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Physical Emergence and Process Ontology

William M. Kallfelz

Committee for Philosophy and the Sciences (CPaS), Department of Philosophy,

University of Maryland, College Park, MD.

Submitted to:

World Futures Journal, special issue on process thought and natural science

Special editors: Franz Riffert and Timothy Eastman

June 3, 2007

Abstract

Alfred North Whitehead introduces in Process and Reality the notion that the “philosophy of organism is a cell-theory of actuality.” I argue here that the most promising venue for a concordance with process ontology vis-à-vis extant physical theory includes the notions of dynamical and ontological emergence in the physical sciences, as described for in Silberstein & McGeever (1999) as well as in Kronz & Tiehen (2002). Here I draw on my previous claims (1997, 2005, 2006) to show in more general terms how process ontology provides a more unified characterization of ontological and dynamical emergence.

Keywords: Emergence, process physics, entanglement, quantum spacetime

I. Overview and History

The notion of emergence has received much renewed attention by scientists and philosophers alike. Robert Bishop (2004) tries to logically secure the notion in terms of certain necessary and sufficient conditions, involving mereology (i.e. the study of the interrelation between properties of parts and wholes), that notions like supervenience[1] do not share. Robert Batterman’s (2002-4) study focuses on aspects of emergence as they occur in critical phenomena associated with phase transition and caustic surface in non-linear optics.[2] Paul Humphreys (1996-7) seeks to give an account of emergence in terms of what he describes as “property fusion.” Andreas Hütterman (2005) distinguishes between “diachronic” versus “synchronic” emergence, in terms of a failure to explain dynamic properties of the whole versus parts, and similarly for instantaneous properties for parts and wholes, respectively. Frederik Kronz and Justin Tiehen (2002) offer their notion of dynamical emergence, as contrasted with radical and programmatic emergence. Silberstein & McGeever (1999) distinguish between what they consider as a weaker or “epistemic” sense of emergence, versus a stronger or “ontological” sense. Still others like Clayton (2004), Gregerson (2003), Kauffman (2000), Lauglin (2005), and Morowitz (2002) offer accounts that are more broadly sketched out in terms of levels of description appropriate to general and the special sciences. Because my study here focuses exclusively on how such a notion appears in certain classes of phenomena pertaining exclusively to quantum physics, I will omit from my discussion Gregerson, Kauffman, Laughlin, and Morowitz. For a concise philosophical overview of emergence, see O’Connor & Wang (2003).

Amidst all these recent studies, “it is not clear whether intuitions about emergence are shared intuitions.” (Hütterman, 2005, 121, italics added) Beginning with J. S. Mill, many nineteenth-century British philosophers held intuitions that contrasted ‘resultant’ from ‘emergent’ mereological properties in the following sense: Resultant properties share an additive parts/whole relationship, i.e. the property of the aggregate is simply the sum of properties of its constituents, whereas emergent properties are non-additive. What was meant by “resultant” was directly inspired by vector and scalar addition in Newtonian mechanics. (Kronz & Tiehen 2002, 331) For example, four hydrogen atoms combine to form a helium atom in the sun.[3] The mass of the helium atom is non-additive or emergent, with respect to the masses of the hydrogen atoms, according to Newtonian mechanics. However, according to special relativity, additivity is restored, insofar as mass and energy are interchangeable (the mass deficit of the helium atom is converted to energy in the fusion of the hydrogen atoms).

As my above survey of the literature may indicate, most scholars consider the traditional distinction along the lines of additivity and non-additivity as at best over-simplifying, if not outright misleading. For instance, Krontz and Tiehen (2002) state:

[I]t appears that a central claim of the British empiricists, that additivity is the mark of resultant (i.e., non-emergent) properties, is wrong…the mark of a non-emergent property of composite systems in quantum mechanics crucially involves a multiplicative operation …[However] [t]he situation is different for evolution.[4] A non-separable evolution is a product rather than a superposition [i.e. addition] … This may provide a way to partially vindicate the British emergentists. (333)

This sets the stage for my discussion concerning the issue of emergent quantum mechanical phenomena, and the role played by operations of the different senses of product and sum as characterized by quantum theory’s mathematical formalism. In this respect, I continue some of the same issues raised a decade ago in Finkelstein and Kallfelz (1997). I specifically point out how certain subsequent advances by David Finkelstein (2001, 2004a-c) in the context of process thought can provide a more comprehensive framework for characterizing emergent quantum phenomena in comparison with what is typically presented in contemporary philosophical and scientific literature.[5]

II. The Varieties of Contemporary Notions of Emergence

Contemporary studies share a common aim of providing accounts of emergence offering fresh insights from highly articulated and nuanced views reflecting recent developments in the fundamental as well as the special sciences. Many in their accounts seek to revise what they consider are misrepresentative and oversimplified abstractions: “It is…possible that…standard divisions and hierarchies between phenomena that are considered fundamental and emergent, aggregate and simple, kinematic and dynamic, and perhaps even what is considered physical, biological, and mental [should be] redrawn and redefined.” (Silberstein & McGeever, 1999, 200) I briefly survey below some recent work in terms of what I consider are the salient conceptual categories conveyed (ontological/epistemic, dynamic, logical/explanatory).

II.1 Ontological Versus Epistemic Emergence (Silberstein & McGever)

Silberstein & McGeever (1999) contrast weaker ‘epistemological’ with stronger ‘ontological’ notions of emergence. Epistemological emergence is best understood as a kind of artifact of a certain mathematical formulation or model arising through a macroscopic or functional analysis of the theory’s ‘higher level’ descriptions or features in its domain (182). This is a weak notion because it connotes practical or theoretical limitations on the resolving and computing power of the theory and, in turn, of its agent.[6] Epistemic emergence is metaphysically neutral. An epistemically emergent property of an object in principle could be reduced to, or determined by that object’s intrinsic properties. However such a property in practice will resist reduction by explanation, prediction, or derivation.

The simplest example of such an epistemically emergent case involves the ‘three-body problem’ in classical mechanics. Such a problem is unsolvable in the practical sense, because no exact solutions of the differential equations exist which determine the trajectories (in 6-dimensional phase space) for the general case of three interacting force centers. Nevertheless, numerical and statistical approximation-schemes aid in giving an account for classes of solutions, to an agreed-upon error.[7]

Ontological emergence, on the other hand, comprises features of systems/wholes possessing capacities (causal, and otherwise) that are in principle not reducible to the intrinsic capacities of the parts, nor among the reducible relations among such parts (Silberstein & McGeever, 1999, 182). Ontological emergence is usually thought to entail epistemic emergence,[8] though the converse never holds: “Epistemological emergence cannot entail ontological emergence, because it is defined to preclude it.” (185)

On a more strongly metaphysical note, Paul Humphreys (1996, 1997) characterizes an ontological notion of emergence in terms of a dynamical fusion of previously two (or more) lower-level properties into a higher-level property. For example, consider a wooden deck comprised of beams that are glued together. Before the glue has dried, each beam x_rⁱ had the property P_rⁱ enabling x_rⁱ some flexing, constrained relative motion with respect to its nearest neighbors. Once the glue has dried, the planks become rigid: their previous properties of relative mobility vanishes,[9] to fuse into the aggregate property P_kⁱ⁺¹ of being able to support the weight(s) of person(s) standing on the deck (1996, 65-66).

Stated more precisely and in general terms, for objects x_rⁱ , x_sⁱ at level i and at time t, endowed with some i-th level n & m -type properties P_mⁱ , P_nⁱ , then during time interval Dt = t’ – t they will fuse in such a manner to form a composite i-th level object x_cⁱ º x_rⁱ Å x_sⁱ = {x_rⁱ , x_sⁱ } such that: P_mⁱ(x_rⁱ,t)* P_nⁱ(x_sⁱ,t) ® [P_mⁱ* P_nⁱ] (x_cⁱ,t’) º P_kⁱ⁺¹(x_cⁱ,t’), where * is the fusion operation.[10] Only the properties fuse to become a higher-level properties. (1996, 60).

II.2 Dynamical versus Prototypical and Radical Emergence (Kronz & Tiehen)

Inspired by Humphreys, Kronz & Tiehen (2002) examine cases of Humphrey’s levels among the sciences according to the following general paradigm involving physics, chemistry, and biology.[11] (Fig. 2, 337):

Physics

Chemistry

Biology

t

[12]

Due to the uniquely quantum mechanical nature of entanglement, a discussion involving at least some reference to the formalism of quantum theory is in my opinion unavoidable. Though I will keep the technical details to a minimum level, the examples discussed below, involving triplet and doublet states, nevertheless involve some technical aspects. The reader may skip over such subtleties without losing the main conceptual points concerning the nature of entanglement, as described in the preceding paragraph.

In the case of quantum mechanics the three state spaces of particles 1,2,3 (described respectively by the two-dimensional spinor spaces (H ₁, H ₂ H ₃) combine to form a tensor product (versus a direct sum Å in the case of classical mechanics) eight-dimensional composite space: H ₁Ä H ₂ ÄH ₃.[13] Similarly, the system Hamiltonian[14] combines via the rules of tensor product and superposition. Now, in principle, can evolve in time to become fully entangled, that is to say, the (8-dimensional) matrix representing cannot be factored into the (8-dimensional) representations of the Hamiltonian matrices representing particles 1, 2, 3 respectively (represented accordingly by ).[15] In other words, in such a case, no such factorization exists, which would allow one to state that . Instead, let us denote the fully entangled triplet (pure state) case with the Hamiltonian: . Other possibilities include evolving into a superposition of partially entangled mixed doublet states, with respect to, say, systems 1 & 2: .[16] Last of all, the system Hamiltonian can evolve into a superposition of (fully non-entangled) Hamiltonians: . Each of the three cases (fully entangled triplet, partially entangled doublet, non-entangled singlet) for Krontz and Tiehen represent different levels (Fig. 3, 338):

t

LE

LP

LU

[17] The notions of emergence are:

1.) Prototypical emergence : Every whole is comprised in terms of contemporaneous parts with independent characterizations.

2.) Dynamic emergence : Emergent wholes have contemporaneous parts, but the parts cannot be independently characterized from their associated wholes.

3.) Radical emergence : Resultant wholes have contemporaneous parts, but emergent wholes do not.

Kronz and Tiehen argue that the weaknesses of prototypical versus radical emergence have to do with a lack of exemplary physical phenomena characterizing 3.) versus excessively inclusive or exclusive criteria characterizing 1.) to the point of rendering the notion trivial in either case. (345) In the case of dynamic emergence, however:

[W]e can say that it does not make sense to talk about reducing an emergent whole to its parts, since the parts are in some sense constructs of our characterization of the whole…there is no genuine mereological reduction of X produced by a description of the parts of X that makes an ineliminable reference to X…since an essential interaction of the parts causes them to go out of existence, as in the second view [Radical emergence], but new parts arise that are dependent on the whole, unlike [Radical emergence]. (345-346)

II.3 Logical and Explanatory Notions of Emergence (Bishop, Hüttermann)

Robert Bishop (2004) seats his notion of emergence in a context defined in terms of logical necessity and sufficiency:

i.) Reduction: When more fundamental properties or descriptions provide necessary and sufficient conditions for less fundamental properties/descriptions.

ii.) Contextual Emergence: When more fundamental properties or descriptions provide necessary but not sufficient conditions for less fundamental properties/descriptions.

iii.) Supervenience: When more fundamental properties or descriptions provide sufficient but not necessary conditions for less fundamental properties/descriptions.

iv.) Strong Emergence: When more fundamental properties or descriptions provide neither necessary nor sufficient conditions for less fundamental properties/descriptions.

In terms of the properties/description division, contextual and strong emergence seem to correspond with Silberstein and McGeever’s ontological/epistemic senses of emergence. However, despite what may appear at the outset as an attractive and simple means of classification, it is precisely this excessive simplicity that may prevent Bishop’s classification scheme from doing much in the way of useful explanatory work when applied to the sciences. The chief problem here concerns the very dispositional nature of properties and characteristics of scientific laws (vis-à-vis explanatory categories).[18] Scientific laws are typically hedged with provisos (Hempel 1988), ceteris paribus conditions (Earman, et. al. 2002), to the extent that traditional notions of implication (which necessity and sufficiency are based on by default) are undermined. Some argue that the conditional structure of scientific laws must be replaced with something far more nuanced and context-dependent, i.e. in the form of modal-logical ‘fainthearted conditionals.’ (Morreau 1997, 1999) Others like Cartwright (1983, 1999) question the very basis of laws and their associated explanatory categories, in their presumably general character.

Hüttermann (2005), on the other hand, responds to Kronz and Tiehen by characterizing emergence essentially in terms of an in principle failure to provide a micro-explanation for which an explanation of the behavior of the compound system in terms of the behavior of its constituent parts is still available. Conversely, a compound system’s behavior is emergent if, in principle, it is impossible to provide such an explanation. Diachronic micro-explanation gives an account of why a compound system is in a certain state at time t with respect to the earlier state of the compound system and the dynamical laws of the system, which in turn are comprised by the dynamics of the system’s parts. Similarly, a synchronic micro-explanation accounts for why a compound system is in a certain state at time t in terms of the states of the system’s constituents at time t. (116)

Most importantly, however, in the case of diachronic micro-explanations:

All the information that goes into the [diachronic] micro-explanation of the …[classical] system is information about laws—laws for the dynamics of the constituents, laws of composition, and, if necessary, laws of interaction. The states of the constituents play no role in the explanation of the dynamics of the compound system. Diachronic micro-explanation does not require the states of the constituents to be specifiable. (119)

In other words, a state of a compound system is diachronically micro-explained if (at least in principle) it is possible to deduce its temporal evolution on the basis of: a.) general laws concerning the temporal evolution (its dynamics) of the components considered in isolation, b.) general laws of combination, c.) general laws of interaction. (122)

Hence while admitting that entanglement does provide a case of synchronic emergence, i.e. the failure to give an account for why a compound system is in a certain state at time t, in terms of the states of the system’s constituents at time t,[19] Hüttermann nevertheless argues that entanglement still exhibits diachronic reduction:

[Q]uantum entanglement, i.e. the failure of synchronic micro-explanation, does nothing to undermine diachronic micro-explanations…the same sort of completely general micro-reductive strategy is available and employed as in classical mechanics. (123)

In other words, in the case of entanglement, as in the case of classical mechanics: a.) General Hamiltonians[20] are still built according to general laws dealing with the temporal evolution (dynamics) of the system, b.) General laws of composition are still employed, c.) General laws of interaction are still specified. “Quantum mechanical explanation of dynamics of compound quantum mechanical systems is just as reductionist as its classical counterpart.” (124)

The case of the three-body problem in classical mechanics, for example, satisfies all three, since: a.) In isolation, each one of the three particles’ trajectories evolve in 6-dimensional phase space according to Hamilton’s equations of motion. b.) The individual phase spaces of each particle combine according to a precise law: apply the direct sum of their phase spaces to produce a resultant 18-dimensional phase space, and the resultant Hamiltonian is formed by a sum of the individual kinetic energy terms and the interaction potential energy terms. c.) The laws of interaction among the three particles (force centers) are in turn described by their interaction potential energy terms. In this respect, according to Hüttermann, classical mechanics is the “paradigm of reduction,” since the theory is always capable of providing synchronic and diachronic micro-explanations for classical systems (even chaotic ones).[21]

Prima facie, Hüttermann’s points concerning entanglement seem intuitively plausible. As a toy example, consider the following model of a doublet state, described by a pair of two-dimensional quantum systems A, B with states spanned by bases: {| -ñ_A, |+ñ_A},{| -ñ_B, |+ñ_B}.[22] Suppose that the initial composite system is in the (separable) state:

|Y_AB(0)ñ = ¹/₂{| -ñ_A + |+ñ_A} Ä {| -ñ_B - |+ñ_B} ) (II.3.1)

= ¹/₂{| -ñ_A |-ñ_B - | -ñ_A|+ñ_B + | +ñ_A|-ñ_B - | +ñ_A|+ñ_B }

º ¹/₂{|–ñ - |-+ñ + |+-ñ - |++ñ}[23]

Consider the system Hamiltonian: H ={ |–ñá–| + |-+ñá+-| + |+-ñá-+| - |++ñá++|}. Then the time-evolution operator U(t, t₀ = 0) = exp(-i2pHt/h) after time t = ^h/₄ becomes:

U(^h/₄,0) = -i{ |–ñá–| + |-+ñá+-| + |+-ñá-+| - |++ñá++|}. (II.3.2)

The initially separable |Y_ABñ now evolves into the state:

|Y_AB(^h/₄)ñ = U(^h/₄,0)|Y_AB(0)ñ = -ⁱ/₂{|–ñ - |-+ñ + |+-ñ + |++ñ} (II.3.3)

It is simple to show that H is non-factorizable, and that (II.3.3) is likewise non-factorizable, and hence represents an entangled doublet state.[24]

Hence, the initially factorizable state (II.3.1) evolves according to the dynamical law |Y_AB(t)ñ º U(t,0)|Y_AB(0)ñ (satisfying condition a.) above). Moreover, the joint state and its associated joint Hamiltonian combine according to the law of combination determined by the tensor product Ä and its superposition (vector addition), satisfying conditions (b.) and (c.)

Hence, similar to the triplet state discussed by Kronz & Tiehen, in the above example an exact solution depicting the composite state of the system is specifiable for all times t, whether or not |Y_AB(t)ñ (linearly) evolves into an entangled state at certain specific times. Kronz and Tiehen however distinguish such special cases from entangled states evolving from multi-electron atoms, and applications in atomic physics and quantum chemistry involving systems (e.g. molecules) with non-separable Hamiltonians and a large number of degrees of freedom. In general, most models in quantum mechanics involving systems more complicated than the hydrogen atom either have finite degrees of freedom in which at least some are specified by vector spaces of infinite dimension, or possess an infinite number of degrees of freedom, where each may be finite or infinite dimensional. (393) None of these systems can be solved exactly, and hence a variety of approximate semi-classical boot-strapping methods must be employed:

[T]he Born-Oppenheimer approximation[25] [for instance] elucidates structure, but does not predict it. Moreover, it is a gross exaggeration to claim that molecular structure may be derived from first principles…no one has a clue how to use quantum mechanics to explain the different isomers…of C₆H₆. (343)

Hüttermann (2005), on the other hand, considers the issue of separability of the Hamiltonian for simple toy systems (as discussed above in the case of the doublet) versus the more complicated cases involving non-separable Hamiltonians as a difference “that makes no difference” to his argument:

[I]f there is a sense of emergence that develops from non-separable Hamiltonians, it does not affect my argument…[W]ith respect to the dynamics of compound systems, there is no analogous difference between classical mechanics and quantum mechanics. The presence of interaction terms leads to time evolutions of the parts that depend on the compound and thus implies what one may call ‘dynamic emergence’…[which] neither undermines the micro-explanation of the dynamics of compound systems, nor does it introduce a distinction that has quantum mechanics and classical mechanics on different sides with respect to the issue. (126-127)

In other words, the same framework applies, as specified in a.) – c.) above, i.e. application of general dynamical laws, combination, interaction. Being forced to employ approximation techniques in the case of quantum mechanics, (whether Born- Oppenheimer, Hartree Self Consistent Field, etc.)[26] is no different from the situation of applying approximation techniques to the three-body problem in classical mechanics. By this reasoning Hüttermann aims to show that quantum entanglement is not diachronically emergent.

I believe this to be a disadvantage for Hüttermann’s approach. Though he writes:

I take emergence to be an ontological notion, which concerns the relation between parts and wholes…meant to capture the intution that there might be some sense in which the behavior of a compound system is independent vis-à-vis the behavior of its parts.(115)

Nevertheless his treatment is by and large epistemic, insofar as couches the notion in terms of micro-explanation. Such a notion is too coarse-grained to distinguish what was discussed above in §III.1, concerning the distinction between ontological and epistemic senses. In this regard, the toy models involving doublet and triplet states exhibit ontological but not epistemic emergence. For the doublet, a simple and direct linear combination of the entangled state |Y_AB(^h/₄)ñ = -ⁱ/₂{|–ñ - |-+ñ + |+-ñ - |++ñ} is expressed in the composite basis for the two systems: {|–ñ, |-+ñ , |+-ñ , |++ñ}. An exact linear solution is the essence of epistemic reduction. The ontological emergence, on the other hand, is clearly represented by virtue of the non-factorizibility of |Y_AB(^h/₄)ñ into product states spanned by the individual bases:{| -ñ_A, |+ñ_A},{| -ñ_B, |+ñ_B}. On the other hand, the more complicated quantum mechanical models involving non-separable Hamiltonians, to the extent that they are in entangled states as well, exhibit both ontological as well as epistemic emergence, and hence are of a different kind than classical systems (like a chaotic system, the three body problem, etc.) which exhibit epistemic emergence only.

III. Disunity in Contemporary Philosophy of Science: Some Brief Remarks

I have reviewed in the above section what I consider are some major trends in the analysis of emergence within contemporary philosophy of physics. All trends seem to exhibit a Janus-faced character, in their nuanced and detailed approaches. No doubt, all studies I briefly surveyed above seem to exhibit in varying degrees what Mathias Frisch (2005) would describe as his “principle of charity:”[27]

As philosophers we might be tempted to think that physicists are simply confused when they speak of an appropriate equation as ‘fundamental,’ ‘correct,’ or even ‘exact.’ This, however would mean imposing a philosopher’s rigid conception of theories on science rather than trying to understand the practice of theorizing…we should [examine]…which sets of equations physicists themselves take to be the most basic and important in a certain domain, and then ask what criteria of theory-choice would allow us to make sense of the physicists’ decisions…[W]e should adopt a principle of charity and interpret the physicists’ claims in a way that makes them defensible…[for instance, a theory’s] internal consistency does not come out as a necessary condition governing theory choice, since considerations of simplicity, mathematical tractability, and conceptual fit appear to be able to override concerns for strict logical consistency. (italics added, 70-72)

On the other hand, however, the above approaches concerning the analysis of emergence seem lacking precisely because of their somewhat selective methodology concerning the level of quality and the quantity of detail they focus on. In my opinion, this seems emblematic of a general trend of excessive particularism and pluralism characterizing much contemporary philosophy of physics.

For instance, inspired primarily by his study of emergent phenomena, Robert Batterman (2002) argues that reduction and explanation should be treated as separate epistemic modes, and argues that a species of ‘asymptotic explanations’ indicate that the superseded (or reduced) theory T still somehow plays a necessary role vis-à-vis the superseding (reducing) theory T’. In other words, in explanations involving critical behavior, the ‘old’ theory T doesn’t get completely reduced by the newly superseding theory T ’, but continues to play an essential role.[28]

In subsequent work (2004, 2005) Batterman further sunders notions of ‘fundamental’ by arguing that ontologically versus epistemically fundamental theories act at cross-purposes: the former seek to give a metaphysically accurate account of the phenomena at the expense of explanatory efficacy, while the latter do exactly the opposite. For example, in the case of fluid droplet formation, one may appeal to the ontologically approximate Navier-Stokes theory which models the fluid as a continuum, to account for the universally regular features of droplet formation shared by all classes of fluids of varying density. The Navier-Stokes theory, in short, is epistemically fundamental: It is able to provide a universal account of scale-invariant features of certain critical phenomena only by hiding the underlying ontology. The fluid basically consists of a discrete collection of molecules. Any ontologically fundamental theory modeling the fluid from this accurate level of description, aside from becoming computationally intractable, would by its very nature of describing the particular ontological details, sacrifice the very possibility to provide universal or scale-invariant descriptions of droplet-formation. Conversely, the epistemically fundamental theory is able to capture universal features so well precisely because of its approximate representation of the fluid as a continuous medium.

I have critiqued the distinctions Robert Batterman (2002, 2004) poses (Kallfelz 2005, 2006) in a manner such that while I share sympathies for Frisch’s ‘principle of charity’, I nevertheless question some of the ramifications of Batterman’s conclusions. Similar to Hoefer’s (2002) and Sklar’s (2003) critiques of Cartwright (1999), I argue that that the activity of theorizing might be more systematically interconnected in a manner that calls into question Batterman’s distinctions. In (Kallfelz 2005b) I argue that Batterman (2002) often subtly equivocates a physical theory’s ontology with its so-called ‘topology,’ or structure of its underlying mathematical formalism. Carefully distinguishing the two, and employing geometric (or Clifford) algebraic methods in a theory’s topology together imply that reduction and explanation may not be so different after all.[29] I refine this point (Kallfelz 2006) by pointing out that Clifford algebraic methods are an active area of research in computational fluid dynamics, and seem to work well for modeling the behavior of droplet-formation in such a manner as to instantiate a ‘methodologically fundamental’[30] approach. I argue that methodologically fundamental procedures subsume the ontologically/epistemically fundamental categories Batterman (2004) presents.

IV. An Engagement of Emergence and Aspects of Process Thought

In this concluding section I offer suggestions in which aspects of process thought can offer a more comprehensive and systematic framework in characterizing some of the recent studies of emergence, as briefly mentioned above. In the most general sense, echoing what was discussed in Finkelstein & Kallfelz (1997), it is certainly safe to assume that the notion, as discussed in some of the aforementioned senses (ontological, epistemic, dynamic) functions more as an explanans (i.e. what does the explaining) as opposed to an explanandum (i.e. what must be explained) in the context of Whitehead’s metaphysical immaterial indeterminism. I will also briefly mention some of the recent progress made by Finkelstein’s research group (2001, 2004a-c) to the extent that it relates to some of the issues in Finkelstein & Kallfelz (1997) vis-à-vis emergence.

In our (1997) paper we concluded:

[W]e believe that something like a cellular organism functions at a still deeper level, and we have expressed this belief in our Quantum Network Dynamics. Whitehead’s prophetic conceptual structure and the more experimentally founded and mathematically structured developments in physics will continue to illuminate each other for some time to come. (291)

Indeed, with respect to mathematical structure, Baugh, et. al. (2003), Finkelstein (2001, 2004a-c) present a unification of field theories (quantum and classical) and space-time theory based fundamentally on finite dimensional Clifford algebraic structures. The regularization procedure fundamentally involves group-theoretic simplification. The choice of the Clifford algebra is motivated by two fundamental reasons:

1. The typically abstract (adjoint-based) algebraic characterizations of quantum dynamics (whether C*, Heisenberg, etc.) just represent how actions can be combined (in series, parallel, or reversed) but omit space-time fine structure.[31] On the other hand, a Clifford algebra can express a quantum space-time.

2. Clifford statistics[32] adequately expresses the distinguishability of events as well as the existence of half-integer spin. (2001, 7)

In 1997 the speculation was to relate particle structure to spacetime structure through an action-based formulation of quantum physics, “in which a hierarchic Grassmann algebra replaces set theory.” (290) Clifford algebra, however, proves itself an even more versatile formulation, based on its regularizability and precise geometric correspondence to all its significant algebraic operations such as addition, external grade raising product, inner grade lowering product, in a manner superseding that of Grassman algebra. (Finkelstein, et. al. 2001, Hestenes 1984, 1986, Lasenby, et. al. 2000)

The recent results of Finkelstein (2001, 2004) indicate that the prime variable is not the space-time field, as Einstein stipulated, but rather the dynamical law. That is to say, “the dynamical law [is] the only dependent variable, on which all others depend.” (2001, 6) This marks a greater refinement and development of the quantum network dynamics’ action principle delimiting forbidden versus allowed processes, discussed in 1997 and associated with Whithehead’s notion of “aim and creative development.” (284) Moreover, in Finkelstein (2001, 2004a-c) the “atomic” quantum dynamical unit (represented by a generator of a Clifford algebra) is the chronon c, with a closest classical analogue being the tangent or cotangent vector, (forming an 8-dimensional manifold) and not the space-time point (forming a 4-dimensional manifold). The chronons, or elementary quantum units of process further refine our 1997 remarks: “Each concresence has an ordering (Process and Reality, IV ch 1) and this ordering constitutes time and process, which are effectively co-extensive for Whitehead.” (284)

In particular, since a natural factoring exists according to the ‘Octad Lemma’ in Clifford statistics gives rise to the chronons (details are summarized in Kallfelz 2006, 12-15 ), the chronon likewise proves a suggestive refinement of Whitehead’s notion that certain “analyses of concrescences yield more ‘concrete’ occasions than others.” (286) The mathematical details summarized in Kallfelz (2006) likewise present a more precise characterization (compared to our 1997 article) of how a spacetime structure emerges from a discrete system of elementary quantum process characterized in this case by Clifford statistics. This further complements Whitehead’s intuitions which in places “seem to be transcending classical thought in order to understand how a continuous topology can arise with discrete tenants.” (288)

The recent advances of Finkelstein vis-à-vis Whitehead’s cellular cosmology provide a systematic framework in which to characterize in very general terms the notions of epistemic, ontological, and dynamical emergence. Clifford algebras are graded structures, with the maximal grade of 2^N, where N is the dimensionality of its algebra’s underlying vector space. Likewise, Whitehead suggests a cellular hierarchy of societies of occasions constituting nature, no doubt influenced by his work in set theory. (Finkelstein & Kallfelz, 1997, 291) The Clifford graded structure and Whitehead’s cellular hierarchy set the general ontological conditions for the possibility of Kronz & Tiehen’s refinement of Humphrey’s levels, as discussed in §III.2 above. Moreover, Whitehead’s Ontological Principle (that the being of objects is constituted by their becoming) and Relativity Principle (the Universe is a sum of actual occasions) lend the appropriately fundamental dynamical element of emergence that Kronz & Tiehen advocate. Last of all, the notion that every actual occasion contains a mental and physical aspect or pole, with differing relative importance, depending on what level of aggregation the society of occasions is based on (Whitehead 1929/1978, 239), presents a picture of ontological and epistemic emergence as dual or complementary aspects, as opposed to qualitatively disjunct metaphysical categories.

Acknowledgement: I thank Timothy Eastman for inviting me to contribute this paper

V. References:

Ayer, A. J. 1956/1998. What is a law of nature? Revue Internationale de Philosophie 36: 144-165. (Reprinted In Eds. Curd M. and Cover J. A. 1998. Philosophy of science: The central issues. New York: W. W. Norton & Co 808-825.)

Batterman, R. 2002. The devil in the details: Asymptotic reasoning in explanation, reduction and emergence. New York: Oxford University Press.

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