All cultural forms arise from and are indicative of some form of organized activity.
This is so also with cultural concept words, including economic concept words and practices that led to the capitalist markets or the concept words and practices that led to, and were included in, the Euclidean formulation of planar geometry. Prior to the (social, economic, natural) sciences, there already existed activities out of which the recurrent use of certain words emerged. Although such concept words may appear abstract later, they directly arose out of mundane exchanges in mundane activities.
It is important to understand the cultural history of concept words, because subse- quent members of the associated cultural practices in a way re-understand what the original geometers, went through in the re-constitution of the conditions within which actions and language make sense.
One genetic analysis of abstract concept words at the cultural-historical level uses geometry as its case; its history is understood in terms of an unfolding tradition of the unity of humans and their environment (Husserl 1939). Thus, geometry emerged from the activities of Greek craftsmen. The things they were concerned with were surfaces, corners, edges, lines, and points (e.g. of intersection). In using building materials such as stones, they were concerned with smoothening the sur- faces to make them fit, and to straighten out lines to bring neighboring stones as close together as possible. Whereas flat surfaces, straight lines, sharp edges, and point-like corners would have been of concern in some activities where rounded surfaces would have bothersome, other activities would have required increasingly perfect curved surfaces and lines, such as in the construction of wheels. Thus, for example, the word cylinder has its origin in the ancient Greek kúlindros [κὑλινδροσ], which translates as “roller,” the perfection of which would have improved the capac- ity of an object to roll evenly; the word cube has its origin in kúbos [κὑβοσ], a die to play with, the perfection of which would be concerned with addressing fairness of all sides to have equal probability; and the term pyramid has its origin in the Greek name puramid [πυραμιδ-, πυραμι𝜍] for the monumental structures of the Egyptian people. Refinement processes involved measuring and counting events. It is out of these and similar concerns that the idea of an ideal geometrical object his- torically emerged. Real objects are not perfect, but ideal objects could be envisioned as the limit objects if the craftsmen could shape their objects perfectly. The possibil-
ity then arose of working with these ideal objects, ideal in a practical sense of per- fect material objects, which then were objects of thinking alone because they did not and could not exist in an inherently imperfect world. Historically, there is thus a move toward the distinction between two very different worlds: one material and imperfect, the other ideal, immaterial, and perfect. Once detached from materiality, these ideal objects appeared to exist outside of time. During the historical period of what will have been upon looking back the first geometers, there was no conceptual language that they could use to communicate. Nor did they have the apparent need for a precise description of the after-the-fact pre-scientific materials that they were working with; and there was no need yet for describing the ways in which geometri- cal identities and the first axiomatic statements emerged during their collaboration.
Emergence of the Ideal: A Cultural-Historical Account
The form of value is ideal, although it exists outside human consciousness and independent of it. (Ilyenkov 2009, 254)
At the heart of the quest for appropriate theories of abstract concepts, that is, of ideals and universals, is the relation of these phenomena to the material world of our everyday concerns. In the cognitive sciences, the failure to theoretically relate ideal, universal, general, and abstract concepts to the world we inhabit is known as the above-noted symbol grounding problem, which may be taken to be another word for the psychophysical problem. The Dutch philosopher Baruch Spinoza had suggested that all attempts to understand the relation by beginning with the material body or with the immaterial mind (thought) would have to fail. Some recent cultural- historical studies of cognition in educational settings draw on the works of the German philosopher Georg F. W. Hegel, who had developed a (phenomenological) theory of the development of mind that focuses on process – Being [Sein]. The choice to ground those studies of cognition in the works of Hegel was justified, for his “intelligent (dialectical) idealism, which is the idealism of Plato and Hegel, is much closer to the essence of the matter than materialism that is popular, superficial and vulgar” (Ilyenkov 2012, 164). Although Hegel’s formulation is superior to sim- ple materialist (biological) accounts, he did not provide the key to the real material- ist solution concerning the relation between the ideal and material aspects of life, between thinking and the body. Hegel did not consider the material as material; he writes “not about nature ‘as it is,’ but exclusively about nature as it is represented”
(172, emphasis added). Recognizing the shortcomings of the best of idealist (Hegel) and materialist epistemologies (Ludwig Feuerbach), Karl Marx and Friedrich Engels developed an event-based approach that actually explains how ideal, univer- sal, and abstract ideas emerge from real, mundane activities. This conceptualization would be the basis of cultural-historical approaches to human activity, mind, and culture in the works of Vygotsky and his student and collaborator Alexei N. Leont’ev.
Both authors extensively refer to and actively engage with the works of Marx and Engels.
In Das Kapital [Capital] (Marx and Engels 1962), a cultural-historical analysis is provided of the emergence of the abstract, suprasensuous (exchange-) value from the concrete, sensuous use-value of a commodity. In the first section of the book, Marx shows how the abstract exchange-value of a commodity historically emerges from what initially only is concrete use-value. That is, he exhibits the emergence of the sensuous–suprasensuous character of commodity from its initial solely sensu- ous form; this sensuous–suprasensuous character is synonymous to saying that it is societal (Mamardašvili 1986). That something has suprasensuous quality should not lead us to think that it cannot be present; instead, we theorize it in terms of everything present in perception but in non-sensuous manner, such as that which is recognized as permanent across occasions (Whitehead 1920). Classical epistemologies treat the suprasensuous as the ideal (e.g. mathematical knowledge). But the noted dichoto- mies are overcome only when the ideal is thought as sensuous–suprasensuous. This means that abstract concepts never are detached from the plenum of the material world, where the sound-words denoting so-called concepts are events among events, having the special function of making some of these events intersect. This chapter works toward such a practical conceptualization of the concept, which is provided after a case study that serves as its material ground.
Marx and Engels begin with considering simple barter exchanges; and they return to the earliest considerations of such exchanges in the recorded history, such as in the works of Aristotle, with whose examples they begin. Thus, the simple value form of a commodity comes as value expression or value relation between things:
Form A
5 beds = 1 house; or 20 yards of linen = 1 coat
Here, in the exchange relation between 20 yards of linen and 1 coat, the actual proportion may be accidental (contingent). Such an exchange relation is an event that also expresses a particular social relation: the exchange, as an event, is a social relation between buyer and seller, which thus also is an event rather than a thing. It may occur in the absence of human society (e.g. in exchange relations observed among animals). The value of the beds is measured in terms of the commodity house such that the five of the former are equivalent to one of the latter. Marx and Engels note that Aristotle stopped in his analytic tracks because he could not over- come the equivocation of dissimilar things: two things such as beds and houses cannot be commensurable. The idea of value thus did not yet exist during that his- torical period even though in practice, people were not only engaging in barter exchanges but also using money to pay for products.
The next step in the conceptualization occurs when the exchange is given general form: z Com[modity]. A = u Com. B, or = v Com. C etc. In this form, value no lon- ger is contingent to the particular, inherently social exchange event, but stands in relation to the world of commodities particular to a society. At a minimum, two living persons – i.e. families of events – are involved. These respective families of events intersect in an event involving giving and taking simultaneously – not unlike
in the event of communication where words are exchanged. That is, the economic exchange, as all exchanges, is transactional; in other words, each action involves at least two people and thus is joint action. Exchange thereby comes to be conceptual- ized in temporal terms, not as an abstract relation-as-thing where the things exchanged stand (thus mediate) between the individuals-as-things. The failure to conceptualize the exchange from a transactional perspective – i.e. as event – leads to a misunderstanding of what the theory of Marx and Engels entails. Thus, for example, commodities are conceived in terms of the unity/identity of use-value and (exchange-) value. Those who take interactional perspectives will say that what a commodity is depends on the perspective of the person involved, such that the 20 yards of linen are exchange-value for one (e.g. the weaver) and use-value for the other (e.g. the tailor). But adding up two individual perspectives does not yield something new. It is when they act jointly that the two eyes can give rise to stereovi- sion; and this binocular “double view is the relationship” (Bateson 1979, 133). In the transactional perspective, the two perspectives cannot be separated but are an integral quality of the relation. In other words, the exchange situation cannot be reduced to the addition of two individual perspectives because the event is one in which giving and taking not only occur simultaneously but also are constitutive of one another. They are phases of one whole event: the exchange.
Continuing with their analysis, Marx and Engels conceive of the total or expanded form of value, which appears in the relative relations of Form B:
Form B
20 yards of linen = 1 coat 20 yards of linen = 10 lbs. tea 20 yards of linen = 40 lbs. coffee
…
In this form, the value of linen remains constant whether it is traded for and expressed in 1 coat, 10 lbs. of tea, 40 lbs. of coffee, etc. In this form, the accidental dimension that characterizes Form A has disappeared so that the value amount determines the exchange relation rather than the other way around. The bodily expression of each of these commodities is realized as a particular equivalent form.
Form B thus expresses the situation where weavers trade their linen for everything else that they need. But even earlier in the evolution of societies, commodities and their relations existed in Form B. Thus, for example, certain tribes on the Northwest Coast harvested eulachon oil, which they then traded for everything they needed but did not produce on their own – e.g. “four blankets, two beaver skins, or two boxes of dried halibut” (Le Dressay et al. 2013, 118). These tribes related to others in terms of what they could get for a certain amount of eulachon oil, which thereby became the basis of a measure for everything else. But to become measure for everything, something else was required.
In Form B, any general expression of value is excluded, because the value of a commodity (e.g. eulachon oil) is expressed in terms of the value of many other com- modities. In contrast, the general form of value, the condition for the money form,
makes one commodity the measure of all the others. That is, it reverses the forms of Form B, which is expressed in a new kind of system of relation:
Form C
1 10
40 1
2 coat
lbs tea lbs coffee quarter wheat
ounces gold etc
=
=
=
=
= .
.
..commodity
yards linen
= ü ý ïï ïï þ ïï ïï
20
Such an expression no longer is merely social, for, as it is valid in society as a whole: it is societal. The value of everything now is measured in terms of the same commodity. It was in this form that the blanket became the basic unit of trade among Northwest Coast aboriginal peoples. Everything traded was expressed in terms of blankets. This was therefore a precursor to the money form; but in addition it had a form that it could be used: to keep warm. Here, then the value of all commodities but that of linen in the Marx and Engel’s example and the blankets in the Northwest Coast First Nations case now is expressed in terms of the value of linen or blankets.
All these commodities express their value simply, in elementary form, and unitarily, that is, with respect to the same commodity linen or blankets. The result of this development is that “value is elementary and the same for all, therefore general”
(Marx and Engels 1962, 79). The value form is general because in every single trade relation of a society or nation, the same references – e.g. linen, blankets – serve to specify the values of all other commodities in the pertinent exchanges. The value- form comes to be articulated in Das Kapital [Capital] as a reified form of human societal life-activity (Ilyenkov 2012). This has the effect that the human production of life “now appears as a twofold relation – on the one hand, as a natural, on the other as a societal relation” (Marx and Engels 1978, 29). It is that same twofold relation that reappears in the introductory quotation from Mead’s Philosophy of the Act. The natural relation here pertains to that between two commodities that are literally present in the hands of buyer and seller simultaneously – e.g. the coat and the 20 yards of linen; and in that same exchange there exists a social relation between the two people. The two relations reflect each other. Whereas this may appear mysterious when we use an object-oriented approach, the appearance of the twofold relation is a consequence of the transactional approach, where it is theo- rized as an event of exchanging. In this event, at least four strands come to be bun- dled; in other words, four strands (person 1, person 2, commodity 1, commodity 2) come to intersect and thus are related in and by the same event. Take out one of these
strands, and the exchange event will disappear. To Marx, therefore, the ideal (here value) is a societal product existing precisely as societal relation (an event) not merely in societal relation.
Value Forms A to C are expressions of a historical process from simple barter exchanges between individuals to complex economies. The ideal (exchange-) value of a commodity is an abstraction that completely differs from its use-value. The value does nothing to keep warm or quench the thirst, but the blankets, linen, and tea are used in the satisfaction of human needs. Value is an ideal form, a form of con- sciousness directly arising from the societal relations that turn 20 yards of linen, a certain number of blankets, and, later, money into universal measures. The ideal first existed as relation – exchanging as a form of Being – before it could be a form of consciousness (ideal, suprasensuous exchange value). We observe here how a com- modity expresses (i.e. manifests) itself in two distinct ways, as material sensuous use-value and as ideal suprasensuous exchange-value. Once the system is estab- lished, every new commodity follows the pattern of all previously existing com- modities and expresses its value in the same general form.
These considerations lead to what is going to be of central importance to psycho- logical considerations: the societal nature of the ideal, general, and generalization.
In the commodity exchange, the value form “has to have societally universal form”
because “the object character of value of commodities,” which is “solely the ‘soci- etal existence [Dasein]’ of these things,” “can be expressed only by means of its general societal relation” such that “the general value form … is the societal expres- sion of the commodity world” (Marx and Engels 1962, 80–81). As a result, “the value form is ideal, although it exists outside human consciousness, independent of it, in the space outside the human head, in things, i.e., in the commodities them- selves” (Ilyenkov 2012, 164). Alternatively, the value form might be said to exist in human life-activity, that is, in the evental forms of economic, mathematical, scien- tific, and any other mundane praxis.
The foregoing analysis shows that exchange-value expresses itself in the relation of people and things. It does not exist otherwise, because as the abstract and general, it cannot be found in the object (commodity, mathematical expression) or in the human (mind). Instead, the ideal is a recurrent form in the {unity | identity} of the forms of (cultural) things such as commodities or mathematical objects, which exist outside the individual, and forms of dynamic life-activity, inherently alive and soci- etal in nature. Thus, “‘ideality’ as such exists only in the constant transformation of these two forms of its ‘external incarnation’ and does not coincide with either of them taken separately” (Ilyenkov 2012, 192). This transformation is a “process by which the material life-activity of societal man begins to produce not only a mate- rial, but also an ideal product, begins to produce the act of idealization of reality”
(158, underline added). Once it exists as a recurrent and thus permanent form, the ideal functions as an essential moment of the material life-activity of the societal person, making possible the opposite process (event) of materialisation and objecti- fication of the ideal. Ideality, appearing in the form of knowing-how, is entirely soci- etal in nature, existing in societal, transactional relations (i.e. events) that cannot be reduced to self-identical individuals interacting such as to produce the merely social.
The ensemble of societal relations – i.e. also society-as-event – is reflected in the relations among commodities generally and in the generalized value specifically.
Value, the result of commodity relations, hides its own origin in generalized human relations. Thus, the “proper societal relationship of people of common labor (value) is presented by consciousness as occurring outside the societal relations of things, as consciousness of the suprasensuous properties of the latter” (Mamardašvili 1986, 109). The ideal, in the way developed here, is a recurrent pattern of the real material human life activity, which, qua activity, is theorized in this book as event. As rela- tion involving material people and things, the ideal exists outside the head, outside the brain and thus accessible to all, as recurrent features of real relations-as-events.
From Commodities to Educationally Relevant Objects
Those objects of school subject matters – mathematics, science, social studies, or reading – are no less material and thus alive than economy. Thus, for example, related to school mathematics, the ideal includes mathematics truths (e.g. 2 + 2 = 4), logical categories, topological structures, imaginary numbers, regularities in natural numbers, and, everything else that mathematicians investigate. It is precisely because of their materiality that these idealities exist as recurrent characteristics of in events, and, therefore, exist for those participating in public forums. This has been shown in the social studies of science even for the most abstract of subject matters, including mathematics and theoretical physics (e.g. Merz and Knorr-Cetina 1997). Thus, for example, mathematics “investigates the real material world, even though it examines it from its own special perspective, from its own specifically mathematical point of view” (Ilyenkov 2012, 183). As a result, if we were to “declare the topological struc- ture to be exclusively a psychological phenomenon, as subjective idealism tends to do” then we would effectively “deny mathematical science, and in the end the whole of mathematical natural science, of the objective and necessary meaning of its con- structions” (183). The ideality of mathematical forms therefore does not come from their mental characteristics, but from the fact that the material forms of events are only external expressions of something very different with which the material form has nothing in common. Thus, according to Marx, ideality is but the form of soci- etal-human activity represented in an abstracted object-thing that reflects living objective and mundane reality. We might also say, conversely, that the form of human life-activity, which reflects objective reality, is represented as an inherently abstracted object-thing. If the ideal were to be approached as something existing in the human head, as “some purely psychological or psycho-physiological, mental phenomenon” then we would already be “helpless before a subjective-idealist understanding of the object of contemporary mathematical knowledge” (183).
Instead, it is real precisely because it is not in the head that the ideal (i.e. “meaning”) can be investigated anthropologically as a quality of real events.