Infinity,infinite processes and limit concepts

Introduction

In order to recover a neglected chapter in the historical and theoretical background of social theory in general and critical theory in particular in a way that allows for a better understanding of the presuppositions of social-theoretical thought and prepares it for meeting the challenges it currently faces, the following paragraphs are devoted to a brief reconstruction of the mathematical-philosophical tradition from ancient to modern times. The focus is on that part of this tradition that is marked by the ideas of infinity, infinite processes and limit concepts which have substructured social-theoretical thought from the start and currently promise to unlock its future development.

The account starts with Aristotle’s broaching of the problem of infinity and its limits as well as his reserved attitude toward an aspect of it, but by far the larger part is focused on the early- and later-modern attitudes toward and treatment of this problematic. It moves from the initial enthusiastic yet naïve embrace of the idea of infinite processes and their limits, via the sobering late 18th-century philosophical summation of previous developments, to the 19th-century critical period in mathematics which refined and eventually legitimated the relevant ideas. Anticipated by Galileo Galilei, the names of Wallis, Newton and Leibniz stand for the spirited early-modern transformation of the problem, but it was with Kant’s critical philosophy that the cultural parameters for subsequent developments were made clear. At that stage, the scene was set for the osmotic absorption of the relevant ideas by social theorists. It is at the very time that Hegelian thought, in addition to Kantian ideas, paved the way for the emergence of the differentiated left-Hegelian response of Marx and Peirce, that Cauchy and Abel inaugurated the critical period in mathematics. From this base proceeded the significant mathematical development of the late 19th century in which Peirce played a seminal role but which is represented especially by Dedekind and Cantor. Against this background, the most basic concepts of social theory come to stand out graphically, available for sharp differentiation and being put in their proper places. Building on this account, the final section collates the references relevant to social theory and points in the direction of their systematic significance.

I Potential infinity, actual infinity and limits

Aristotle’s most important contribution to the development of mathematical philosophy and the formal principles of the sciences is his penetrating formulation of the problem of infinity. He was the first to appreciate the need to draw a basic distinction between ‘potential infinity’ and ‘actual infinity’ (2015: book III, part 6). Although this distinction became quite differently evaluated in the modern period, it has proved to be of great significance not just for mathematical thought but also for philosophical thought – both of which served as sources of social-theoretical thinking. For example, the distinction was central to Kantian philosophy which laid the groundwork for reflection on the whole range of scientific disciplines, including the social and cultural sciences or humanities, but its impact is also visible in theoretical concepts employed by Hegel, Comte, Marx and other social and critical theorists.

In accordance with his identification of ‘infinity’ and ‘continuity’, Aristotle (2015: book VII, part 8; book III, part 6) sought to make potential infinity comprehensible by contemplating a sequence that continues without end. It can proceed in one of two directions: either progressing by the addition of another unit – for example, the sequence of natural numbers 1, 2, 3…which never reaches its end; or by making another subdivision of, for instance, a line between two points and thus engaging in a process which confirms its infinite divisibility. By contrast with the never-ending progression characteristic of potential infinity, either toward the biggest or toward the smallest, Aristotle described actual infinity as ‘the infinite in the full sense’, as being ‘complete’ [teleion], having an ‘end’ [telos], where ‘the end is a limit’ (ibid.: book III, part 6). In the philosophical lexicon in his Metaphysics (1961: 32), Aristotle included this circumscription: ‘“Limit” denotes the last point of anything, i.e. the point beyond which it is impossible to find any part of it, but within which all of its parts are found.’ Applied to infinity, it means that actual infinity would amount to all the elements of an adding sequence or of a subdividing sequence being available in their completeness or totality. On contemplating this possibility, however, Aristotle, beholden to the characteristic ancient Greek concern with the concrete, summarily rejected the notion of actual infinity as a logical impossibility.

It is against Aristotle’s evaluation of actual infinity that a prominent line of thinkers would subsequently take a diametrically opposed position. Since the Greeks suffered from so-called horror infiniti, it had been left to mathematicians, scientists and philosophers in the context of the early- modern revival of learning to explore the relation of infinite sequences and their limits. Not only is modern mathematical analysis based on the theory of infinite processes which depend on the idea of limit (Dantzig, 2007: 133), but modern thought more generally, both philosophical and scientific, including the social-theoretical, is pervaded by these very same assumptions. ¹

II Theory of infinite series: divergent, convergent and limits

While the idea of an infinite sequence was known to the ancient Greeks, it was only between the 12th and the 17th–18th centuries that the core aspect of the theory of infinity became established (Kline, 1990; Clegg, 2003; Dantzig, 2007; Elwes, 2013; Alexander, 2015). This was made possible by the fact that the moderns adopted a positive attitude to infinity and accepted the validity of infinite processes, instead of sequestering the problem of infinity, as did the Greeks, and instead of arresting endless sequences scholastically or theologically by assuming the absolute nature of the unlimited, as did Christianity. Having been opened by the medieval businessman and mathematician Fibonacci through contact with Hindu–Arab ideas, the field was explored over the following centuries by mathematicians (Oresme, Wallis, Mengoli, Euler), scientists (Galileo, Kepler, Newton) and philosophers (Leibniz, Kant) who prepared the ground for important developments in the 19th century. Most basically, it emerged in response to the probing question of what follows from an attempt to add together infinitely many numbers: that the historically and theoretically important ‘geometrical sequence’ can be divided into two distinct series – an increasing or ‘divergent’ and a decreasing or ‘convergent’ series (Dantzig, 2007: 150; Kline, 1990: 1,110).

It should be noted that these variants of the geometrical sequence are of central importance in the present context since they not only link with Aristotle’s distinction between potential and actual infinity, but also paved the way for the establishment of the parametric assumptions of social theory. Their conceptualization made possible the consolidation of thinking about two distinct yet interrelated dimensions of the socio-cultural form of life and its conditions: corresponding to the convergent, such notions as time, natural history, human history, historical action, practices, socio-cultural articulation and elaboration, and transformation and change; and corresponding to the divergent, such notions as differentiation, diversification, increase in complexity and the evolutionary stabilization of a variety of forms.

A ‘divergent series’ (Dantzig, 2007: 150; Elwes, 2013: 97–8) is one in which the addition of more and more numbers leads to a total that outgrows any boundary one might try to impose. For example, in the case of adding 1 + 2 + 3 + 4 + 5 +…the total grows exponentially, that is, larger and larger without limit – the total exceeding 100 after 14 steps and surpassing 1,000 after 45 steps. In the case of simply counting 1 + 1 + 1 + 1 + 1 +…the total may be growing much slower yet it does grow without end, the first step reaching 1, the second 2 and the 100th 100.

A ‘convergent series’ (Dantzig, 2007: 150–1; Elwes, 2013: 98) presents an entirely different picture. If instead of whole numbers one adds rational numbers such as decimal fractions, the resulting total does not grow bigger and bigger, but rather tends toward a finite limit. For example, adding 0.9 + 0.09 + 0.009 + 0.0009 +…gives rise to a sequence of totals of 0.9, 0.99, 0.999, 0.9999, and so forth that gets ever closer to 1 yet without ever reaching it, not to mention surpassing it. The best-known example of a convergent series from mathematics is the transcendental number pi (π). When π is algorithmically derived, a number is obtained which grows ever longer in decimal places and ever closer to π, the growth being infinite and the value of π in principle remaining just out of reach. While for convenience π is accepted as 3.14159, by the late 19th century the derivation had yielded more than 700 digits after the point and in our own time the computer has added vastly more, yet still approaching but never reaching the exact value of π. The latter is a finite but ideal limit that cannot be captured perfectly by a number.

III From Leibniz to Kant

Mathematics served as the source from which Leibniz (1965: 223; 1968: 152) extrapolated what he called the ‘law of continuity’ which he adopted as the basis of his philosophical and scientific work. In this context, he dealt extensively with the problem of infinity which, in a certain respect, allowed him the claim to have co-founded the modern mathematical branch of calculus together with Newton. Through a series of highly suggestive but also truncated and erroneous ideas, he was instrumental in a number of crucial respects in pointing the way for Kant. One of the most important ideas that Kant appropriated is Leibniz’s distinction (1965: 235–6; 1968: 182) between ‘truths of fact’ and ‘truths of reason’ which was conditioned by his close study of Galileo (1914) who combated Aristotle’s stance on actual infinity and thus anticipated modern mathematics. Despite having used this distinction in his criticism of Locke, however, Leibniz nevertheless saw fit to revert to classical Aristotelianism. This is one significant respect in which Kant departed decisively from him.

Kant’s critical philosophy is replete not only with the assumption of the basic nature of infinite or continuous series, but also with articulating issues in accordance with it. A crucial instance from the Critique of Pure Reason (1968) is where he addresses the question of reason as ‘the faculty of inferring, i.e. judging mediately’ (ibid.: B386 = A330 ff.). Here he distinguishes two infinite series: ‘a series of inferences that…can be continued…[or]…prolonged indefinitely on the side of the conditions [per prosyllogismos] or of the conditioned [per episyllogismos]’ (ibid.: A331 = B387–8). Of these two, the former ‘ascending series’ stands ‘in a different relation to the faculty of reason from that of the descending series’ (ibid.: A331 = B388). It is the case that although ‘we can never succeed in comprehending a totality of conditions, the series must none the less contain such a totality, and the entire series must be unconditionally true if the conditioned…is to be counted true’ (ibid.: A332 = B389). Thus Kant is able to submit that it can be assumed that

…all the members of the series on the side of the conditions are given (totality in the series of premises); only on this assumption is the judgment before us possible a priori; whereas on the side of the conditioned, in respect of consequences, we only think a series in process of becoming, not one already presupposed or given in its completeness, and therefore an advance that is merely potential.’ (Kant, 1968: B388 = A331–2)

The designation ‘idea or concept of reason’ (Kant, 1968: A320 = B377) applies to ideas that are unrelated to perception and construction, neither having been derived from nor being directly applicable to sense-experience, and in this sense transfinite or ‘transcendental’. Despite the fact that such transcendental concepts are beyond the substantive concern of any system or discipline dealing with concrete objects, Kant argued that they could nevertheless amplify such domains as long as they could be shown to be logically coherent and internally consistent. Indeed, he was convinced that they were not merely useful, but actually necessary for achieving adequate amplification of this kind. Kant not only employed such concepts in his philosophy of mathematics and science, but also extended them appropriately modulated to his practical and aesthetic philosophies.

In the Critique of Judgement where he compares the mathematical and the aesthetic estimation of magnitude, for instance, of natural phenomena, Kant (1972: section 26) elaborates on the relation between the descending or convergent and the ascending or divergent series. On the one hand, he depicts the inferential operation of the imagination in its constructive activity as ‘go[ing] on without hindrance to infinity’ (ibid.: 93). On the other, he stresses that the mind requires a totality to limit this unbridled progression by appealing to reason: ‘Reason consequently desires comprehension in one intuition, and so the joint presentation of all the members of the progressively increasing series. It does not even exempt the infinite…from this requirement; it rather renders it unavoidable to think the infinite…as entirely given (according to its totality)’ (ibid.). Another aspect of the relation between Kant and Aristotle thus becomes apparent. Like Aristotle, Kant accepts that the concept of limit is a necessary concomitant of infinite processes, but he differs decisively from his predecessor by forging a link between limit and actual infinity. Central to this step is the concept of idea of reason, for in his view this type of idea is necessary to capture actual infinity, complete infinity or infinite totality.

What Kant in an earlier quotation calls ‘the conditioned’ includes the relevant appearances of the object of inference, while what he conceives as the ‘totality of conditions’ is expressed by ideas of reason. Since he associates the former with the convergent and the latter with the divergent series, this means that ideas of reason qua the totality of conditions represented by the divergent series provide the infinite ideal limit framing any convergent series. These ideas serve as ideal limit concepts, and as such they render the convergent series qua processes of becoming intelligible and manageable. Through them, the power of the mind is brought to bear on the process of becoming and rendered a suitable object of experience, knowledge and action. Considering all three of Kant’s critiques (1956, 1968, 1972), the most important ideas of reason he identifies include: ‘world’ (the totality of objects of possible experience), ‘immortal soul’ or ‘freedom’ and ‘autonomy’ (the subject of experience, knowledge and action), ‘beauty’ (the cognitive mode of imagination and reflection) and, finally, ‘God’ or later ‘the faculty of judgement’ (validity). These ideas all designate totalities that operate as infinite ideal limit concepts, some as forms of thought and others as forms of elementary practical concepts and of aesthetic experience. However, whereas Kant tended rather strongly toward dualistic thinking, it became clear in the wake of a variety of post-Kantian developments in philosophy and mathematics that ideas of reason as ideal limit concepts can be regarded as involving ‘idealizations’ (Körner, 1968: 19) in the sense of a mode of procedure that occupies a position between Platonic idealism and Aristotelian empiricism, being as they are neither Forms independent of concrete reality nor simply parts of empirical objects. Of them it could now be said that they are idealizing abstractions that emerge through mediating reflexivity from human activities of all sorts and become stabilized over long periods of time as the humanly most significant conceptual (linguistic) and quasi-conceptual (logical and mathematical) foundations of the socio-cultural form of life. ²

While the ideas of reason qua infinite ideal limit concepts provide the overall framework of intelligibility and practical and aesthetic orientation relevant to ongoing processes of becoming, this still leaves the finite yet also ideal limit concepts associated with the convergent series – that is, limits like the value of π toward which such processes tend but never reach. For Kant is concerned not just with the transcendental ideas of reason corresponding to the divergent series, but also with their ‘immanent…employment’ (1968: B383 = A327) regulating the theoretical and practical concepts and aesthetic projections corresponding to the convergent series which give effect to them at the lower level. In this respect, he submits that reason involves not exclusively ideas, but also ‘ideals’, ‘archetypes’, ‘models’ or ‘examples’ (ibid.: A569–71 = B597–9). Ideas have the task of ordering and unifying concepts ‘with a view to obtaining totality in various series’, whereas ideals depend on concepts of the understanding, moral concepts and aesthetic sensibility which in turn unify the elements of objects in a way that means ‘such series of conditions come into being’ (ibid.: A643 = B672). As with π as the limit value of the mathematical convergent series, Kant stresses that any attempt to realize the ideal contained in a model or an example, whether by knowledge production, action or experience, is ‘impracticable’ (ibid.: A570 = B598) – and that in the sense that while ‘a great goal…is set before it…it can never of itself reach’ that ideal (1956: 152). Knowledge, action and experience are always incomplete and perfectly emulating the completeness of the ideal of, say, ‘the wise man’ is an ‘illusion’.

From the above, the following preliminary consolidation emerges (see Figure 1).

OPEN IN VIEWER

Figure 1.

Infinity, infinite processes and limit concepts in Kant.

IV Intermediate social-theoretical reflections

Reflection on this diagrammatic icon from the perspective of social theory leads to the identification of historical-theoretical precedents of familiar social-theoretical constructs as well as contact points for the proposal of basic concepts for a newly conceived cognitive social theory – one presupposed by critical theory.

Constructs that have become settled as standard theoretical assumptions in social theory have been inspired by the distinction between the convergent and divergent series deriving from the mathematical-philosophical tradition. These constructs are the basically assumed parameters of social reality expressed by the conceptual pair of dynamics and statics or, differently, by the notions of the process of the creation, construction or production of society and of the structure, structuration or reproduction of society. This takes care of the two series, but still leaves the question of the types of limit concept necessarily accompanying the two infinite series.

On this matter, contributions have indeed been made by classical and contemporary social theorists insofar as they stress generality, yet without thinking the problem of generality through and without recognition of the status of their contributions as being explicable in terms of limit concepts. Here a great lack of clarity and much equivocation prevail in social theory – indeed, to a degree that could be judged as having been detrimental to it. A major desideratum under contemporary conditions demanding a reconfiguration of the relation to self, society and nature is the elimination of this debilitating state of affairs. The proposal is to conceptualize the two distinct types of limit concept in cognitive theoretical terms – that is, the infinite ideal limit concepts as context-transcendent or transcendental cognitive order principles, and the finite ideal limit concepts as context-immanent, cognitively structured cultural models of different levels and scales. This issue is resumed in the final section of the article.

V The post-Kantian situation

When interest in the problem of infinity was rekindled in the Renaissance, the moderns embraced it enthusiastically yet quite naïvely, certainly without the critical rigour of the Greeks. On all historical accounts (Dantzig, 2007; Clegg, 2003; Alexander, 2015), a veritable orgy of the reckless use of confusing terminology, rough and ready methods and wild and woolly calculations prevailed for some two centuries, implicating figures from Kepler, through Wallis, Leibniz and Newton, to d’Alembert. This exuberance was brought to a halt by the cultural change marked by Kant’s critical philosophy which was followed by the rigour of the 19th-century critical period in mathematical thought. It was embodied by mathematicians like Weierstrass, Dedekind, Schröder and Cantor as well as the outstanding philosopher, logician and mathematician Peirce, the founder of pragmatism, who is also of great social-scientific significance (Apel, 1995; Strydom, 2011a). This critical disciplining of argumentation involved rendering the ideas of infinite processes and limit concepts more precise so as to bring the endless progression of such processes under control. Kant offered a philosophical example of such disciplining with his critical philosophy aimed at curtailing the speculative or dogmatic employment of the cognitive capacities and reason, but the mathematicians pursued it in their own terms – terms centrally implicating the convergent and divergent series and their limits.

VI The critical period in mathematics

That the concern with infinite processes and their limits had immense intellectual significance was confirmed by a number of key achievements during the critical period in mathematics. Cauchy inaugurated this period in the early 19th century by showing in precise terms that the geometrical sequence can indeed be classified into a divergent and a convergent series (Kline, 1990: 1,110; Dantzig, 2007: 150). The foundations of calculus deriving from Leibniz and Newton’s struggle with infinity remained ambiguous, however, which inspired Weierstrass to resolve its lack of soundness in the early 1840s and to refine it in subsequent years. By the 1880s, the question of infinity and limit came to a head in the intricate set of relations involving Peirce, Schröder, Dedekind and Cantor. Toward the close of the critical period, these men conducted a searching analysis of the whole problem with a view to eliminating the vagueness and ambiguity still marring these concepts. While Peirce’s (1992, 1998) logical analyses contributed decisively, this important development can be briefly stated with reference to Cantor and Dedekind whose seemingly conflicting contributions nevertheless productively complemented each other (Dantzig, 2007: 181–2). For the purposes of his astounding transfinite mathematics, Cantor adopted a dynamic theory according to which the limit value is generated by the motion of a point on a continuous or infinite line, while Dedekind adopted a static theory according to which the power of the mind by means of a cut [Schnitt] imposes a special classificatory scheme on an infinite process to generate its limit value.

In this debate, both the infinite convergent and divergent processes and their limit concepts moved to the foreground. But most significant is that it was unequivocally established that the concepts of infinity, infinite processes and limits endow mathematics with generality, with the implication that reality can by no means be restricted to the immediate experience of the human senses. There is a context-transcendent or transfinite dimension beyond the immanent that Kant earlier sought to pinpoint philosophically under the title of ‘the transcendental’ – albeit without establishing an adequate version of the necessary relation of mediation between the two dimensions. Such mediation is what Peirce (1998), in contact with the leading mathematicians and abreast of the cutting-edge mathematics of his time, sought to accomplish by means of his triadic semiotics which possesses both general and social-scientific significance.

VII Social-theoretical relevance of infinite processes and limit concepts

The question arising now is precisely what the significance of the submerged mathematical-philosophical background is for the development of social theory – not just for extant social theory, but especially for its future development and potential contribution to critically facing the mounting challenge of having to reconfigure the relation of humans to nature, to others and to themselves. In the intermediate reflections above, it was suggested that this neglected background contains historical-theoretical precedents of what have become standard parametric constructs as well as fecund anchor points for the formulation of certain key concepts for a newly conceived cognitive social theory which is both critical and compatible with a weak-naturalistic ontology. Here these two matters need some elaboration.

The first set of considerations turns on the notion of infinite processes, particularly the differentiation between the convergent and divergent series. Significance can be ascribed to this distinction insofar as it represents a thoroughly thought-through conceptual achievement attained and legitimated over an extended period of time that provided the basis for the conceptualization of a variety of intellectual domains, including the social sciences. In the case of social theory, it allowed basic conceptual constructs that became transposed into theoretical assumptions operating as parameters of social reality. Among the most basically assumed is the conceptual pair of dynamics and statics which corresponds directly to the two series. Round about 1840, Comte (1974) was able to formulate this social dynamics–social statics distinction for the new domain of sociology on the basis of his knowledge as a mathematician, undoubtedly in the wake of Cauchy’s contribution, and to correlate it with progress and order. Noteworthy is that he regarded statics as the conditions and pre-conditions of social order and, further, that all of these were knowable, apparently informed by his understanding of the divergent series and its infinite ideal limit concept. This was the first authoritative step toward the general acceptance in social theory up to the present of the closely allied concepts of process and structure or, more fully, of the process of the creation, construction or production of society and the structure, structuration or reproduction of society. ⁶ It would be relatively easy to provide a broad survey of social theory in which the adoption and operation of this fundamental assumption are demonstrated in the writings of the major social theorists past and present – for example, Marx’s praxis–general concepts, Durkheim’s division of labour–obligatory categories, Weber’s rationalization–value spheres, Mead’s evolution/social interaction–generalized other/universal society, Piaget’s developing structures–completed structures, Habermas’ communication/evolution–formal pragmatic world concepts and Luhmann’s autopoiesis–world. But being more interested in making a proposal that could possibly benefit the development of social theory at the point of decision where it stands today, I turn instead to the second set of considerations concerning the problem of limit concepts and its potential for the elaboration of social theory.

The fundamental insight from which a social-theoretical consideration of the problem of concepts applicable to infinite processes has to proceed is the one regarding reality going beyond immediate sense-experience that Leibniz suggested, Kant cast in transcendental terms and the mathematicians eventually established in the form of complete infinity – the insight, namely, that the concepts of infinity and limits endow thought, whether mathematics, philosophy or science, including social science and social theory, with generality. The central problem here, however, is to think through the problem of generality in a philosophically defensible manner relevant to social theory. This is something social theorists have not done properly, which accounts for the conspicuous lack of clarity and much equivocation on the matter in social theory. The key to the solution of this problem is the vital distinction between generality and universality. ⁷

In terms of what has thus far been discussed, this distinction can be aligned with that between the convergent and divergent series and, therefore, can be made intelligible with reference to the associated limit concepts. Accordingly, generality and universality are respectively captured by the finite ideal limit concepts punctuating the convergent series and the infinite ideal limit concepts punctuating the divergent series. Given this mathematical-philosophical basis, it becomes possible, second, to give theoretical content to the abstract concepts of generality and universality. To begin with, let it be noted that both types of limit concept, as their designation as ‘ideal’ indicates, necessarily involve idealization – a characteristic stressed by both mathematical and philosophical thought (Dantzig, 2007; Körner, 1968; Habermas, 1984). Although both generality and universality are modes of idealization, they nevertheless differ in a theoretically decisive sense. The former is context-immanent and contingent, while the latter is context-transcendent and necessary in the sense of being constitutive of making the human socio-cultural form of life possible. ⁸ The proposal for the purposes of a cognitive enhancement of social theory, particularly critical theory, is accordingly as follows: first, to conceive of the context-immanent, contingent, finite ideal limit concepts as taking the form of cultural models of different levels and scales; ⁹ and, second, to conceive of the context-transcendent, necessary, infinite ideal limit concepts as transcendental cognitive order principles. The key corresponding cognitive social-theoretical concepts are thus cultural model and the cognitive order of society, and treating them in this way allows them, by contrast with the existing confused and opaque state of affairs, to be unambiguously located at their distinct levels. Before providing an overview of both types of limit concept and the infinite processes they punctuate, a brief characterization and clarification of their status are required.

Insofar as finite and infinite limit concepts are idealizations, they are dependent on the rule structure of language. Idealization in the medium of ordinary everyday language takes the form of meaning which implies that, in the first instance, idealization is a semantic matter; more specifically, a manifestation of semantic generality. ¹⁰ Both types of limit concept thus possess meaning or semantic generality. Examples of finite ideal limit concepts in the sense of context-immanent cognitive structures or forms exemplifying generality of this kind would include, among many others of different levels and scales, such cultural models as democracy, Irishness, French republicanism, Roman Catholicism, communism and so forth. They are all structures or forms that generalize certain values, norms, symbols and orientations so that all those involved share a particular culture which gives direction to and guides their engagements and practices, even in cases where the engagements and practices diverge. In terms of the previous analysis of convergent series, then, cultural models can be understood as limit concepts that represent finite ideals – context-immanent ideals toward which practices tend but which can never be reached or fully realized, like π as an example of the limit value of potential infinity.

As already indicated, infinite ideal limit concepts in the sense of context-transcendent cognitive order principles, such as truth, right, justice, equality, beauty, truthfulness, authenticity and so forth, indeed also exemplify semantic generality insofar as they are concepts meaningfully articulated in language. But since these limit concepts above all possess universality, they cannot be conceived exclusively in terms of semantics or symbolically packaged meaning. Most characteristically, in fact, they are seats of validity, or what philosophically is considered under the concept of ‘unconditionality’ (Kant, 1968: B367 = A311; Peirce, 1998: 457; Habermas, 2003: 99). Beyond meaning, they have validity. While meaning is tied to linguistic rules, validity is anchored cognitively in the brain–mind – that is, in the organic endowment and the culturally articulated, phylogenetically acquired mind of the human species. ¹¹ By contrast with meaning as an instance of semantic generality, validity is one of cognitive universality. It is therefore not just a matter of appreciating the difference between generality and universality and between meaning and validity, but also the distinction between the semantic and the cognitive. Besides language or concepts, the cognitively rooted universality of infinite ideal limit concepts depends also on what one might perhaps call quasi-conceptual media represented, for instance, by logic and mathematics – that is, cognitive rule systems given with the human organic endowment other than language that nevertheless provide meaning with a structure and underpin the correct use of language. Given their semantic and especially basically cognitive nature, the components of the cognitive order qua infinite ideal limits can be regarded as the conceptual and quasi-conceptual underpinnings of the human socio-cultural form of life. ¹² Rather than simply complexes of culturally specific values, norms, symbols and orientations that are generally shared by a particular group in distinction to other comparable groups, cognitive order principles are commonly presupposed by human beings qua human beings who have grown up and live in the characteristic human socio-cultural form of life. In terms of the previous analysis of divergent series ending in actual or complete infinity, then, cognitive order principles can be regarded as limit concepts that represent infinite ideals providing classification and ordering schemes which are the means for constructing cultural models, lay down their parameters, structure them from on high and make possible the regulating of their embodiment and realization.

Systematizing the preceding discussion, Figure 2 below starts from the reconstructed mathematical-philosophical conceptualization and proposes a social-theoretical interpretation of the world for us humans as consisting of three sets of infinite processes, each of which consists of an ongoing convergent process of becoming and a structurally accumulating divergent process. The three sets of infinite processes are the objective, the socio-cultural and the subjective.

OPEN IN VIEWER

Figure 2.

Infinite processes and limit concepts in social theory.

First, the objective set covers the ongoing natural-historical process of which the socio-cultural form of life is a continuation and extension in a specifically human form. It should be stressed that the conceptualization of this dimension is intentionally geared toward admitting a weak-naturalistic ontology ¹³ as a necessary part of the proposed cognitive twist of social theory. The objective world of all possible objects of experience and reference embraces neither just socio-cultural reality nor just nature as seen by the natural sciences, but both of these as well as nature-in-itself or natura naturans in the sense of the infinite process that spontaneously brings forth a plethora of diverse forms, including humans and their form of life. Second, the socio-cultural set houses the ongoing process of the creation, construction, production, formation and organization of the socio-cultural form of life internally and in relation to nature. Finally, the subjective set consists of the ongoing process of self-cultivation and subject-formation, whether under objective or socio-cultural pressure or, alternatively, initiating change. In turn, these three ongoing processes are over-determined by the concomitant complementary processes of the long-term accumulation and evolutionary stabilization of rational potentials that take on an incursive and recursive structuring role in respect of the ongoing processes – which leads to a consideration of limit concepts in the form of cultural models and cognitive order principles.

In all three cases, Figure 2 identifies the limit concepts relevant to each of the infinite processes. For the ongoing convergent processes, a selection is made from the vanishingly large number of possible cultural models or finite ideal limit concepts of ones that have only very recently emerged as, perhaps, the most urgently required directing and guiding cultural models under the currently rather challenging conditions. This particular choice is made since the aim here is not to present a typology of extant cultural models, but rather to make a contribution to the enhancement of social theory with a view to being relevant to the challenges humankind is facing at present on the brink of the emerging world society and the new age of ‘the Anthropocene’ (Crutzen, 2002; Strydom, 2017).

In view of the sobering realization in the age of the Anthropocene that society forms part of nature and that humans have become a geophysical force contributing to global warming and climate change, the limit value or cultural ideal toward which the ongoing objective process since the 1970s tends is that of a sustainable human–nature relation. In the wake of globalization, mondialization and the emergence of world society, second, the ongoing socio-cultural process in the 1990s had begun to tend toward the limit value or cultural model of a democratic-cosmopolitan society. As regards the ongoing subject-formation process, finally, the pressures emanating from both objective and socio-cultural processes reinforce and further structure the changes in the formative process which were inaugurated in the 1960s, so that it is beginning to become clear today that the fitting limit value or cultural model in this case is that of a cognitively fluid subject appropriate to a democratic-cosmopolitan existence in a cared-for planetary biosphere (Strydom, 2011b, 2015b).

In respect of the divergent processes in the sense of the long-term accumulation of rational potentials, infinite ideal limit concepts are listed in Figure 2 in the objective, socio-cultural and subjective cases that reflect those evolutionarily stabilized potentials which take the form of the most basic cognitive order principles of the human socio-cultural form of life. They cover not just the main principle directly related to the internal organization of that form of life, namely right, but also the main principles respectively related to its subjective component, truthfulness, and to its objective nature and conditions, that is, truth. While the principles of truth, right and truthfulness tower over the three sectors of the cognitive order, the latter of course contains a vanishingly extensive number of principles, ¹⁴ any one of which could function as an infinite ideal limit concept if the situation so requires. In any concrete case, the identification of these principles and the incursive and recursive roles they play immanently is of the essence for social-scientific analysis led by clear cognitive social-theoretical thinking.

Conclusion

The brief social-theoretical reconstruction of the mathematical-philosophical tradition opened a way for seeing more clearly the direction that social theory, critical theory and, more generally, the social sciences are compelled to take and, to be sure, have already begun to take under contemporary conditions – that is, an integrated cognitive social science embracing a weak-naturalistic and a socio-cultural component. Not only did it allow the two most basic conceptual parameters of social reality corresponding to the convergent and divergent infinite processes to be backed up by historical and theoretical evidence deriving from centuries of hard research and design work done by philosophers, mathematicians and scientists. At the same time, it made possible also the foregrounding of the important notion of limit concept, indeed, two types of limit concept which provide anchor points for much-needed conceptual clarification and development. While the notion of a limit concept due to its dependence on the power of the mind has a direct bearing on the cognitive upgrading of social theory, one of the three sets of infinite processes, the objective process which generates the world of all possible objects of experience and reference, brings the need for the adoption of a weak-naturalistic ontology within the cognitive social-theoretical purview.

The proposal for introducing the new concept of the cognitive order of society and its unambiguous differentiation from the concept of cultural model feeds into the enhancement of social theory. It enables the employment of cognitive structures and structuration emanating from both the cognitive order and cultural models for the purposes of hitherto either haphazard or lacking conceptual and substantive analyses. That such a cognitive direction is unavoidable for social theory today is given with the fact that long-standing, taken-for-granted assumptions and ideas regarding the relations of humans to nature, to others and to themselves have to be brought to the level of conscious reflection and, further, that central cherished ideas have to be subjected to penetrating evaluation, reconsideration, critique, transformation and even replacement where necessary. In all these cases, we have to do with long-term accumulated and stabilized rational potentials such as schemata, forms and formats stemming from the power of the mind, some context-immanent and others context-transcendent, that require cognitively inspired theorizing and analyses, with reference both to their structural properties and their dynamics. ¹⁵

The acknowledgement of the ongoing infinite objective process opens the possibility of building a weak-naturalistic ontological assumption into social theory as a complement to and support for its articulation in cognitive terms. This is the case since the process’s continuous and multidimensional nature draws attention to the fact that social theory, as far as is relevant, has to take into account its full range. The objective process is multidimensional insofar as it embraces all possible objects of experience and reference which cover the ongoing spontaneous natural-historical and evolutionary process in its diversity; and it is continuous insofar as it is an infinite process of which the socio-cultural form of life is a continuation and extension in a specifically human form. Far from such a weak naturalism entailing a naturalistic reduction of the socio-cultural world, however, it distinguished itself from strong naturalism precisely by making space for the integrity of a relatively independent socio-cultural world. Especially important is that it dovetails perfectly with a socio-cultural cognitive approach by virtue of the fact that nature-in-itself or natura naturans spontaneously gives rise to natural cognitive forms. They come in the elementary or primitive social forms exhibited by our primate ancestors as well as many animal species, like kinship, group membership, dominance, cooperation, competition, sharing, playing, fighting, reconciling and so forth – all forms that humans presuppose yet then socially and culturally elaborate into a variety of domains directed and guided by cognitive order principles and cultural models and further articulate by corresponding social arrangements and practices.

The conclusion that follows, finally, is that a rethinking of social theory by recourse to a recovery of its mathematical-philosophical background, particularly as captured by the notions of infinity, infinite processes and limit concepts, enables its channelling in an integrated cognitive – both socio-cultural and weak-naturalistic – direction. And it is exactly such a departure that is urgently called for by our currently fraught situation in which we are compelled to begin to transform ourselves so as to be able to think in new terms and to relate both to others and to nature in an appropriate way.