THEOLOGICAL APPLICATIONS OF CATASTROPHE THEORY
Catastrophe
Theory is a mathematical formalism that was developed during the 1970's and
applied to explain various types of discontinuities in nature. Our related chapter Formalism and Application
of Catastrophe Theory (referred to below as
Catastrophe theory was popular thirty years ago, but now is no longer in vogue for explaining natural phenomena. This makes the intrinsic value of the present article somewhat dubious, but it was fun to write at the time.
CATASTROPHE THEORY EXPLANATION OF
VITALIZATION,
HOMINIZATION,
Charles P. Poole, Jr.
Department of Physics and Astronomy
(1985, unpublished)
ABSTRACT
A formalism called catastrophe theory (CT) is presented and then applied to three problems of theological interest. It is explained how life can evolve from non-life and how intelligent man can emerge from primate ancestors in manners that are consistent both with the usual biological emphasis on the continuity of the process, and with the traditional theological emphasis on the enormous changes in kind that are involved in these transformations. The formalism is then applied to the case of abortion to provide a deeper understanding of the process of animation.
CONTENTS
1. Introduction
2. Discontinuities in Nature
3. Catastrophe Theory
4. Appearance of Life and the Vitalization Catastrophe
5. Genesis of Man and the Hominization Catastrophe
6. Abortion and the Fertilization Catastrophe
7. The Implantation Catastrophe
8. Discussion
References
1. INTRODUCTION
Down through the ages there has been a commonly held theo1ogica1 position that an enormous gap exists between the living and the non-living, and another enormous gap exists between man and lower animals. In recent years biological phenomena have been explained more and more in terms of chemistry and physics (1,2). Consequently many scientists claim that the gap between life and non-life is a small one, and they believe that their research endeavors will continue to narrow the man - primate gap. During a previous decade a new branch of mathematics called Catastrophe Theory (CT) (3,4) has been developed which is capable of resolving these apparent conflicts between theology and science in a manner which preserves the main tenets of each viewpoint. In this article we will provide an introduction to catastrophe theory which is a means for treating discontinuities in nature, and then we will apply it to three theological situations.
2. DISCONTINUITIES IN NATURE
When we reflect upon the world ordinarily we concern ourselves with its continuities. For example we find that the more we heat an object the more its temperature increases, (6), the longer we study the more we learn, and the heavier the object that we place on the end of a diving board the more the board bends (6). These are three commonplace events which can be explained by simple theories of thermodynamics, learning, and mechanics, respectively. Nature is in many ways quite continuous, and elementary theories of nature tend to emphasize these continuities.
A little more reflection upon the world will reveal that it is not entirely continuous. For example, if we persist in heating an object to higher and higher temperatures it will eventually burst into flame, or will melt. Long periods of concentrated study sometimes lead to sudden insights which enormously increase our understanding of a subject. In the diving board example we will eventually reach the point where the board breaks instead of bending further. These sudden discontinuous changes reveal an aspect of nature which is missing from our theories which treat its continuities. New properties appear and new types of behavior become manifest.
In Physics we are familiar with what are called changes of phase, such as melting and boiling, the conversion of iron to a magnetized form, the establishment of a superconducting state, etc. In Chemistry there are chemical reactions, in geology we study earthquakes, in biology we observe the metamorphosis of a butterfly through its egg, larva and pupa stages to its adult form (7), and in psychology we encounter nervous breakdowns. All of these transformations may be given the geneic name of catastrophes (3), and in the next section we will present a mathematical technique for describing them.
3. CATASTROPHE THEORY
The mathematical formalism of
catastrophe theory (3-6) treats a type of behavior in terms of its dependence
upon certain variables called control parameters. In a simple form of this
theory (3) the behavior depends upon two control parameters which tend to
produce opposing effects. For certain
sets of values of these parameters the behavior will be of one type, for other
sets of values it will be of an opposite type, and for a third set of values it
can be of either type depending upon the previous history of the system. This third case is called bimodal since
either behavior can occur. Just knowing
the values of the control parameters will not tell us the behavior; it is also
necessary to know their sequence of reaching the final state. See Figures 1 to
4 of the
As an example of this bimodal characteristic let us take the model of a dog whose behavior pattern of aggression or flight depends upon two factors or control parameters, namely rage and fear (3). If the dog feels no fear then a steady increase rage will eventually cause the dog to attack. If the dog feels no rage then a steady increase of fear will cause the dog to flee. In each case the behavior is predictable
These two behavior patterns are simple to describe when only one control parameter varies. Now let us examine what happens when both rage and fear are present simultaneously. A fearful, retreating dog will commence to attack if it is sufficiently enraged and an enraged attacking dog will flee if rendered sufficiently fearful. The changes from retreating to attacking and from attacking to fleeing are sudden changes but their causes are not. For example the fearful dog may be gradually enraged as it retreats until a critical point is reached so a small additional increase in rage induces the attack. The same is true of the enraged dog where a continual increase in fear. changes its behavior from attack to retreat.
We have seen that there is a point where
a very small increase in rage will cause a retreating dog to attack. Once the attack starts a small decrease in
rage will not stop it. Instead a very large decrease in rage is needed to
induce the behavior to revert to a retreat.
A mere knowledge of the magnitude of the fear and the magnitude of the
rage is not enough to enable us to predict the behavior of the dog. Its behavior is in the bimodal region, as
shown on Fig. 1 (and in Fig. 5 of the
There is another property that is
important in respect to the bimodal region, and that involves the concept of hysteresis. If we keep fixed the value of fear and
gradually increase the rage from point “A” on Fig. 1 the
flight continues until a point “C” is reached where retreat turns to
attack. Now continuing to maintain the
same level of fear we can decrease the rage, and eventually we reach a value “B" where the attack ceases and flight
begins. The range from “B” to “C” on the figure is bimodal, and must be
traversed completely to reverse the behavior. This is called hysteresis. The
concept of hysteresis is applicable to many
systems. For example, sometimes a fairly
small provocation is enough to start a war, but much more then a removal of the
provocation is needed to stop it. The
reason for the hysteresis is easily visualized from
the three dimensional plot of Fig. 5 of
4. APPEARANCE OF
To apply catastrophe theory to the origin of life we select biochemical complexity and environmental severity as the control parameters. The evolution of life is favored by an increase of biochemical complexity, and by a decrease in the severity of the environment. The corresponding catastrophe theory diagram is shown on Fig. 2. Consider the situation at point "A" on this figure. The biochemical complexity and environmental severity are unfavorable so life cannot start.. If we keep increasing the biochemical complexity we can pass through point "B" to point "C" where life can emerge or evolve from non-life. As an alternative possibility, after reaching point “B” we could improve the environment by proceeding to point "D" where life can form. The important point is that the bimodal region must be crossed.. Non-life does not simply transform to life with very little else happening. The transformation at "C" is a large change to a new state of existence, and may be called the vitalization catastrophe. Thus catastrophe theory applied to life explains the scientific conclusion that the biochemical difference between life and non-life is small, and the Christian tradition that the fundamental difference is quite pronounced.
There is another important factor about life which is also explained by catastrophe theory and that is the property of hysteresis. A living organism can considerably reduce its biochemical complexity through injury or disease and still not die. A species formed in a particular environment can withstand a considerably more severe environment before it becomes extinct. Both of these factors are examples of hysteresis, For example on Fig. 2 life formed at point “C” requires a considerable increase in the severity of the environment until it reaches point "E” before it can die out. The foregoing is an oversimplified version of the application of catastrophe theory to life's origins. Thom (3) gives a more sophisticated version in terms of negative entropy expressed mathematically as topological complexity, but we will not elaborate upon it here.
5. GENESIS OF
Anthropologists sometimes associate
the intellectual level of fossil man with his brain size as determined by his cranial
capacity (8-10), and this is chosen as a control parameter. Modern man averages 1350 cubic centimeters,
which is a little more than twice the 600cc range of modern large apes. The Java and
The straightforward interpretation of the above is that once point "C" is reached then hominization is able to take place. This may be unacceptable to some as limiting the role of God in creating man (1l) although Christian evolutionists like Teilhard de Chardin (12) adhere to similar positions. A more theologically acceptable position is to maintain that at point "C" nature has reached a state of readiness for hominization to occur if God so chooses to will it. Since man now exists it is clear that at some epoch in the past God did so will it.
6. ABORTION
Abortion is an important moral issue (13-l6) to which catastrophe theory should be applicable. The basic theological problem concerns the point at which the ovum or fetus transforms into being a human person, an event called animation or ensoulment. In the present context the scientific consideration is to describe all catastrophe theory type transformations that occur during the progress of the events associated with a pregnancy. We suggest that it is after a CT transformation that a human person emerges with all of his or her the rights and privileges, as detailed in the Catechism.
It is important to point out at the outset that the occasion of birth does not constitute a CT event since no anatomical or physiological change takes place in the baby at that time. The only changes that occur are changes in the mode of respiration and of food assimilation. The lungs and the digestive system are already fully formed before birth and ready to perform their functions. Our legal system treats the event of birth as a CT type transformation of a so-called fetus having an animal nature to a baby having a human nature, but this reasoning is totally contrary to the scientific facts. It is important for society to acknowledge the scientific facts and make judgments based on them.
Returning to the earlier discussion, there are two CT type transformations that take place during the events associated with the pregnancy. These events are the fertilization of the ovum or egg, and the implantation of the embryo in the placenta, and we will discuss each in turn from a CT viewpoint. We shall refer to these as the fertilization and the implantation catastrophes, respectively. Neither event matches a CT explanation very well because it is problematic to select control parameters, and the transformations involved are irreversible, that is they cannot go backwards. Nevertheless they can be looked upon as constituting substantial changes in kind.
In the case of fertilization we will consider the ovum as the persistent entity in the process since it changes so little in overall structure. The dominant change that takes place is the merger in the nucleus of the haploid chromosomes of the individual sperm and egg gamete cells into the final combined diploid chromosome configuration of a body cell. The cytoplasm surrounding the nucleus is not appreciably affected by what happens in the nucleus. The progress of this merger or of the associated overall process called meiosis might be selected as a control parameter. Other possible control parameters are the distance of the nearest sperm from the ovum and the rate at which the ovum undergoes cell division. This latter process can occur only after fertilization.
7. THE IMPLANTATION CATASTROPHE
What we call the implantation catastrophe occurs about two weeks after fertilization. Diamond (17) has given a good summary of the events that occur during the early stages of embryonic development. He mentions that fertilization is followed by an initial period of cell cleavage during which the fertilized ovum divides to form two cells, which subsequently divide to form four cells, then eight cells, etc. During this stage each cell is totipotent, i.e. capable of forming any tissue, and in addition the cell mass (morula) may divide to form identical twins, or two fertilized ova may fuse by reconjunction to eventually form a single morula. The early part of this cell doubling stage takes place while traveling to the site of the placenta. After about six days the first cells differentiate to form the placenta, and in another week the second differentiation occurs to form brain tissue, then heart tissue, etc. Henceforth each cell is pluripotent, or capable of forming only a particular tissue type. At this point in time the embryo or zygote has become firmly implanted, and is developing into an individual with specialized cells which can no longer break off to form twins. Hitherto the zygote underwent cell division without growth, henceforth it will both divide and grow, receiving nourishment from the mother's bloodstream. At this point the mother misses a period, thereby becoming aware of the implantation and the presence of her child. We suggest that the complex of events at this stage of fetal development, and in particular the implantation itself, constitutes a CT transformation. This is an overall CT type transformation involving the entire fetus. Subsequent transformations during the gestation are much more localized, limited to particular tissues or organs, and can in no sense be considered as CT types. From the viewpoint of the unborn child the act of birth itself is definitely not a CT type transformation. .
8. DISCUSSION
In this chapter we have shown that the formalism of catastrophe theory provides an explanation for the emergence of life which accounts for the smallness of the chemical differences between life and non-life, and also accounts for the large magnitude of the behavior difference between the living and the nonliving. In the second application of this formalism we explained how primates can be anatomically and physiologically quite close to man while at the same time separated from man by an enormous gap in intelligence and volitional capabilities. As a result we have an explanation of each of these two events which is self-consistent and at the same time in conformity with both contemporary science and traditiona1 theology. In Sections 6 and 7 we explained how the formalism of catastrophe theory can enhance our understanding of the important contemporary moral problem of abortion. We suggested that the points of fertilization and of implantation are the only two events that have the characteristics of catastrophes. Between the time of implantation at two weeks and the birth at nine months there are no other events that can qualify as catastrophes. Birth itself certainly cannot qualify since at that time no substantial change takes place in the infant except the change in his or her mode of respiration and food intake. Abortion should be unthinkable during this catastrophe-free period. We accept the judgment of the Church that from the moment of conception abortion is not morally justifiable, so the fertilization catastrophe is the theologically significant one.
It is possible that other theological applications of this theory may found in the future. If explanations of the type presented in this article are accepted then Christian scientists do not have to worry about compartmentalizing their scientific and their religious lives to avoid conflicts between the two. They can have one overall point of view which is in conformity with the data of science and the dogmata of theology. Theologians, on the other hand, no longer need shy away from science as a potential disrupting influence, but rather can embrace it as an ally. This could help bring to an end the alleged state of conflict between science and religion which has influenced Western thought since the Renaissance.
REFERENCES