Today's buzzword: nonrenormalizable

 kw: science, philosophy of science, explanations

In a post last week I reviewed The Grand Biocentric Design: How Life Creates Reality, by Robert Lanza, MD and Matej Pavšič, PhD, with Bob Berman. Much of the post expressed my objections to the theory that "life creates reality", a theory based on a dramatic over-extension of the Copenhagen Interpretation of quantum theory (I disagree also with Neils Bohr, who concocted CI). 

I decided now to add a brief discussion of a point made several times in the book, that the version of quantum gravity theory these authors espouse is "non-renormalizable". The word is never explained. Perhaps I can at least give the public some understanding of "normalization" and "renormalization".

A problem arose in the early 1900's when scientists began to solidify the mathematics of quantum electrodynamics (i.e., the dynamics of charged particles in the quantum realm). If equations for the behavior of an electron (for example) were to be solved, partway through the solution one had to contend with expressions that divided by zero, leading to "infinities". Eventually, mathematical methods were developed that "renormalized" these equations, so that the term(s) leading to division by zero could be removed before one actually had to calculate the result.

To understand renormalization, it helps to first understand normalization, which is used to derive the basic equations of differential calculus. The term "derivative" means a function, derived from another function, that expresses the slope at any point. It is the basis for a large family of functions that are needed to optimize functions and, used "in reverse", to calculate areas and volumes (among many other useful summation operations). 

I will illustrate by deriving the "first derivative" of two functions. The first is a basic parabolic function, y = 5x². This states that, in an x-y coordinate system, the value of y is found by squaring x and multiplying by 5. Thus, for x = 3, y = 5*3*3 = 45. The second is a simple quartic, y = x⁴. For this one, when x = 3, y = 3*3*3*3 = 81. Both functions are very useful in mechanics: the position of an object falling in a uniform gravitational field is expressed by a parabolic function, and the energy radiated by a heated body is related to the temperature by a quartic function.

This illustration shows both functions graphed in the domain x = [-1,2].

NOTE: There are several notations used in calculus, because of the complicated history of its discovery three-plus centuries ago. Here I use notation based on that of Gottfried Liebniz (d. 1716).

The mathematical slope of either of these curves is dy/dx, and one can approximate it by calculating the functions at x and x+dx, for very small dx. But the exact value can only be found when dx = 0!

Below is the derivation for the parabolic equation as I learned it in high school.

"Lim" means the limit of the expression in brackets, as the term in parentheses is satisfied. We do the derivation using dx and dy as algebraic variables.

The three lines in the middle expand the function. The following line is the "normalization": dividing by dx removes dx from one of the terms in the equation.

Then, as shown in the last line, when we set dx to zero, the term that contains it vanishes, leaving us with a function in x only. This function is the first derivative of function y1.

Let's drive the point home by performing the same operation on the quartic function:

This time, multiplying out the function results in a larger number of terms. As before, the calculations are shown in the three-line cluster in the middle. The next step, normalization, divides by dx to yield one term that doesn't include dx, and three others that include it. Setting dx to zero yields the first derivative of function y2.

To summarize: Normalization is a mathematical process of deriving expressions that would contain a division by zero if carried out with numbers, but then dividing out the term that will become zero, so that a perfectly calculable function remains.

We can do a simple verification to see how these derivatives perform. Here is a table of the two functions, calculated from the values of x as given. The graph above was produced from these numbers.

In the graph above the green lines are anchored to x = 1.5. Let's start with dx = 0.1 and calculate y1 and y2 at x = 1.6, subtracting values found in this table at 1.5:

y1(1.6) = 5*1.6*1.6 = 12.8; dy1 = 12.8 - 11.25 = 1.55; dy1/dx = 15.5.

y2(1.6) = (1.6)⁴ = 6.5536; dy2 = 6.5536 - 5.0625 = 1.4911; dy2/dx = 14.911.

We can shrink dx to 0.01, so we calculate at 1.51. The results are then 

dy1 = 11.4005 - 11.25 = 0.1505; dy1/dx = 15.05.

dy2 = 5.19885601 - 5.0625 = 0.13635601; dy2/dx = 13.635601.

These numbers seem to be closing in on 15 and something more than 13.

Now let's find exact slopes using the two derivatives:

Slope #1 = 10*1.5 = 15.0

Slope #2 = 4*(1.5)³ = 13.5

It's possible to use smaller and smaller values of dx to get a converging series, from which we can get pretty accurate results. However, the mathematics of the derivative ensure that the values calculated by them are exact.

Based on all that, what is Renormalization? Here I must get conceptual because the functional expressions are enormous, and people got Nobel prizes for figuring them out. Calculus, which is already based on Normalization, was used to determine the functions needed by quantum electrodynamics. The equations were labored over until a version of each derivation could be performed that "divided out" the expressions (analogous to dx but much more elaborate) that caused trouble. This second dividing-out process is renormalization.

In one of his books, Richard Feynman tells of being asked, at the Nobel Prize ceremony, "What did you do?" He answered, "Buddy, if I could tell you that in one minute, it wouldn't be worth a Nobel Prize." Renormalization was part of it.

So what is Non-renormalizable? The authors of The Grand Biocentric Design make much of their contention that a quantum theory of gravity cannot be renormalized. No such theory has yet been fully developed, as the authors admit. Since we don't know what form such a theory will take, we (and they) don't have a way to determine if their contention is anywhere close to being correct.

On a plane ride several years ago I sat with a young man who, it developed, is a cosmologist. I said, "I have a question I've been hoping to ask a cosmologist, if you're willing to give it a crack." He agreed. I asked, "If gravity is a quantum phenomenon, that means there is a gravitational quantum, a 'graviton'. The most powerful sources of these gravitons are black holes. What is it, for a black hole or for any other massive body, that emits the gravitons? If they start 'inside', where the mass is, how do they get out?" He said, "That's two questions, but it's worth some thought." After a half hour of silence he said, "By analogy to photons, which are emitted when charged particles such as electrons are accelerated or make a quantum jump from one orbital to another: The motion of an electron may cover at most a nanometer or two, particularly in an orbital transition. The photon that is emitted has a wavelength of a few hundred nanometers or more. We don't yet know what is the wavelength of a typical graviton. But I think gravitons are emitted over an area larger than the typical black hole, not from inside it. That's the best I can say on short notice." That was pretty good! I reckon it will be a while before a better answer is in the offing.

The Man Who Shaved the Universe

 kw: book reviews, nonfiction, science, astronomy, philosophy of science

I was a developer of scientific software for forty years. One bit of my "Coder's Credo" is, "A complex system that works began as a simple system that works." In practical terms, this meant that I had to first "get the science working", which was usually simple, at least conceptually. The complications that had to be added all derived from the user interface (making the software usable for humans) and the data interface (coupling it to the database or knowledge base). I built my career on a minimalist approach: Add new stuff only when there is a clear advantage.

Millennia ago, the Universe seemed simple compared to the Earth. In the night sky, stars were thought of as distant lamps stuck to a "firmament". The Sun, Moon, and five rather bright "wanderers" (in Greek, πλανόδιοι, which became "planets") were a complication that most folks ignored. But certain curious ones began to theorize; they wanted to figure out how the sky worked.

Fast-forward to a mere 21-22 centuries ago. The prevailing theory of the sky, at least in Europe and north Africa, was a nest of concentric, "crystalline" spheres. The outer sphere held the "fixed stars", and the seven wanderers were each ensconced in its own sphere. Over time, observations of the motions of these "planets" showed something odd: they didn't all march across the face of the "fixed stars" at a steady rate, and some looped back on themselves. Also, the Moon's apparent size changed a little. By about 150 AD, a system of epicycles attached to the spheres had been developed to better model the movements of the planets, including the Moon and Sun.

This illustration from an Arabic document of the 1300's shows the epicycles needed to model the motion of Mercury, shown at four times during a particular year. This image is from Alamy (a commercial site), where its epigraph says,

"Ibn al-Shatir's model for the appearances of Mercury, showing the multiplication of epicycles using the Tusi couple, thus eliminating the Ptolemaic eccentrics and equant."

This shows that Arabian astronomers went beyond Ptolemy. At its height in the first half millennium of the Christian era, about 80 epicycles were needed for a "good" model, and the notion of "crystalline" spheres was politely ignored. Here, I count six epicycles needed to produce motions for Mercury that matched astronomical observations.

We all know that Copernicus tried to simplify the Solar system by recognizing the Sun as its center. However, he also needed epicycles to model planetary motions accurately, because he thought all orbits were perfect circles centered on the Sun…or, at least, the rotational center of a cluster of circular epicycles followed a circle about the Sun.

Leaving behind circles in favor of ellipses, Kepler, using Tycho Brahe's data for positions, produced a greatly simplified model of the Solar system, such as that seen here (this one leaves out Saturn, at twice the distance as Jupiter).

This particular image also shows the orbits of several major asteroids and three comets. Comet Halley's ellipse extends to 35 AU, seven times as far as Jupiter. The orbit that just brushes past Jupiter belongs to Comet Kopff, one we never hear of because it is visible only with a telescope at least 4" in diameter.

The older tradition of natural philosophers, exemplified by Ptolemy, resulted in models of natural phenomena with steadily increasing complexity. Something happened about the time that Ibn al-Shatir began writing his astronomical manuals, that began to turn the study of nature from natural philosophy to science as we know it.

Here I turn to a better authority on science history, Johnjoe McFadden. In his book Life is Simple: How Occam's Razor Set Science Free and Shapes the Universe, Professor McFadden traces the progressive simplification of science and scientific theories, based on a 14^th Century meme we call Occam's Razor. This is expressed in several ways, as it was by William of Ockham in the early 1300's. I like, "Do not multiply entities beyond necessity." This statement does not disallow complexity, it discourages unneeded complexity. Einstein's version is, "Make things as simple as needed, but no simpler," which looks at the matter from the other end.

Either way one looks at it, the principle known as Occam's Razor slices away unnecessary encrustations from scientific models. Before reading Life is Simple, that's about all I knew of the matter. I didn't even know that William, born in Ockham, lived in the early 1300's, about 700 years ago. This was just before the era of Geoffrey Chaucer (Canterbury Tales), who was born just a few years before William of Ockham died. The "English" of the day was Middle English, when the use of "thee" and "thou" and "doest" for "does", still found in the King James Bible, were at their height. But William wrote in Latin, which requires just a tad more translation than Middle English.

Neither did I know how the Razor grew and spread among the literate people of Europe and the Middle East. By the time of Kepler, 300 years later, and Newton, a generation later, simplification of theories was accepted throughout the world of the Enlightenment. The thread of the Razor through history is followed in all its excursions, leading to its dominance today.

It has become the ambition of many scientists to determine a Theory of Everything, which can be expressed on a T-shirt as a single equation that unifies not just the Weak and Strong and Electromagnetic forces, but also Gravity and Quantum Mechanics. Such a theory would not be a theory that "explains" everything, for a corollary to the Razor is, "That which explains everything explains nothing." The prolific clusters of epicycles in cosmology are an example. The more cycles you add, to account for refinements in astronomical observations, the less you actually know about them. The laws of orbital areas derived by Kepler, and the three laws of motion of Newton, as modified by Einstein, allow us to calculate exactly where each planet, moon, asteroid, comet, and artificial satellite is going, for decades or centuries into the future, and where they were at any time in the past. The calculations are tedious, but not difficult, and modern computing machinery shoulders the load of the tedious part.

Sadly, many (most?) modern theorists have gotten bogged down in String Theory. Somehow, these mathematical models require calculations in at least 10 or 11 dimensions (some versions, as many as 26 dimensions). None of the string theories so far proffered can be tested experimentally, and the number of possible string theories is a gigantic number with about 500 digits. And we thought 80 epicycles are too many! At the moment, this is a lot more "hair" than the Razor can manage to tame.

I was quite enthralled by the stories, the history, of how modern science developed once it was freed from the cosmogony of Aristotle and Ptolemy, which somehow became the foundation of Roman Catholic cosmology (for the curious: cosmogony is about "what is there", and cosmology is about "how it goes"). In effect, the Razor removed God's hand from the tiller of the Universe, at least so far as science is concerned. William of Ockham was also far ahead of his time in political understanding, which is probably a consequence of his revolutionary understanding of nature: he insisted that rulers' legitimate power came through the consent of everyone. His understanding of natural rights is an embryo of the Bill of Rights in our Constitution.

While I recommend this book for its historical perspective, I have a few quibbles about statements made by the author when he stepped outside his area of expertise, which is molecular genetics. Those who think my objections are TMI can stop here. What follows touches on three items that surprised me the most:

• On p 271, discussing the Planck Law for the spectrum of a heated blackbody, he writes that such bodies "emit light in a narrow band that depends only on the black body's temperature." Not quite. The actual spectrum of a blackbody (note the absence of a space) covers all wavelengths, and the width-at-half-height of the spectrum is about 2.8:1. For example, for a blackbody at a temperature of 7,250K (~12,600°F), the half-height spectrum ranges from 240 nm to 680 nm. The peak of the spectrum for this temperature is at 400 nm. The location of peak radiation depends on temperature, and the relative shape of the spectrum follows. An analogy about whacking a piano and somehow getting only a single note is quite bogus. The range of "notes" so emitted is strongest over more than an octave (18 half-tones), and there is some resonance from every string on the "piano".
• On p 293, about symmetry, "…time symmetry implies energy conservation, translational symmetry implies conservation of momentum, and Newton's third law, that every action has an equal and opposite reaction, is a consequence of rotational symmetry." About the last phrase: Where did that come from? Newton's third law is equivalent to time symmetry, and has nothing specific to do with rotation.
• On p 327, regarding the Bayesian likelihood of a particular combination of numbers being thrown in ten tosses of a 60-sided die, he states correctly that this is the tenth power of 60, or 60¹⁰, but then he evaluates it as 600 million to one. Hardly! 60¹⁰ = 6.05 x 10¹⁷, or 600 million times about a billion, or 600 quadrillion. Really! Don't any of his editors and readers know enough math to punch this out on a calculator?

That's enough of that. I can't blame him too much. Although I strive to be a generalist, I admit I know woefully little about molecular genetics, at least compared to Prof. McFadden. So, if I ever write a book that happens to wander into that arena, I'll see if he's willing to give it a read, and after he stops laughing, make the odd correction here or there.

Small book on a large subject

kw: book reviews, nonfiction, essays, memoirs, religion, philosophy of science

He claims to be an agnostic, one who prays and studies the Talmud daily and keeps kosher. Now age 98, Herman Wouk has written novels that span three generations of readers. While he still can, he has written of his exploration of the boundaries of science and religion, based in part on three conversations with Richard Feynman, and a great deal of his own experience. In their first conversation, Feynman asked him in parting if he knew calculus. He confessed he didn't, and Feynman responded, "You had better learn it. It is the language God talks."

The Language God Talks: on Science and Religion is Herman Wouk's memoir of how his stories developed in the context of his own religious convictions, and the people whose influence meant so much to the way they came together. He confesses that all attempts to learn calculus defeated him. Yet he has persisted nearly a full century in learning the original language God talks, the Hebrew of the Torah and its ongoing commentary, the Talmud.

The Hebrew itself is not really the language of which he speaks, but rather the incisive reasoning behind these great books. As he imagines himself saying to Feynman in a fourth conversation that he wished had happened, study of the Talmud, and earlier the debates and reasoning that produced it, represent the mental recreation of generations of thoughtful Jews during centuries with no technology of entertainment, no telephones or TVs or iPods (I wonder what the great sages would make of Wikipedia, whether they would contribute to it enthusiastically, or shun it. Probably the former!).

There is more insight into what it means to be Jewish in this little book than in any other I have read, of any size. I look upon the continued existence of the Jews as the foundational proof of God's existence. As a prophet wrote (here I paraphrase), God chose Israel not because they were greater or stronger or better or more numerous than the nations around them, but because He desired them for a testimony to His name. While He promised repeatedly to bless Israel, this "blessing" has been rather backhanded: they have endured continual attempts to exterminate them, for more than 3,000 years. Indeed, the first mention of Israel outside the Torah is an inscription in Egypt, dated about 1210 BCE, announcing that Israel had been "totally defeated". That long-dead pharaoh Merneptah spoke too soon.

But Wouk is the great writer here, not this poor scribbler, and dwells for a third of the book on his fictional character Aaron Jastrow, whose sermon "Heroes of the Iliad" is included as a coda to the book. That sermon, which morphed into a meditation on the meaning of Job's suffering, is the core of Wouk's belief and a most powerful statement of the Jewish understanding of the delicate choreography of God and His people.

What Wouk says indirectly I will state more frankly. God's own writing, His Bible, states that humans, "male and female", are in God's image and after His likeness. And just as we begin as infants and grow through a long process, so does God! If you read through the Torah, the Songs and the Prophets with a view to God as a juvenile, growing to maturity, you can reliably sort the books in time. The great figures of Genesis, particularly Abraham and Moses, sometimes had to talk God out of doing something rash. The man for whom the nation is named, Israel, formerly Jacob, was so named because he "wrestled with God, and prevailed". This view will be seen as rank heresy by my Christian colleagues, but I think a Talmud scholar would understand.

Readers in a FOG

kw: book reviews, nonfiction, philosophy, philosophy of science

Let us briefly discuss the Gunning Fog Index. It measures the level of education one needs to fully comprehend a document. Beginning with a sample of at least 100 words (200-300 is better), one counts three things: the number of words, the number of "hard" words, and the number of sentences. "Hard" words are words of three or more syllables except proper names and words whose third syllable is an inflection such as the "-ly" in "vividly" or "-er" in "reformer".

From the three counts, calculate (1) Number of words per sentence (Sw), and (2) percent of hard words (Hw). For example, we might find 187 words, 19 hard words, and 12 sentences. Sw = 187/12 = 15.6; Hw = 100*19/187 = 10.2. Then calculate 0.4*(Sw+Hw); for the example Fog = 0.4*(15.6+10.2) = 10.3. This means someone who finished the 10th grade ought to have no trouble reading the document...or at least that sample.

Few popular texts have a Fog Index greater than 10. Most "newspaperese" is found to be in the range 7-9, though online technical news such as I read from CNet and Yahoo can range from 12 to 16. Comic book captions (there is little other text) fall in the range 3 to 6 (Just for the record, my first paragraph above has a Fog Index of 8.5; if "vividly" and "reformer" are counted as "hard" words, it is instead 9.5).

What does a number like 16 mean? On the face of it, it means you'd find it hard to read unless you've completed a BA or BS in college. However, an interested amateur with a high school education and several years work or hobby experience will be able to read "tougher" text than just education might indicate.

Now, imagine getting a book in hand, and finding that the Fog Index is more than 30! When I began to (attempt to) read Exceeding Our Grasp: Science History, and the Problem of Unconceived Alternatives by P. Kyle Stanford, I found myself quite in a fog. I checked a few paragraphs in the first couple of chapters. Their Fog Indices ranged from 27 to 39!

Guess what...I ain't reading this book! I have fourteen years of college and graduate school. Peter Gunning would tell me I ought to be able to handle text that "fogs out" at 26. What does it take for text to attain a Fog Index of 39? About 300 words in four sentences (Sw = 75), and almost 25% hard words (that's more than 70 words such as "unrepresentative", "idiosyncrasy", "underdetermination", and "nonskeptical"). I find nearly every sentence to be sufficiently difficult to parse (I can't recall the antecendent when I get to the end of a 90-word sentence) that I simply don't have enough years left in my life to devote to comprehend the text in full. My opinion is that this is the worst-written technical treatise I've ever encountered.

With a little hopping around and puzzling out sections here and there, I can summarize the thesis thus: "Underdetermination of Theories" means that the evidence does not admit of any single, comprehensive theory, of anything. Many philosophers of science posit that, for any accepted theory, such as General Relativity, Quantum Electrodynamics, or Darwinian Natural Selection, another theory could be found that explains all known phenomena and makes the same predictions. They just don't say how hard it will be to find that new theory.

The enormous body work that Einstein, Lorentz, and others did to produce first Special Relativity then General Relativity, to replace Newtonian Mechanics, indicates that "how hard" is often "almost impossible". Regardless, Dr. Stanford is a leader among the anti-Realists in scientific philosophy. He points out that Maxwell's equations were produced based on the Ether model of electromagnetic propagation. He neglects to say that the genius of Maxwell was to produce a theory sufficiently robust that the overturning of Ether didn't invalidate his work. Just to say this at a Fog Index of 1: I don't believe him. Should he reply that I don't understand, I'd reply the onus is on him to write readable text.

I foresee primarily libraries obtaining the book, and few individuals. People who succeed in reading it will be those who are already conversant in the field. The opaque diction will minimize the number of non-philosophers whom he could reach were he a better writer.