The term isomorphism literally means equality or sameness (iso) of form (morphism). In mathematics an isomorphism between two systems requires a one-to-one correspondence between their elements (that is, each element of one system corresponds to one and only one element of the other system, and conversely), which also preserves structures. Referring to isomorphism as one of the most important and general mathematical concepts, R. Duncan LUCE and Patrick SUPPES (1968, p. 72) characterize it as "a one-to-one mapping of a system A onto a system B in which the operations and relations of A are preserved under the mapping and have the same structure as the operations and relations of system B." In Gestalt psychology, the one-to-one correspondence between elements is not required; similarity of structures is required.
What does isomorphism mean in Gestalt theory? To answer this question, we attempted to survey some of what had appeared in the Gestalt psychological literature (mainly in English) about isomorphism and related concepts. We cite the views of the founders of Gestalt theory and of a sample of other psychologists.

We begin with a historical remark by Kurt KOFFKA (1935), recalling his conversations with Max WERTHEIMER in 1911, shortly after the completion of experimentation on apparent movement in which Wolfgang KÖHLER and KOFFKA were the chief subjects. We do not know precisely what WERTHEIMER said, but he might have mentioned his hypothesis that the apparent movement, which he called the phi phenomenon, resulted from "a kind of physiological short-circuit" in the brain (1912b). KOFFKA was impressed by "the relation between consciousness and the underlying physiological processes, or, in our new terminology, between the behavioural and the physiological field." He noted that the statement in these new terms was made possible only by WERTHEIMER's idea. After referring to WERTHEIMER as the one who "first pronounced" the theory and KÖHLER as its elaborator, KOFFKA mentioned the principle of isomorphism, "according to which characteristic aspects of the physiological processes are also characteristic aspects of the conscious processes."
We then cite KÖHLER's references to isomorphism in some of his writings (e.g., 1920, 1929, 1938) and note his acknowledgement of the ideas of the co-founders of Gestalt psychology. His studies of physical Gestalten culminated in the hypothesis of psychophysical isomorphism.

Turning to Max WERTHEIMER, we first describe his work on the phi phenomenon and its significance (A.S. LUCHINS, 1968). Then we discuss lectures that WERTHEIMER gave in a 1937-1938 seminar at the New School for Social Research. He related isomorphism to perception of feelings, emotions, and expressive movements. He also pointed to differences between his and KÖHLER's conceptions of isomorphism. Our sources were the first author's notes on WERTHEIMER's lectures and our reconstruction of the seminars (1973; 1991-1993). [fn 1]

Next we turn to Martin SCHEERER (1954) who, in a section on Gestalt psychology in a chapter on cognitive theory, raised the question of what determines the organizational character of a percept. He pointed to the Gestaltists postulate of a dynamic self-distribution of nervous excitations triggered off by the proximal stimuli; this "culminated in KÖHLER's theory of isomorphism." SCHEERER noted that for the Gestaltist the total field consists of the geographic environment, which includes the psycho-physical organism; he also characterized the phenomenal field and the behavioural environment. Additionally, he pointed to some deficiencies or gaps in Gestalt psychological research, for example, the focus on the "palpably present behavioural environment" to the neglect of the environment which one imagines or thinks about. KOFFKA (1935) also had agreed that there were gaps in the research. Since 1935, there have been attempts to close the gaps, for example, by research and exposition on Gestalt principles applied to emotions, imagery, music, art, language, and thinking.

An example is Rudolf ARNHEIM's work on Gestalt theory applied to perception and art (1969). Another example is George HUMPHREY's writing in Thinking (1951) about psychoneural processes and isomorphism in Gestalt theory.

We then refer to two survey articles. In his encyclopedia article on Gestalt theory, Solomon ASCH (1968) discussed perceptual organization, as well as physical and physiological Gestalten. He also referred to WERTHEIMER's apparent movement study but not to the physiological short-circuit hypothesis; the only reference to isomorphism was to KÖHLER's psychophysical isomorphism.

Then we turn to the historian of psychology, Edwin G. BORING (1942, 1950), to consider what he wrote about the phi phenomenon, and about isomorphism and its relation to projection. BORING also described some criticisms of the isomorphism concept in Gestalt psychology and suggested that the future might show the validity of the criticisms, or put otherwise, the worth of the concept. We suggest that the future has arrived and that it is time to discuss the concept of isomorphism in Gestalt psychology.

A section entitled "Isomorphism, Phenomenology, and Beyond Phenomenology" refers to Giovanni VICARIO's description of his mentor, Gaetano KANIZSA, as a Gestaltist and experimental phenomenologist. We suggest that WERTHEIMER, who might have been influenced by phenomenology, was more oriented than KÖHLER to experimental phenomenology and less interested in physiological hypotheses. Such differences might help account for differences in their conceptions of isomorphism.

In a section entitled "Relation Between Behavioural and Physiological Field Crucial," Koffka (1935, pp. 53-54) wrote about a conversation that

remains in my memory as one of the crucial moments of my life. It happened at Frankfort on the Main early in 1911. WERTHEIMER had just completed his experiments on the perception of motion [phi phenomenon] in which KÖHLER and I had served as the chief observers. Now he proposed to tell me the purpose of his experiments ... [O]n that afternoon he said something which impressed me more than anything else, and that was his idea about the function of a physiological theory in psychology, the relation between consciousness and the underlying physiological processes, or in our new terminology, between the behavioural and the physiological field. To state it in these new terms, however, is not quite fair, because this very statement was only made possible by WERTHEIMER s idea; before, nobody thought of a physiological or, for that matter, of a behavioural field.

The assumption of merely molecular physiological processes is erected on much too slender an empirical basis; it results either in a molecular interpretation of behaviour, and consciousness, which is contradicted by the facts, or it severs completely the two series of processes, physiological and behavioural or conscious. (p. 56)
WERTHEIMER s Solution. Isomorphism. And now the reader can understand WERTHEIMER s contribution; now he will see why his physiological hypothesis impressed me more than anything else. In two words, what he said amounted to this: let us think of the physiological processes not as molecular, but as molar phenomena. If we do that, all the difficulties of the old theory disappear. For if they are molar, their molar properties will be the same as those of the conscious processes which they are supposed to underlie. And if that is so, our two realms, instead of being separated by an impossible gulf, are brought as closely together as possible with consequence that we can use our observations of the behavioural environment and of behaviour as data for the concrete elaboration of physiological hypotheses. (Ibid.)

if B stands for the behavioural world, G for the geographical, and P for the physiological processes, BP(G shows the relationship .... [If] B and P are essentially alike, then it only depends upon the G-P relation when and how we can gain about G from P. And if it is so, then surely observation of B reveals to us properties of P. This theory, first pronounced by WERTHEIMER, was carefully elaborated by KÖHLER. In his book on the "Physische Gestalten" (1920) he has gone deeply into physics and physiology to prove the compatibility of the theory with physical and physiological facts; in his "Gestalt Psychology" [1929] he has formulated this theory of isomorphism in a number of special axioms [and] the general principle in these words: "Any actual consciousness is in every case not only blindly coupled to its corresponding psychophysical processes, but is akin to it in essential structural properties" (p. 193). Thus, isomorphism, a term implying equality of form, makes the bold assumption that the "motion of the atoms and molecules of the brain" are not "fundamentally different from thoughts and feelings."

KÖHLER acknowledged the contributions of WERTHEIMER and KOFFKA. Referring to the close approach between general biology and psychology in the theory of nervous functions, particularly in the doctrine of the physical basis of consciousness, he wrote in his book on physical Gestalten (1920; abridged translation in ELLIS, 1938):

Here we have an immediate correspondence between mental and physical processes and the demand seems inescapable that at this point organic functions be thought of as participating in and exhibiting essentially Gestalt characteristics. The import and extraordinary significance of this was first recognized by WERTHEIMER who thereby attached to Gestalten a degree of reality far beyond any they had previously possessed. This implies, as KOFFKA emphasized, that central physiological processes cannot be regarded as sums of individual excitations, but as configured whole-processes. (1920/1938, p. 6)
The work of WERTHEIMER and KOFFKA has proceeded... in conformity with our earlier remarks about physical systems... It is the aim of this essay to support the WERTHEIMER hypothesis on physical grounds. (p. 20)

The law exemplified in cases of this sort may be called the tendency towards simple Gestalten, or the law of Prägnanz... This designation comes from WERTHEIMER, not as a description of inorganic physical behaviour, but of phenomenal and therefore also of physiological process-structures. Nevertheless it is possible to apply the terms to physical phenomena also, for the general tendency and line of development observed by WERTHEIMER in psychology and designated by him as the law of Prägnanz is obviously the same as we have here been discussing. (p. 54)

There is no reason at all why the construction of physiological processes directly underlying experience should be impossible, if experience allows us the construction of a physical world outside, which is related to it much less intimately... I should have ever so much difficulty in trying to relate definite experience to definite processes so long as I failed to assume one specific relationship between the two orders, viz., that of congruence or isomorphism in their systematic properties. (1929, p. 61)

... the most essential traits of experimental or perceptual contexts are the same as those of their physical counterparts. With respect to these traits the perceptual and the physical structures are isomorphic. If they were not, we could have no physics. (p. 162)

in all these cases it is really structure in which the world of percepts and the physical world have so much in common. Resemblance as to the demarcation of definite objects, and therefore to their number, means in fact similarity in the gross structure of the two worlds. And then inside such particular objects there is again structural resemblance between the perceptual and the physical world. (p. 166)
Physics, it was stated, proceeds on the assumption that certain structural traits of percepts agree with the structure of corresponding physical situations. It is, however, only macroscopic structures which can be common characteristics of the perceptual and the physical world. And this statement has sense only if the notion of macroscopic objects is found to refer to definite physical entities. We have, I believe, been able to show that it does. It is therefore a meaningful thesis that perceptual and physical contexts are isomorphic in essential macroscopic respects, and that to this extent there is resemblance between the phenomenal and the physical world, (p. 184)

Concerning the emotional sphere, he wrote: "I propose to consider the nature of cortical processes although many philosophers dislike to hear much about the brain when philosophical problems are being discussed" (p. 185). "The cortical correlates of mental life or, as we may also call them, the psychophysical processes, are more interesting for our purposes than any other biological facts" (p. 194). "[It is not] a plausible assumption that cortical processes consist of independent events in individual cells. In the following paragraphs psychophysical correlates will, therefore, be considered from a macroscopic point of view" (p. 212). "Practically any part of human experience might be taken as an example of the fact that molecular events in the brain do not as such show much resemblance with phenomena" (p. 215).
Continuity is a structural trait of the visual field. It is also a structural fact that in this field circumscribed particular percepts are segregated as patches, figures, and things. In both characteristics, we have found, the macroscopic aspect of cortical processes resembles visual experience. To this extent, therefore, vision and its cortical correlate are isomorphic. In the last chapter the same term has been used. There, however, it applied to the relation between visual organization on the one hand and the macroscopic structure of situations in physical space on the other. The fact which mediates between the physical and the perceptual structure is now found to be cortical organization, which, as a rule, resembles both... .Where perceptual organization does not agree with facts in physical space, cortical organization seems to agree with perception rather than with physics. (1938, pp. 217-218)
... .Our present discussion is mainly concerned with the question of isomorphism between the visual field and its psychophysical correlate... Not for a moment should we forget, however, that isomorphism, thus considered, is a relation between visual experience and dynamic realities. (1938, pp. 218-219)

The following comes from LUCHINS (1968):

WERTHEIMER sought examples from the field of perception, an area of psychology with a high reputation for exactness. He had little success until 1910, when he went on a trip, and while on the train, he thought of an optical phenomenon that seemed suitable. At Frankfurt he got off the train and bought a toy stroboscope. In a hotel room he set up the experiment by substituting strips of paper on which he had drawn series of lines for the pictures in the toy. The results were as he expected: by varying the time interval between the exposure of the lines, he found that he could see one line after another, two lines standing side by side, or a line moving from one position to another. This "movement" came to be known as the phi phenomenon.
WERTHEIMER asked SCHUMANN, his former teacher at Berlin and now at the Frankfurt Psychological Institute, if he could provide someone to act as an experimental subject. SCHUMANN s laboratory assistant, Wolfgang KÖHLER came. For the next experimental session, KÖHLER brought his friend Kurt KOFFKA, who also served as a subject. KÖHLER persuaded SCHUMANN to visit WERTHEIMER and to invite him to conduct his experiment at the Frankfurt Institute. A simple apparatus to demonstrate the phi phenomenon was constructed, and the now classical experiment was conducted (WERTHEIMER, 1912b).
WERTHEIMER explained the significance of the experiment as follows: "What do we see when we see the movements of a hand or a light? Is it appropriate to say that we have a sensation in different places on the retina from which movement is inferred? Is it appropriate to cut the phenomenon of movement in this way into a number of static sensations?" (1937). Although there had been psychologists and philosophers before him who believed that movement was not an inference from static sensations on the retina but was a sensation sui generis, they had not demonstrated this in a scientific manner. WERTHEIMER now presented the thesis in a way which made experimental decisions possible.
It was not merely WERTHEIMER s experiment but his formulation of the underlying problem and the way to proceed to solve this problem that launched Gestalt psychology. Through experimental variations, he tested, one by one, various possible explanations of the phi phenomenon and found them wanting. According to WERTHEIMER, the essential features of the phi phenomenon are the following: it is a counter example to the assumption that piecemeal and summative approaches to psychological phenomena are universally adequate; it belongs to a category of genuine dynamic experience which must be understood in terms of dynamics rather than reduced to static events; finally, it is an example of a structure that is not an arbitrary arrangement of events but has inner connectedness (1937).
WERTHEIMER felt that there was a need for a model of such dynamic experiences, and he hypothesized a possible physiological process: "The motion is due to a field of activity among cells... not excitation in isolated cells but field effects" (1937). This model applied concepts of field-theoretical physics to a neurological event. (LUCHINS, 1968, pp. 523-525)

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[ * ] Paper prepared for the 11th Scientific Convention of the GTA, March 11-14, 1999. Thanks are offered to Dr. Gerhard STEMBERGER for his interest in the topic and for encouragement in the preparation of this report. We are grateful to Lorraine PISARCZYK, Administrative Secretary of the Department of Mathematical Sciences of Rensselaer Polytechnic Institute, for her careful typing of the manuscript. [Note that our parenthetical comments are usually enclosed in square brackets.] [back to text]

[fn 1] Of the authors we cite, ARNHEIM, ASCH, SCHEERER, and the present first author, attended WERTHEIMER s seminars at the New School for Social Research. [back to text]

3rd part

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