this: even though I became fairly fluent in discussing all sorts of quantum paradoxes, and could recite quite a bit of quantum philosophy (thanks to reading books such as Max Jammer’s Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective ), this was basically a quantum bluff.

And now, my confession: I still didn’t really understand the quantum stuff in any satisfying way. I was in the odd position of believing and knowing what I was saying, but not really understanding the universe in terms of physical laws. I guess it’s just that I always struggled with the fact that the universe was extremely wacky and random, and any attempt to understand it in terms of simple particles and interactions was basically a waste of precious time and energy. I had to change my view of what was knowable, and totally re-visit my beliefs of what type of stuff that makes up the universe.

Now the Copenhagen interpretation is supposed to help out here, and it does, but with an anti-epistemology twist. According to this interpretation, the measurement act is crucial in establishing reality. Therefore, certain pre-measurement questions cannot be asked - not because the answers can’t be known, but because concepts such as the position of a particle before a measurement is effectively meaningless. In other words, the Copenhagen Interpretation is the don’t ask, don’t tell policy of the atomic world.

There have been many other interpretations of quantum mechanics that have been proposed: ensemble, consistent histories, many-worlds, pilot waves + several more. In addition to these, the internet is full of amateur ontologists positing their own& view of reality.

I certainly enjoy reading and wondering about the other interpretations but, if pressed, I would have to say that I am more in the Copenhagen camp. Whether this is because I truly believe that it is the best may not be as important as the fact that it is the first one I learned. The fact that the Copenhagen Interpretation is the canonical interpretation really annoys a number of physicists who believe other interpretations are much better descriptions of reality. In a recent (July 2006) Physics and Society article, Marlboro College physicist Travis Norsen delivers an over-the-top diatribe against the doctrinal teaching of the Copenhagen interpretation in almost all physics curricula. He is not arguing that all of the basic interpretations should be taught - but he does argue strongly for the Bohm-de Broglie pilot wave idea. (Norsen basically describes the teaching of the Copenhagen Interpretation over Bohm-de Broglie as tantamount to the teaching of something as "non-scientific as Intelligent Design". This is not surprising - he does list Ayn Rand and Objectivism as a "hobby" on his web site.)

Click for larger view of the Neils Bohr Institute
Actually, the reason I’m a Copenhagener is much simpler: when I last traveled to Denmark I had my picture taken with my best friend Eric in front of the Neils Bohr Institute, and imagined myself as one of Bohr’s or Heisenberg’s students in 1920 Denmark, helping them to work out the mysteries of the quantum world.

Over a few glasses of single-malt scotch.

Underwater silica streamers in New Zealand. Click to enlarge.
I recently came across an interesting quote by Robert Root-Bernstein, a MacArthur Fellowship "genius" teaching physiology at Michigan State, and the controversial author of Rethinking AIDS.

The quote is from a letter to the editor in the July 2006 Physics Today written by Kent Eschenberg

Most eminent scientists agree that nonverbal forms of thought are much more important in their work than verbal ones. This observation leads me to propound the following hypothesis. The most influential scientists have always nonverbally imagined a simple, new reality before they have proven its existence through complex logic or produced evidence through complicated experiments. ...I suggest that this ability to imagine new realities is correlated with what are traditionally thought to be nonscientific skills—skills such as playing, modeling, abstracting, idealizing, harmonizing, analogizing, pattern forming, approximating, extrapolating, and imagining the as yet unseen—in short, skills usually associated with the arts, music, and literature. (Click here for full quote.)

Root-Bernsteininvestigates creativity and is a champion of the essential nature of the art-science interface. (He himself is something of a digital artist.) In his Music, Creativity and Scientific Thinking, he goes even further than the above quote, clearly putting scientists and artists at the same level:

We will therefore be able to recognize the greatest breakthroughs in the use of the human imagination precisely by their inability to be subsumed into the existing categories of either science or art.

Root-Bernstein’s work on the linkage between science and the arts has always sounded right to me, although I must admit that I suffer from a common malady among scientists: the belief that we are somehow more musically or artistically inclined than those in non-science disciplines. (I admit to being a fledgling blues piano player - mainly repeating blues riffs in the Key of C for the past 40 years or so) The advent of fractals, and the appearance of photoshop-like softwarefor fractal manipulation has removed this perceived asymmetry , which really is nothing more than basic elitism. Scientists and artists regularly produce fractal "art" , and compete head-to-head in juried competions (see my earlier post on the Mandelbrot Fractal Art Competition) It doesn’t matter whether the producer of the image knows the mathematics behind the images, or doesn’t know to make a simple sketch of a figure or still life.

What matters is the ultimate object of art and that ever-present eye-of-the-beholder rating scale.

Of course, it’s not just fractals that have helped forge the science-art divide. Photograph is also a rich source of science/art interplay. A great example of this is the figure at the top of this post. Taken by Duncan Graham, a geothermal scientist in New Zealand and titled An Angels Wing, the photo was part of the 2004 New Zealand exhibit Unseen Worlds – New Dimensions.

From the press release for the exhibit:

" Unseen Worlds – New Dimensions explores the shared ground between art and science, and is designed to give New Zealanders an opportunity to see the amazing images that scientists create in the course of their everyday work. "

Here’s a description of Duncan’s everyday work: he took the picture using a digital camera and glass-bottomed bucket … in a wastewater drain at the Wairakei Power Station, near Taupo…These deposits form when silica dissolved in the water precipitates on micro-organisms shaped like long filaments. As the micro-organisms grow and multiply, they build up thick deposits of silica which trail out in the surrounding water currents.

And now a lesson for artist-scientists: don’t dump out that wastewater.

The ubiquity of fractal art can be observed with a simple web search, where a recent google query yielded 640,000 hits. Some will claim that thisfact does not convey any artistic status to fractal images, only that an awful lot of folks name their works fractal art.

Which is why I’m happy to see that some big-time mathematics groups are finally recognizing the potential for creative artists to produce captivating art - even if they don’t know the mathematics.

Earlier this year (August, 2006)the International Congress of Mathematicians (ICM) - the group that decides therecipient of the Fields Medal (the world’s top mathematics prize) includeda fractal art exhibit. The exhibitors were chosen after competing in the2006 Benoit Mandelbrot Fractal Art Contest. Please visit the site where you’ll be able to see the winners as well as a number of other entrants.

The promo for the exhibit does a great job of arguing for the existence of fractal art that is truly art:

The exhibition is formed by a collection of computer generated images by a group of artists and/or scientists specialized in fractal art. The mathematical expressions and the parameters used confer a unique and distinctive colour and aesthetics to every image. Much like painters and sculptors transmit their personality and sensibility to their works by means of their technique, the authors of this exhibition express themselves by means of formulae and algorithms, modifying them progressively until the desired goal is obtained; reaching the frontiers between Art and Mathematics. The synthetic computer generation of every picture may look cold and mechanical, but behind every picture there are hundreds of hours of work in the formulae, algorithms and parameters generating the picture. These artists use computer and fractals (and mathematics, in general) like a painter uses canvas and paint-brushes: creating a piece of art allowing us to capture sensibility and emotions; just like any artwork.

I actually like one of the non-winners’ works the best: Pere Soto, whose Alien Images of Coca-Cola appears at the top of this post. Soto is both a guitarist and a digital artist. His web site is definitely worth exploring - while most of it is dedicated to his music, his web design is beautiful. (You will also find a gallery of his digital artworks.)

With the ICM held in Madrid (the contest was sponsored by the Fundación Española para la Ciencia y la Tecnología (FECYT)-i.e. the Spanish Foundation of Science and Technology), perhaps it’s understandable that there are quite a few Spanish participants in the contest. Or maybe it’s just that there’s something endemic to the culture that produced Goya and Dalí that fosters the embracing of new formsand techniquesthat push the limits of artistic representation.

Click to enlarge mapThe Arctic National Wildlife Refuge is a flash pointfor environmental and energy issues, combined in a volatile atmosphere of anti-terrorism. If only the U.S. could start drilling for oil there, the up to 8 billion estimated recoverable barrels of oil would surely lower our dependance on foreign oil, especially from the Middle East.

Or so proclaim those in favor of opening the refuge for drilling. (See ANWR.org - with a tag line of "Jobs and Energy For America" - for a collection of pro-drilling arguments.)

Currently, ANWR drilling is not permitted by federal statute. Over the past few years there have been several attempts in Congress to allow drilling - on some occasions from the House, only to be turned down by the Senate, and then from the Senate, shot down by the house. (Click here for more details.)

Even though it appears as if those against drilling are currently holding their own, the no-end-in-sight Iraqi situation combined with the explosion of jihadist sentiment and activity could tip things enough so that both the Senate and House finally agree to open up the slopes for drilling. (Which is just one of the reasons that the upcoming mid-term elections, and possilbe change in majority party in both the Senate and the House, are extremely significant for the ANWR.)

The Sierra Club is one of the more public groups leading the charge against ANWR drilling. In addition to the obvious environmental arguments against drilling, Sierra also claims that, even with massive driliing, the price at the pump will only drop approximately a penny a gallon, i.e. the amount available is so small, and the cost of extracting and refining the oil so large that very little will happen to the retail cost.

I had not seen anyanalyis that could shed light on just how ANWR production would affect U.S. oil supplies untilI read a recent article by Richard Wiener in the July 2006 edition of Physics and Society (vol. 35, no. 3). Wiener describes the use of Hubbert modeling for prediction of oil production. Hubbert models oil production logistically, and Wiener extends this by using multiple logistic curves. With very plausible values for the parameters in his logistic model, Wiener predicts that large-scale drilling at ANWR will not help lower the U.S. reliance on foreign oil. Instead, it will only slow down the rate of our dependance on foreign oil! This result sound very much like the typical election-year bantering about reducing the deficit: most of the time, what is being talked about is reducing the rate of deficit growth.

It is often the case that modeling predictions are picked apart by those on the opposite side of the ideological aisle, but in Wiener’s case, his conclusion is too obiously true to really be refuted: dependance will only go down with significant drop in demand on the part of the U.S.

And no modeler out there is ever going to predict that.

Iceberg in Jacobshavn Isfjord, Greenland. Photo by Evert Wesker.Greenland holds an important place on the frontier - both of the habitable world, and of  global warming.  Greenland is where sudden climate changes have been mapped, a phenomenon that was one of the first markers of global warming to be widely accepted.  (Click here for a  previous post.)   

Greenland is back in the global warming news:  the rate at which ice is leaving Greenland (through meltwater and ice shearing) is apparently accelerating. 

The study was done by Eric Rignot (NASA’s Jet Propulsion Laboratory) and and Pannir Kanagaratnam (Univ. of Kansas Center for Remote Sensing of Ice Sheets.) 

Ten years ago Greenland was losing ice at a rate of 22 cubic miles/year.  It has now increased to 53 cubic miles/year.  The size of this number is staggering. (22 cubic miles/year was already incomprehensible.)   53 cubic miles has a weight of approximately 2.4 trillion tons.   To put this number in perspective, an asteroid that is totally iron would be over a mile in diameter to have the same weight!

Glaciers at the edge of Greenland have picked up the pace to move this amount of ice,  and some are moving up to 8 miles/year.

While some dispute the magnitude of the data, very few doubt that Greenland is losing ice. 

The main culprit is most likely ocean warming leading to faster glacier movement.  Because the increased rate of ice loss was not expected, it is clear that climate models must again be modified to account for this.  This is not particularly easy given that the more ice lost, the higher the ocean level rises, but also there is a potential for lower water temperatures near the edge of Greenland.  The lower temperatures may then slow down the ice loss.

The call for better modeling is urgent, according to one of the researchers (Rignot):

"The Greenland ice sheet's contribution to sea level is an issue of considerable societal and scientific importance... These findings call into question predictions of the future of Greenland in a warmer climate from computer models that do not include variations in glacier flow as a component of change. Actual changes will likely be much larger than predicted by these models."

Having had a chance many years ago to travel to Thule , Greenland, I am saddened by the thought that this great oasis of ice and tundra - the most austere and beautiful place that I have ever seen - is the biggest loser in our civilization’s increasing negative effect on the planet. 

The fundamental constants (e.g. the mass of the electron, the universal gravitation constant, Planck’s constant) are given that name for good reason: all computations used to understand or model the physical universe rely on their values.In addition to the fundamental constants, there are other constants that define our universe. These numbers are really the statistics of the world, e.g. the heights of mountains, lengths of rivers, masses of the planets, etc.There is a very odd, hard-to-believe-at-first- sight mathematical law that describes the distribution of these constants. In 1938, Frank Benford analyzed over 20,000 numbers taken from the fundamental constants of physics and far-removed areas such as sports stats and street addresses. Benford wanted to measure the frequency distribution of the starting digits for these numbers because of another odd fact - in antique tables of logarithms, the first few pages are often more worn, indicating that the log user was thumbing through the first pages much more frequently than latter pages, i.e. the logs with a "1" as a leading digit.Now here’s the amazing part. Instead of determining that starting digits from 1-9 appeared with approximately the same frequency, Benford found that the numbers he was studying began with a "1" a disproportionate 30% of the time. Benford went further, measuring the entire frequency distribution of starting digits, and developed a formula that predicts this distribution - a formula known as Benford’s Law. The Law predicts that the frequency of occurrence of a digit drops off logarithmically with increasing digit size, and therefore numbers starting with 9 appear less frequently than all other numbers .The explanation for the law comes from the fact that the numbers analyzed have units, and the important fact that the law is independent of the system of units. For a mathematical derivation of Benford’s Law based on this scalability between units, click here. (And for a Java simulation, click here.) Benford’s Law holds true in a surprising number of situations, and has been described as showing that "natural processes can be remarkably resistant to complete randomness. "What we have here is a nice example of a different type of universality. Where Feigenbaum showed that all functions of a certain type behaved in the same quantitative way, Benford has found that all fundamental and statistical constants follow the same frequency distribution.So we live, apparently, in a universe that only parcels out numbers in a certain way. Even though there are an infinite number of numbers (and, according to Cantor, an infinite number of different size infinities of numbers), this fact seems terribly limiting. Which is why I was happy to come across a fascinating site called The Secret Life of Numbers, a site that describes a very interesting project: what are the most popular integers?Sort of like an American Idol for numbers, the authors designed software that worked in concert with search engines to find the popularity ranking of every integer between 0 and 1 million.OK. A strange project. Why did they do it?

The resulting information exhibits an extraordinary variety of patterns which reflect and refract our culture, our minds, and our bodies.For example, certain numbers, such as 212, 486, 911, 1040, 1492, 1776, 68040, or 90210, occur more frequently than their neighbors because they are used to denominate the phone numbers, tax forms, computer chips, famous dates, or television programs that figure prominently in our culture. Regular periodicities in the data, located at multiples and powers of ten, mirror our cognitive preference for round numbers in our biologically-driven base-10 numbering system. Certain numbers, such as 12345 or 8888, appear to be more popular simply because they are easier to remember.Humanity’s fascination with numbers is ancient and complex. Our present relationship with numbers reveals both a highly developed tool and a highly developed user, working together to measure, create, and predict both ourselves and the world around us. But like every symbiotic couple, the tool we would like to believe is separate from us (and thus objective) is actually an intricate reflection of our thoughts, interests, and capabilities. One intriguing result of this symbiosis is that the numeric system we use to describe patterns, is actually used in a patterned fashion to describe. We surmise that our dataset is a numericsnapshott of the collective consciousness. Herein we return our analyses to the public in the form of an interactive visualization, whose aim is to provoke awareness of one’s own numeric manifestations.

The applet at the Secret Lives of Numbers Site is beautiful to use: you can select broad ranges of numbers with the mouse, zooming in and out at will The displays are dynamic, and it’s probably no surprise by now that the pattern of peaks is fractal-like (see the image on the left). As you zoom in on a smaller range you will get a very self-similar distribution.When you click on a peak in the frequency distribution corresponding to a "popular" number, an info screen presents the important stuff: the number’s ranking, and the reasons for its popularity. For example, in the chart on the left, the 7500 peak is characterized by a ranking of 1196/100K, which puts it in the 98.8th percentile. The reasons for number’s monster performance? 7500 appears as a model number for a number of different items (and manufacturers): Motorola 7500, hp7500, Dell Inspiron 7500…So break free from Benford’s Law and try out the Secret Life of Numbers .Check your birthdate, your address, your telephone number, and become aware of your "numeric manifestation."

  "Ikebana" by Tim Fadden     

Recently, the following provocatively simple set of questions was added to an earlier post describing Jackson Pollock and fractals - one of a series of posts that debate whether technology & mathematics -produced art can actually be ART:

why does the mathematics have to be producing art? why can't the mathematics be art itself? would that actually solve any of the problems discussed here.

This question is from Jonathan "Fish" Fisher - co-winner of the 2005 Duke poetry award. Fish has posted a number of interesting comments and questions, adding his voice to the dialogue of whether or not art can be produced using mathematics and technology. He recently posted his Poem For Benoit Mandelbrot, A Connoisseur of Chaos, a moving poem that ends that concludes with his own awareness of the beauty and mystery of mathematical forms and structure via Mandelbrot’s fractals:

Briefly I taste the salt sting Of equations I'll never understand, But a wave of awe Sweeps me up as if divine Artistry had finally Conformed to a function of Some rigid geometry. Can any man be more than an artist?

I am always heartened when a non-mathematician senses the beauty of mathematics, which Fish makes evident in his poem, and his suggestive question: why can’t the mathematics be art itself?

There are mathematicians who will go into full rapture mode when describing the beauty, in their eyes, of mathematics. This is especially true when they describe mathematical proofs that are considered "elegant" - proofs whose logical path leads inexorably from axiom to conclusion using an economy of steps, with leaps to other areas of mathematics hitherto unrelated, demonstrating a dizzying web of connections among the farthest reaches of mathematics.

This view is typified by G.H. Hardy, an early-mid-20th-century British number theorist, and the author of A Mathematician’s Apology, his 1941 memoir and a book that should be read by all mathematicians. Hardy writes…

In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

Applying some logic here, mathematics must be art because it has an art-essence that is independent of the observer. If you don’t see it, it must be there.

The quotes from G. H. Hardy in the previous post are well known - in mathematical circles at least. Their succinctness is a hallmark of Hardy’s desired economy of words in mathematical proofs.Hardy was a number-theorist, and it seems appropriate to follow up the posting of his quotes on the beauty of mathematics with a very interesting fictional passage about the nature of numbers, and the connection with human life.The following passage is from Smilla’s Sense of Snow, a 1992 novel by Danish author Peter Høeg. A strange mystery that involves a child’s murder and an eery trip to Greenland, Smilla is a fascinating heroine, at home in the worlds of mathematics, intrigue, and, obviously, snow…

Do you know what the foundation of mathematics is? … The foundation of mathematics is numbers. If anyone asked me what makes me truly happy, I would say: numbers. Snow and ice and numbers. And do you know why?Because the number system is like human life. First you have the natural numbers. The ones that are whole and positive. The numbers of a small child. But human consciousness expands. The child discovers a sense of longing, and do you know what the mathematical expression is for longing?The negative numbers. The formalization of the feeling that you are missing something.And human consciousness expands and grows even more, and the child discovers the in-between spaces. Between stones, between pieces of moss on the stones, between people. And between numbers. And do you know what that leads to? It leads to fractions. Whole numbers plus fractions produce rational numbers. And human consciousness doesn’t stop there. It wants to go beyond reason. It adds an operation as absurd as the extraction of roots. And produces irrational numbers.It’s a form of madness. Because the irrational numbers are infinite. They can’t be written down. They force human consciousness out beyond the limits. And by adding irrational numbers to rational numbers, you get real numbers.It doesn’t stop. It never stops. Because now, on the spot, we expand the real numbers with imaginary square roots of negative numbers. These are the numbers we can’t picture, numbers that normal human consciousness cannot comprehend. And when we add the imaginary numbers to the real numbers, we have the complex number system. The first number system in which it’s possible to explain satisfactorily the crystal formation of ice. It’s like a vast, open landscape. The horizons. You head towards them and they keep receding.

Greenland icecap. Photo by Evert Wesker.Luckily for all of us, the madness hasn’t stopped. The complex number system is essential to fractals, and forms the mathematical plasma in which algorithms for determining the Mandelbrot set are calculated. This set is like a vast, open landscape with horizons replaced by fractal basin boundaries. As you head towards them, they don’t recede. Rather, they get more complex. Without complex numbers, then, we’d be living in world bereft of color and self-similarity - an uninteresting world of snow.

Notes - Thanks to Sharon Armstrong, of La Salle’s Psych department, for reminding me of Smilla’s mathematical musings…Julia Ormond played Smilla in the movie version. Click here for a film clip of the mathematics scene…And be sure to try this excellent fractal basin applet.

A fractal-like prayer rugMy post on March 19, 2006, described the Templeton Foundation, whose mission is to

“pursue new insights at the boundary between theology and science through a rigorous, open-minded and empirically focused methodology.”

Prayer research is a fascinating topic and may well continue in additional modes to that presented as the outcome of the STEP project. However, the null results obtained by the methodologically rigorous STEP experiment appear to provide a clear and definitive contrasting result to an earlier published finding (Byrd study) of a positive effect for patient-blind distant intercessory prayer in a prayer experiment involving recovery of patients in a cardiac care unit.* Result: The STEP project did not confirm these findings.*Note: The Byrd study also involved randomization to receive or not receive patient-blinded distant intercessory prayer

Clearly, the null result is troubling, and I assume that the Foundation will fund studies looking for those “additional modes.” And, at a funding level of $2.4 M - the cost of STEP - there should be many more research proposals coming in.Naturally, there are many bloggers on both sides of the science/religion chasm who have weighed in on the results. Most are predictable, polarized responses, but they do share a common theme: in addition to the staggering amount spent on the study, the question of the efficacy of prayer is just not researchable, and will never be.To understand why, just read Jeff Mullin’s commentary for the Enid News & Eagle in Oklahoma:

One of the study’s co-authors, Dr. Charles Bethea, a cardiologist atIntegris Baptist Medical Center in Oklahoma City, told the Associated Press “Intercessory prayer under our restricted format had a neutral effect.” To which I say, how do we know? How do we know some of the people being prayed for wouldn’t have died without pleas for divine intervention? How do we know some of the people who developed complications despite prayer wouldn’t have been much worse off without it?The point is, we don’t. Prayer and its effects are not something scientists can measure, study or accurately quantify. But one thing I know is, prayer works.