The issue of isomorphism

Roughly speaking, an isomorphism is said to exist between two entities or events when one can recognize a system of connections in both that tends to indicate a similarity of structure or function¹⁹ with respect to the collection of elements that constitute the system of connections in such entities or events. To use a sports analogy, in order to help illustrate what is meant here, one might say that hockey and lacrosse were, in many respects, both structurally and functionally isomorphic because the system of connections (both in terms of the rules and in terms of the actual activity which constitute these sports, respectively, and represent the elements within the system of connections) indicate a similarity of form and process without being identical, whereas the isomorphism between, say, hockey and basketball is much 'weaker - although one could still talk in terms of 'partial' isomorphisms since there are certain similarities, on both a structural and functional level between the two sports.

Biological examples of isomorphism are readily identifiable as are sport isomorphisms. For instance, one might talk of the structural isomorphism between' a human skeletal system and that of a gorilla or a dog - the degree of isomorphism being weaker in he latter case than in that of the gorilla. Or, one could compare the functional isomorphism existing between a human capacity for the conserving of information (i.e., memory) and such a capacity in, say, a paramecium.

Piaget is committed to rooting epistemology in a biological context - while, simultaneously, differentiating between strictly organic structures or functions, on the one hand, and, on the other hand, cognitive structures or functions. Consequently, the notion of isomorphism is of considerable importance to his overall theoretical position since Piaget believes his perspective represents a means of showing how epistemological structures and functions could be based on - and, therefore, be linked to various organic processes and organs - without being identical, or reducible, to such processes and organs.

In essence, the foregoing is a fundamental hypothesis or theme running throughout Biology and Knowledge, in particular, and many other written works of Piaget. Thus, for Piaget:

"... cognitive mechanisms constitute both the resultant of general auto-regulatory processes in the living organization and are also specialized regulatory organs in exchanges with environment. If there is a good foundation for this hypothesis. it will mean that, from the functional point of view, certain general functions common to both organic and cognitive mechanisms do exist, but that, in the case of cognitive mechanisms, a progressive specialization of functions also exists."²⁰

In short, what Piaget wants to show is that organic and cognitive mechanisms are, functionally and structurally, isomorphic and that this isomorphism is the result of cognitive mechanisms having developed, over evolutionary time, out of general, organic auto-regulatory processes. At the same time, Piaget wants to contend that cognitive mechanisms, once having been generated, are capable of constructing (again, over time) their own specialized organs of regulation that can become involved in complex interchanges with the environment - which, as far as the organism is concerned, tend toward a progressive specialization of functions.

According to Piaget:

"... knowledge comprises first and foremost an organization function, and that is our first fundamental analogy with life."

The meaning of "organization function" in the above quote refers to the way in which functioning as a whole - that is, the dynamic aspects of structures taken as a whole - acts upon the various substructures.

This is another way of saying that the organization function plays a reciprocal role in the auto-regulation of the organism (or an organ) relative to the part played by the functioning of a given substructure and its (i.e., the substructures) action upon the total structure ("structure" referring to either the organism as a whole or the particular structure of which the substructure is a part). Thus, in attempting to establish an isomophism between knowledge and life (which he later hopes to cash in on as a basis for arguing that epistemology is rooted in biological functioning, in general, and auto-regulatory processes, in particular), Piaget is claiming that, among other things, both life and knowledge share this feature of an organization function in.

The functioning encompassed by both totalities (i.e., life and knowledge) acts upon the various substructures such that, in each case, the whole is not merely a simple collection of individual elements but has an effect greater than the sum of the parts in terms of the interrelationships made possible by the whole's existence. On the other hand, the whole is not something which is other than, and distinct from, its various functional and structural elements.

In further elaborating the idea of "organization function", Piaget stipulates that:

"... the organization qua functioning is not transmitted by hereditary as are characteristics such as shape, color, etc., it continues and succeeds itself qua functioning as a condition necessary to every transmission and as a transmitted content."

From this perspective, Piaget points out that one of the primary characteristics of the organization function is its quality of conservation in which the organism or organ is observed to maintain its form and/or function despite the fact that various transformations are taking place within the organism or organ on a continuous basis.

This leads to something of a problem for Piaget, however - especially in terms of its implications for his evolutionary theory. Once again, an issue is raised, discussed previously, concerning the question of origins.

If, as the previous quote indicates, organization qua functioning is said to be a necessary condition for hereditary transmission in addition to being a transmitted content, then, in terms of accounting for the first appearance of life on earth, one would like to know how such an organization function came about since it seems to presuppose itself as a necessary condition. If it didn't already exist in some sense - which Piaget needs to assert due to the preformationist ramifications implicit in accepting the prior existence or organization - then, it couldn't have presupposed itself and, therefore, is not a necessary condition for hereditary transmission.

This logical, if not empirical, question not only has the effect of leaving unexplained how the organization function subsequently develops, but, as well, of creating something of an ontological chasm between organization and evolution since that which remains unexplained leaves open the possibility that the two are not as dependent on each other as Piaget would like them to be, and, as a result, drags one back toward something like a mutationist account in which random, unorganized occurrences, somehow, bring about evolutionary change - at least, until the sort of cybernetic auto-regulatory system which Piaget is proposing was somehow generated. Yet, this is the gap which Piaget cannot account for - namely, how this generation comes about.

Piaget's difficulties do not end here, however, since, quite ironically, the foregoing problem has a persistent tendency to carry over into the issue of epistemology in an ‘isomorphic’ fashion. For instance, Piaget already has stipulated, on several occasions in Biology and Knowledge (e.g., see p. 145, toward the beginning of Section 3), that he considers it "ridiculous" for anyone to contend that intelligence exists at every level of organic life - which obviously means that some levels of organic life manifest organization without the organization possessing the quality of intelligence.

However, given that Piaget maintains that:

"Any act of the intelligence presupposes the continuity and conservation of a certain functioning,"

one is left with a disturbing puzzle. If one is to assume that the nature of "certain functioning" does not manifest a quality of intelligence - although it is organized - and, thereby, avoid the problem (for Piaget) of preformation, and if one is to assume that this "certain functioning" possesses, through its very activity, the capacity to modify structure, one would not be unreasonable if one were to ask, at this point, how such functioning (which is, by Piaget's own admission, non-intelligent but organized) is - merely through its dimension of activity - able to either generate structures and functions of an intelligent sort or lead toward a context which will be able to do so.

Up until now, the focus has been on the possibility of functional isomorphism (or correspondence) between organic and cognitive contexts. Piaget, however, in a bid to lay the foundations for his contention that cognitive functions developed out of organic functions, also attempted to establish a case for the existence of structural isomorphisms.

Nonetheless, and one might have anticipated this, there are some problems lying in wait for him. For instance, Piaget maintains that all cognitive systems - from the simplest to the most complex - are characterized by two broad features: namely, the tendencies to differentiate and integrate, which are features that cognitive systems hold in common or share with all biological systems.

Furthermore - and leaving aside, for the moment, the issue of integration in order to be able to concentrate on the dimension of differentiation - one notes that Piaget claims (see p. 158) that every differentiation of an organization contains an hierarchical order with respect to the structures and functions which are involved in the differentiation. In addition, according to Piaget, one of the most basic forms which hierarchical orders tend to manifest - a hierarchy in this instance being a matter of structure, ranging from very general to quite specific - is that of "inclusion" in which a structure incorporates into itself either a substructure of some kind or part of a structure.

On many different levels of biological activity, the notion of inclusion could represent an important means of linking-up cognitive and organic structures, especially since the inclusion property is a prominent theme of many logical structures. While discussing the notion of inclusion in different biological contexts, Piaget points out that on every step of the evolutionary ladder there exists a wide range of assimilatory processes through which the organism is engaged in interchanges (e.g., food cycles) with the environment and that all these various processes involve "discriminations of a type not unlike classificatory inclusion" (p. 161).²¹

In other words, since different organisms have varying requirements with respect to, say, sustenance (e.g., what will nourish a beetle will not necessarily nourish a bee, and so on), then, each organism is required to make the kind of discriminations among the variety of possibilities in the environment which will provide appropriate nourishment and avoid that which will not. Moreover, even on a physiological level, Piaget describes how different cells have different requirements and manifest a certain kind of discrimination with respect to their internal conditions and the external circumstances most immediate to them, which result in complex interchanges.

Apparently, classificatory schemata (which are forms of inclusion) are very prevalent and very important in all biological organisms. Indeed, they are so important that Piaget insists:

"... there can be no behavior without some elementary form of classification. Every act of perception is "categorical", as J. Bruner has demonstrated; this means that it tends to identify the object perceived in relation to previous action schemata, and this presupposes some classification. The exercise of instinct like-wise presupposes classification..."

The last sentence of the above quote is an interesting one, as well as being obscure. It is interesting because of the questions it tends to elicit.

For example: what kind of classification does instinct presuppose? Or, how does such a classification system lead to the establishment of instinct - especially since Piaget has insisted, again and again, that instinct is a transindividual (e.g., see pages 277-278) phenomenon and, therefore, cannot be a function of just the classification schemata that appear in a single individual?

Besides being interesting, the above quote is, as indicated previously, also obscure. This is because the questions raised remain unanswered and, consequently, shroud Piaget's meaning in a certain amount of theoretical darkness.

Once again, he is faced with some very embarrassing queries. This time the problem concerns the origins of the classificatory inclusion structures which all behavior presupposes. One could agree with Piaget when he says:

".. the classification function seems to be found in every organization structure, and this fact constitutes a remarkable structural isomorphism between biological and cognitive organizations. Of course, we are not talking of the same kinds of classification."

Yet, saying this does absolutely nothing to establish Piaget's position. If anything, it tends to bring out, under examination, certain problems for Piaget, as well as to point, increasingly, to the need for some other kind of explanatory approach since his epigenetic approach to evolution seems to be floundering on a beach of unanswered (and, perhaps, unanswerable) questions - not the least of which concerns the problems surrounding the questions of origin and etiology of (and here one couldn't be more in agreement with Piaget) the differences in classificatory structures.

Although the idea of differentiation is, in principle, a very important one, nevertheless, as it stands, it is far too vague. Supposedly , this vagueness was to have been eliminated by Piaget's epigenetic approach, but, as it has turned out so far, what has happened is that one merely ends up substituting one brand of vagueness for the condition of vagueness inherent in various theories of evolution proposed by, say, the mutationists, upon which, Piaget is seeking to improve.

In a sense, what Piaget has provided is a sort of surface structure clarity. However, this has been purchased at the expense of an understanding of the deep structure which generates and shapes many, if not most, of the surface features - or, at least, which establishes the limit boundaries within which, and through which, the surface features may take on their different values. Indeed, Piaget as much as admits this is the case when he notes that:

"... Bertalanffy says rightly: "what we would like to know is not merely a few equations of measurable vectors but the law which integrates them..."

Repeatedly throughout Biology and Knowledge, Piaget makes claims similar to the following one:

"Between a hereditary system and some acquisition imposed on the subject by the environment and its regular sequences, there does, in fact exist a tertium quid, which is exercise. Thus, it seems almost certain by now that maturation of such a sector of the nervous system is allied to some functional exercise."

One can agree that maturation of the nervous system is intimately connected to functional exercise without having to be committed to saying, therefore, that such exercise represents a tertium quid or is a sui generis phenomenon (as Piaget does, among other places, on page 321 of Biology and Knowledge). What Piaget often seems to overlook or ignore is that the aforementioned process of exercise is, itself, dependent on what is structurally and functionally possible in an organism, and what is possible in this sense is a function of the potential built into the genetic givens.

A human being is different from a bird not because they become engaged in different patterns of exercise (although, of course, they do become so engaged) but because the allowable possibilities concerning such patterns have already been set down in what has been transmitted genetically.

Piaget seems to want to establish a sort of semi-pre-suppositionless philosophy in which, for example, the organization function - in conjunction with, say, the cyclic open²² system - are related to organic givens but, somehow, independent of them as well, such that they can combine to construct new functions and structures (or new sub-functions and new sub-structures) which were not present, even in principle, in what existed prior to the construction.

The phrase: "even in principle", in the foregoing sentence, is crucial. Otherwise, Piaget, implicitly, would be harboring a preformationist position which is logically inconsistent with his epigenetic perspective.

Consequently, what one is left with is a neutralized sort of exercise principle ("neutralized", because to have anything built into it would permit some form of preformation to slip in the back door, so to speak). This principle appears, mysteriously, to be suspended in the midst of the organism and represents the means of transformation from: merely organic functions or structures, to: cognitive functions or structures and which, thereby, also is intended to explain - the existence of isomorphism between organic and cognitive forms of activities since the latter have developed out of the former and carry with them something of their ancestry without being limited by it.

At one point, Piaget states:

"If one can pass from schemata made up of forms that are both organic and sensorimotor, such as reflex and instinct schemata, to schemata that are sensorimotor, properly speaking, such as "habit" schemata, it becomes clear that such a transition is equally natural if made between habit schemata and schemata of representational intelligence. The intervening stages are supplied in this case by the many schemata of sensorimotor intelligence which are initially mere co-ordinations of habit schemata but which eventually set up schemata astonishingly isomorphic to those of representational intelligence. For example, a certain number of partial displacements, each one of which can correspond to one habit schema only, finally coordinate into a wider system, corresponding to a "displacement group... Now this sensorimotor "group" schema, however limited it may be in its functioning nonetheless constitutes a substructure, on which, at some time between seven and twelve years, the thinking will build a corresponding operational structure - a structure that is still unreflective, in the sense that it remains internal to the functioning of the intelligence (but as a representation now, no longer merely as an action) and is not an object of the intelligence. After this, reflective abstraction of a mathematical kind will build up a structure qua object of reflection, in the same way that it builds up all other elementary operational structures (groupings, inter-sections, orders, connections, etc.) from structures inherent in the functioning of thought and action."

Piaget has been quoted at length, here, because the excerpt is 'vintage' stock, so to speak. It is fairly representative of any number of passages in his various books and articles that attempt to describe how the various transitions from instinct to mathematical/logical operations occur.

The passage also displays the sorts of characteristics which give so many of his descriptions the appearance of an explanation without having, upon examination, the substantive qualities of a true explanation. For example, contrary to what Piaget maintains, it does not become "clear that such a transition (i.e., from instinct and reflex schemata to 'true' sensorimotor schemata) is equally natural if made between habit schemata and schemata of representational intelligence". At least, this isn't clear as long as Piaget contends that the transition is effected by neutralized epigenetic principles of a sui generis nature (see Biology and Knowledge, p. 321).

Moreover - and despite Piaget's confidence in the alleged significance of the astonishing isomorphism between the schemata set up by sensorimotor co-ordinations and those of representational intelligence - one still is at a loss concerning exactly how the intervening stages that are supplied by sensorimotor intelligence are able to bridge the gap between instincts and representational intelligence (or, how a number of partial displacements corresponding to certain habit schemata become 'coordinated' into a wider system) unless one were to assume that the coordinating capacity, and so on, were already present in a sense that goes far beyond what Piaget is willing to agree to in the way of innate givens.

Finally, notwithstanding Piaget's assurances concerning the ‘sameness’ of the construction or building process, one remains mystified as to how, in epigenetic terms, thinking is to, first, build unreflective operational structures from non-operational sensorimotor substructures and, then, is somehow capable of generating reflective abstractions from structures that are neither reflective nor abstract.

Seemingly, for Piaget, there, still, are unexplained evolutionary gaps between the organic and the cognitive. In addition, there are unexplained gaps between cognitive stages as well - gaps which, despite the isomorphisms between stages and functions, do not seem capable of being filled in by Piaget’s sui generis, epigenetic tertium quid.

Piaget is very adept at talking in terms of: "integration", "differentiation", "building up", "recombining", "spontaneous exercises",²³ "setting up", "generalizing", "abstracting", "reconstruction", "organizing", "adjusting to" and a number of other similar expressions which give the appearance of explaining, without saying anything sufficiently specific to allow one to understand what it is that is actually going on in any deep sense. In fact, Piaget closes out his discussion of isomorhpism in the following way:

"The analysis we have striven to make throughout this chapter remain, neverthe-less, incomplete and fragile, for partial isomorphisms have no meaning... unless transformation laws can be produced such as will allow a transition from one of the compared terms to the other, and unless proof is furnished that these transformations can actually - and in this case biologically - be realized;"

Precisely these transformation laws are what are missing from Piaget's epigenetic position. Without such laws, the idea of making isomorphic comparisons may be suggestive, but this idea is extremely problematic as well, as has been indicated in the previous 30-plus pages of discussion.

19. Although Piaget uses the term "structural isomorphism" without any misgivings with respect to whether, or not, the structural features of two given entities can be compared in a meaningful or legitimate fashion, he has certain reservations concerning the use of the term "functional isomorphism" because one often is unable to identify a one-to-one correspondence between the elements of different organs which manifest or carry out the same kind of function. Consequently, Piaget prefers to use the term "functional correspondence" when attempting to compare the similarities of functioning in different frameworks and tends to focus, therefore, on the dynamic roles played by various organs (or substructures of such organs) with respect to the overall functioning of the organism of which the organs are part. As long as this shift in emphasis is kept in mind, speaking of "functional isomorphism" seems perfectly appropriate.[Back to Text]

20. While, clearly, Piaget believes that the inter-relationship between structure and function is a very intimate, cybernetic sort of phenomena, he tends to assign slightly more significance to the functional dimension (in the sense that many different situations may fulfill or serve the same function) and, therefore, in a sense, are more likely to be stable across evolution.[Back to Text]

21. This claim is based, to a large extent, on his agreement with certain aspects of Ludwig von Bertanlanffy's approach to examining organizational systems and their implications for, and applications to, a biological framework. Piaget especially keys in on the notion of an "open system" and when given the added dimension of the 'cyclic' order, Piaget develops the idea of a cyclic open system that represents a set of structures within all organisms which is said to be capable of systematic interchanges with the environment, as well as being capable of an 'upwardly mobile' sort of transformation process. But, since the idea of a 'cyclic open system' tends to encounter many of the same problems pointed out in the discussion on function isomorphism, this will not be examined further.[Back to Text]

22. In addition to inclusion structures and cyclic open system structures, Piaget also speaks of order structures. However, given such statements as: "...order structures do seem at the outset (from the DNA stage) to be inherent in every biological organization and its functioning" (p. 166), then, seemingly, one might encounter additional difficulties which are similar to the ones noted in the discussion of functional isomorphism.[Back to Text]