Download here: Ultraquantum Relativity

Comments welcome.

————–

Dear Michael,

An update. The path is getting clearer, so I am traveling faster; but it can still peter out.

What on earth did Peter do to deserve this verb?

I could not resist coining merology to indicate indebtedness to and departure from the mereology of Lukasiewicz, and to shed his otiose -o-.

Do pass it on.

Best,

David

Comments welcome–

Please do not circulate outside group (until published, Spr 2009)

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):

[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):

[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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In his latest post, Henry writes:

“Mike’s position has been that W’s full scheme does entail an absolute serial order. My position is that W’s words do not entail such an absolute serial order of the coming into being of the occasions (or of their initiation and termination “dates”), and implicitly that Nobo’s supersessional ordering involves adding essential content to what W said (at least in PR). Whitehead’s words permit, I believe, a “relativistic” notion of coming into beingness in which the ordering of the parts of occasions whose standpoints are wholly spacelike is not ontologically or metaphysically or logically significant.”

I believe this to be an accurate summary of our respective views.  More accurately, my position is that Whitehead’s metaphysics and cosmology do imply a fundamental serial ordering of concrescences, as described above–but not one that is spatiotemporal.  The serial ordering is strictly mereological-logical and described fully in Whitehead’s Theory of Extension (Part IV of Process and Reality).

NB: This post has been converted into a paper pending publication in Process Studies, and in the forthcoming book Whiteheadian Physics, eds. David Ray Griffin, Michael Epperson & Tim Eastman.  The pre-publication version can be read here:

Download PDF - v8.30-Logical Causality in Whitehead’s Theory of Extension

The issue here is whether Whitehead’s theory of nature, particularly as described in Process and Reality (PR), entails the existence of a universal/absolute/objective linear ordering in which all occasions (or at least their initiation and termination “dates”) can be placed. Nobo’s theory contains such an ordering (supersessional time), and one question is whether the existence of such an ordering is entailed by Whitehead’s actual words.

This issue has been the subject of a discussion involving Mike Epperson, Jorge Nobo, and myself. These exchanges have served to define the issues, and our respective understanding of W’s scheme. Mike’s position has been that W’s full scheme does entail an absolute serial order. My position is that W’s words do not entail such an absolute serial order of the coming into being of the occasions (or of their initiation and termination “dates”), and implicity that Nobo’s supersessional ordering involves adding essential content to what W said (at least in PR). Whitehead’s words permit, I believe, a “relativistic” notion of coming into beingness in which the ordering of the parts of occasions whose standpoints are wholly spacelike is not ontologically or metaphysically or logically significant.

This present message takes of from the communication of Nobe to Epperson of July 31 (2008), in which Nobe says to Mike: “After skimming your exchanges with Henry, I am beginning to think that Henry is right and Whitehead cannot provide for an absolute order of occasions. This makes my differences with Whitehead all the more important.”

To simplify the discussion, I am considering a special case of our own epoch in which each actual occasion has a “standpoint” in a 4-d spacetime continuum.

Whitehead (W) describes the process of creative advance in terms of the coming into being of actual occasions, each of which has its own actual word (its causal past). The full set of eventually completed actual occasions are classified, relative to an occasion X, as belonging to X’s causal past, its causal future, or its causal elsewhere, the later set of occasions being its “contemporaries”.

It is tempting at this point to identify an occasion’s past, future, and contemporary occasions by means of the relative locations of their standpoints (or some particular “physical point of causation” within each standpoint) with respect to lightcone conditions. I do not think what Whitehead’s word definitely preclude this possibility. The way he says things seem to me to be constructed so as to not rule out this possibility. If this possibility is not definitely ruled out by W’s actual words, then if one is proposing a conception that differs from this simple “relativistic” construal, one must say how it differs from this relativistic construal. Doing so will add to what is implied by W’s own words themselves.

W says (PR 21) “The many become one and are increased by one”. The context in which these words appear seem clearly to be talking about the coming into being of an occasion, in relationship to the many occasions in its own causal past. This is compatible with a “relativistic” construal in which there is no absolute serial ordering of the ‘dates” of contemporary occasions with standpoints that are completely spacelike separated. The assertion (PR 35) “In these lectures the term ‘creative advance’ is not to be construed in the sense of a uniquely serial advance.” reinforces to idea that W intended to accommodate the “relativistic” possibility that the order of coming into being of occasions whose standpoints are completely space-like-separated is not specified. This intent seems so clear and unequivocal at this point (PR 35) that one is justified, in cases where possible ambiguities may exist, in interpreting all other statements to be compatible with it. Later (PR 65) W says: “Curiously enough, even at the early stage of metaphysical discussion the influence of the ‘relativity theory’ of modern physics is important. According to the classical ‘uniquely serial’ view of time ” W is here reinforcing the idea that he intends to accommodate a lack of any absolute serial order for the coming into being of contemporaries.

The faster-than-light aspects of nature entailed by quantum theory came to be widely appreciated only in the 1960’s, long after PR was written. Nobo is attempting to accommodate this new development, which contravenes certain ideas of relativity theory. To do so, Nobo (or anyone else who wants to accommodate violations of normal relativity) must add ideas that go beyond Whitehead’s apparent intentions to accommodate “relativity”, in the sense not ascribing to contemporaries any absolute serial order of coming into being.

The way that “relativity”and faster-than-light (FTL) transfer of information are consistently accommodated in Relativistic Quantum Field Theory (RQFT) instructive.

The quantum state of the universe—which one can conceptualize as a collection of (very large dimensional square) matrices, one for each tiny cell in a cellularization of the 3-d spatial continuum, can be thought of as one giant matrix of an infinite number of dimentions. It evolves via a repetitive sequence of a sequence of three different processes.

Process One divides the cellularized 3-d space into a set of non overlapping regions, each containing a set of cells. The collection of matrices associated with the cells in each region can be represented as a giant matrix. A local process associated with each region “poses a Yes/No question”. This action is represented by an operation that reorganizes this matrix and divides the rows (and, correspondingly, the columns) into two disjoint part, labeled “Yes” and “No” respectively. It is important that this Process One leaves unchanged the “Traces” of all matrices, where a Trace of a matrix M is the sum of the “diagonal” elements of M. Process Two can be visualized as a local deterministic process that advances a spacelike surface “Now” , divided unto regions by Process One, a short distance into the future, but keeping all boundaries of regions fixed (in space-time). Process Three then collapses the state associated with the new temporally advances spacelike surface by choosing for each region either the Yes/Yes or the No/No part of the associated matrix. This Process Three is supposed to conform “statistically” to rules specified by quantum theory. This is what gives QM its predictive power.

Because of the Trace condition mentioned above no prediction in one region can depend directly on which questions were asked in space-like-separated regions: there is no possibility of sending a message chosen by an agent in one region to a witness in a space-like-separated region. This is in accord with the classical idea of “relativity”.

However, the way in which Process Three choices of outcomes made in two space-like-separated regions are correlated entail that the Process Three must in some cases be able to access the information inputted in spacelike separated regions. The Process Three acts like a global witness/agent in which witnesses and acts over the space-like “Now” described earlier. Process one acts like a set of independent question posers, but Process Three BEHAVES like a global answerer. The Process Three response, as represented by the behavior of quantum state of the universe, is the collapse behavior appropriate for a representation of the collective state of knowledge of all the local probers. Heisenberg calls it “our knowledge”.

“The many become one, and are increased by one” acquires new meaning with respect to the global process. At each stage there is “One” new global state at a new global space-like surface “Now”, which is then decomposed into many specific possibilities by the actions of the many local question-posers, who act independently on the basis of local conditions. These “Many” are then converted by The Process into a new “One”.

This way of combining QM with W is a variation of the one given in Chapter 13 of Mindful Universe. There each step was connected to a local advance of the surface “Now”: here each such advance is global, but with local inputs from local IGUSes.

(cf. Gell-Mann)

Stapp’s Comments of 7/30/08 on the first part of Epperson’s:

Whitehead’s Theory of Extension as it Pertains to Henry’s Question: Part 2

Dear Mike,

Many thanks for sending your more concise attachment, which focuses on the particular passages in PR that you believe lead to the logical conclusion that W’s scheme, as described in PR, entails a single universal time in which the coming into being of the ALL occasions can be plotted.

MGE: Well, nothing in my attached document said anything about time ordering. The ordering he talks about in his Theory of Extension is purely mereological. Again, the thesis in W’s Theory of Extension, which I attempted to summarize in the attached document, is that Spatiotemporal ordering is a ‘more specialized’ form of ordering embedded in a more fundamental, purely mereological, serially ordered, inclusively related regions—regions that are NOT spatiotemporal! These regions are, for Whitehead, purely first-order regions describing the internal relatedness of actualities. They are, in other words, metaphysical regions, not physical regions. They are denumerable. They are serially ordered.

“It is to be noticed that each abstractive set is to be conceived with its members in serial order, determined by the relation of inclusion.

[i.e., 'The many become one and are increased by one.']

The series starts with a region of any size, and converges indefinitely towards smaller and smaller regions, without any limiting region…”

An abstractive set is one where “(i) any two members of the set are such that one of them includes the other non-tangentially and (ii) there is no region included in every member of the set.”

Perhaps we can look at it another way. A more fundamental ordering such as the mereological scheme of extensive abstraction proposed by Whitehead in Part IV may not be wholly revealed in some more specialized ordering, such as spatiotemporal ordering.

HPS: My understanding of Part IV is that it was W’s attempt to deduce Geometric properties from logical properties. In our epoch physics is imbedded in a 4-dimensional space time filled with straight lines. W makes it clear at the outset of Part IV that he finds the usual approaches to the introduction of straight lines Into the spacetime of physics unsatisfactory. Part IV is his attempt to get this undergirding of physics from what he regards as more satisfactory assumptions. These assumptions are weaker than just a blatant assumption of a metric spacetime continuum. He uses weaker topological assumptions, plus an assumption of the existence of a class of ovate regions. From these weaker assumptions he is able to get the straight lines that in our Epoch undergird physics. Thus his work in Part IV when applied to our own special case gives a geometric structure to the 4-d spacetime continuum that underlies our lives, and in which each occasion has its standpoint. The construction of these straight line depends upon the nesting property that you emphasize above. It is part of the construction of the geometry, which in our own special case is exactly the 4-d spacetime continuum of present day physics. There is not, in our own world, also some other pertinent geometry, or at least there need not be any other. The most direct interpretation of W is that the geometry that he erects in Part IV is, for our lives, exactly to spacetime upon which our physics is based.

“The many become one and are increased by one” in context (PR 21) refers to the process of the coming into being of a single actual occasion. There is no presumption that it entails in any way an absolute serial ordering of contemporary occasions. Indeed, he endorses “relativity” which essentially denies such an absolute serial ordering for contemporaries. The central thrust of his whole work is to provide an ontological foundation that explains “creative advance” and reconciles it with the existence of contemporaries in conformity with the relativistic notion that contemporaries do not come into being in a serially ordered manner..

.

MGE: The most fundamental extensive order for Whitehead, as he lays out in his theory of extensive abstraction, is DENUMERABLE. Any more specialized super-denumerable extensiveness, such as relativistic space-time extensiveness, must be shown to derive from that more fundamental denumerable mereological order. I’m not saying his argument is flawless; in fact, it’s a bit counterintuitive. Normally, one considers the set Q of rational numbers (denumerable) to be a dense subset of the set R of all real numbers, which is super-denumerable.

HPS:This nestedness pertains, I believe, in our universe, to regions of spacetime: it pertains to nested open sets in spacetime. It pertains to geometry, not to some serial order of coming into beingness. I believe, at present, that your construal of Part IV is a misapplication of the content and intent of that part. In short, I see Part IV as W’s attempt to deduce an aspect of the geometric structure of the spacetime in which we live from essentially logical premises, including his essential-for-this-purpose assumption of the existence of a preferred nested set of ovate regions.

PR 35: “In these lectures the term ‘creative advance’ is not to be regarded in the sense of a uniquely serial advance.”

PR 65 “Curiously enough, even at this early stage of metaphysical discussion the influence of the ‘relativity theory’ of modern physics is important. According to the ‘uniquely serial” view of time two contemporary actual entities define the same actual world. According to the modern view, no two actual entities define the same actual world. Actual entities are called ‘contemporary’ when neither belongs to the actual world of the other.”

As regards the meaning of ” The many become one, and are increased by one.” the context (PR 21) makes clear, I think, that this refers to the coming into being of a single occasion: the many in its actual world become integrated into itself, which actual world now includes also itself. Its contemporaries are not incorporated. Indeed, the description of the incorporation of a created IMAGE of a present world now (PR 121-127) makes it clear, I think, that the creation of this image is rather analogous to the creation in our own consciousness, on the basis of delayed data arriving from our receptors, and of past experiences, of an ‘imagined best guess’ for a ‘contemporary now’ that we cannot yet know or access. I do not see in this account by W  any suggestion of any serial order of coming into being of contemporary occasions. Any such suggestion must, I think, be regarded as going what W’s words entail or suggest.

All the best,

Henry

Books

Bedau, Mark and Paul Humphreys, eds., 2008. Emergence: Contemporary Readings in Philosophy and Science. Cambridge: MIT Press.

Bub, Jeffrey. 1997. Interpreting the Quantum World. Cambridge: Cambridge University Press.

d’Espagnat, Bernard, 2006. On Physics and Philosophy. Princeton: Princeton University Press.

Flanagan, Owen, 2007. The Really Hard Problem: Meaning in a Material World. Cambridge: MIT Press.

Green, H. S., 2000. Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process. Berlin: Springer-Verlag.

Griffin, David, 1998. Unsnarling the World-Know: Consciousness, Freedom, and the Mind-Body Problem. Berkeley: UC Press.

Griffin, David, 2007. Whitehead’s Radically Different Postmodern Philosophy: An Argument for its Contemporary Relevance. Albany: SUNY Press.

Kauffman, Stuart, 2008. Reinventing the Sacred: A New view of Science, Reason, and Religion. NY: Basic Books. [see also Kauffman's earlier works: Investigations, At Home in the Universe, The Origins of Order]

Laughlin, Robert, 2005. A Different Universe: Reinventing Physics from the Bottom Down. NY: Basic Books.

Pred, Ralph, 2005. Onflow: Dynamics of Consciousness and Experience. Cambridge: MIT Press.

Rescher, Nicholas, 2000. Process Philosophy: A Survey of Basic Issues. Pittsburgh: University of Pittsburgh Press [see also Rescher's Process Metaphysics, SUNY, 1996].

Rosenblum, Bruce and Fred Kuttner, 2006. Quantum Enigma: Physics Encounters Consciousness. Oxford: Oxfor University Press.

Teller, Paul, 1995. An Interpretive Introduction to Quantum Field Theory. Princeton: Princeton University Press.

Walleczek, Jan, ed., 2000. Self-Organized Biological Dynamics & Nonlinear Control: Toward Understanding Complexity, Chaos and Emergent Function in Living Systems, Cambridge: Cambridge University Press [includes papers by Paul Gailey and Jan Walleczek].

General Resources
Resource Guide for Physics and Whitehead, Eastman and Keeton, eds. (PS Supplements, 2004, Issue 6) available online at http://www.ctr4process.org/publications/ProcessStudies/PSS/
(includes many relevant references up to 2004).

Hi everyone–

In his 7/28 post, Henry makes a great point here re: my last post, and it brought up what might be an important question:  Could PNC taken as an a priori metaphysical first principle (implied by Whitehead’s desideratum of logical coherence), writ large and within the context of the metaphysical system given in Process and Reality, yield a well-ordered series of occasions in a scheme where:

1. The physical-causal efficacy of every actual occasion is restricted to its forward light cone.

and

2. Each and every becoming actual occasion is logically, internally related to each and every being.

I believe that Part III and IV in PR together posit both of these, respectively. With respect to 2 and its relationship to extensiveness, for example, Part IV.II.II Assumption 2 (PR 295) states that any two regions A and B are mediately connected such that both A and B are connected with some region C. (Fig. iii, p.296) The question is, if we bring 1 and 2 together, with added attention to governance of internal relations by PNC, will that yield an interpretation of “The many become one and are increased by one” that is free of the problem Henry raises?

Here is our exchange:

——–

Dear Mike,

Regarding Continua, absolute order, and causal objectification: Part III

I have doubts about your concept:

“the order of occasions ABCDE in I’s actual world”

Because we are interested here in the metaphysical structure, not the spacetime structure of our epoch, we should, I believe, consider your diagram as representing the past, future, and contemporary structure at the *metaphysical* level.

At this metaphysical level we do not want to start with an absolute ordering of occasions, as that would beg the question.

MGE: Yes, it is intended to represent Whitehead’s metaphysical conceptions of objectification (given in Part III) combined with his metaphysical conception of extensiveness (given in Part IV). Indeed, these conceptions contain a number of explicit presuppositions (question begging). The Ontological Principle and the Principle of Relativity are 2 such presupposed principles explored in Part III; and the axioms of his Theory of Extension in Part IV are similarly presupposed. Insofar as Whitehead’s work, and ours, are examples of ’speculative philosophy,’ I have always operated under the impression that starting with presupposed principles or axioms or desiderata is fine for us, as it was for Whitehead, so long as these are empirically applicable, adequate, and logically coherent.

In any case, the purpose of my posts was the same as Jorge’s–simply to demonstrate that the kind of objective ordering we’re talking about “follows from the logic of the theory of causal objectification, which is a wholly metaphysical theory.”

I agree with Jorge’s assessment here, and my posts were intended to substantiate this agreement–not to offer a metaphysical proof that there is an objective ordering among all actual occasions. More specifically, as discussed in my posts, I believe that this ordering follows when the theory of prehension/objectification presented in Part III is combined with the theory of extension in Part IV. Whitehead says that the concepts of extensive connection / ‘extensive abstraction’ are to be read as part of his discussion of the order of nature–i.e., the order of actual occasions. (PR 96-7) And in Part IV he defines this ‘order’ as a mathematical/logical/mereotopological order–as ordered sets with hierarchical relations. That is what I attempted to show, in shorthand, in the graphic.

The question I would like to pursue with the physicists in the group is whether or not Whitehead’s Theory of Prehensions/Objectification together with and in the context of his Theory of Extension, and all the metaphysical presuppositions contained therein, are empirically applicable in the light of modern physics. Both of these theories contain many metaphysical presuppositions / first principles / categoreal obligations, etc–some explicit, and some implicit. I want to explore their scientific applicability and adequacy.

So in my interpretation of your diagram, one is supposed to consider it to be a representation at the metaphysical level of a logically possible set of metaphysically possible causal connections, where the absolute time ordering is not significant —that would beg the question being examined here.

Instead, we are to imagine that the metaphysical past-future-contemporary causal relationships are such that they conformto the usual light-cone conditions.

MGE: Yes, this is the kind of test for empirical adequacy that very much interests me.

Any Lorentz transformation will leave all light-cones unaffected.

But some Lorentz transformations will change the relative time ordering CDE to ECD. So the ordering ABCDE that you cite is, in the context that is pertinent here, is a feature of the diagram that will not be captured in I or J, because it is a purely conventional ordering that is not part of the intrinsic causal structures captured, within this logically conceivable model of the metaphysical connections, by light-cone relationships.

MGE: To say that neither CDE or ECD is ‘captured’ in I or J would be problematic for Whitehead’s model, so I want to make sure I understand what you mean. The causal order of CD is captured. But the relationship of E to CD is not captured, correct? This makes it difficult to correlate ‘capture’ with Whitehead’s term ‘objectify.’ According to the metaphysics, I objectifies E *with* C and D *in relation*–integrating them and recreating them as data for subsequent subject-superjects. H does the same with respect to E, F, and G. And J does the same with I and H. Referring to the PR 65 quotation in my post, J’s potential integrations of I and H are partially reproductive (of the ‘given’ actual world in the back-facing lightcone) and partially creative (the data for creativeness being the ‘real’ potentialities for novel integrations–data that are ‘beyond’ that ‘given’ standpoint bounded by the lightcone)–i.e., Lorentz transformations giving either CDE or ECD. That kind of ‘extensive’ creativeness is, as I understand it, akin to the ‘intensive’ creativeness similarly enjoyed by a becoming occasion. (The former pertains to the coordinate division of satisfaction, the latter pertains to its genetic division.) The only a priori principle governing this creativeness is Whitehead’s desideratum of logical coherence–namely, that no integration/reproduction of data during concrescence will violate PNC. This is the key desideratum in his metaphysics (in my opinion) because it keeps the actual world unified. The only alternative would be a ‘many universes’ model where PNC can be safely violated.

Either way, the physics (and the metaphysics) requires a presupposed first principle: One where PNC is never violated (Whitehead); or one where PNC can be safely violated because of presupposed parallel universes. I believe the former to be the more economical choice.

Perhaps the question at hand is really: To what extent does PNC in Whitehead’s metaphysics yield a single, objectively ordered continuum of settled occasions? Perhaps in my graphic, PNC doesn’t adequately govern the choice between CDE and ECD for J. But might the reason be that there aren’t enough data represented in the graphic? If there were a trillion-trillion occasions added, wouldn’t there then be sufficient logical stricture via global satisfaction of PNC to force I to capture *either* CDE *or* ECD? (Akin to the way coarse-graining gets rid of superpositions in the decoherence approach to QM)? Would that be a viable approach to the problem?

Thus I do not see that the partial ordering implicit in the *metaphysical* concept of Past-Future-Contemporary is converted to a well orderedness by the fact that past occasions, and their connections, are recreated in later occasions. In this counter-example the light-cone representation of the causal relationships are re-presented of later occasions without their defining the relative order of (even) the beginnings and endings of the standpoints of metaphysically contemporaneous occasions.

W’s claim that “The many become one, and are increased by one”

is, it seems to me, hard to reconcile with any idea other than a serial single-file ordering of the complete occasions. The “many” that become X is only the set of occasions in occasion X’s causal past, But the occasions in X’s causal past is not increased by one by the completion of X.

And if the statement refers to some totality of completed (fixed and settled) entities, then by the time that X is completed, many occasions (some of X’s contemporaries) besides the occasions recreated by/in X should have been added to the totality.

MGE: Yes, I believe he meant the latter–that the ‘many’ refers to the totality of occasions. And you have definitely hit on the problem! The question for me is whether or not PNC taken as an a priori metaphysical first principle, writ large, will yield a well-ordered series of occasions in a scheme where:

1. The physical-causal efficacy of every actual occasion is restricted to its forward light cone.

and

2. Each and every becoming actual occasion is logically, internally related to each and every being.

I believe that Part III and IV in PR together posit both of these, respectively. With respect to 2 and its relationship to extensiveness, for example, Part IV.II.II Assumption 2 (PR 295) states that any two regions A and B are mediately connected such that both A and B are connected with some region C. (Fig. iii, p.296) The question is, if we bring 1 and 2 together, with added attention to governance of internal relations by PNC, will that yield an interpretation of “The many become one and are increased by one” that is free of the problem you raise?

So I wonder whether this claim by W, though very neat and beautiful, is rationally compatible with the rest of his claims.

Henry

One and all:

I am attaching a file with selected passages from my Whitehead’s Metaphysics of Extension and Solidarity. I will add a second file in a few days with passages having more to do with the theory of metaphysical extension. I hope these passages will make my meaning clearer and enable the discussion of supersession and absolute order of initiation and termination dates significantly to advance beyond its present point.

I apoligize for the length of the file, but the book is over 400 pages long and I am shooting for quoting no more than 10 % of it.

Best to all,

Jorge

…There is an objective well-ordered series of alpha and omega dates that can be read from the well-ordered causal objectifications found within the regional standpoint of every occasion. That follows from the logic of the theory of causal objectification, which is wholly a metaphysical theory. Therefore, the relations it attributes to occasions cannot depend on the contingent features of our cosmic epoch [see my previous message]. They would obtain even in epochs in which phrases like space-like and time-like separations are meaningless.

Such an order, in other words, is necessarily implied by Whitehead’s theory of causal objectification. The latter simply doesn’t work without such an underlying objective ordering of occasions.  It would lose all coherence as a theory given that its signature feature is that every becoming is internally related to all beings antecedent to it.

The question at hand is whether Whitehead’s use of the word ‘antecedent’ here was meant to be understood as strictly:

1. temporal (and therefore relativistic), such that all ‘antecedent’ beings are those restricted to the becoming’s backward light cone, and therefore no invariant ordering among spacelike separated occasions is implied;

or

2. logical (and therefore objective), such that all ‘antecedent’ beings are those defined according to Whitehead’s self-described ‘first order,’ invariant mereotopological order, where every becoming occasion is a ‘whole’ internally and genetically related to its antecedent dative ‘parts.’  It is only in this way that Whitehead’s phrasing,  “The many become one and are increased by one” maintains coherent meaning.

As I have argued before, attempting to cast this distinction as an ‘either-or’ proposition is a misstep in the context of Whiteheadian metaphysics and cosmology for two reasons:

First, a becoming occasion is internally related to past occasions–i.e., that relations of past occasions are reproduced within the constitution of future occasions.  “The many become one and are increased by one.” (PR 21). By Whitehead’s theory of prehensions and his theory of objectification, this process of ‘becoming one’ involves a vector-genetic inheritance and logically governed integration of the antecedent dative world within the concrescing actual occasion.  Dative causes are not merely ‘represented’ in the new occasion; they are internally reproduced and recreated.  “The actual world is the ‘objective content’ of each new creation.” (PR 65) Each concrescing occasion “is the cumulation of the universe and not a stage-play about it.” (PR  237)

Given that every becoming occasion is internally related to every occasion antecedent to it, it simply doesn’t matter whether the word ‘antecedence’ here is restricted to the backward lightcone.  Consider the following graphic (click to enlarge):

Occasion E objectifies the data A and B, as these lie within its past light cone. Relative to Occasion E, the orders CD and FG are spacelike separated from E–that is, contemporaneous with E and therefore not causlly efficacious within E. (This is per the demands of both relativity theory and Whiteheadian metaphysics.)

Occasion H includes within its data F and G, but also E; therefore H mediately objectifies A and B via E.  In this way, H is internally related to all the data of its actual world.  Similarly, Occasion I objecifies C and D, but also E–and via E, A and B.  Also, I and H are spacelike separated from one another and therefore causal contemporaries.

Occasion J objectifies all these data; and by the internality of its relations with these data, it also objectifies their ordering within their own individual histories.  Again, this is because J’s ‘internal relationship’ with its data means that within J, both the order of occasions ABCDE in I’s actual world, and ABEFG in H’s actual world will be included and logically integrated in J.  The explicit desideratum of logical coherence in Whiteheadian philosophy as a first principle of his metaphysics guarantees that within J’s integration of H and I, the principle of non-contradiction (PNC) will not be violated.  This means that the integrations (and reproductions) of H’s actual world and I’s actual world within J’s actual world must entail relations of mutual and non-contradictory ordering among the occasions constituting the actual worlds of E, H, I, and J.

Whiteheadian cosmology, in other words, is not of the ‘many universes’ genre; if Occasion E is associated with the radioactive decay that triggers the death of Schroedinger’s Cat associated with Occasion F, then the cat is dead for both Occasion H and I–even though H and I are contemporaries.  It’s also dead for Occasion J, which subsumes H and I; and if that is the case, there must be an ordering of occasions comon to all occasions by virtue of the fact that each future occasion is internally related to every occasion antecent to it. Thus the past world in Whiteheadian philosophy maintains its character as a unified, objective actual world–not a confused and logically incoherent patchwork of mutually exclusive possible subjective worlds.  This is, as Jorge points out, simply a necessary implication of Whitehead’s theory of prehensions and objectification, without which his metaphysics loses all coherence.

Whitehead’s awareness of this is evinced by his devotion of Part IV of Process and Reality to an accounting of how PNC is preserved in this way–i.e., within J’s internal relationship to ABCDEFGHI described above.  The internality of these relations and the logical ordering of the relata is accounted for by a ‘first order’ mereotopological scheme of extension (PR 65-82 and Part IV)–that is, the logical relations of part-to-whole as a presupposition for any more specialized concept of geometrical extension including spatiotemporal extension in relativity theory.  As applied to his theory of objectification, ‘whole’ pertains to the concrescing occasion, internally related to the ‘parts’ of its actual world.  In line with relativity theory, the causal efficacy of these parts upon the concrescing whole is restricted to the backward lightcone; but at the same time, and crucial to the discussion above, every ‘whole’ (subject) is also a ‘part’ (superject) for subsequent ‘wholes.’ This is Whitehead’s Principle of Relativity: “The potentiality for being an element in a real concrescence of many entities into one actuality is the one general metaphysical character attaching to all entities…that every item in its universe is involved in each concrescence.  In other words, it belongs to the nature of a ‘being’ that it is a potential for every ‘becoming.’” (PR 22).

Thus, relatively to any actual entity, there is a ‘given’ world of settled actual entities and a ‘real’ potentiality, which is the datum for creativeness beyond that standpoint. This datum, which is the primary phase in the process constituting an actual entity, is nothing else than the actual world itself [i.e., in contrast to the relativistically restricted 'given' world] in its character of a possibility for the process of being felt. (PR 65)

Again, “every item in its universe” from PR 22, and “‘given’ world” from PR 65 reflects the intended compitiblity with relativity theory.  However, per the discussion above, the relativistic privacy of contemporary occasions H and I and their particularly prehended actual worlds in no way obviates or undermines or is incompatible with the heart of Whitehead’s theory of objectification: Namely, the requirement that the orders of the actual worlds of H and I be logically integrated within subsequent occasion J such that J is internally related to the actual worlds of H and I, both logically and mereotopologically–i.e.,  as whole is related to part. Thus, for Whitehead, it was possible to preserve both 1. relativistic privacy among contemporary occaions, in line with Einsteinian relativity theory; and 2. an objective ordering among all  ‘individual past worlds’ by virtue of the fact that every past world is logically integrated (i.e., without violations of PNC)  within every future occasion.  The ‘first order’ mereotopology described in Part IV is the foundational accounting of this logical order.

The real potentialities relative to all standpoints are coordinated as diverse determinations of one extensive continuum. This extensive continuum is one relational complex in which all potential objectifications find their niche. It underlies the whole world, past, present, and future. Considered in its full generality, apart from the additional [relativistic] conditions proper only to the cosmic epoch of electrons, protons, molecules, and star-systems, the properties of this continuum are very few and do not include the relationships of metrical geometry. An extensive continuum is a complex of entities united by the various [mereotopological] allied relationships of whole to part, and of overlapping so as to possess common parts, and of contact, and of other relationships derived from these primary relationships. [i.e., the logical order of antecedent/subsequent, cause/effect, etc, where a subsequent occasion is internally related to antecedent occasions as whole is related to part.] This extensive continuum expresses the solidarity of all possible standpoints throughout the whole process of the world. [i.e., the solidarity that provides for the objective immortality of all settled facts and their logical relations of whole/part, cause/effect.] It is not a fact prior to the world; it is the first determination of order [i.e., logical order]-–that is, of real potentiality [cf. 'real potentiality' in the quote above]–arising out of the general character of the world. (PR 65)

The second reason that it is a misstep to attempt to define ‘antecedence’ in Whiteheadian philosophy as either

1. temporal (and therefore relativistic), such that all ‘antecedent’ beings are those restricted to the becoming’s backward light cone, and therefore no invariant ordering among spacelike separated occasions is implied;

or

2. logical (and therefore objective), such that all ‘antecedent’ beings are those defined according to Whitehead’s self-described ‘first order,’ invariant mereotopological order, where every becoming occasion is a ‘whole’ internally and genetically related to its antecedent dative ‘parts.’  It is only in this way that Whitehead’s phrasing,  “The many become one and are increased by one” maintains coherent meaning.

is, perhaps, even more fundamental. The problem is that expecting to find, in Whitehead, a strict reduction of ‘antecedent’ to either 1 or 2 exclusively is very much akin to expecting to find a strict reduction of ‘reality’ to either mentality or physicality exclusively.  The question itself attempts to impose a Cartesian duality upon the metaphysics.  Whitehead admits, as Jorge points out, that the physical aspects of the actual occasion are dominated (and restricted) by relativistic spatiotemporal extensiveness. But this does not mean that for Whitehead, all extensiveness is to be reduced and assimilated to relativistic spactiotemporal extension.  This is because the actual occasion is not sheerly physical; it is dipolar, with both physical (relativistically causal) and conceptual (objectively logical) features.

Indeed, the dipolarity of the actual occasion–its a) conformal and b) conceptual/comparative phases–demands that 1 and 2 above be understood in the same dipolar framework.  The conformal features of concrescence (the physical pole) entail 1.–i.e., the temporal and relativisitc aspects of ‘antecedence’;  and the conceptual/comparative features of concrescence (the mental pole) entail 2.–i.e., the logical aspects of ‘antecedence.’

One cannot attempt to reduce or assimilate 2 to 1, or 1 to 2, and say that by ‘antecedent’ Whitehead ‘really’ meant 1, or ‘really’ meant 2.  That would be akin to attempting to reduce the actual occasion to either a physical entity (trumped by relativistic spatiotemporal relations) or mental entity (trumped by logical-mereotopological relations)–and, by implication, to assimilate its mental features to its physical features, or its physical features to its mental features.

Whitehead’s entire theory of objectification has, built into it, the logical order of 2 above.  And as Henry points out, it also has, built into it, the relativistic spatiotemporal order of 1 above.  And these orders are brought together in dipolar fashion in each concrescence, where 1. is dominant in the physical pole, and 2. is dominant in the mental pole.

One could make the argument that the logical order of the latter is similarly restricted by the backward light cone; but one would be hard-pressed to argue that Whitehead suggested that the relativistic restriction of causal efficacy be extended to the logical and mereotopological features of his theory of objectification. Indeed, he explicitly denied such a blunt imposition of what he called the ‘more specialized’ forms of spatiotemporal extension and their relativistic restrictions upon the more fundamental logical-mereotopolgical ‘first order’ foundation of the extensive continuum.  (See Part IV of PR).

As Jorge emphasizes, this is because the latter order is, for Whitehead, a metaphysical ‘first order’ extensiveness that is necessarily presupposed by more specialized relativistic concept of physical spatiotemporal extension.  It is a presupposition, I have argued elsewhere, that is akin to the more general presupposition of an objective and invariant logical-mathematical order by physics. In Part IV, Whitehead argues that without this presupposed order, there could be no principle of congruence, and therefore no internally consistent geometrical ordering of any kind would be possible–whether it be Euclidian spatial geometry (PR 283) or relativistic spatiotemporal geometry:

…The ambiguity as to the relative importance of competing definitions of congruence, are determined in favour of one family and one congruence-definition. The transformations into an indefinite variety of coordinates, to which the ‘tensor theory’ refers, all presuppose one congruence-definition. The invariance of the Einsteinian ‘ds’ expresses this fact.” (PR 98)

Physicist Roland Omnes makes a similar point with respect to the necessary presupposition of the logical principles of Non-Contradiction and the Excluded Middle in quantum theory.  For Whitehead, these two principles are intimately bound up with his ‘first-order’ logical-mereotopological extensive scheme.  For me, this is one interesting point of intersection between Whitehead’s metaphysics & cosmology, and quantum mechanics & relativity theory.  It could even be that the key to bringing together the latter pair in a truly coherent manner will be by making explicit connections to the former.

The broader point in this regard is that since physics cannot presume to ‘account for’ that which it necessarily presupposes, it seems unlikely that a fundamentally objective logical ordering of actualities can be sufficiently or meaningfully ‘explained’ solely by reference to some physical process; this is because any such process given in the language of physics would necessarily presuppose the very order it attempts to account for.  Certainly physics can provide important novel descriptions of this order as operative in physical processes; but it clearly cannot provide fundamental explanation of such order–i.e., an explanation that answers the questions, “Where does this order come from? And why is our universe characterized by this order, rather than by a disorder that is equally well-described by physics?”