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Abstract

In this article we address two key questions in the application of dynamical systems theory (DST) to second language acquisition (SLA) that have not been resolved in recent debates about this issue. The first question relates to reductionism. Is an antireductionist position a necessary element of DST? We show that the radical antireductionist stance put forward by key movers of the application of DST to SLA neither follows from the mathematics of DST nor from the application of DST to science and that radical antireductionism results in an impasse for empirical research. In contrast, we argue that reductionism offers ways in which theory-derived hypotheses can be formed about subsystems that can be studied empirically using DST mathematics. The second question relates to intentionality. Are physical systems and mental systems similar enough to justify applying DST principles based in the physical sciences to processes in the human mind? Following Tschacher, we argue that current evidence suggests that there is a limited class of mental phenomena that can arise from physical phenomena. It is only this class of mental phenomena that can currently be modeled using DST mathematics. We offer a discussion of these perspectives to indicate how important it is to resolve these fundamental questions. In our view, these fundamental issues need resolution to put the application of DST to SLA on a solid conceptual and empirical footing.

Keywords

antireductionism dynamic systems theory intentionality naturalization of intentionality reductionism

I Introduction

As Feldman (2019: ix) writes, ‘[a] dynamical system is any mathematical system that changes in time according to a well specified rule.’ Dynamical systems theory (DST) is an area of mathematics that describes and predicts the behavior of complex dynamical systems. Applying DST to the study of second language acquisition raises at least two fundamental issues, reductionism and intentionality. The first issue we consider concerns the role of reductionism, i.e. whether it is possible to consider parts of a given dynamical system in order to study them in detail without having to consider all parts of the system simultaneously. A debate relating to this issue was initiated by Pallotti (2022) in this journal and followed up by Dyson (2023). Pallotti argues against a holistic anti-reductionist stance and in favor of a rationalist form of reductionism. Dyson argues that Processability Theory (PT) (Pienemann, 1998, 2005; Pienemann and Lenzing, 2020) meets the criteria of the reductionist and dynamic approach to second language acquisition (SLA) demanded by Pallotti. An anonymous reviewer pointed out that ‘PT makes a similar point to the authors’ [i.e. the authors of the current article: MP, AL and HN], i.e. that the complexity of SLA can be modeled in procedural, processing terms via reductionism.’ We agree with this point. However, the dynamics inherent in PT are limited to linguistic dynamics as described in Pienemann (2007). The current article and our current research focus go beyond that state as we integrate DST as a mathematical theory. In this article we adopt a different approach from those of Pallotti and Dyson. Rather than considering possible contributions from different theories, in this article we will examine the role of reductionism exclusively within DST principles.

The second issue associated with the attempt to apply DST to SLA that we consider is the issue of intentionality (Brentano, 1874). This issue derives from the view that the natural world and the mental world of the human mind are fundamentally different. As DST focuses on natural phenomena such as celestial mechanics, this difference motivates the in-principle question of whether DST can be applied to mental dynamical systems.

In a series of publications we have made a case for DST to be applied to SLA in the same manner as has been done in various natural sciences and engineering, namely by acknowledging that DST is a mathematical theory and applying its mathematical core to the new field (Lenzing et al., 2023; Nicholas et al., 2022; Pienemann et al., 2022).

Our proposal stands in stark contrast to the ‘complex dynamic systems theory’ (CDST) approach advocated by Larsen-Freeman (e.g. Larsen-Freeman, 2011), de Bot et al. (2007) and their associates in what Dörnyei et al. (2014) labeled ‘the dynamic approach’. That ‘dynamic approach’ is based on DST-inspired metaphors rather than the mathematical core of DST, and it includes a strong antireductionist stance exemplified in the following quote from Larsen-Freeman (2011):

Because everything is interconnected, it is problematic to sever one component from the whole and single it out for examination. By doing so, one is likely to get findings that do not hold up when the whole is considered. (Larsen-Freeman, 2011: 60)

Accompanying its metaphoric and antireductionist positioning, a third feature of the ‘dynamic approach’ is its focus on the unpredictability of aspects of dynamical systems (e.g. de Bot, 2016: 130; Larsen-Freeman, 1997: 143–144) that is seen as limiting the testability of dynamical systems in language.

Historically, the two contrasting (respectively, mathematical and metaphorical) approaches to applying DST to SLA follow a conceptual development that goes back three decades when attempts began to be made to apply DST to cognition. At that point in the history of DST, one strand continued the scientific tradition of applying the mathematical core of DST to the new field (e.g. Elman, 1995), whereas the other strand developed DST-inspired metaphors (for a detailed account, see Eliasmith, 1996).

Focusing on the science strand, we note in Lenzing et al. (2023) that authors such as Feldman (2019) and Strogatz (2015) have shown that DST can in principle be applied to fields that are amenable to mathematical modeling. We have also created a DST-based and mathematically explicit model of specific linguistic dynamics inherent in SLA (Pienemann et al., 2022). This computational model of a narrow domain of SLA dynamics shows that these aspects of SLA are amenable to mathematical modeling. The formal predictions derived from this model were tested successfully in large sets of natural data, demonstrating their empirical validity.

In other words, we have been able to show that specific aspects of SLA are amenable to the type of mathematical modeling that is inherent in DST. Nevertheless, this leaves us with the two fundamental and unresolved issues that we mentioned above. The first fundamental issue that needs to be clarified when seeking to construct a DST-based theory of SLA relates to reductionism, particularly to the question of whether it is really the case that an application of DST necessarily implies an antireductionist position.

Feldman (2019: 167) summarizes reductionism as follows:

In broad strokes, reductionism is the belief that the best line of attack on a scientific problem is to try to understand something by learning about its constituent parts. [emphasis in the original]

Pallotti (2022) critiques Larsen-Freeman’s antireductionist stance from a philosophical position, arguing in favor of reductionism and generating testable hypotheses. In contrast, our own critique of Larsen-Freeman’s stance is argued from a DST-internal perspective. We will show that antireductionism is not an inherent part of DST and, indeed, that an antireductionist approach to applying DST to SLA would create an impasse for empirical research.

The second issue (intentionality) concerns the relationship between physical and mental systems: Can we be sure that phenomena in the mental world of the human mind will obey the same dynamics as phenomena in the natural world that have been successfully modeled using DST formalisms?

In this article we align ourselves with Tschacher’s (2014: 2) argument that there is no way to avoid addressing this issue. Either it must be shown that the mental phenomena that are to be modeled using DST are sufficiently similar to material phenomena to warrant the use of DST or ‘a solution is demanded that can elucidate how mental phenomena may be explained avoiding intentional language [i.e. avoiding concepts that assume a fundamental difference between mental and natural phenomena; MP, AL and HN].’ In attempting to resolve these issues, Tschacher (2014) argues that there are only limited mental domains that are amenable to formal DST modeling.

In what follows, we consider the issue of reductionism before exploring the issue of intentionality.

II Reductionism

As we pointed out above, the application of DST to SLA proceeded in two contrasting strands, one based on the mathematical core of DST – as advocated by Lenzing et al. – and one based on a set of DST-inspired metaphors – as advocated by Larsen-Freeman (2011, 2017) and de Bot et al. (2007). These two conceptual strands also imply different views on reductionism, resulting in fundamental differences for the kinds of empirical research possible within each strand.

Briggs and Peat (1989) place reductionism in its historical context and describe how the natural sciences made considerable progress after 1600 by using reductionism. Mitchell (2009: x) considers more recent historical contexts and argues:

Many phenomena have stymied the reductionist program: the seemingly irreducible unpredictability of weather and climate; the intricacies and adaptive nature of living organisms . . . The antireductionist catch-phrase ‘the whole is more than the sum of its parts,’ takes on increasing significance as new sciences such as chaos, systems biology, evolutionary economics, and network theory move beyond reductionism to explain how complex behavior can arise from large collections of simpler components.

Proponents of the ‘dynamic approach’ construe the history of science in relation to dynamism as one of disruption in which reductionism was replaced by antireductionism and argue that antireductionism follows categorically from DST. Consistent with this line of thinking, Larsen-Freeman (1997: 142) states the following objective of her interpretations and applications of dynamical systems theory:

It is my hope that learning about the dynamics of complex nonlinear systems will discourage reductionist explanations in matters of concern to second language acquisition researchers.

The issue that arises from this position is how one can empirically study a complex system without focusing on and thereby reducing any of its constituent parts.

Larsen-Freeman¹ (2017: 14) aligns her antireductionist position with postmodern thinking:

Cameron and I (Larsen-Freeman and Cameron, 2008) . . . take Bahktin’s dialogism as something quite compatible with Complexity Theory [an earlier label for CDST; MP, AL and HN] and post-modern sociolinguistics.

Larsen-Freeman’s (2017) view can also be seen in Kramsch’s (2012) evaluation of Larsen-Freeman’s work on DST and SLA and in de Bot et al. (2007: 8), who assume that:

all variables [of a dynamical system; MP, AL and HN] are interrelated, and therefore changes in one variable will have an impact on all other variables that are part of the system.

Based on this assumption, de Bot et al. conclude that:

any account that focuses on one aspect only cannot but provide a gross oversimplification of reality. (de Bot et al., 2007: 18)

Larsen-Freeman (2017) quotes Cilliers (2001) and Byrne and Callaghan (2014) among others as witnesses of a postmodern outlook on DST that views the discovery of areas of unpredictability in science as a massive rupture in the development of science from complete predictability to vast uncertainties. She aligns this position with her argument that earlier approaches to SLA allegedly saw second language acquisition as a monolithic process.

Larsen-Freeman (2011) criticizes the traditional scientific method and proposes replacing it by a new approach based on ‘retro-diction’ rather than prediction:

the usual scientific method, which calls for making predictions and then testing them, is fraught with problems from a complexity theory perspective. Systems and behaviors can of course be described retrospectively – once change has happened; in fact, this is the central work of complexity theory. (Larsen-Freeman, 2011: 61)

This radical postmodern position creates a conundrum for empirical research that Larsen-Freeman (2011, 2017) recognizes but fails to solve. Larsen-Freeman (2017: 32) states that:

[t]he late philosopher Cilliers (2001) made the dilemma clear: ‘Because everything is always interacting and interfacing with others and the environment organically, the notions of ‘inside’ a system and ‘outside’ a system are never simple or uncontested (Cilliers, 2001: 142). Of course, it is humanly impossible to study everything at once.

As Cilliers (2005: 612) later observed, ‘Boundaries are still required if we want to talk about complex systems in a meaningful way – they are in fact necessary.’ In her deliberations about the issue of defining subsystem [b]oundaries Larsen-Freeman (2017: 34) arrives at the following conclusion:

Deciding where to draw the lines to define a focal system of interest always involves ‘an element of choice which cannot be justified objectively’ (Preiser, 2016).

In other words, on the one hand Larsen-Freeman views all parts of a dynamical system as relevant at all times and insists that leaving out one of them in an analysis of the system may jeopardize the validity of the analysis. At the same time, she acknowledges that it is not ‘humanly possible’ to include all parts of a system in an analysis and therefore proposes to ‘draw the lines to define a focal system of interest’, assuming that this kind of line-drawing cannot be based on objective criteria. In effect, she accepts reductionism in the research process despite her vocal antireductionist stance. However, she does not offer any principles for the proposed reduction.

De Bot (2016: 130) acknowledges the same conundrum when he states that:

[a] somewhat problematic characteristic of a DST approach is that language development cannot really be explained: the factors playing a role in development may all be known, but how their interaction will change in the course of development cannot be predicted and may in that sense be chaotic. . . . For the moment it seems wise to accept that we cannot explain development and to focus on adequate descriptions of developmental processes.

Leaving aside the validity of de Bot’s view of the relationship between predictability and explanation, it should be noted that de Bot (2016) is willing to forgo what VanPatten et al. (2020: 4) describe as the centerpiece of a theory of SLA, namely its ‘ability to explain observed phenomena’.

It is important to realize that the variety of antireductionism championed by Larsen-Freeman and de Bot does not follow from DST. These authors infer their antireductionism primarily from their dictum that all parts of a dynamical system are interconnected and that the smallest changes in any part can have massive repercussions for the overall system at any (unpredictable) point in time, including resulting in an unpredictable state of chaos or in Larsen-Freeman’s (1997: 143–144) words:

While . . . chaos may seem predictable . . . the onset of the randomness of complex nonlinear systems is in fact unpredictable. That the randomness will occur is predictable, what is not is exactly when it will occur . . .

A major reason for the unpredictable behavior of complex systems is their sensitive dependence on initial conditions.

Contrary to Larsen-Freeman’s claims, Feldman (2019: 83) shows in his mathematics-based introduction to dynamical systems that the onset of chaos is predictable with extreme precision as it depends on the evolution of a deterministic function. He also shows that complex systems have sensitive dependence on initial conditions only within a narrow range of the function, and that this range is fully predictable (for further details, see Lenzing et al., 2023).

In a way similar to Larsen-Freeman, de Bot et al. (2007: 8) also misconstrue basic DST concepts, especially those relating to the allegedly ever-present unsteadiness of dynamical systems, as evident in the following quotations:

dynamic sub-systems appear to settle in specific states, so-called ATTRACTOR STATES, which are preferred but not necessarily predictable [emphasis in the original]. (p. 8)

Attractor states are by definition temporary and not fixed. (p. 8)

In contrast to this claim, complexity scholars such as Feldman (2019), Devaney (2003) or Hasselblatt and Katok (2003) show that dynamical systems can have stable fixed points and stable cycles (with different periods); attractor states include fixed points; stable cycles and fixed points are predictable, and the corresponding dynamical systems continue producing their stable output to eternity.

The misconceptions in the work of de Bot and Larsen-Freeman mean that they have significantly overstated the overall role of chaos and unpredictability in dynamical systems. However, predictability by far outweighs chaos in a dynamical system; order by far outweighs chaos, and even chaos follows universal patterns (Feldman, 2019: 129ff).

Building on these recognitions, Feldman’s (2012, 2019) differentiated treatment of reductionism clearly shows that DST does not necessarily imply an antireductionist stance.

Reductionism in its extreme is surely a bad idea, but the same can be said about almost any ‘ism’ in science. While reductionism should be approached with caution, I nevertheless think there is much good that can be said about reductive approaches. . . ., I think that all knowledge is reductive. In order to try to understand our world, it is inevitable that we need to choose some portion of the world to focus on that is smaller than the world itself . . . To me, the question is not whether or not to be reductive, but what sort of reduction to do and how to do it. (Feldman, 2012: 148–49)

These points show that the alignment of the ‘dynamic approach’ with postmodern antireductionism is not motivated by the original work on DST.

One additional way of showing this is to trace the DST sources that some of the prominent advocates of the ‘dynamic approach’ to SLA have used in order to substantiate their claims in relation to DST. Table 1 lists the key sources relating to DST used by key proponents of the ‘dynamic approach’. The entries in the table show that the ‘dynamic approach’ publications are not based on original work on DST. Instead, these authors have used secondary sources, including the popular bestseller by the journalist Gleick (1987).

Table 1.

Works on dynamical systems theory (DST) used by advocates of the dynamic approach to second language acquisition (SLA).

	Math/science works	Other works
de Bot et al. (2007)	–	van Geert
Lowie (2013)	–	van Geert
Lowie and Verspoor (2015)	–	van Geert
Larsen-Freeman (1997)	–	Gleick
Larsen-Freeman (2011)	Brief, nontechnical reference to von Bertalanffy (1950)	Gleick, Thelen and Smith
Larsen-Freeman (2017)	–	Gleick, Thelen and Smith
Herdina and Jessner (2002)	–	Gleick, van Geert, Briggs & Peat
Hiver and Al Hoorie (2020)	See footnote ^a
Lowie et al. (2011)^b	Global reference to Verhulst	van Geert

Notes. ^a Hiver and Al-Hoorie (2020) quote several mathematics-based sources including Abraham (2011); Abraham and Shaw (1992); Johnson (2009); Haken (2006) and Strogatz (2003). However, in these quotations the key mathematical concepts underlying DST are not explained in any detail. In fact, Hiver and Al-Hoorie (2020: 6) distance themselves from DST mathematics: ‘we became convinced, like our mentors, that uncritically importing mathematical methods wholesale from complexity science would be of limited productivity, given that the objects of interest in applied linguistics sometimes differ radically from those in the natural sciences, and because the social sciences do not have the sort of data that the physical sciences use to model complexity mathematically.’ ^b Lowie et al. (2011) discuss key mathematical aspects of DST. However, unlike Feldman (2019) they neither utilize any of the original sources nor do they derive the relevant equations systematically from their conceptual point of departure, Verhulst’s logistic map. As a result, the reader is unable to follow the logical development of the underlying ideas.

None of the above publications by advocates of the ‘dynamic approach’ gives an account of DST that is sufficiently explicit to be able to identify in these texts (without previous knowledge) the original DST definitions of key DST notions (as detailed in, for instance, Feldman, 2012, 2019) including the following: attractor, fixed point, oscillation, trajectory, initial condition, chaotic (strange) attractor, bifurcation and chaos. Instead, these authors use examples and metaphors in an attempt to illustrate these notions but also resort to reductionism in their own empirical work (see Table 2).

Table 2.

Research focus in complex dynamic systems theory base (CDST-based) empirical studies.

Researcher	Focus	Method
de Bot et al. (2007)	Morphemes	Distributional statistics
Caspi (2010)	Lexical acquisition
Dong (2016)	Listening strategy	Distributional statistics
Evans and Larsen-Freeman (2020)	Non-finite adverbial clause
Penris and Verspoor (2017)	Lexical and syntactic complexity in written texts	Statistical analysis (frequency, type/token, accuracy, t-tests)
Lowie et al. (2018) Lowie et al. (2017)	T-unit length, lexical diversity	Statistical analyses
Plat and de Bot (2018)	Word naming	RT experiment; Statistical analysis; Spectral analysis
Lowie et al. (2014)	Word naming	Statistical analysis of RT data; Spectral analyses
Lowie and Verspoor (2019)	Individual differences (Motivation and aptitude)written language production	Holistically (relative complexity, fluency, accuracy, 5-point scale, 10 raters) and analytically (syntactic complexity, lexical complexity, lexical sophistication); statistical analysis (linear regression analysis)
Sun et al. (2016)	Coordination of verbal and non-verbal behavior	Recurrence quantification analysis and cross recurrence quantification analysis
Chan et al. (2016)	Lexical development (written data)	Logistic model
Hou et al. (2016)	Development in writing	Complexity measures (Synlex Analyzer); CAF measures
Lowie et al. (2011)	Vocabulary growth	Simulation using logistic map
Freeborn et al. (2023)	Motivational constructs:4 factors	Network analysis: bootnet R-package

Note. RT = Reaction Time; CAF = Complexity, Accuracy, Fluency.

In Table 2 we provide evidence that, despite the antireductionist positioning characterizing the ‘dynamic approach’, reductionism continues to play a role in SLA research framed in relation to DST-inspired metaphors. We demonstrate, first, that the antireductionist positioning of the ‘dynamic approach’ is not carried through in their research practice. Second, we show that in order to resolve the impasse that results from their antireductionist positioning, researchers reduced the scope of their investigations to parts of language and reduced the complexity of their data analysis.

The sources listed in Table 2 do not reveal a principled approach to resolving the dilemma of how to create a focus for SLA research that would make such research humanly possible. As noted above, DST itself does not create this dilemma. Instead, it was brought about by positioning DST close to postmodern thinking and replacing the mathematical core of DST by metaphors.

III Intentionality

Attempting to understand the distinguishing features of nature and the human mind brings us straight into the philosophy of mind, more specifically to the ‘issue of whether mental phenomena are distinguished from physical phenomena as a result of possessing a property known as intentionality’ (Bechtel, 1988: 39). The term ‘intentionality’ was introduced into modern philosophy by Franz von Brentano (1838–1917) (Brentano, 1874). In this context, ‘intentionality’ is a technical philosophical term that needs to be differentiated from the everyday usage of the word derived from the verb ‘to intend’ (Fitch, 2008: 2). As Bechtel (1988: 59) points out:

[o]ne of the distinguishing features of the intentionality of mental states is that they can distort the real situation in the world and be about things that do not exist.

Bechtel (1988: 40 ff.) explains that it is typical of mental states – such as beliefs – that they are about something and that ‘[t]his characteristic of being about something is what philosophers call “intentionality” ’. For instance, mental states can refer to lies, to events that never occurred, to processes that are physically impossible or to events that one desires to occur. In other words, mental states can be representations of things regardless of whether they (currently) exist in the real world or not, and it is this quality of ‘aboutness’ that ‘intentionality’ refers to in philosophy and enables language to capture more than ‘the real world’.

Tschacher (2014: 2) describes the philosophical and cognitive issues arising from intentionality as follows:

When viewing the constituents of the mind (the cognitive system) in this [Brentano’s; MP, AL and HN] intentionalist manner, we stand in stark contrast to scientific descriptions of physical systems. These latter systems are material things, which are sufficiently described without reference to objects they would ‘be about’, or to states they might desire to realize. Therefore, are mental and physical systems qualitatively different with respect to intentionality? If yes, we are confronted with a dualist . . . view of the mind-body problem. If no, a solution is demanded that can elucidate how mental phenomena may be explained avoiding intentional language or, conversely, how physical systems may show or mimic the features of intentionality.

If there is no one-to-one relationship between mental states and the real world, is it possible to study mental states using DST? Can DST capture such mental processes in the first place?

In de Bot et al.’s (2007) application of DST to SLA, they do not address the relationship between mental states and the real world. Instead, they argue that DST can be applied to SLA because ‘the four DST constructs’ (de Bot et al., 2007: 15) can be found in SLA: (1) an initial state; (2) attractor states; (3) variation and (4) non-linearity. These constructs do indeed correlate with a subset of the features of DST. However, Lenzing et al. (2023) point out that in DST each of these constructs is operationalized mathematically. No aspect of such an operationalized approach is transferred to the SLA context in de Bot et al.’s paper. Instead, de Bot et al.’s (2007) approach rests on the argument that SLA displays some of the features of dynamical systems rather than demonstrating that these features operate in SLA in the manner defined in DST.

Larsen-Freeman (2011: 58) insists that DST can be applied to systems in the mental world in a 15-line argument about intentional actions of individuals and their dissipating effect on the speech community. This way of arguing rests on the everyday meaning of ‘intentional’ rather than the philosophical concept, as shown here:

it is not contradictory that, at the same time as individuals are operating in intentional ways in the moment, their personal language resources and those of their speech communities are being transformed beyond their conscious intentions.

In effect, Larsen-Freeman (2011) and de Bot et al. (2007) follow Thelen and Smith’s (1994) assumption that DST is applicable to the human mind without explicitly considering the detail of Thelen and Smith’s argument. van Geert (2003: 662), whose work is central to the development of approaches to cognition that are compatible with DST, summarizes Thelen and Smith’s position as follows:

Thelen and Smith’s dynamic systems approach to knowledge . . . lies in the tradition of non-symbolist approaches to the nature of cognition. They defend the position that concepts and representations do not function as mechanisms of human action and, hence, that they do not exist.

Focusing on ‘[h]ow . . . the organism continually adapt[s] and create[s] new solutions to new problems’, Thelen and Smith (1994: 42) state the following: ‘The answer we present . . . makes no use of representations or representation-like processes. The yardstick by which we measure our theory is thus not rule-like nor symbol-like behavior.’²

Recently, Stelma and Kostoulas (2021) attempted to integrate different readings of intentionality into a CDST-based ‘model of the intentional dynamics of TESOL’. Stelma and Kostoulas (2021) explicitly refer to Brentano's account of intentionality and mention the necessity of mental representations to model the issue of intentionality as a distinct feature of the human mind. However, they do not elaborate further on the core of Brentano's argument. Brentano argues that the fact that the human mind exhibits intentionality serves as a key feature distinguishing the mental and the physical worlds. This fundamental difference between mental and physical states leads to the question of whether it is possible to ‘study mental states using the tools of physical sciences’ (Bechtel, 1988: 42) – a possibility that Brentano rejects. Instead, ‘Brentano’s treatment of intentionality provides support for the dualist view that the mind is distinct from the body’ (Bechtel, 1988: 44). However, the dualist position is explicitly rejected by proponents of the CDST approach (see Larsen-Freeman, 2019: 63). Instead of addressing this apparent contradiction and attempting to resolve the problem that arises in combining these incompatible positions, Stelma and Kostoulas go on to connect their perspective on Searle's notion of intentionality to their ideas about different kinds of intentionality in order to ‘undercover [sic] the “generative processes” that constitute TESOL as an ecology and complex dynamic system’ (p. 55). How Brentano's account relates to this model and how the contradictory views on intentionality and their underlying assumptions about the mind-body problem can be merged is not addressed.

In contrast to mainstream CDST (see de Bot et al., 2007; Larsen-Freeman, 2011), van Geert (2003: 662) had implicitly taken Tschacher’s ‘yes’ option (i.e. mental and physical systems are qualitatively different) and proposed a view of cognition that includes representations and a mechanism by which representations (and by implication intentionality) can emerge from the dynamics of the physical world. He believes that:

concepts and representations and similar notions are indispensable in the description of complex dynamic processes such as human action and they are perfectly compatible with a dynamic systems view if correctly interpreted.

Endeavoring to include representations (and by implication intentionality) when applying DST to the mental world, van Geert (2003: 663) alludes to the difference between so-called ‘control parameters’ (that drive a system) and ‘order parameters’ (that emerge from and in turn constrain the behavior of parts of the system). These parameters are key constructs in synergetics (see Haken, 1977, 1996, 2000), where they are part of the synergetic approach to naturalization, i.e. an account of how some form of intentionality can emerge in physical systems.

In synergetics, naturalization refers to ‘[e]xplaining mental phenomena using concepts and models derived from the natural sciences’ (Tschacher, 2014: 1). The approach to naturalization in synergetics is an explicit and formalized account of the way in which intentionality arises from complex dynamical systems in the physical world. In the synergetic approach to naturalization (see also Tschacher et al., 2003), the emergence of intentionality starts from elementary perception–action cycles that develop into more abstract cognition, based on self-organization. The latter, in turn, is based on the interplay between control parameters and order parameters. In this context, naturalization relies:

largely on the interdisciplinary modeling approach of synergetics. Synergetics deals with complex systems, i.e. systems composed of multiple components (Haken, 1977, 1996, 2000). By way of their interactions, these components can produce new qualitative features on macroscopic scales. (Tschacher, 2014: 5)

Haken (2007: 1) exemplifies control parameters and order parameters as follows:

systems . . . are subject to control parameters which may be fixed from the outside or may be generated by part of the system considered. An example for an external control parameter is the power input into a gas laser by an electric current. An example for an internally generated control parameter is hormones in the human body or neurotransmitters in the brain [emphasis in the original].

When control parameters reach specific critical values the system may become unstable and adopt a new macroscopic state. Close to such instability points, a new set of collective variables can be identified: the order parameters. They obey . . . low dimensional dynamics and characterize the system macroscopically. . . . [T]he order parameters determine the behavior of the individual parts [emphasis in the original].

The synergetics approach applies to a large class of dynamical systems that is often illustrated with reference to the Bénard system. This system occurs in a layer of fluid heated from below. The temperature at the upper surface is lower than the heat added from below. As long as there is only a minimum difference between the temperature at the upper surface of the fluid and the temperature at the bottom of the (shallow) container, the molecules of the liquid are in a state of ‘Brownian motion’. This means that there is no preferred direction for the random oscillations of the molecules, and the liquid does not display any patterns. When the difference between the surface temperature and the heating temperature reaches a critical value, coordinated motion appears in the fluid, creating patterns known as Bénard cells, as shown in Figure 1. As Tschacher (2014: 5) points out, ‘these patterns are an example of the emergence of the new qualities’ focused on by synergetics.

Figure 1.

View of the surface layer of a liquid showing regularities in self-organizing patterns (‘Bénard cells’).

The synergetics claim is that the dynamically-induced emergence of new qualities visible in the structured motion of the fluids is also present in what Tschacher (2014: 10–14) refers to as the naturalization of intentionality. As Mitchell (2009: 298) pointed out:

Hermann Haken’s Synergetics . . . ha[s] attempted to integrate ideas from thermodynamics, dynamical systems theory, and the theory of ‘critical phenomena’ to explain self-organization in physical systems such as turbulent fluids and complex chemical reactions, as well as in biological systems [emphasis in the original].

Tschacher’s and Haken’s works have added aspects of the human mind to this list. In proposing a synergetics-based naturalization approach, these authors have expanded the scope of synergetics to include aspects of the human mind.

Figure 2 illustrates the dynamics underlying the pattern formation process mentioned above. As illustrated in Figure 2, control parameters that may be external to the complex system – as in the case of the heat source – drive the dynamics of the system. Order parameters and the individual parts of the system are related by ‘circular causality’: ‘[T]he cooperation of the individual parts enables the existence of order parameters’ (Haken, 2007: 1), and the order parameters in turn determine the behavior of the individual parts. In other words, order parameters are ‘generated’ by changes in the system (that may be due to changes in the control parameters) and the newly generated order parameter ‘rules’ the new form of the system. For our argument, the key point is that all of these interrelationships are formally modeled using differential equations.

Figure 2.

Schematic illustration of the relationship between control parameters and order parameters.

Tschacher (2014) argues that the processes in Figure 2 also apply to the naturalization of intentional states. He makes his case by showing that the features of intentionality may emerge as properties of self-organizing complex systems. These features are:

Aboutness: the intentional system’s state must be about something in the system’s environment;

Functionality: intentional states should be functional or instrumental with respect to what they are about;

Mental-likeness: in the sense that apart from being intentional model systems should have properties that resemble the properties of mental states (Tschacher, 2014: 2)

Tschacher shows that at least two of these features of intentionality are present in self-organizing complex systems as exemplified – at a basic level – by the Bénard system. In the case of Bénard systems, aboutness can be recognized as being present as:

the complex system ‘represents’ an external object by the generation of an order parameter. The intentional object in this case is the external control parameter c.

In relation to functionality, Tschacher (2014: 11) quotes Schneider and Sagan (2005) who pointed out that (1) self-organizing systems maintain their overall structure by discontinuing processes that lead to non-equilibrium and that (2) the fewer such discontinued processes, the more efficient are the dynamics at work in the given system. Tschacher (2014: 11) illustrates these principles with reference to research in animal locomotion showing principled relationships between gait and metabolic cost³ that have been modeled as self-organizing complex systems. Tschacher (2014: 11) concludes that ‘[t]he association of pattern formation with gradient reduction makes pattern formation functional’.

Tschacher (2014) points out that the first two features of intentionality discussed above are necessary but not sufficient conditions for intentionality and that they may also be associated with non-intentional systems. Therefore, the third feature (mental-likeness) is of crucial importance. In relation to the third feature, Tschacher (2014) discusses key aspects of mental/cognitive systems including complexity reduction, stability, autonomy and intentional content as components of mental-likeness that have been identified and empirically supported in neural networks in biological organisms. The latter have been modeled using the formal principles of complex self-organizing systems.

Tschacher (2014: 11) refers to the processes shown in Figure 2 as the ‘naturalization of intentional states’, i.e. the emergence of intentional states out of the dynamics of the natural world.⁴ However, Tschacher (2014: 11) points out that the scope of such naturalized intentional states is limited to a basic type of intentionality:

we found that . . . naturalization of intentional states is achievable when these states comprise motivation, goals . . . or drives; these may be viewed as basic intentional states related to behavioral strivings. At the moment, however, the intentionality problem appears intractable when these intentional states are of a symbolic and propositional nature.

Tschacher (2014: 11) explains that by ‘intentional states of a symbolic and propositional nature’ he is referring to Fodor’s (1975) language of thought, i.e. to higher mental functions rather than basic intentional states. This limitation has repercussions for the applicability of DST to mental states. The consequence of this limitation suggests that non-propositional (procedural) components of language can be modeled using DST, whereas propositional components cannot be – as long as there is no known formally operationalized mechanism that is capable of deriving higher intentional states from lower intentional states.

The arguments in Tschacher (2014) align with such a contention. The line of thinking presented by Tschacher explores the assumption that mental systems can arise from physical systems through a process of naturalization that results in basic intentional states. Tschacher’s line of thinking shares some basic ontological assumptions with an evolutionary proposal made by Fitch (2008), an evolutionary biologist and cognitive scientist. Both Tschacher’s (2014) and Fitch’s (2008) accounts of the emergence of intentionality rely on self-organizing systems.

However, the fundamental mechanisms driving the two approaches to the emergence of intentionality are placed in quite different explanatory frameworks with different starting points. These different frameworks mean that it cannot be taken for granted that any parts of the two approaches can be interchanged. Whereas Tschacher’s model starts at the molecular level and is mathematically explicit, Fitch’s approach starts at the level of living cells and reasons within a broad evolutionary framework. Fitch’s fundamental thesis is that a very basic form of intentionality, dubbed ‘nano-intentionality’, is present in one of the earliest forms of life (on earth), the eukaryotic cell that evolved over a period of two billion years and formed the basis of life on earth. Fitch argues that the properties of full intentionality developed by assembling – through evolution as outlined below – the properties of nano-intentionality and higher intermediate forms of intentionality to shape a conscious mind that can represent things in the physical world, that has self-identity and that can have wishes, dreams and desires relating to real and unreal things.

In Fitch’s (2008) analysis, full intentionality emerges in five major evolutionary stages, starting with the two billion-year period of the eukaryotic cell. In stage two, these nano-intentional cells formed multicellular organisms in which cell types were able to specialize. Stage three is marked by the development of the neuron, a quintessentially specialized cell type.

Fitch (2008: 23) characterized the relevance of this stage by focusing on the emergence of an information-processing capacity:

As far as we know, neurons have always worked together in groups [i.e. they never occurred as isolated cells; MP, AL and HN], as exemplified by the nerve net of a jellyfish or hydra, which I call micro-intentional by virtue of this information-processing specialization: the purpose of nervous tissue is to process information, in the same way that the purpose of a flagellar or heart cell is to pump fluid. I see this third stage as proto-mental because it lacks representations.

Fitch (2008: 23) summarized the final two stages (4 and 5) as follows:

The fourth stage simply expands upon the information processing of nervous systems by adding the capacity for representations of the body and the world–the hallmark of the truly mental. Finally, the necessity for choosing among various options represented, and ‘tagging’ the one actually implemented, leads to the serial awareness that we subjectively experience as consciousness.

The approach to the emergence of intentionality developed by Fitch (2008) focuses on intrinsic intentionality that is not driven from outside. This feature of intrinsic goal-directedness is present in all living cells. Fitch (2008: 1) characterized his approach as follows:

The form of intrinsic intentionality I propose is thoroughly materialistic, fully compatible with known biological facts, and derived non-mysteriously through evolution.

The phrase ‘derived non-mysteriously’ characterizes Fitch’s approach as one that does not require recourse to non-material notions such as ‘spirit’ or ‘soul’ – or more explicitly in Fitch’s (2008: 3) own words:

I . . . assume a non-vitalist stance . . .: a cell is ‘just a machine’. There is no non-physical élan vital that sets off living things from other complex physical objects, and all of a cell’s functions can, in principle, be derived from the chemistry and physics of its components, and thus ‘reduced’ to physics in the same way as the functions of carburetors and laptops . . . However, the ‘cell = machine’ truism obscures an equally important fact: that a eukaryotic cell is unlike any machine ever produced by humans.

In fact, this materialistic perspective is one feature that Fitch’s and Tschacher’s approaches have in common.

The concepts behind Fitch’s nano-intentionality as well as Tschacher’s naturalization of intentionality offer potential solutions to the problem of the evolution of intentionality and hence to the issue of whether DST can be applied to SLA. However, the concepts are – at least for the time being – limited to lower order mental states such as basic procedural aspects of language processing. Obviously, it would be extremely important for the application of DST to language acquisition to define this dividing line as precisely as possible, but that is not a task that we attempt here. However, this task should be high on the agenda of researchers who intend to apply DST to non-procedural aspects of language.

IV Conclusions / Summary

In this article we have examined two fundamental issues relating to the question of whether DST can be applied to SLA.

We have shown that the antireductionist stance of the ‘dynamic approach’ does not follow from DST and that strong antireductionism creates a deadlock for empirical research and also contrasts with the actual practice of researchers working within the ‘dynamic approach’. Empirical research within DST requires some form of reductionism. As Feldman (2012: 149) pointed out, ‘it is inevitable that we need to choose some portion of the world to focus on that is smaller than the world itself.’

Given that DST was developed to account for dynamical processes in the natural world, one needs to ensure that DST mechanisms that were formalized using DST mathematics also apply to the mental world, including SLA. This issue has so far not been adequately addressed. We have provided some evidence that for basic procedural aspects of the mind DST mathematics may be applicable in the analysis of the human mind’s activities. Confirming the scope and limits of this finding is a precondition for applying DST to SLA since it strongly limits the domains of application.

Resolving these two issues needs to be a priority for SLA researchers who wish to make effective use of dynamical systems theory.

Footnotes

We would like to thank two anonymous reviewers for their insightful comments on an earlier version of this article.

Author Note

As we note in the opening lines of this article,dynamical systems theory (DST) is a mathematical theory. However,in the context of SLA and TESOL it has also been referred to in non-mathematical ways using different labels including CDST. In order to keep the terminology used in this article transparent,we use the term ‘dynamical systems’ to refer to theorizing in mathematics. In contrast,we use the term ‘dynamic systems’ to refer to the approach in SLA that does not adhere to this mathematical framework. In this article,we use DST to refer to the mathematical dynamical systems theory.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship,and/or publication of this article.

ORCID iD

Manfred Pienemann

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Can dynamical systems theory be applied to second language acquisition? The issues of reductionism and intentionality

Abstract

Keywords

I Introduction

II Reductionism

III Intentionality

IV Conclusions / Summary

Footnotes

Author Note

Declaration of Conflicting Interests

Funding

ORCID iD

References