Whitehead and Peirce’s Synchronicity with Hegel’s Capital Error. Thought of the Day 97.0


The focus on experience ensures that Whitehead’s metaphysics is grounded. Otherwise the narrowness of approach would only culminate in sterile measurement. This becomes especially evident with regard to the science of history. Whitehead gives a lucid example of such ‘sterile measurement’ lacking the immediacy of experience.

Consider, for example, the scientific notion of measurement. Can we elucidate the turmoil of Europe by weighing its dictators, its prime ministers, and its editors of newspapers? The idea is absurd, although some relevant information might be obtained. (Alfred North Whitehead – Modes of Thought)

The wealth of experience leaves us with the problem of how to cope with it. Selection of data is required. This selection is done by a value judgment – the judgment of importance. Although Whitehead opposes the dichotomy of the two notions ‘importance’ and ‘matter of fact’, it is still necessary to distinguish grades and types of importance, which enables us to structure our experience, to focus it. This is very similar to hermeneutical theories in Schleiermacher, Gadamer and Habermas: the horizon of understanding structures the data. Therefore, we not only need judgment but the process of concrescence implicitly requires an aim. Whitehead explains that

By this term ‘aim’ is meant the exclusion of the boundless wealth of alternative potentiality and the inclusion of that definite factor of novelty which constitutes the selected way of entertaining those data in that process of unification.

The other idea that underlies experience is “matter of fact.”

There are two contrasted ideas which seem inevitably to underlie all width of experience, one of them is the notion of importance, the sense of importance, the presupposition of importance. The other is the notion of matter of fact. There is no escape from sheer matter of fact. It is the basis of importance; and importance is important because of the inescapable character of matter of fact.

By stressing the “alien character” of feeling that enters into the privately felt feeling of an occasion, Whitehead is able to distinguish the responsive and the supplemental stages of concrescence. The responsive stage being a purely receptive phase, the latter integrating the former ‘alien elements’ into a unity of feeling. The alien factor in the experiencing subjects saves Whitehead’s concept from being pure Spirit (Geist) in a Hegelian sense. There are more similarities between Hegelian thinking and Whitehead’s thought than his own comments on Hegel may suggest. But, his major criticism could probably be stated with Peirce, who wrote that

The capital error of Hegel which permeates his whole system in every part of it is that he almost altogether ignores the Outward clash. (The Essential Peirce 1)

Whitehead refers to that clash as matter of fact. Although, even there, one has to keep in mind that matter-of-fact is an abstraction. 

Matter of fact is an abstraction, arrived at by confining thought to purely formal relations which then masquerade as the final reality. This is why science, in its perfection, relapses into the study of differential equations. The concrete world has slipped through the meshes of the scientific net.

Whitehead clearly keeps the notion of prehension in his late writings as developed in Process and Reality. Just to give one example, 

I have, in my recent writings, used the word ‘prehension’ to express this process of appropriation. Also I have termed each individual act of immediate self-enjoyment an ‘occasion of experience’. I hold that these unities of existence, these occasions of experience, are the really real things which in their collective unity compose the evolving universe, ever plunging into the creative advance. 

Process needs an aim in Process and Reality as much as in Modes of Thought:

We must add yet another character to our description of life. This missing characteristic is ‘aim’. By this term ‘aim’ is meant the exclusion of the boundless wealth of alternative potentiality, and the inclusion of that definite factor of novelty which constitutes the selected way of entertaining those data in that process of unification. The aim is at that complex of feeling which is the enjoyment of those data in that way. ‘That way of enjoyment’ is selected from the boundless wealth of alternatives. It has been aimed at for actualization in that process.

Rhizomatic Topology and Global Politics. A Flirtatious Relationship.



Deleuze and Guattari see concepts as rhizomes, biological entities endowed with unique properties. They see concepts as spatially representable, where the representation contains principles of connection and heterogeneity: any point of a rhizome must be connected to any other. Deleuze and Guattari list the possible benefits of spatial representation of concepts, including the ability to represent complex multiplicity, the potential to free a concept from foundationalism, and the ability to show both breadth and depth. In this view, geometric interpretations move away from the insidious understanding of the world in terms of dualisms, dichotomies, and lines, to understand conceptual relations in terms of space and shapes. The ontology of concepts is thus, in their view, appropriately geometric, a multiplicity defined not by its elements, nor by a center of unification and comprehension and instead measured by its dimensionality and its heterogeneity. The conceptual multiplicity, is already composed of heterogeneous terms in symbiosis, and is continually transforming itself such that it is possible to follow, and map, not only the relationships between ideas but how they change over time. In fact, the authors claim that there are further benefits to geometric interpretations of understanding concepts which are unavailable in other frames of reference. They outline the unique contribution of geometric models to the understanding of contingent structure:

Principle of cartography and decalcomania: a rhizome is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or deep structure. A genetic axis is like an objective pivotal unity upon which successive stages are organized; deep structure is more like a base sequence that can be broken down into immediate constituents, while the unity of the product passes into another, transformational and subjective, dimension. (Deleuze and Guattari)

The word that Deleuze and Guattari use for ‘multiplicities’ can also be translated to the topological term ‘manifold.’ If we thought about their multiplicities as manifolds, there are a virtually unlimited number of things one could come to know, in geometric terms, about (and with) our object of study, abstractly speaking. Among those unlimited things we could learn are properties of groups (homological, cohomological, and homeomorphic), complex directionality (maps, morphisms, isomorphisms, and orientability), dimensionality (codimensionality, structure, embeddedness), partiality (differentiation, commutativity, simultaneity), and shifting representation (factorization, ideal classes, reciprocity). Each of these functions allows for a different, creative, and potentially critical representation of global political concepts, events, groupings, and relationships. This is how concepts are to be looked at: as manifolds. With such a dimensional understanding of concept-formation, it is possible to deal with complex interactions of like entities, and interactions of unlike entities. Critical theorists have emphasized the importance of such complexity in representation a number of times, speaking about it in terms compatible with mathematical methods if not mathematically. For example, Foucault’s declaration that: practicing criticism is a matter of making facile gestures difficult both reflects and is reflected in many critical theorists projects of revealing the complexity in (apparently simple) concepts deployed both in global politics.  This leads to a shift in the concept of danger as well, where danger is not an objective condition but “an effect of interpretation”. Critical thinking about how-possible questions reveals a complexity to the concept of the state which is often overlooked in traditional analyses, sending a wave of added complexity through other concepts as well. This work seeking complexity serves one of the major underlying functions of critical theorizing: finding invisible injustices in (modernist, linear, structuralist) givens in the operation and analysis of global politics.

In a geometric sense, this complexity could be thought about as multidimensional mapping. In theoretical geometry, the process of mapping conceptual spaces is not primarily empirical, but for the purpose of representing and reading the relationships between information, including identification, similarity, differentiation, and distance. The reason for defining topological spaces in math, the essence of the definition, is that there is no absolute scale for describing the distance or relation between certain points, yet it makes sense to say that an (infinite) sequence of points approaches some other (but again, no way to describe how quickly or from what direction one might be approaching). This seemingly weak relationship, which is defined purely ‘locally’, i.e., in a small locale around each point, is often surprisingly powerful: using only the relationship of approaching parts, one can distinguish between, say, a balloon, a sheet of paper, a circle, and a dot.

To each delineated concept, one should distinguish and associate a topological space, in a (necessarily) non-explicit yet definite manner. Whenever one has a relationship between concepts (here we think of the primary relationship as being that of constitution, but not restrictively, we ‘specify’ a function (or inclusion, or relation) between the topological spaces associated to the concepts). In these terms, a conceptual space is in essence a multidimensional space in which the dimensions represent qualities or features of that which is being represented. Such an approach can be leveraged for thinking about conceptual components, dimensionality, and structure. In these terms, dimensions can be thought of as properties or qualities, each with their own (often-multidimensional) properties or qualities. A key goal of the modeling of conceptual space being representation means that a key (mathematical and theoretical) goal of concept space mapping is

associationism, where associations between different kinds of information elements carry the main burden of representation. (Conceptual_Spaces_as_a_Framework_for_Knowledge_Representation)

To this end,

objects in conceptual space are represented by points, in each domain, that characterize their dimensional values. A concept geometry for conceptual spaces

These dimensional values can be arranged in relation to each other, as Gardenfors explains that

distances represent degrees of similarity between objects represented in space and therefore conceptual spaces are “suitable for representing different kinds of similarity relation. Concept

These similarity relationships can be explored across ideas of a concept and across contexts, but also over time, since “with the aid of a topological structure, we can speak about continuity, e.g., a continuous change” a possibility which can be found only in treating concepts as topological structures and not in linguistic descriptions or set theoretic representations.

Conjuncted: All Sciences Are Still Externalist.


Constructor Theory seeks to express all fundamental scientific theories in terms of a dichotomy between possible and impossible physical transformations (Deutsch). Accordingly, a task A is said to be possible (A✓) if the laws of nature impose no restrictions on how (accurately) A could be performed, nor on how well the agents that are capable of approximately performing it could retain their ability to do so. Otherwise A is considered to be impossible (A✘). Deutsch argues that in both quantum theory and general relativity, “time is treated anomalously”. He sees the problem in that time is not among the entities to which both theories attribute “objective existence (namely quantum observables and geometrical objects respectively), yet those entities change with time”. Is not this an interesting appraisal? According to him “there is widespread agreement that there must be a way of treating time ‘intrinsically’ (i.e. as emerging from the relationships between physical objects such as clocks) rather than ‘extrinsically’ (as an unphysical parameter on which physical quantities somehow depend).” However, Deutsch reckons, it would be difficult to accommodate this in the prevailing conception, “every part of which (initial state; laws of motion; time-evolution) assumes that extrinsic status”. According to Constructor Theory it is “both natural and unavoidable to treat both time and space intrinsically: they do not appear in the foundations of the theory, but are emergent properties of classes of tasks …”. Note that the differentiation between “intrinsic” and “extrinsic” has different meaning than the one used in phenomenological philosophy.

Other projections, such as the Circular Theory (Yardley) and the Structural Theory of Everything (Josephson) also question the laws underlying the foundations of physics. However, as long as they consider placing events “over time” extrinsically, they are all of the same kind: non-phenomenological. (Note that duration has two definitions: endo and exo). This is a very interesting conclusion in Steven M. Rosen’s words:

The point may be that, in physics, time is treated ‘extrinsically’ because it does not lend itself to being formulated in objectivist terms, i.e., as something that is limited to the context of the objectified physical world.

Therefore, the terms “intrinsic” and “extrinsic” in Deutsch’s objectivist usage are both “extrinsic” in the phenomenological sense: both are limited to a physicalistic paradigm that excludes the internal perspective of a lived (bodily) subjectivity or an agent-dependent reality (Rössler), a continuing durée (Bergson). This appears to be the key to all the “trouble with physics” (Smolin), and not only with physics.  In their 2012 paper No entailing laws, but enablement in the evolution of the biosphere Longo, Montévil and Kauffman claim that biological evolution “marks the end of a physics world view of law entailed dynamics” (Longo et al.). They argue that the evolutionary phase space or space of possibilities constituted of interactions between organisms, biological niches and ecosystems is “ever changing, intrinsically indeterminate and even (mathematically) unprestatable”. Hence, the authors’ claim that it is impossible to know “ahead of time the ‘niches’ which constitute the boundary conditions on selection” in order to formulate laws of motion for evolution. They call this effect “radical emergence”, from life to life. In their study of biological evolution, Longo and colleagues carried close comparisons with physics. They investigated the mathematical constructions of phase spaces and the role of symmetries as invariant preserving transformations, and introduced the notion of “enablement” to restrict causal analyses to Batesonian differential cases. The authors have shown that mutations or other “causal differences” at the core of evolution enable the establishment of non-conservation principles, in contrast to physical dynamics, which is largely based on conservation principles as symmetries. Their new notion of “extended criticality” also helps to understand the distinctiveness of the living state of matter when compared to the non-animal one. However, their approach to both physics and biology is also non-phenomenological. The possibility for endo states that can trigger the “(genetic) switches of mutation” has not been examined in their model. All sciences are (still) externalist.

Philosophy of Mathematics x ‘Substance Versus Body Metaphysic’. Note Quote.


An obvious initial dilemma that faces the prospective Structural Realist (SR) theorist is the ontological status of the spacetime structures themselves. Do they exist as a sort of Platonic universal, independent of all physical objects or events in spacetime, or are they dependent on matter/events for their very existence or instantiation? This problem arises for the mathematician in an analogous fashion, since they also need to explicate the origins of mathematical structures (e.g., set theory, arithmetic). Consequently, a critic of the SR spacetime project might seem justified in regarding this dispute in the philosophy of mathematics as a re-emergence of the traditional substantivalist versus relationist problem, for the foundation of all mathematical structures, including the geometric spacetime structures, is once again either independent of, or dependent on, the physical.

Nevertheless, a survey of the various positions in the mathematical ontology dispute may work to the advantage of the SR spacetime theory, especially when the relevant mathematical and spacetime options are paired together according to their analogous role within the wider ontology debate. First, mathematical structuralism can be classified according to whether the structures are regarded as independent or dependent on their instantiation in systems (ante rem and in re structuralism, respectively), where a “system” is loosely defined as a collection of “objects” and their interrelationships. Ante rem structuralism, as favored by Resnik and Shapiro, is thus closely akin to the traditional “absolute” conception of spacetime, for a structure is held to “exist independent of any systems that exemplify it” (Shapiro). Yet, since “system” (and “object”) must be given a broad reading, without any ontological assumptions associated with the basis of the proposed structure, it would seem that substantivalism would not fit ante rem structuralism, as well. The structure of substantivalism is a structure in a substance, namely, a substance called “spacetime”, such that this unique substance “exemplifies” the structure (whereas ante rem structure exists in the Platonic sense as apart from any and all systems that exemplify it). The substantivalist might try to avoid this implication by declaring that their spacetime structures are actually closer in spirit to a pure absolutism, without need of any underlying entity (substance) to house the structures (hence, “substantivalism” is simply an unfortunate label). While this tactic may be more plausible for interpreting Newtonian spacetimes, it is not very convincing in the context of GTR, especially for the sophisticated substantivalist theories. Given the reciprocal relationship between the metric and matter fields, it becomes quite mysterious how an non-substantival, “absolute” structure, g, can be effected by, and effect in turn, the matter field, T. For the ante rem structuralist, mathematical structures do not enter into these sorts of quasi-causal interrelationships with physical things; rather, things “exemplify” structures (see also endnote 8). Accordingly, one of the initial advantages of examining spacetime structures from within a mathematical ontology context is that it drives a much needed wedge between an absolutism about quantitative structure and the metaphysics of substantivalism, although the two are typically, and mistakenly, treated as identical.

In fact, as judged against the backdrop of the ontology debate in the philosophy of mathematics, the mathematical structures contained in all spacetime theories would seem to fall within a nominalist classification. If, as the nominalists insist, mathematical structures are grounded on the prior existence of some sort of “entity”, then both the substantivalists and relationists would appear to sanction mathematical nominalism (with in re structuralism included among nominalist theories, as argued below): whether that entity is conceived as a unique non-material substance (substantivalism), physical field (metric-field relationism), or actual physical objects/events (relationism, of either the modal (R2) or strict (R1) type), a nominalist reading of mathematical structure is upheld. This outcome may seem surprising, but given the fact that traditional substantivalist and relationist theories have always based spatiotemporal structure on a pre-existing or co-existing ontology – either on a substance (substantivalism) or physical bodies (relationism) – a nominalist reading of spacetime structure has been implicitly sanctioned by both theories. Consequently, if both substantivalism and relationism fall under the same nominalist category in the philosophy of mathematics, then the deeper mathematical Platonist/nominalist issue does not give rise to a corresponding lower-level substantival/relational dichotomy as regards the basis of those spacetime structures (e.g., with substantivalism favoring a Platonic realism about mathematical structures, and relationism siding with a nominalist anti-realism). This verdict could change, of course, if a non-substantival “absolute” conception of spacetime becomes popular in GTR; but this seems unlikely, as argued above.

As there are a number nominalist reconstructions of mathematics, a closer examination of their content reveals that the different versions can be paired to different substantival and relational theories. For instance, a reductive (R1) spacetime relationism can be linked to some strict nominalist reconstructions of mathematics, as in Field’s attempt to treat mathematical objects and structures as entirely dispensable, or “fictional”. Field posits a continuum of spacetime points, conceived physically in the manner of a manifold substantivalist, in his effort to rewrite Newtonian gravitation theory along mathematically anti-realist lines. Modal (R2) relationists, like Teller, would not constitute the spacetime analogue of Field’s program, accordingly, since this form of relationism sanctions modal spacetime structures that can transcend the structures exhibited by the actually existing physical objects: e.g., the affine structure ∇ instantiated by a lone rotating body. Whereas Field requires an infinity of physical spacetime points (isomorphic to ℜ4 in order to capture the full content of the mathematician’s real numbers, the (R2) relationist can allow modal structures to serve this function, thus releasing the ontology of such extravagant demands. More importantly, if Field’s nominalist program is committed to manifold substantivalism, M then it is susceptible to the hole argument. All spacetime theories that utilize the metric in the identity of spacetime points, such as sophisticated substantivalism and metric-field relationism, would thereby incorporate a divergent set of structural scenarios (since they are not susceptible to the hole argument). Therefore, Field’s nominalist mathematics entails a spacetime structure that comprises a different SR theory than sophisticated substantivalism and metric-field relationism.

The mathematical equivalent of both sophisticated substantivalism and (R2) relationism is, rather, any of the less stringent nominalist theories that rejects Field’s strict nominalism, as in, e.g., Chihara in re structuralism. Much like the modal (R2) relationist theories surveyed above, the “minimal nominalists” do not allow structures to exist independently of the systems they exemplify, yet they do not believe that these structures can be dismissed as mere fictions, either. Contra Field, the minimal mathematical nominalists deny a purely instrumentalist construal of mathematical structures (while simultaneously rejecting a Platonic absolutism): they all insist, for instance, that mathematical structures cannot be excised from scientific theories without loss of valuable physical content. In re structuralism, moreover, employs “possible structures” as a means of avoiding a commitment to an infinite background ontology, a feature that helps to explain its common nominalist classification. The minimal nominalist theories often differ on how the mathematical structures are constructed from their basic ontology, as well as how to construe the truth-values of mathematical statements about mathematical structures. But these differences, such as in the kind of modality sanctioned, etc., can be quite subtle, and do not lend themselves to any drastic distinctions in type or basic intent. Hence, any attempt to rekindle the substantival/relational distinction among the competing minimal nominalist theories would seem implausible. A firm reliance on some form of modality and a non-instrumental construal of mathematical structures is the common, and crucial, similarity among these theories; and it is these aspects that are most important for the spacetime structuralist, whether of the sophisticated substantivalist or relationist variety.

These last observations are not meant to downplay the importance of the ongoing research in the philosophy of mathematics on the origin of structures, for it is always possible that substantial problems will arise for some of the minimal nominalist theories, thus eliminating them from contention. From the SR standpoint, in fact, the philosophy of mathematics would likely be considered a more proper arena for assessing the structures employed by spacetime theories, at least as opposed to the apparently unverifiable metaphysics of “substance versus body”. Not only has the traditional spacetime dichotomy failed to explain how these mathematical structures arise from their basic ontology, but, as we have seen, the underlying structures advocated by the sophisticated versions of both substantivalism and relationism are identical when judged within the wider philosophy of mathematics framework. Whether that foundational entity is called a substance or a physical object is irrelevant, and probably a conventional stipulation, since the real work, as judged from the mathematical perspective, concerns how the structures are constructed from the underlying entity—and the competing claims of substance or physical existent do not effect this mathematical construction. In essence, the only apparent difference between the sophisticated substantivalists and (R2) relationists are where those mathematical structures are located: either internal to the substance or field (for the substantivalists and metric-field relationists, respectively), or external to bodies/events (for non-field formulations of (R2) relationism, such as Teller’s). Needless to say, this internal/external distinction does not provide any information on how the mathematical structures are built-up; rather, it reveals the pervasive influence of the age-old substance/property dichotomy within the philosophy of science community, an unfortunate legacy that the SR theorist regards as hindering the advancement of the debate on spacetime theories.

Finally, since the competing minimal nominalist constructions are not being judged solely from a mathematical perspective, but from a scientific and empirical standpoint as well, a few words are in order on the relevance of empirical evidence in assessing spacetime structure. This issue will be addressed further in the remaining sections, but, in brief, it is unlikely that any physical evidence could provide a strong confirmation of any one of the competing nominalist theories. In effect, these nominalist constructions are only being utilized to explain the origins of the spacetime structures, such as M or g, that do appear in our best physical theories, with the important qualification that these nominalist constructions do not commit the physical theory to any problematic or meaningless physical outcomes (e.g., Field’s nominalism and the hole argument, as noted above). As for the sophisticated brands of both substantivalism and relationism, all of the minimal nominalist construction are apparently identical as regards their implications for possible spacetime scenarios and meaningful physical states. Unless other reasons are brought forward, the choice among the competing minimal nominalist constructions could thus be viewed as conventional, since it is difficult to conceive how empirical evidence could reach the deep mathematical levels where the differences in nominalist constructions of spacetime structures come into play.