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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 in CitationSilberstein and McGeever (1999) as well as in CitationKronz and 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.
Acknowledgments
I thank Timothy Eastman for inviting me to contribute this article.
Notes
1. That is, asymmetric covariation. In other words a supervenient property will change if the properties of its constituent parts are altered, but not vice versa.
2. A caustic surface is also an example of a singularity: under certain conditions reflected wavefronts “bunch together” and constructively add up to produce a spike in the associated light intensity. Indeed, in more general terms, Batterman primarily focuses exclusively on studies of emergent phenomena in the context of singular behavior, that is, when a certain quantity spikes or appears to become arbitrarily large. However, as I demonstrate here, there are many important cases of emergence that do not necessarily involve singularities.
3. Two of the hydrogen atoms “generously” give up their electrons to the helium atom's orbital electron “account,” and subsequently reside in the helium nucleus as protons. The other two hydrogen atoms “greedily” swallow up their orbital electrons, to become neutrons dwelling in the helium nucleus.
4. In the physical sense, “evolution” usually implies “unicity.” That is to say, one presupposes in this concept that a system's dynamical properties are isolated from those of the macrosystem the system is embedded in, or the environment. This notion has nothing in common with the biological idea, presupposing natural selection.
5. In CitationFinkelstein and Kallfelz (1997) among other issues we focused extensively on the similarities and contrasts of CitationWhitehead's (1929/1978) recursive concept of concrescences and the discrete network structure of Finkelstein's Quantum Network Dynamics (1996). For instance, CitationFinkelstein (1996) relies heavily on Grassmann algebra, a mathematical system that Whitehead was well familiar with, in his treatise of Universal Algebra, most notably of the important logical, geometrical, and physical meanings of Grassmann's progressive (exterior) product v, regressive (inner) product 
, and extensor addition (superposition) +. Nevertheless, Whitehead seemed to have built his concrescence cosmology primarily on one mode of composition, most likely based on Peano's ι = … set-formation operation. (CitationFinkelstein and Kallfelz, 1997, p. 284) Perhaps Whitehead's omission of notions of composing different occasions with different operations (despite his being conversant with Grassmann algebra) was due to his not distinguishing the composition of classical and quantum vectors (p. 286). In other words, Whitehead appealed primarily to the early (pre-Copenhagen) quantum theory, “a classical theory with ‘quantum rules’ tacked on” (p. 285).
6. Strong and weak notions of emergence find an interesting counterpart in CitationHumphrey's (2000) distinctions of theoretical limits versus practical limitations. “A limit is an ‘in-principle’ epistemological constraint, whereas a limitation is an epistemological or pragmatic constraint” (Carrier et al., 2000, p. 2). There remains the rather daunting task to establish a more precisely metaphysical notion of principled epistemic limits. For instance, do such limits reflect some objectively irresolvable features in the world, or do they merely represent some fundamentally limiting aspect concerning the cognitive capacities of the agent? In any case, the limits/limitations distinction serves as a cautionary reminder against ascribing literal truth to a theory. “[W]e [can be] … committed to the claim that a theory is reliable, but … not committed to the literal truth of its empirical consequences. This does not mean we have to be instrumentalists … a scientific realist [for instance] might be committed to the reality of electrons and fields, yet demand only that electromagnetic models represent the behavior of these ‘unobservables’ reliably, while an empiricist could be content with the fact that the models are reliable as far as the theory's observable consequences are concerned” (CitationFrisch, 2005, p. 42).
7. For details, see Goldstein (Citation1980 pp. 61, 540).
8. An exception includes some simple cases involving entanglement in quantum mechanics, as shall be discussed later.
9. There certainly remains (literally) some relative motion or flexing, hence couched in more precise terms one can say that the property of relative motion to a degree greater than certain tolerances (for instance, of the order of one centimeter) vanishes.
10. The ⊕ represents the conventional procedure of forming a direct sum, that is, the operation involved in the formation of a simple aggregate or set. On the other hand, the * operation is Humphrey's depiction of property fusion, which, as demonstrated earlier, has nothing to do with its usual representation as a product.
11. For the sake of simplicity, in their discussion they omit levels of description above biology, like psychology, sociology, anthropology, and so on.
12. For a simple example involving a preparation of an ensemble of entangled electrons violating Bell's Inequalities, and a more detailed description thereon, see for instance Appendix VI (pp. 29–31) in CitationKallfelz (2002).
13. There is no analogue for “spin” in the classical sense. Spin represents an “internal degree of freedom” whereas classical canonical coordinates (position, momentum) are “external” degrees of freedom, insofar as they are in principle characterizable purely in terms of “external” spatiotemporal coordinates. This is one instance of the general distinction between Classical and Contemporary science, as depicted by CitationEastman (1997) in Table 2, p. 245 (specifically the distinction between “External Source of Order only” versus “Both External and Internal”). Also, in the example discussed earlier, I am temporarily leaving out the subtleties posed by the case of identical (indistinguishable) particles. Krontz and Tiehen (2005) bring up this case in their concluding discussion (p. 347). Moreover, particles like in the earlier example, characterized by only two possible values of spin number (± 1/2 or “up”/“down”) have spin state space or “spinor” space that, as opposed to the classical case consisting of a set of two points, consists of a two-dimensional vector space H consisting of complex valued coefficients. Such state spaces combine via the tensor product ⊗ to produce higher-dimensional vector spaces. For example, two three-dimensional vector spaces combine via ⊗ to form a six-dimensional vector space. One can think of this procedure as the quantum mechanical analogue of the classical mechanics case, insofar as in the classical case two sets consisting of three points each combine via ⊕ to form a set consisting of six points.
14. This is an operator (represented by an n × n matrix for state spaces of dimensionality n) representing the energy “observable” (i.e., measurable quantity) of the system.
15. The lower-case Hamiltonians represent the 2-dimensional matrices representing the Hamiltonians of each of the three particles 1, 2, 3, when isolated from one another. The operator I i (i = 1 or 2 or 3) represents the identity operator (a 2-dimensional matrix) for isolated systems 1, 2, 3 while I j + k (j,k = 1 or 2 or 3, and j ≠ k) represents the 4-dimensional identity matrix on composite system(s) 1 + 2, 2 + 3, 1 + 3.
16. The lower-case Hamiltonian represents the 4-dimensional reduced Hamiltonian matrix for composite system 1 + 2. Of course, this matrix is not factorizable for the same reason why H 1 + 2 + 3 was not.
17. A hopelessly “ontic” notion, as it inevitably “smuggles in the false doctrine that all physical entities have states of being that describe them completely, determined uniquely by their past and uniquely predicting their future” (CitationFinkelstein and Kallfelz, 1997, p. 279).
18. The inevitably dispositional nature (i.e., characterizable in terms of ability or propensity) of basic scientific terms continues to present significant challenges regarding formulating a precise logical characterization of scientific theories. For example, barring the niceties of gauge field theories, quantities like “mass” or “charge” are usually defined dispositionally, that is, in terms of the ability or capacity of a system to possess mass or charge. This results in impredicative definitions; the terms to be defined comprise part of the predicate of the definition. Such problems were extensively analyzed by logical empiricists from the 1950s onward (see, for instance, A. J. CitationAyer, 1956/1998, pp. 821–822). On the other hand, the problem with opting for a more theory-driven approach, that is, by relying for instance on what gauge field theory tells us what mass or charge “is” has to do with relying exclusively on the meaning of a term as constituted by a particular theoretical framework. As was well argued by those in the historical tradition (Kuhn, Hansen, Feyerabend, etc.) in response to the logical empiricists, it altogether remains unclear whether the same term has similar or even overlapping semantic or referential meaning from the standpoint of different theoretical frameworks. “Mass” can mean different things, depending on which theoretical framework one chooses (Newtonian, relativistic, gauge field theory, etc.).
19. This seems apparent enough: the non-factorizability of a state vector |Φ ⟩ describing a system in an entangled mode prevents the extraction of any information from the system's sub-constituents, which would otherwise be made available in their correspondence with the factors of |Φ ⟩ (if |Φ ⟩ could be factored).
20. Recall note 14. In classical mechanics, Hamiltonians also describe the energy of a system, but are defined not as operators in a vector space, but rather as functions in phase space coordinates (i.e., in terms of the position(s) and momenta of the system's constituents).
21. He argues, for instance, in the case of deriving the specific heat for a classical crystal through its Hamiltonian and other purely dynamic considerations, that the “diachronic seems the more important case” (p. 120).
22. Recall note 13. Again, the reader may skip the technical details of this example without loss of the conceptual points being made here. For convenience, I am adopting the Dirac notation for the spin state vectors.
23. The last line in (1) adopts the shorthand representation for denoting the ordering of base elements in the composite system.
24. For details demonstrating the non-factorizibility, see Kallfelz (Citation2002, pp. 31–32).
25. “The Born-Oppenheimer ‘approximation’ … is not simply a mathematical expansion in series form. … It literally replaces the basic quantum mechanical descriptions with a new description, generated in the limit ϵ → 0, [for the governing parameter ϵ = (m e /m N )1/4 where m e is the mass of the orbital electron, and m N the mass of the nucleon, i.e. one forms an asymptotic series S(ϵ) = ∑ k a k ϵ k .] This replacement corresponds to a change in the algebra of observables needed for the description of molecular phenomena. … The Born-Oppenheimer approach amounts to a change in topology—i.e., a change in the mathematical elements modeling physical phenomena—as well as a change in ontology—including fundamental physical elements absent from quantum mechanics” (Robert CitationBishop, 2004, p. 4).
26. As discussed in detail by Krontz and Tiehen, pp. 339–341.
27. In homage to W. V. O. Quine's pragmatic holism.
28. “The superseding theory T′, though ‘deeply containing T’ (in some non-reductive sense) cannot adequately account for emergent and critical phenomena alone, and thus enlists T in some essential manner” (CitationKallfelz, 2006, p. 3).
29. For highly accessible overview of Clifford algebra, see CitationLasenby et al. (2000).
30. I precisely define a “methodologically fundamental” procedure in CitationKallfelz (2006), pp. 11–12. Briefly, if a theory T is methodologically fundamental, then its underlying mathematical formalism, suitably characterized by any multilinear algebra, would exhibit: (a) A simple relativity group (i.e., the group describing all its covariant symmetries would not contain any invariant subgroups), (b) A stable Lie algebra (i.e., the algebra describing the class of all the infinitesimal transformations in the theory varies smoothly, or “contracts” smoothly, in the zero limit of any one of its structure constants. I borrow these notions from CitationSegal (1951), CitationInonu and Wigner (1952), and Finkelstein (Citation2001, Citation2004a–Citationc) who greatly expands on Segal, Inonu and Wigner's original work.
31. The space-time structure must be supplied by classical structures, prior to the definition of the dynamical algebra (CitationFinkelstein et al., 2001, p. 5).
32. That is, the simplest statistics supporting a 2-valued representation of S N , the symmetry group on N objects.