CHAPTER 13
HOW MUCH BEAUTY IS THERE IN A BALLET?
Pure mathematics is, in its way, the poetry of logical ideas. —ALBERT EINSTEIN
Professor Zhong-shen Liu, a close friend of mine, is a pharmaceutical expert at Heilongjiang University of Traditional Chinese Medicine in Harbin, China. In 1986 he made the long trip to Hangzhou, my hometown, for a conference. I took the opportunity to invite him to participate in an interdisciplinary discussion group at Zhejiang University and to present a talk to the group.
Like many of his colleagues, his research involved performing chemical analyses of herbal medicines. This usually involves using liquid and gas chromatography to separate an herbal remedy into its thousands of constituent chemical components, then testing each of them individually to determine which one is the key component. Once the decisive component has been identified, the next step is to determine the chemical structure of the relevant molecule in order to develop a way of synthesizing it in laboratories, and then economically in factories.
These steps represent the typical successful approach to studying herbal medicines at that time and still apply today in China and many other countries. The process is consistent with reductionism, which has dominated biology and medicine for hundreds of years, particularly in the twentieth century.
Natural Medicine Is Music instead of Mechanics
Liu shared his personal research experience with us. In some cases the reductionist approach was successful in finding the decisive component, but in other cases they could not find the decisive component at all. This shows that the potency of some herbal medications lies in the combined effects of many components rather than a single one.
His opinion resonated strongly with the members of our group, particularly since we had just been discussing the problems of reductionism in biology and medicine. A young student of the arts, Jin Fu, even likened this approach in studying Chinese herbal medicine to trying to study music by means of reductionism—trying to determine which note is most important and decisive in the music of Mozart, and considering how to discard all the other less important notes. Thus one could attempt to compose a purified symphony with only a single note, even with only a single frequency, which, in this method, would be the essence of Mozart’s music.
While this satirical idea is extreme, it does reflect the essential nature of herbal medicine research and conventional medicine in general. Millions of scientists spend their whole lives looking for the decisive factors of herbal remedies or the singular causes of diseases, like tirelessly looking for the decisive note in a piece of music.
The reductionist approach is not completely misguided, and has proved extremely successful in biological and medical research. Almost every infectious disease is caused by a unique species of bacteria or virus, and Western medicine has been successful in conquering epidemic deceases. However, medicine has changed, and acute bacterial diseases are relatively rare. Instead, people, particularly those in developed countries with good medical systems, suffer from chronic diseases and functional disorders. Many have not found relief in Western medicine and have looked to alternative therapies like homeopathy, acupuncture, ayurveda, massage, Qigong, and other natural forms of medicine.
The underlying principle of many natural forms of medicine is holism rather than reductionism, and the key point of holism is coherence. The word coherence is used in music and other arts but rarely in science because it is difficult to comprehend from a rational, quantitative perspective. Fortunately, studying coherence is not as hopeless as generally believed, as shown in the following discussion.
Two Levels of Problems in a Symphony
The first precondition for performing a successful symphony is that every musical instrument is well made and in good condition. A single faulty instrument immediately damages the coherence of the symphony. In such a case, to fix our orchestra we must discern which instrument is out of order and repair it. At this level, which we can refer to as single-factor thinking, reductionism works perfectly.
The second precondition is that all the musicians cooperate and coordinate well. Even if every instrument is in good condition and each musician plays the piece properly, a lack of coordination will ruin the symphony. At this level, single-factor thinking and reductionism no longer work.
It is evident that Western medicine has focused on the first precondition for creating a beautiful symphony. The modern challenge for medicine is to move toward being able to fix the symphony when the second precondition is not being met, when coordination and cooperation are not functioning properly. This poses a second-level challenge, a major challenge to the human intellect. It is a much more difficult task for scientists to objectively and quantitatively analyze something as complicated as the body-mind system.
The Role of Order in Music
As the challenge facing medicine shares some common characteristics with music, let us consider whether it is possible to scientifically and quantitatively evaluate the degree of coherence in music and dancing. The first step is to contemplate order.
In the 1970s, the concept of order became fashionable in both science and art. Interestingly, the antonym of order, chaos, has also become fashionable. Musicians believed that there is some implicit order in music that makes it pleasing to the ear. Therefore, the higher degree of order presented, the better the music is. Similarly, medical doctors believed that living systems are always highly ordered, and that a higher degree of order denotes a healthier individual. These beliefs are not far from the truth, but when we start to consider coherence, the analysis becomes more complicated.
The 1990s saw more and more scientists studying the concept of coherence or the coherent state. Since scientists’ understanding of coherence is constantly evolving, it is difficult to categorically define exactly what coherence is. However, the following three examples of idealized states of coherence illustrate the coherent state.
The first example is the perfectly chaotic state, such as a group of very young children in a nursery without adult supervision. Too young to participate in cooperative play, there is no coordination in their actions. This state of coherence represents an almost perfectly chaotic state. In such a state, one hundred children have one hundred degrees of freedom, meaning one hundred parameters that can vary independently.
Soldiers in an honor guard represent an example of the opposite of a chaotic state, exhibiting the highest degree of order, all acting exactly the same, like a single person. This state is a perfectly ordered or “crystal” state, and the one hundred soldiers have only one degree of freedom.
The third state is approximated by dancers in a ballet. They are not acting chaotically, like children in a nursery, nor are they perfectly ordered, like soldiers in an honor guard. Their cooperation is both dynamic and coherent. Participants are free to move independently, but their actions are perfectly coordinated. Scientists view this as a coherent state, neither highly ordered nor chaotic.
Coherence: Neither Highly Ordered nor Chaotic
Music represents the combination of many acoustic frequencies. In addition to being dissected into a combination of many individual notes, music can also be dissected into a combination of many frequencies (fig. 13.1). When frequencies are combined, the fundamental frequency designates the note. For example, the fundamental frequency of C is 532 hertz, while A has a fundamental frequency of 440 hertz. As discussed in chapter 8, the fundamental frequency is complemented by many higher frequency overtones (fig. 8.2). The frequencies of the overtones are usually a whole number of times larger than the fundamental frequency, so A, with its fundamental frequency of 440 hertz, has a first overtone that is twice the fundamental frequency, or 880 hertz, a second overtone three times the fundamental frequency, or 1,320 hertz, and so on. However, the combination of the strengths of overtone frequencies varies by instrument and from person to person. This is what gives each person and each instrument its distinctive sound, even when they are playing the same note.
Music in a state of highest order exhibits only one frequency, the fundamental frequency. In a chorus, the highest order occurs when all singers sing the same note and frequency, similar to the movement of soldiers in an honor guard. This does not represent coherent music, however. Few would want to listen to a completely random combination of frequencies that would typify a perfectly chaotic state of sound, and it would be difficult to classify it as music; most would call it noise. Science goes one step farther, classifying it as ideal noise, or if the strengths of all the frequencies are similar, white noise. In coherent music, the relationship between the different notes and frequencies is in a coherent state, similar to the relationship among dancers in a ballet. Thus the question becomes, what is it about the combined actions of dancers in a ballet that creates coherence?
Figure 13.1. Frequency analysis of one note of music.
Figure 13.2. How many dancers are in the picture?
1 + 1 = 3: The Miracle in Coherence
The task of calculating the degree of coherence is not as daunting, and the mathematics involved not as painful, as one would suppose. The fundamental principle behind the calculation can be initiated with a simple count. The following example illustrates the concept. Let us start with the question, “How many dancers are there in figure 13.2?”
On a basic level, it is a straightforward question, and short of some disguised dancers lurking in the background, the answer, two, is evident. If, however, instead of thinking in terms of the number of humans on stage and instead consider the number of possible forms, the answer is more variable. For instance, when the two are dancing in perfect cooperation, it is sometimes hard for us to tell which arm and leg belong to whom. It begins to appear that a new creature, a third dancer with four arms and four legs, emerges from such perfect cooperation. The most appropriate mathematical formula to describe the situation is 1 + 1 = 3, instead of the ordinary 1 + 1 = 2.
In fact, there are at least four possibilities or four different answers to the most basic mathematical formula 1 + 1. For instance, a duel could result in 1+1=0, if both parties fire at the same moment and hit one another. 1 + 1 = 0 can also be observed in wars, in business battles, and even within families.
Of course, most try to avoid an outcome of 1 + 1 = 0. In addition to this and to 1 + 1 = 2, there is also the situation of 1 + 1 = 1, which is also quite common in history, as when a victor successfully occupies and eventually annexes a losing country. Similarly, in business, when a company outcompetes and eventually merges with a rival, 1 + 1 = 1 applies, providing the companies are lucky enough to avoid the 1 + 1 = 0 scenario.
The question then becomes, is there a situation that is not only better than 1 + 1 = 0, but also better than 1 + 1 = 1 and 1 + 1 = 2? There is: 1 + 1 = 3 is also possible. So three different answers to 1 + 1 are possible, depending on the type of mathematics used.
Three Ideal States
The formula 1 + 1 = 0 is not relevant to this discussion, as it describes the state of dying. The three different formulas that can be used to describe the three idealized states in living systems are:
For a chaotic state, like children in a nursery without supervision: 1 + 1 = 2
For a crystal state, like soldiers in an honor guard: 1 + 1 = 1
For a coherent state, like ballet dancers: 1 + 1 = 3
Let us discuss the three ideal situations individually.
Ideal Chaotic State: 1 + 1 = 2
In this state, the participants act completely independently, like the behavior of young children in a nursery without any supervision in a perfect chaotic state. The basic mathematical formula is the everyday 1 + 1 = 2, and the extensions for situations with more participants is also straightforward:
For two children: 1 + 1 = 2
For three children: 1 + 1 + 1 = 3
For four children: 1 + 1 + 1 + 1 = 4, and so on.
In terms of physics, in a perfect chaotic state, one hundred children have one hundred degrees of freedom.
Ideal Crystal State: 1 + 1 = 1
The second state is like soldiers in an honor guard (fig. 13.5), in the highest order and acting exactly the same, like a single person. The basic mathematical formula is also simple: 1 + 1 = 1. Extension of 1 + 1 = 1 is also very simple:
For two soldiers: 1 + 1 = 1
For three soldiers: 1 + 1 + 1 = 1
For four soldiers: 1 + 1 + 1 + 1 = 1, and so on.
The situation is as simple as it appears. In terms of physics, in a perfect ordered state, one hundred soldiers have only one degree of freedom. While it sounds counterintuitive that one hundred individuals could collectively only have one degree of freedom, this approximates the goal of dictators, and many of them come close to achieving it. It is also undeniable that this crystal state works very well for a war, even though it is predicated on sacrificing human freedom and human rights.
Ideal Coherent State: 1 + 1 = 3
The third state is like dancers in a ballet (fig. 13.2). They are neither as chaotic as the children in the nursery, nor as highly ordered as the soldiers in the honor guard. Each dancer moves independently, but at the same time coordinates their actions. In the case of two dancers, in addition to their individual forms, their cooperation produces a third form, which can be thought of as another dancer. Scientists call it a coherent state: it is neither highly ordered nor chaotic.
Figure 13.7. How many dancers are in these pictures?
How Many Degrees of Freedom in a Coherent State?
How Many Dancers in a Ballet?
The basic idea behind the calculation can start with a simple count. In figure 13.6, the calculation is a little complicated, but not too difficult if you start with the simplest case, and then incrementally increase the complexity. The mathematics involved is called combinatorics. Two dancers acting coherently produce a third form, thereby allowing three degrees of freedom. This outcome can be calculated using the following general equation:
For n number of dancers, the number of possible combinations = 2n − 1
Thus in the case of two dancers, the number of possible combinations = 22 − 1
As shown in figure 13.8, if three dancers perform together with excellent cooperation, there are seven possible combinations, or 1 + 1 + 1 = 7. If we apply the general equation, for three dancers, the result is 7, from 23 − 1 = 7. Similarly, for four dancers, 1 + 1 + 1 + 1 = 24 − 1 = 15, and so on. By applying the formula it is relatively straightforward to calculate the extensions of 1+1=3.
Figure 13.8. The number of combinations with three dancers.
Extensions of 1 + 1 = 3
For two dancers: 1 + 1 = 22 − 1 = 3
For three dancers: 1 + 1 + 1 = 23 − 1 = 7
For four dancers: 1 + 1 + 1 + 1 = 24 − 1 = 15
For five dancers: 1 + 1 + 1 + 1 + 1 = 25 − 1 = 31
For six dancers: 1 + 1 + 1 + 1 + 1 + 1 = 26 − 1 = 63
For seven dancers: 1 + 1 + 1 + 1 + 1 + 1+1 = 27 − 1 = 127
So what about one hundred dancers? The extraordinarily large number is:
2100 − 1 = 126,750,600,228,229,401,703,205,375
This means that instead of having just one hundred degrees of freedom, as if they were operating completely independently, in a perfectly coherent state they possess over 126 billion quadrillion different combinations or degrees of freedom. This is the miracle of coherence.
In 1991, just five years after my discussions with Liu about music in medicine, my team in Germany developed the key to analyzing coherence in music and in living systems. The first academic paper presenting rigorous mathematical proof of this idea was published in the United Kingdom in 1994 in the journal Medical Hypotheses, and then republished in Switzerland in a special issue of the International Journal of Modeling, Identification and Control on physiotherapy.1
The principles and discoveries discussed in this chapter were applied to develop an instrument, called a coherence meter, which measures the state of the electromagnetic body in individuals. The following chapter looks at how this mathematical thinking can be practically applied to medicine and health care.