Isomorphism in Science and The Unity of Science

1. Isomorphism in Science

The present study merely intended to briefly point out the general aim and several concepts of general system theory. Further tasks on the one hand would be to express this theory in a logico-mathematically strict form; on the other hand the principles holding for any type of systems would have to be further developed. This is a concrete problem. For example, demographic dynamics may be developed homologous to mechanical dynamics (Volterra, cf. d’Ancona, 1939). A principle of minimum action may be found in various fields, in mechanics, in physical chemistry as Le Chatelier’s principle which, as may be proved, is also valid for open systems, in electricity as Lenz’s rule, in population theory according to Volterra, etc. A principle of relaxation oscillations occurs in physical systems as well as in many biological phenomena and certain models of population dynamics. A general theory of periodicities appears as a desideratum of various fields of science. Efforts will therefore have to be made towards a development of principles such as those of minimum action, conditions of stationary and periodic solutions (equilibria and rhythmic fluctuations), the existence of steady states and similar problems in a form generalized with respect to physics and valid for systems in general.

General system theory therefore is not a catalogue of well- known differential equations and their solutions, but raises new and well-defined problems which partly do not appear in physics, but are of basic importance in non-physical fields. Just because the phenomena concerned are not dealt with in ordinary physics, these problems have often appeared as metaphysical or vitalistic.

General system theory should further be an important regulative device in science. The existence of laws of similar structure in different fields makes possible the use of models which are simpler.or better known, for more complicated and less manageable phenomena. Therefore general system theory should be, methodologically,.Rn important means of controlling and instigating the transfer of principles from one field to another, and it will no longer be necessary to duplicate or triplicate the discovery of the same principles in different fields isolated from each other. At the same time, by formulating exact criteria, general system theory will guard against superficial analogies which are useless in science and harmful in their practical consequences.

This requires a definition of the extent to which “analogies” in science are permissible and useful.

We have previously seen the appearance of similar system laws in various sciences. The same is true of phenomena where the general principles can be described in ordinary language though they cannot be formulated in mathematical terms. For instance, there are hardly processes more unlike phenomenologically and in their intrinsic mechanisms than the formation of a whole animal out of a divided sea-urchin or newt germ, the reestablishment of normal function in the central nervous system after removal or injury to some of its parts, and gestalt perception in psychology. Nevertheless, the principles governing these different phenomena show striking similarities. Again, when we investigate the development of the Germanic languages, it may be observed that, beginning with a primitive language, certain sound mutations occurred in parallel development in various tribes, though these were geographically located far apart from each other; in Iceland, on the British Isles, on the Iberian peninsula. Mutual influence is out of question; the languages rather developed independently after separation of the tribes, and yet show definite parallelism.* The biologist may find a corresponding principle in certain evolutionary developments. There is, for instance, the group of extinct hoofed animals, the titanotheres. During the Tertiary, they developed from smaller into gigantic forms, while with increasing body size formation of ever larger horns took place. A more detailed investigation showed that the titanotheres, starting from those small, early forms, split up into several groups which developed independently of each other but still showed parallel characteristics. Thus we find an interesting similarity in the phenomenon of parallel evolution starting from common origins but developing independently—here: the independent evolution of tribal languages;’ there: independent evolution of groups within a certain class of mammals.                                                         ‘

In simple cases, the reason for isomorphism is readily seen.

For example, the exponential law states that, given a complex of a number of entities, a constant percentage of these elements decay or multiply per unit time. Therefore this law will apply to the pounds in a banking account as well as to radium atoms, molecules, bacteria, or individuals in a population. The logistic law says that the increase, originally exponential, is limited by some restricting conditions. Thus in autocatalytic reaction, a compound catalyzes its own formation; but since the number of molecules is finite in a closed reaction vessel, the reaction must stop when all molecules are transformed, and must therefore approach a limiting value. A population increases exponentially with the increasing number of individuals, but if space and food are limited, the amount of food available per individual decreases; therefore the increase in number cannot be unlimited, but must approach a steady state defined as the maximum population compatible with resources available. Railway lines which already exist in a country lead to the intensification of traffic and industry which, in turn, make necessary a denser railway network, till a state of saturation is eventually reached; thus, railways behave like autocatalyzers accelerating their own increase, and their growth follows the autocatalytic curve. The parabolic law is an expression for competition within a system, each element taking its share according to its capacity as expressed by a specific constant. Therefore the law is of the same form whether it applies to the competition of individuals in an economic system, according to Pareto’s law, or to organs competing within an organism for nutritive material and showing allometric growth.

There are obviously three prerequisites for the existence of isomorphisms in different fields and sciences. Apparently, the isomorphisms of laws rest in our cognition on the one hand, and in reality on the other. Trivially, it is easy to write down any complicated differential equation, yet even innocent-looking expressions may be hard to solve, or give, at the least, cumbersome solutions. The number of simple mathematical expressions which will be preferably applied to describe natural phenomena is limited. For this reason, laws identical in structure will appear in intrinsically different fields. The same applies to statements in ordinary language; here, too, the number of intellectual schemes is restricted, and they will be applied in quite different realms.

However, these laws and schemes would be of little help if the world (i.e., the totality of observable events) was not such that they could be applied to it. We can imagine a chaotic world or a world which is too complicated to allow the application of the relatively simple schemes which we are able to construct with our limited intellect. That this is not so is the prerequisite that science is possible. The structure of reality is such as to permit the application of our conceptual constructs. We realize, how- ever, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality; just as the blueprint of a building isn’t the building itself and by no means represents it in every detail such as the arrangement of bricks and the forces keeping them together, but nevertheless an unequivocal correspondence exists between the design on paper and the real construction of stone, iron and wood. The question of ultimate “truth” is not raised, that is, in how far the plan of reality as mapped by science is correct or in need or capable of improvement; likewise, whether the structure of reality is expressed in one single blueprint —i.e., the system of human science. Presumably different representations are possible or even necessary—in a similar way as it is meaningless to ask whether central or parallel projection, a horizontal or a vertical plan are more “correct.” That the latter may be the case is indicated by instances where the same physical “given” can be expressed in different “languages”—e.g., by thermodynamics and statistical mechanics; or even complementary considerations become necessary, as in the corpuscle and wave models of microphysics. Independent of these questions, the existence of science proves that it is possible to express certain traits of order in reality by conceptual constructs. A presupposition for this is that order exists in reality itself; similarly—to quote the illustration mentioned—as we are able to draw the plan of a house or a crystal, but not of stones whirling around after an explosion or of the irregularly moving molecules in a liquid.

Yet there is a third reason for the isomorphism of laws in different realms which is important for the present purpose. In our considerations we started with a general definition of “system” defined as “a number of elements in interaction” and expressed by the system of equations (3.1). No special hypotheses or statements were made about the nature of the system, of its elements or the relations between them. Nevertheless from this purely formal definition of “system,” many properties follow which in part are expressed in laws well-known in various fields of science, and in part concern concepts previously regarded as anthropomorphic, vitalistic or metaphysical. The parallelism of general conceptions or even special laws in different fields therefore is a consequence of the fact that these are concerned with “systems,” and that certain general principles apply to systems irrespective of their nature. Hence principles such as those of wholeness and sum, mechanization, hierarchic order, approach to steady states, equifinality, etc., may appear in quite different disciplines. The isomorphism found in different realms is based on the existence of general system principles, of a more or less well-developed “general system theory.”

The limitations of this conception, on the other hand, can be indicated by distinguishing three kinds or levels in the description of phenomena.

At first, there are analogies—i.e., superficial similarities of phenomena which correspond neither in their causal factors nor in their relevant laws. Of this kind are the simulacra vitae, popular in previous times, such as when the growth of an organism was compared to the growth of a crystal or of an osmotic cell. There are superficial similarities in the one or other respect, while we are safe to say that the growth of a plant or an animal does not follow the pattern of crystal growth or of an osmotic structure, and the relevant laws are different in both cases. The same applies to the consideration of a biocoenosis (e.g., a forest) as an “organism,” with the obvious difference between the unification of an individual organism and the looseness of a plant association; or the comparison of the development of a population with birth, growth, aging and death of an organism where the comparison of life cycles remains highly dubious.

A second level are homologies. Such are present when the efficient factors are different, but the respective laws are formally identical. Such homologies are of considerable importance as conceptual models in science. They are frequently applied in physics. Examples are the consideration of heat flow as a flow of a heat substance, the comparison of electrical flow with the flow of a fluid, in general the transfer of the originally hydrodynamic notion of gradient to electrical, chemical, etc., potentials. We know exactly, of course, that there is no “heat substance” but heat is to be interpreted in the sense of kinetic theory; yet the model enables the stipulation of laws which are formally correct.

It is logical homologies with which the present investigation is concerned. We may express this as follows: If an object is a system, it must have certain general system characteristics, irrespective of what the system is otherwise. Logical homology makes possible not only isomorphy in science, but as a conceptual model has the capacity of giving instructions for correct consideration and eventual explanation of phenomena.

The third level finally is explanation—i.e., the statement of specific conditions and laws that are valid for an individual object or for a class of objects. In logico-mathematical language, this means that the general functions / of our equation (3.1) are replaced by specified functions applicable to the individual case. Any scientific explanation necessitates the knowledge of these specific laws as, for example, the laws of chemical equilibrium, of growth of an organism, the development of a population, etc. It is possible that also specific laws present formal correspondence or homologies in the sense discussed; but the structure of individual laws may, of course, be different in the individual cases.

Analogies are scientifically worthless. Homologies, in contrast, often present valuable models, and therefore are widely applied in physics. Similarly, general system theory can serve as a regulatory device to distinguish analogies and homologies, meaningless similarities and meaningful transfer of models. This function particularly applies to sciences which, like demography, sociology, and large fields in biology, cannot be fitted in the framework of physics and chemistry; nevertheless, there are exact laws which can be stated by application of suitable models. The homology of system characteristics does not imply reduction of one realm to another and lower one. But neither is it mere metaphor or analogy; rather, it is a formal correspondence founded in reality inasmuch as it can be considered as constituted of “systems” of whatever kind.

Speaking philosophically, general system theory, in its developed form, would replace what is known as “theory of categories” (N. Hartmann, 1942) by an exact system of logico- mathematical laws. General notions as yet expressed in the vernacular would acquire’the unambiguous and exact expression possible only in mathematical language.

2. The Unity of Science

We may summarize the main results of this presentation as follows:

  1. The analysis of general system principles shows that many concepts which have often been considered as anthropomorphic, metaphysical, or vitalistic are accessible to exact formulation. They are consequences of the definition of systems or of certain system conditions.
  2. Such investigation is a useful prerequisite with respect to concrete problems in science. In particular, it leads to the elucidation of problems which, in the usual schematisms and pigeonholes of the specialized fields, are not envisaged. Thus system theory should prove an important means in the process of developing new branches of knowledge into exact science—i.e., into systems of mathematical laws.
  3. This investigation is equally important to philosophy of science, major problems of which gain new and often surprising aspects.
  4. The fact that certain principles apply to systems in general, irrespective of the nature of the systems and of the entities concerned, explains that corresponding conceptions and laws appear independently in different fields of science, causing the remarkable parallelism in their modern Thus, concepts such as wholeness and sum, mechanization, centralization, hierarchical order, stationary and steady states, equifinality, etc., are found in different fields of natural science, as well as in psychology and sociology.

These considerations have a definite bearing on the question of the Unity of Science. The current opinion has been well represented by Carnap (1934). As he states, Unity of Science is granted by the fact that all statements in science can ultimately be expressed in physical language— i.e., in the form of statements that attach quantitative values to definite positions in a spacetime system of co-ordinates. In this sense, all seemingly nonphysical concepts, for instance specifically biological notions such as “species,” “organism,” “fertilization,” and so forth, are defined by means of certain perceptible criteria—i.e., qualitative determi- nations capable of being physicalized. The physical language is therefore the universal language of science. The question whether biological laws can be reduced to physical ones—i.e., whether the natural laws sufficient to explain all inorganic phenomena are also sufficient to explain biological phenomena—is left open by Carnap, though with preference given to an answer in the affirmative.

From our point of view, Unity of Science wins a much more concrete and, at the same time, profounder aspect. We too leave open the question of the “ultimate reduction” of the laws of biology (and the other non- physical realms) to physics—i.e., the question whether a  hypothetico-deductive system embracing all sciences from physics to biology and sociology may ever be established. But we are certainly able to establish scientific laws for the different levels or strata of reality. And here we find, speaking in the “formal mode” (Carnap), a correspondence or isomorphy of laws and conceptual schemes in different fields, granting the Unity of Science. Speaking in “material” language, this means that the world (i.e., the total of observable phenomena) shows a structural uniformity, manifesting itself by isomorphic traces of order in its different levels or realms.

Reality, in the modern conception, appears as a tremendous hierarchical order of organized entities, leading, in a superposition of many levels, from physical and chemical to biological and sociological systems. Unity of Science is granted, not by a utopian reduction of all sciences to physics and chemistry, but by the structural uniformities of the different levels of reality.

Especially the gap between natural and social sciences, or, to use the more expressive German terms, of Natur- und Geisteswissenschaften, is greatly diminished, not in the sense of a reduction of the latter to biological conceptions but in the sense of structural similarities. This is the cause of the appearance of corresponding general viewpoints and notions in both fields, and may eventually lead to the establishment of a system of laws in the latter.

The mechanistic world-view found its ideal in the Laplacean spirit— i.e., in the conception that all phenomena are ultimately aggregates of fortuitous actions of elementary physical units. Theoretically, this conception did not lead to exact sciences outside the field of physics—i.e., to laws of the higher levels of reality, the biological, psychological and sociological. Practically, its consequences have been fatal for our civilization. The attitude that considers physical phenomena as the sole standard of reality has lead to the mechanization of mankind and to the devaluation of higher values. The unregulated domination of physical technology finally ushered the world into the catastrophical crises of our time. After having overthrown the mechanistic view, we are careful not to slide into “biologism,” that is, into considering mental, sociological and cultural phenomena from a merely biological standpoint. As physicalism considered the living organism as a strange combination of physico- chemical events or machines, biologism considers man as a curious zoological species, human society as a beehive or stud-farm. Biologism has, theoretically, not proved its theoretical merits, and has proved fatal in its practical consequences. The organismic conception does not mean a unilateral dominance of biological conceptions. When emphasizing general structural isomorphies of different levels, it asserts, at the same time, their autonomy and possession of specific laws.

We believe that the future elaboration of general system theory will prove to be a major step towards unification of science. It may be destined in the science of the future, to play a role similar to that of Aristotelian logic in the science of antiquity. The Greek conception of the world was static, things being considered to be a mirroring of eternal archetypes or ideas. Therefore classification was the central problem in science, the fundamental organon of which is the definition of subordination and superordination of concepts. In modern science, dynamic interaction appears to be the central problem in all fields of reality. Its general principles are to be defined by system theory.

Source: Bertalanffy Ludwig Von (1969), General System Theory: Foundations, Development, Applications, George Braziller Inc.; Revised edition.

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