From Crisp to Confounded: An Introduction to George Conger’s “Doctrine of Levels”
Metaphysical and Emergentist Levels
A professor of philosophy at the University of Minnesota, George Conger was loosely associated with the movement known as Emergent Evolutionism, the proto-phase of emergence-based thought in philosophy, the sciences, and even theology existing roughly from 1915 to 1940 (see, Blitz, 1992; Stephan, 1999; Goldstein, Forthcoming; and various Classical Papers published in E:CO since 2004). Conger’s paper on the idea of emergent levels, published in the prestigious Journal of Philosophy in 1925, was a careful and thought-provoking examination of the idea coming right at the intellectual zenith of proto-emergentism. Conger explicated what he found conceptually helpful in the notion of emergent level, but did not fail in taking-on what he found conceptually troublesome about it as well. Conger’s views on emergent levels are not only important for their insights into what Emergent Evolutionists in general held about levels, but also because of their display of striking prescience in foreseeing critical issues about levels that would arise later when the phenomena of emergence became a key construct within the sciences of complex systems. Conger’s same prescience also provides perceptiveness into a central aspect of the notion of emergent level, a kind of paradox associated with the idea, a subject I’ll return to in the conclusion.
Emergent Evolutionism held that the Darwinian theory of evolution needed to be supplemented by a series of saltations or jumps in novelty, the coming into being of a series of new emergent levels constituted by new emergent phenomena with new emergent properties. In the words of one of its chief proponents, the animal behaviorist and philosopher Conwy Lloyd Morgan (1923:15), “Each higher entity in the ascending series is an emergent ‘complex’ of many entities of lower grades, within which a new kind of relatedness gives integral unity...each higher com-plex takes on the role of a com-plex in virtue of its integral unity.” Morgan emphasized the prefix “com- “ because of its meaning of “together” in order to highlight how the emergent level was, in Morgan’s words, supervenient upon whatever was coming together. An emergent level, accordingly, was a concept that called attention to the importance of considering the novel dynamics of a level operationally “higher” than that of the parts that were being newly related. As Conger understood it, each level was thought to be composed of structures sharing major attributes and interacting with one another. The beginning signs of the paradox of emergent level mentioned above can be gleaned in the simultaneous positing of a discontinuous crispness distinguishing one level from another and the need to attend to the entanglements and interactions of said levels in order to account for emergence.
The stratification of reality, nature or the cosmos into a hierarchy of levels, each representing a separate domain with unique entities, dynamics, and laws, is an ancient notion. Aristotle (cited in Lovejoy, 1936), for instance, proposed several different hierarchical schemes, one of the most influential being his vertical partitioning of the “powers of soul”: the nutritive soul of plants; the animated soul of animals; and the rational soul of humans. Such layered cosmologies were typically tied closely with sequenced cosmogonies, e.g., the Neo-Platonic hierarchy of Plotinus (1991) which conceived the phenomenal world as a lower level in a series of “hypostases” emanating downward out of the highest, the One beyond Being Itself.
The idea of such metaphysical levels need not have had a religious basis as is shown in both modern philosophical, but non-theological, conceptions of metaphysical cosmic stratification such as those developed respectively by Nicolai Hartmann (see Werkmeister, 1990) and James Feibleman (1951) as well as the prevalent use of the construct of level found in various scientific theories. In the latter, different justifications have been offered for the particular allotment of levels, which entities exist on which levels, and so forth. Perhaps the most sophisticated and trenchant account of the promise and perils of appealing to the construct of theoretical levels in scientific thinking, especially that found in theoretical biology including its many recent complexity-based inspirations, is the one offered by the philosopher of biology William Wimsatt (many of whose papers were pulled together in his magnum opus of 2007), a name we will be returning to below.
What Emergent Evolutionists bestowed on antediluvian hierarchy scenarios was an emphasis on an ascension of novel strata, not as somehow pre-given and thereby built-into the very substratum of reality, but instead as emerging dynamically over time in a series of discontinuous emergent levels. This bestowal can be seen, for instance, in how the proto-emergentists reworked the philosopher Walter Marvin’s (see Blitz, 1992) notion of “logical strata” that covered everything from ordinary things, to events, to constituents of these things, to constituents of these events, as well as logical, mathematical, physical, biological, and mental categories. Although in Marvin’s layered reality, there were novel characteristics at each new level, proto-emergentists emphasized that the dynamical emergence of each new level would continuously supplement whatever antecedent levels were pre-existing. Accordingly, for proto-emergentists, the novelty of the new emergent levels should be considered a radical1 type of novelty in the sense that the structures, patterns, dynamics, and properties of each new level composed of novel emergent phenomena were held to be neither predictable nor deducible from the lower level components alone, nor were they reducible to the latter.
The claim that nature is arranged according to a hierarchy of levels has been linked to various rationales for the existence of “special” sciences whose conceptual apparatuses are only deemed appropriate for “special” levels, e.g., biology aims at the level of living beings, chemistry at the level of the chemical reactions by which living beings function, physics at the level of atomic (as well as supra-atomic ensembles and sub-atomic particles) by which chemical reactions are determined, and so on. Conger’s own commitment to such a scheme was along the lines of the Emergent Evolutionist take on the correspondence between levels and the “special sciences”, a position which they had adopted and adapted from their immediate intellectual precursors, the English philosopher John Stuart Mills and the French philosopher and mentor of Henri Bergson, Emile Boutroux (Blitz, 1992). However, as we’ll say more about below, Conger also insightfully queried just how many levels there would need to be in such a correspondence between natural strata and the domain specific to each special science.
In modern times, one of the most interesting versions of this kind of level correspondence was that offered by the late eminent biologist Sir Peter Medawar (1974) who formalized this correspondence according to the particular geometry operationally most appropriate for each level— “geometry” here referring to the famous classification founded on group theoretic invariances put forth by the great German mathematician Felix Klein in the last quarter of the nineteenth century2. The preceding footnote lays-out Medawar’s main elements. In Medawar’s hierarchy, for example, the “geometry” of topology corresponded to the level of physics because of this “geometry’s” invariances of sidedness and ordinality on a line thereby enabling physics a wider capacity for abstraction while affine geometry matched the level of biology since shape as such was not invariant whereas living organisms do possess homographic invariance.
Conger’s Prescient Probings
Whereas Medawar’s hierarchical conceptual edifice was undoubtedly rigorous and sophisticated, it was also metaphoric in its speculating on the usefulness of group theoretically defined geometries. I find it quite interesting that Conger had called attention to what he thought of as the metaphoric status of the concept of an emergent level when this idea was employed in psychology or sociology or metaphysics (notice which disciplines he groups together!) and he did this way before all the modern squabbling about metaphor versus “hard science” among present-day complexity advocates.
Although agreeing in principle with the Emergent Evolutionist doctrine of levels, Conger pointed to several problematic aspects with how the notion of level was being employed, e.g., the issue of just how many emergent levels there were. Even among the more strident proponents of Emergent Evolutionism, one can find an uneasiness with how literally and precisely the idea of emergent level should be taken. The New Zealander/English philosopher and experimental psychologist Samuel Alexander (1966), for one, contended that to try to be complete or exhaustive about listing all the multifarious levels was to miss the point since there was a certain relativity to the apparent crispness of each level. The aforementioned Morgan (1923) had conceded it was beyond the “wit of man to number instances of emergences” and presumably this immeasurability extended to the number of different levels on which these innumerable instances dwelt.
Alexander also proposed there existed many intermediary stages such as the complex chemical complexes between matter and life, a proposal along the same vein as that of another Emergent Evolutionist, the American entomologist William Morton Wheeler (1927) who, by the way, was one of E. O. Wilson’s professors and mentors. Wheeler argued that the assignment of any specific number of levels was arbitrary since emergence was observed in a manifold of spatio-temporal events. Wheeler went so far as to criticize the proto-emergentist emphasis on such weighty emergent level novelties as life and mind which he thought were conceived in such a wholesale and broad-brushed fashion, they represented only the final accumulative stages of very long series of more minimal emergences. He asked whether it really made that much sense to view life as a “higher” emergent level in respect to a “lower” level of inorganic matter, or, was it more accurate to speak about innumerous intermediary emergent levels. But then just how many intermediary levels were there? In such a context, the contemporary emergentist-oriented philosopher William Hasker (1999) has keenly suggested that the weaker the definition of emergence, the more emergent levels that would appear. This should have some bearing on the typological distinction between so-called “weak” and “strong” emergence offered by the contemporary complexity-oriented philosopher Mark Bedau (Bedau & Humphreys, 2008).
Conger himself speculated, after considering varied scientific sources in which the notion of level had been theoretically utilized, that there was indeed a precise number of levels, viz., 25, each of which agglomerated together an unusual congeries (pun intended) of constituents ranging from electrons, spiral nebulae, families, perceptual reflexes, sentiments, and other oddly associated units within levels. For Conger, all 25 levels were distinguishable according to the way empirical investigation, by means of each of the “special” sciences was supposed to proceed on that level. For example, there were different investigational strategies for organic compounds, for unicellular organisms, for multi-cellular organisms, and so forth. Furthermore, Conger warned that all of the levels interact and therefore should be considered, simultaneously, as both independent as well as interdependent with the implication that the designation of a specific emergent level didn’t indicate a sharp distinctiveness between them after all, anticipating, as we’ll soon see, later remarks coming from within complexity theory and adding more hints about the paradox of emergent levels alluded to earlier.
Conger also made it a point to criticize what he believed were the misleading spatial and valuational implications involved in using the terms “higher” and “lower” to describe emergent levels in contrast to their substrate levels. In this context, Conger cited Morgan’s use of the “higher-”connoted term “supervenient” (a term which has been revived in recent years by the philosopher Jaegwon Kim to describe the status of emergent levels—see Bedau & Humphreys, 2008). For Morgan, the emergent level of “new relatedness” is “supervenient” on the “lower” level but, unlike in Kim’s disavowal of any causal efficacy on the part of the emergent level and its emergent entities upon the lower level and its entities, Morgan’s “supervenience” did imply that the events at the lower level would run their course differently as a result, a preliminary statement of what would later be known under the appelation of macro-determination or downward causation, another subject to which we’ll return in the conclusion.
Other, later emergentists have voiced similar concerns of discourse utilizing “higher” and “lower”. For example, the mid-phase emergentist biologist and philosopher Paul Weiss (1969) attributed thinking in terms of spatially conceived level hierarchies as an undesirable vestige of the hard-to-change habit of visualizing a complex system as a whole on a “higher” plane and then dissecting this whole into components displayed on another “lower” plane. In a similar critique of such “higher/lower” descriptors, another mid-phase emergentist, the physicist and philosopher of science Mario Bunge (1979), who incidentally developed a rigorous formalism for emergence and emergent levels mostly unknown to current complexity theorists, pointed to the militaristic insinuation of rank and superiority by which “higher” levels command “lower” ones—the etymology of “hierarchy” includes the Greek “hieros-” meaning “sacred” and “-archein” “to rule.”
Another insight into the less-than-salutatory connotations of spatial ascendance in describing emergent levels can be found in the formalism of emergence offered in a previous issue of E:CO by the mathematician and psychiatrist William Sulis (2004) who suggests that there is as much conceptual justification for understanding emergence in a horizontal as in a vertical fashion, e.g., two entities on the same horizontal level coming together to generate the conditions for the emergence of a third still on that same level such as is found in biological symbiogenesis (my example, not Sulis’s, about which I’ll say more below). Of course, “higher” can be taken in a non-spatial sense, for instance, of wider inclusiveness, although such a move could open the door to the misleading conclusion that emergence takes place by way of some mathematical/logical operation involving the erection of type hierarchies and similar ideas. Conger himself proposed interpreting “higher” in the sense of being further along a developmental pathway, emergent levels then understood from the perspective of a process of development (Salthe, 1993, being the prime representative of allying developmental with evolutionary scenarios in the determination of hierarchical schemes).
In regard to the question of how emergent levels arise, a subject about which Emergent Evolutionism was conspicuously reticent, Conger ventured forth with several possible scenarios. One involved processes of integration or creative synthesis which he understood as originating not only in the earlier work of Herbert Spencer, famous in the mid-nineteenth century for his pre-Darwinian speculations on evolution, but also in intimations on the part of the proto-emergentists E. G. Spaulding and R. W. Sellars on how parts were brought together into novel “organizing relations” exhibited on the new level. Another conjecture offered by Conger along these lines, although, in my opinion, not sufficiently different from the first to be considered an authentic alternative, concerned combining relations or mutuality of relations which referred to the critical importance of the conception of inner relations, a philosophical category somewhat typifying metaphysics in Britain and the US at the end of the nineteenth century, e.g., in the philosophies of F. H. Bradley and William James (see Sprigge, 1993) or what Morgan had identified as “new terms in new relations”.
Interestingly, in this context, Conger cited the work of Mary Parker Follett, otherwise known as one of the main forerunners of modern organizational theory. Follett’s quotation was very prescient itself in its dynamical systems, and Habermas-sounding analysis of human relations (e.g., see Habermas, 1985, where the influence of another proto-emergentist thinker, George Herbert Mead, is overtly evident; see also Mead, 2007, in E:CO): “...I never react to you but to you-plus-me; or to be more accurate... it is plus the-interweaving-between-you-and-me meeting you plus the-interweaving-between-you-and-me, etc. If we were doing it mathematically we should work it out to the nth power... We should obtain a differential equation or a set of differential equations...” In this context, Conger conjectured about universal processes whereby individuals in a group could adjust themselves according to mutual dependence, perhaps something like what happens in synchronization.
Conclusion—Crispness versus Interaction: Flirting with Paradox in the Idea of Emergent Levels
Conger also suggested, in another manifestation of foresight, that much of the “mystery” surrounding emergence could be cleared-up if the “cumulative interaction between units of various levels” and not just from “higher” to “lower” levels was investigated since the potentialities of the novel units on the novel levels were “unknown until its interactions reveal them”. Conger further claimed, in an almost eerie prognostication, that entities at the same level could have different rates of interaction resulting in them being grouped together according to these differing rates and that, accordingly, boundaries would form with the units having slower rates inside the “inner portions” of the system and those with faster rates at the outer portions of the emergent level. An almost identical formulation can be found in one of the most sophisticated contemporary mathematical formalisms of emergence in the work of Ehresmann and Vanbremeersch (2007): “The evolution of the [emergent] system will mostly depend on the interactions between agents of various levels of complexity, acting with different time scales.”
Similarly, Wimsatt (2007) has for many years now been pushing for a similar, more sophisticated treatment of the complexity of complex systems, in particular, the need to consider the crucial fact that as complex systems become more complex, the operative levels themselves get more complex in the sense of being more diffuse, overlapping, and capable of being decomposed in more numerous ways. Thus, a researcher might decompose a multi-cellular organism into various systems of parts making-up different levels, or into levels made-up of cells with common developmental fates, or according to phenotypic features determined by common sets of genes, or in accord with biochemical reactions, or... Similarly, cells can integrate into a multi-cellular organism, the latter being at a “higher” level yet chemical radicals can enter into the structure of some molecules without there being a difference between the level of the radical and the molecule into which it adheres. Consequently, the existence of an integrated entity neither guarantees the presence of crisp level distinctions nor there being just one sure way to analyze it. As the complexity-oriented biologist Richard Levins (cited in Wimsatt, 1972) put it, it’s even harder to distinguish levels when their component subsystems have evolved together. And the mid-phase emergentist theoretical biologist Robert Rosen (1969) emphasized that biologists employ the term “hierarchy” only in an inexact, non-etymological sense since biological functions tended to be distributed over much if not all of a system.
In a similar respect, Emmech et al. (1998) pointed out that levels in complex systems possess non-homomorphic inter-level relations meaning that the nature of the relation between levels varies for each, a point that is not generally incorporated into grand schemes of emergent levels, each emerging sharply and consistently out of the one beneath, all towards the creation of ever increasing complexity in the universe. There’s also the related issue so cogently put forward by Stanley Salthe (1993) that a “higher” emergent level is better thought of as a boundary condition within which “lower” emergent entities integrate so that, strictly speaking, at any one time the notion of an emergent level is an occurrence at level ln out of substrate components at level ln-1 in the context of boundary conditions at level ln+1.
The great importance of interactions in defining levels, particularly emergent levels, was also one of the main themes brilliantly articulated by Douglas Hofstadter (1979) in such works as Gödel, Escher, Bach and I Am a Strange Loop. Many readers may not have realized that, although these book dazzle with their mind-stretching Gödelian logic interposed with Escher’s startling artworks and Bach’s mathematically recursive music, one of Hofstadters main objectives all along was to build-up to his own unique take on emergence, specifically how the mind should be considered an emergent level of the brain, thesis which is spelled out in the last pages of Gödel, Escher, Bach. For Hofstadter, emergence is not a simple matter of crisply demarcated strata but, instead, is a far more interwoven, entangled, messier process and novel emergent levels express precisely this interwovenness, entanglement, and messiness, features which Hofstadter termed “tangled hierarchies” and “strange loops” since the “higher” and “lower” level constituted each other in a cross-referential fashion.
With this entangled/interactive view of an emergent level, a whole new slant became possible on such issues as the downward causation or macro-determination of the “higher” emergent level (e.g., of the mind) on the “lower” level (e.g., of the brain) and vice versa that the neuroscientist Roger Sperry (1986) had argued for in his emergentist theory of consciousness. Indeed, the very idea of downward causation, as it is typically formulated, presumes level distinctions crisp enough to make the idea of an influence of one on the other thinkable at all. In this regard, it perplexes me that in the face of so much prattle being written these days about downward causation in relation to emergence, few putatively complex systems-oriented thinkers are familiar with territory already plowed so rigorously and creatively as Hofstadter has accomplished.
To be sure, although hierarchical descriptions do tend to be of help to researchers in navigating through often impenetrable thickets of complex systems by isolating and then focusing on the dynamics at a crisply defined level, a focus made possible by heuristically ignoring details at lower or higher levels (see, e.g., Salthe, 1993; and O’Neill, et al., 1986), the above remarks of Conger and later complexity theorists reveal that things in nature are most often not so crisp. In this context, it is important to keep in mind an insight into theory-making in biology (and I would add: in complexity science as a whole) put forward the philosopher of biology Morton Beckner (1974), namely, that there is a critical difference between the hierarchies of levels constituting natural systems and the hierarchies of levels devised in our theories of systems.
I propose that the discrepancy between understanding the idea of a “higher” emergent level in the sense of a crisp, sharply defined demarcation and the theoretical cautions enumerated above concerning the arbitrariness and thicket of interactions involved in conceptions of an emergent level indicate that the very concept of an emergent level is one approaching a genuine paradox. On one horn of this paradox is the decided sense that an emergent level is a crisp category, much of this crispness resulting from what I (Goldstein, 2002) have elsewhere referred to as the “singular” or “privileged” status of emergent levels in distinction to the arbitrary designation of a particular scale. Scale is arbitrary in the sense that it depends on an arbitrarily set power of resolution used in observing a system, e.g., the power of resolution of different kinds of microscopes determines the specific scale at which an investigation utilizing such microscopes can proceed. However, an emergent level is not arbitrary since it is just that level at which, phenomenologically, emergent phenomena are observed ostensively.
It is the supposed obviousness of this crispness of emergent level that supports the artificial life researcher Andrew Wuensche (1998) in his contention that much of the promise of artificial life lies in our computational capability for tracking and measuring multiple, sharply-delineated levels simultaneously for the purpose of ascertaining the laws relating one level to another. At first impression, it does seem to be the case that the emergent levels of such artificial life standbys as the Game of Life are glaringly apparent, at least, in our quick perception that the movement of “gliders”, for example, is on a “higher” emergent level than that of the substrate constituted by the on/off status of the cells. Yet, more recent research into artificial life, specifically the Game of Life, reveals that the issue of exactly which levels there are as well as the crispness of these emergent levels is certainly much more complicated and also more open for different interpretations than previously assumed (see, e.g., Griffeath and Moore, 2003). This implies, for me at least, that, if, even in such artificially devised scenarios of emergence as computational simulations where everything, in an important sense, is potentially measurable (if we only know, that is, where exactly to point our measuring instruments at and what exactly to use for appropriate metrics), the nature of emergent levels are not crisp but an admixture and commixture with other levels, then there is in general much more room for equivocations on how many levels there are, ambiguity about how these levels form by interacting among themselves, their transience, and so forth.
Although I think that Hofstadter with his labyrinthian tangled hierarchies and Wimsatt with his causal thickets have come the closest to exhibiting how the idea of an emergent level flirts with paradox, I don’t think they have gone far enough for it is the case that entangled knots can be untangled (as I’ve seen methodical fisherman do with their long lines in a little fishing village north of Viña del Mar, Chile) and thickets of briar thatches can even be traversed (with sharp enough machetes and a willingness to be inflicted with superficial flesh wounds)! That is why, in contrast to tangles and thickets, I am proposing a more fundamental shift in our understanding of what it is that happens when an emergent level emerges: the lower level is so taken-up into the new emergent level by varied complexification operations that the former no longer exists as the separate echelon it had previously, but rather gets confounded up into the new emergent level according to the literally etymological sense of “confounded,” that is, a “pouring together” so thoroughly, it is not possible even in principle to detect what was separate before the pouring together.
An image I have of this confounding is of traveling on a bridge which strangely curls -p behind itself as one traverses it, a bridge operating in just the opposite and reverse manner of the bridge of reductionist explanations which also curl-up behind itself as it traverses from the higher to the lower level. In the bridge of confounding, once the higher level is reached, the lower level has been subsumed into it so radically that lower level components are now not the same separate parts they were before but instead are themselves reconfigured within the confoundedness of the higher level coherence. Confounding, then, transgresses those level distinctions crisp enough to even allow for entanglement. The example of emergence that comes to mind as the most illustrative of confoundedness in a visualizable manner is that of symbiogenesis, one instance being the remarkable theory propounded by Lynn Margulis of the emergence of the first eukaryotic cells through endosymbiosis between primitive prokaryotes (see Reid, 2007). The first eukaryotic cells paradoxically were crisp in their boundaried constitution as separate entities yet their internal constitution was also a confoundedness of their pre-endosymbiotic parts.
Understanding an emergent level as confounded in this sense is definitely not meant to add greater mysteriousness to the construct of emergence such that Samuel Alexander’s call for “natural piety” in the face of emergence should be taken as the most appropriate response to it. However, “confounded” does connote something obdurate to easy understanding as shown in the following quotation from Shakespeare’s “The Phoenix and the Turtle”:
Reason, in itself, confounded,
Saw division grow together,
to themselves yet either neither,
Simple were so well compounded...
By qualifying emergent level as confounded in this sense, I might be judged guilty of the logical imprudence of obscurum per obscurius, that is, trying to explain something obscure by appealing to something even more obscure, a conceptual ruse not infrequently found even more the most eminent of complexity thinkers! However, I believe that qualifying emergent levels as confounded can escape this charge by indicating how to follow through on another of Conger’s prescient insights, namely, his contention that much of the discourse on emergence in his day and age was way too general and indefinite. His remedy was to call for testable hypotheses to initiate promising avenues for detecting processes of emergence. Similarly, by considering an emergent level as essentially confounded, I am also calling for greater definiteness in our probing of emergence, probing deeper and more perspicaciously into those processes and operations of nature and society which possess the potency of generating emergent outcomes that are confounded in the manner I have proposed. That is, confoundedness is being put forward as a goad for greater probity. I have suggested a new logic for emergence along such a line called the logic of self-transcending constructions (see Goldstein, 2002, 2003, 2004, 2006, Forthcoming). This new logic flirts with but does not embrace paradox, just as I am claiming the notion of an emergent level flirts with but does not embrace the paradox of simultaneous crispness and entanglement. But also like the place of emergence in a scientific or philosophical explanation, an appeal to which I (Goldstein, 1999) have suggested functions more as the beginning and not the terminus of an explanation, so the logic of self-transcending constructions is more like an indexical pointing to where explanatory strategies employing the notion of emergence could start looking.
Originally published as Conger, G.P. (1925). “The doctrine of levels,” The Journal of Philosophy, 22(12): 309-321. Reprinted with kind permission.
The term “radical emergence” has been used to describe the position taken by the philosopher Michael Silberstein (2001; Silberstein & McGeever, 1999; and comments on Silberstein’s work in Freeman, 2001; and Feinberg, 2001) regarding his argument for considering consciousness as an emergent phenomenon. The word “radical” presumably connotes two related aspects of consciousness: 1. The claim that consciousness as an emergent phenomenon possesses an ontological and not mere epistemological status, i.e., it is non-deducible in a much stronger sense than the claim that only epistemic limitations on the part of observers restrict its deducibility; 2. Consciousness as emergent is completely irreducible to the dynamics and laws operative on its neuronal substrate in the brain. My use of the term “radical novelty” is a qualification about the kind of novelty an emergent phenomenon is thought to possess. Using “radical” to qualify emergent novelty affords insight into the specific nature of the type of novelty needed for a doctrine of emergence without claiming that some emergences are radical, while some are not. It seems to me that “radical emergence” is qualifying the wrong thing since, in my opinion, the various typologies of emergence that have been offered, e.g., so-called “weak” versus “strong” emergence and other classifications (see Bedau & Humphries, 2008) are not particularly illuminating. But this is a long story that I don’t have the space here to get into—see Goldstein (Forthcoming).
It should be noted that although Medawar’s system of levels was not directly emergentist, it did include what could be called a kind of “conceptual emergence” in that new concepts could emerge in a level which have not appeared in the geometry above (Medawar’s hierarchy goes vertically downwards from the most abstract to the least abstract). Medawar’s hierarchy was not emergentist since it didn’t provide an account of how new levels may come about besides appealing to a repeated restriction of possible transformations (Klein’s transformation groups) which, again, was only a formalistic account of “conceptual emergence.” It also should be recognized that Medawar’s scheme did not consider what might take place between levels such as via the interactions of entities on different levels.
Here is Klein’s classification of group theoretic invariances of different geometries: Metrical geometry: a group of operations including translations of figures on a plane, rotations, inversions or any combination of these so that ordinary properties associated with our everyday meaning of shape do not change, e.g., distance remains unchanged; Euclidian geometry: operations allowing for symmetrical magnification in 3-dimensional space, e.g., shape is not changed but distance is not invariant; Affine geometry: new points or new coordinates are related to old points by linear integral functions; these affine transformations are similar to Euclidian transformations except that the degree of extension or diminution in all three dimensions are not the same; e.g., shape is not invariant but retains a homographic relationship; Projective transformations: the mapping functions are fractional linear functions; among the properties invariant with respect to projective transformation is linearity, i.e., straight lines stay straight lines whereas anharmonic ratio and conic section properties are invariant but parallelism is not; Topology: mapping functions are singled valued both ways and thus bring new points into a one-to-one correspondence with the old ones, plus they are continuous functions; e.g., figures are transformed but remain homeomorphic, i.e., general properties of the space remain such as insideness and outsidedness as well as the order of points on a line.
Each geometry, and by implication, each “special” science is a special case of the one above (since the group of operations it defines is a subgroup of the group related to the one above it; the theorems then becoming more particular).