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  • 12 IWC

5. Whitehead, Mathematics, Logic, and Natural Sciences

Updated: Aug 18, 2019

Section heads: Vesselin Petrov, Tatiana Roque


On the Physical Adequacy of Whitehead's Cosmology (pdf) Francesco Maria Ferrari, Brazil

My purpose is to argue for the adequacy of Whitehead's original metaphysical

insight w.r.t. the actual picture of fundamental physics, i.e., Quantum Field

Theory (QFT).

According to Whitehead, the process of Nature's construction is twofold, as

to based on the interplay of (i) a (formal) process of structural uni_cation and

emergence of novelty { concrescence { and of (ii) a (material-agent) non-linear

process towards phases of increasing complexity { phase transition [1, Part II,

Ch. X, Sec. 2]. By such a process scheme Whitehead reinterpreted the tra-

ditional foundation of the necessary character of natural laws shaped on the

distinction between necessary and su_cient conditions, under the light of the

rising irreversibility (Thermodynamics) and indeterminism (Quantum Mechan-

ics): causal necessity and determination without determinism and reversibility

I argue that Whitehead's process is an intrinsic dual notion: that concres-

cence, the su_cient condition, and phase transitions, the necessary one, inter-

play or interact dually toward the irreversible and indeterministic constitution

of new emerging structures.

Of particular relevance for my reading is that, according to Whitehead, the

concrescent `moment' may be interpreted in terms of the _nal cause [1, Part

II, Ch. X, Sec. 4]. Indeed, within Whitehehead's cosmology this notion is open

to non-epistemic interpretation, for it is \blind" and such `blindness' is to be

understood in the sense that \no intellectual operations are involved".

How are to be read such `dual moments' of the process? My suggestion is

to interpret them in terms of the \minimal conditions" and of the \maximal

conditions" of a physical system evolution, respectively. I'll argue that such an

interpretation is coherent with the formal notion of categorical duality between

algebra(s) and co-algebra(s). I suggest, we may reinterpret in algebraic terms

the necessary and maximal conditions of nature genetic evolution { for any

algebra, inductively, provides an initial structure { and in co-algebraic terms its

su_cient and minimal conditions { for any co-algebra, co-inductively, provides

_nal structures.

The adequacy of Whitehead's dual process scheme can, thus, be proved by

showing it is coherent w.r.t. QFT model:


_ QFT is the best (empirical) theory at hand [2] and _elds are processes [3];

_ QFT assumes the necessary and active agency of the Quantum Vacuum

(QV), of the environment [4, 5].

_ Two are the basic sorts of _elds interacting dually for the emergence of

each and all in_nite and non-equivalent phases or domains of coherence:

the ensemble of microscopic components, i.e., the matter _eld { i.e., the

system { and the extended long-range (i.e., formal) `gauge _elds' { con-

straint by the environment [4].


[1] Whitehead, A. N. (1978). Process and Reality. An Essay in Cosmology,

corrected edition ed. by D. R. Gri_th and D. W. Sherburne, NY: The Free

Press- Macmillan Publishing Co.

[2] Royal Swedish Academy of Sciences (2015). Neutrino Oscillations, Scienti_c

Background on the Nobel Prize in Physics, available online.

[3] Bickhard, M. H. (2011). Some Consequences (and Enablings) of Process

Metaphysics. Axiomathes 21:3{32.

[4] Del Giudice, E., Pulselli, R., & Tiezzi, E. (2009). Thermodynamics of

irreversible processes and quantum _eld theory: an interplay for under-

standing of ecosystem dynamics, Ecological Modelling 220:1874-1879.

[5] Umezawa, H. (1993). Advanced Field Theory. NY: American Institute of



A Whiteheadian perspective on the organism. Whitehead’s foreknowledge of the theories of self-creation

Federico Giorgi, Italy

The aim of my talk is to highlight that Whitehead’s philosophy of organism anticipates the self-creation perspective in biology. The self-creation perspective defines organisms as operational autopoietic unities. According to this perspective, organisms are self-maintaining systems that recursively self-produce their chemical components, organs, functional parts and their own physical borders in a circular causality process. The self-creation perspective presents three aspects one can also find in Whitehead’s philosophy of organism. The first aspect is the autopoietic character of organisms. According to the theory of the self-creation, organisms self- maintain their own circular organization. Living units preserve their own identity through successive interactions in an always changing environment. The second aspect is the processual character of an organism. The behaviour of the organism is subservient to the maintenance of its basic circularity only. The optimal functional state of the organism can only be determined a posteriori, as the result of its actual behaviour. The third aspect is the vindication of the central role played by the notion of organism in biology. According to the theory of self-creation, the organisation of organisms plays a central role in the impressive capacity of life to proliferate, to create an enormous variety of forms, to adapt to completely different environments, and to increase its own complexity. The organisation of living units is the base of all possible ontogenetic and evolutionary transformation in biological systems.

As Beaulieu observes in Whitehead précurseur des theories de l’auto-création, Whitehead’s philosophy of organism anticipates the theory of self-creation in the three above-mentioned aspects. With regard to the first aspect, Whitehead conceives organisms as constituted by self-creative actual entities. Actual entities, in Whitehead’s philosophy, play diverse roles in self-formation without losing their self-identity. With respect to the second aspect, Whitehead attributes an autonomy in their self-realization to the living occasions in the living cells. Living occasions are endowed with a rich development of conceptual feelings of eternal objects in their mental pole. Some of these eternal objects may be novel in the sense that they are not in the original objective datum of the living occasion. Among such novel eternal objects, the living occasion selects those which are more appropriate to the fulfilment of its subjective aim. This allows the living organisms to adapt themselves to their environment by originative action. With regard to the third aspect, Whitehead conceives organisms as autonomous entities. In contrast to the evolutionary theories, he doesn’t explain the ontogenesis of organisms by appealing to a historical supra-individual dimension. According to Whitehead, the ontogenesis of the living units is a non-progressive and multi-serial process of becoming. The discontinuous, multiserial, and «événementielle» character of the self-creative process of the living units provides them with individuation.


The many faces of logicism: a place for Alfred Whitehead

As muitas faces do logicismo: um lugar para Alfred Whitehead

Gregory Carneiro, PhD student – Department of Philosophy - University of Brasília

The logicism is a well-known line of thought in the hard task of creating solid foundations for mathematics. The crisis that took the peace of many mathematicians in the first half of the last century, in the logicism’s view, would be solved with an increase of rigor in all activities involving mathematics. In other words, logic would save mathematics from any crisis. Of course, this solution did not take long to earn a few powerful “enemies”, the most famous one was Jules Henri Poincaré.

For Poincaré, the mathematics would not gain much in being reduced to logics, nor in increasing any kind of rigor (e.g., the D. Hilbert’s formalism), since logics could not add anything new to mathematics, only intuition could. Russell, the paladin of logicism, opposed Poincaré in what resulted in a series of articles for both sides. Even though Alfred Whitehead was the author of one of the most representative works on logicism, Principia Mathematica, he is still often omitted in the discussion “logics vs intuition. And that is not without a reason. Whitehead’s position is not easy to put in a frame, and part of our work was to show why.

First, Whitehead’s opinion about the role of intuition is definitely not the one of Russell’s, what puts him next to Poincaré on this subject. At the same time, his conception of rigor in mathematics, and his own conception of what logics are all about, is much more developed and optimistic than Poincaré’s, what drags Whitehead back to logicism. The conclusion that goes after the study of the details of Whitehead’s ideas is surprisingly new to this important chapter of the history of mathematics: while being a logicist, one does not need to ignore intuitions. It is all a matter of where to place them. And this new place of intuitions helps us to show a whole new face for logicism.

Keywords: Whitehead; logics; logicism; mathematics.

Palavras-Chave: Whitehead; lógica; logicismo; matemática.


DESMET. Poincaré and Whitehead on Intuition and Logic in Mathematics in Process Studies Supplements, Issue 22: 1–61. 2016.

WHITEHEAD, Alfred. Adventure of ideas. New York: Macmillan Company, 1967.

____________. The organization of thought. London: Williams and Norgate, 1974.

____________. An introduction to mathematics. Dover Publications, 2017.

POINCARÉ. Intuition and logic in mathematics In The Mathematics Teacher, v. 62, n. 3, p. 205-212, 1969.

___________. On the nature of Mathematical Reasoning in Mathematics News Letter Vol. 3, n. 8, p. 3-5, 1929.

RUSSELL, B; WHITEHEAD, A. Principia Mathematica. Cambridge: University Press, 1912.

RUSSELL. My Philosophical Development, London: George Allan and Unwin, 1959.


1920’s Books: Whitehead’s Natural Philosophy and its Relation to Natural Sciences

Antonio Catalano (Italy). PhD candidate (second year, with scholarship) at Università Vita-Salute San Raffaele of Milan, Italy

The present abstract aims to analyse some questions inherent to the Whitehead natural philosophy of the 1920s. The theoretical study carried out by Whitehead on the foundations of mathematics first, then physical sciences, is also indispensable for understanding the specific density of the future process philosophy. With direct reference to the works of the twenties, the following questions will be addressed:

1. What is the relationship between natural philosophy and natural sciences? Whitehead’s goal is to determine the most general characters, which are necessary to understand things observed by the senses, and that will later be applied by particular sciences. «This philosophy exists because there is something to be said before we commence the process of differentiation».

2. What is meant by natural knowledge? In An Enquiry Concerning the Principles of Natural Knowledge (1919), Whitehead specifies the difference between natural philosophy and metaphysics: natural philosophy investigates nature as directly in contact with perceptual knowledge, and not as the synthesis operated by a knower of a known. No question concerning nature can be resolved by reference to a mind that should know it. If our perception is in direct contact with nature, it follows that natural knowledge does not contemplate nature as an external object, but it is a «knowledge from within nature», an awareness of the natural relations of one element in nature to the rest of nature.

3. Why do some interpretations of the theory of relativity risk awakening the «Berkeleyan Dilemma»? The Berkeleyan Dilemma is the following: Perceptions are in the mind and universal nature is out of mind, and thus the conception of universal nature can have no relevance to our perceptual life. Whitehead is concerned that Einstein's brilliant mathematical method is nullified by unconvincing philosophical systematizations; he thinks of some interpretations of relativity which rehabilitate a certain form of Berkeleyan idealism. The risk that Whitehead sees is the following: that the Berkeleyan observer, not satisfied with having absorbed the objects of sensible experience in his mental process, does the same with the uniformity of space-time.


Whitehead and Sheffer’s “incompatibility”:

An investigation of the relationship between metaphysics and logic

Naoki Arimura, Ristumeikan University, Kyoto, Japan

In this presentation, I will examine the relationship between Whitehead’s metaphysics and logic by considering the concept of “incompatibility.” A revolution in logic occurred between the mid-19th century and the early 20th century. This revolution implied a freedom from the bias of Aristotelian metaphysics. Traditional logic had presupposed Aristotelian metaphysics, and the bias had inhibited the development of logic. Therefore, logicians attempted to disconnect logic from Aristotelian metaphysics and succeeded in improving it. After the 1930s, philosophers and logicians became more radical, and excluded all types of metaphysics from logic. Ernest Nagel’s “logic without ontology” or Carnap’s conventionalism are typical examples of that movement. However, logicians who were in a period of transition from traditional logic to modern logic were not too radical to abandon metaphysics. Rather, they felt the need to make new metaphysics in harmony with modern logic. For example, B. Russell constructed metaphysics called “logical atomism.” I believe Whitehead also felt the need to connect modern logic with his metaphysics. To prove this, I will focus on a concept called “incompatibility” that is useful for understanding the relationship between his metaphysics and logic. Henry Sheffer invented the concept in 1913, and Russell used it in the second edition of Principia Mathematica as a primitive notion to define other logical notions. The symbol that represents the idea is well-known by the name of “Sheffer stroke.” Whitehead evaluated Sheffer’s idea, and used incompatibility in his own works. Whitehead did not just see the concept as a tool for logic, but one with metaphysical implications to it as well. In this presentation, I will discuss the point where Whitehead’s metaphysics meets the foundation of logic using the concept of incompatibility. Moreover, if time permits, I will also compare his metaphysics with other philosophies that were influenced by Sheffer (such as that of Russell and Wittgenstein).


‘Now’ as Instant, ‘Now’ as Interval: Examples from the Process of Music

Michael Heather & Nick Rossiter , Computer and Information Sciences, Pandon Building, Northumbria University, Newcastle NE2 1XE, UK

What is ‘now’? Now is more precisely defined by Alfred North Whitehead, OM (1861–1947) as an actual occasion that gives rise to a fact and is then gone. There are no facts in the past for all history is myth and the past is always therefore a matter of interpretation. There are no facts in the future for they have yet no actual occasion. The Universe itself may be defined as just one big NOW— but an extremely complicated notion for as Einstein explained simultaneity nowhere exists between any of its parts.

We can dig up artefacts from the past but their existence is now. We may have memories of the past but these memories only exist presently. Every entity in the World has a fleeting (covariant) capacity to act on and a (contravariant) perception to receive from its immediate surroundings. That is a local ‘now’ in both time and space giving rise to the synchronicity of the actual occasion where all the separate loci of each entity meet. These local ‘nows’ all compose to form the big NOW. Curiously Whitehead’s view of time as no more than the non-linear ordering of the World is really more in tune with Einstein’s intuitive notion of space-time-mass than his (Einstein’s) own solution of curving the absolute space of Newton. It also accords with Whitehead’s understanding of quantum mechanics.

Whitehead uses the technical term of ‘concrescence’ to describe this process of formation and composition. It is the process that brings into existence matter or value in Whiteheadian analysis. Matter is inexplicable in the standard Yang-Mills model of mainstream particle physics without a hypothesis of the existence of the Higgs-Boson and its subsequent controversial observation to fill the gap. It should be noted that that is bottom up and does not satisfy true topos theory which has no initial object. The Higgs-Boson like the inverse square law of Newton’s theory of gravity is a mathematical creature born out of Euclidean space which Whitehead rejected after the intense study of logic in his early years. However a very good example of topos process can be found in the role of ‘now’ in the forging of neurological paths in the brain.

To date it is only the emergence of category theory and the topos that enables the life sciences to escape the clutches of Euclidean space. It was Robert Rosen (1934–1998) who first proposed the use of category theory as a formal approach to the study of life itself but beware for mainstream is still running on a legacy version of category theory inherited with the failings of a Euclidean space founded in set theory. So the ‘topos’ of current literature is merely a reductionist model and only a shadow of the true topos that has no (so-called) ‘natural number’ object and therefore free of the constraints of Euclidean space. Although he did not live to see it developed, true topos theory is a full and faithful formal representation of Whitehead’s cosmology. It is also the appropriate ‘space’ to support quantum processes on which surely the living brain relies. It should be noted that the current mainstream version of quantum mechanics is a hysteron proteron construction of Whitehead’s thinking. The structure of a true topos is determined by all the potential relationships within it. All possible connections are a priori available. Whitehead uses the term ‘non-separable’ to describe this feature. That applies to the cosmos as a whole where relationships arise from the laws of physics. The same applies to any sub-cosmos within it such as the living brain where the relationships may be described as neural paths.

The existence and prime importance of limits and co-limits as universal was not really recognised until the 1970’s and not explicitly recognised by Whitehead although implicit in some of his writings. They need to be appreciated as operating at the level of metaphysics as recommended by Whitehead and therefore difficult to handle without category theory. Unlike physics which is never exact but only approximate, metaphysics can be precise for it potentially contains everything contrary to scientific models which are always reductionist. Co-limits are co-exact in that they are identifiable individually while remaining non-separable: a characteristic not easily representable in set theory, if at all.

Exactness arises from the unique relationships of adjointness. Thus for medical practice it is the difference between determining the proper remedy for the specific characteristics of a patient in personal medicine by contrast with relying on the results of statistical inferences from a wider population. At first sight this would seem even more critical for psychiatry and mental illness than in general medicine. However it is a much wider overarching problem for science as a whole. Whitehead’s later work is a broadside attack on the confidence that science places on number.

Statistical models only hold at first order and their application to higher order phenomena should always be treated with great caution. This includes the use of measurement as pursued by physicists with almost religious fervour as contrasted with their scepticism for ‘anecdotal evidence’. Measurement is a projection on to Euclidean space. For higher order phenomena anecdotal evidence may be the only evidence because it is evidence of now and the only statement of fact ever available. For every ‘now’ is unique in exactness. Reproducibility in scientific method is not repeatability. The Universe never repeats itself exactly. Whitehead does not include among his fallacies the false confidence in number but in reality it is allied to his fallacy of misplaced concreteness.

With the benefit of hindsight it is not surprising that category theory and the cerebral hemispheres follow equivalent relationships although historically their theories developed quite independent of one another. Left means rational; right means emotive. Left and right categories in a topos are related by a unique pullback f* functor: left and right cerebral hemispheres are connected by a nerve tract in the central cortex of the brain termed the corpus callosum. By the adjoint functor theorem the pullback functor can connect dynamically every relevant object in the left-hand category with every relevant object in the right hand category and vice versa. However, that vice versa is no mere simple ‘equal and opposite reaction’ in the limited sense of Newton’s third law. Rather each relationship takes account of every relevant relationship between every other object in the category. This is the fundamental structure of intuisionistic logic as introduced by Brouwer and developed formally by his student Heyting.

The key feature is relevant immediacy, the ‘now’ determined by the adjointness of theory and by the physics in the real world that give rise to the actual occasion which in formal terms is a monad as an object in a topos.

This is a recursive instance where the method becomes the subject for there is evidence that learning and applying category theory requires the use of both cerebral hemispheres of the brain and the spotlight is on communication between the two hemispheres. Another area requiring such extensive co-communication is music and it is reported that professional musicians are found to have an enlarged corpus callosum. Musical performance turns out as fine example to illustrate the operation of lateralisation in the brain. One hemisphere controls the operation of the other at a lower level but different functions control different operations while maintaining coherence as a whole. This coherent process is coordinated with the outside world as an actual occasion. In an orchestra there is a higher level of coherence in the one actual occasion of the whole. The conductor of the orchestra may be imagined as a personification of the actual occasion semiotically exhibited by the tap of the baton.

Take the first violinist. The left hand physically produces the pitch through intonation and the right hand physically performs articulation through bowing. On account of the cross level control the intonation on the violin is handled by the right-hand side of the brain and the articulation is handled by the left-hand side of the brain. Keeping rhythm is an example of an activity requiring the co-ordination of both hands in playing the instrument and delivering the musical performance is achieved by the coordination of both sides of the brain— an actual occasion.

The front portion of the human corpus callosum, has been reported to be significantly larger in musicians than in non-musicians and musical training has been shown to increase plasticity of the corpus callosum during a sensitive period of time in development. The implications are an increased bimanual coordination, differences in brain structure, and amplification of plasticity in motor and auditory faculties which would serve to aid in future musical training. Thus detailed studies of magnetic resonance in children who practised regularly for at least 2.5 hours a week between the ages of 6 and 9 were found to have a corpus callosum larger by about 25% relative to the overall size of their brain.

Learning to play a musical instrument by practicing is a process action with perception by the senses of the surrounding circumstances and therefore an ordered sequence of ‘nows’ or actual occasions able to train and hence enhance the communication between the exactness and co-exactness of the brain. While Whitehead rails against the bifurcation of nature this is shown to be distinguishable from bifurcation within nature.

Music is always a process and exhibits Whitehead’s very different concept of time from the Euclidean and provides examples of that different structure in the view of time. A single musical note is at a ‘now instant’ while the whole performance of a musical work is at a ‘now interval’ thus illustrating different levels of process space-time.


Whitehead’s Point Free Geometry analysed by Cantor Theorem perspective and its implications on internal event complexity relations

Rafael Martins, philosophy student at the University of Brasília and a Physics student at Paulista University

This presentation aims to demonstrate the implications of interpreting Whitehead's Point Free Geometry from Cantor's Theorem perspective. Whitehead presents, in his work The Concept of Nature, his Point Free Geometry, based on the method of Extensive Abstraction, establishing the event as a primordial and unitary factor that retains in itself the passage of nature. An event, naturally, occurs extensively in relation to other events, since as principles of relational extension of events, in verticality, Whitehead defines that each event contains other events as parts of itself, as well as each event is a part of other events. The conclusion reached by the types of vertical relations required between events is that, starting from any event, it is possible to advance infinitely to the smaller or larger extremity of the cluster of events, without finding a universal or atomic one. As an operational principle of Point Free Geometry, seeking exactitude within this infinity, Whitehead postulates that if A and B are two events and A' is part of A and B' part of B, the relations between parts A' and B' will be simpler than the relations between A and B. However, A' is only one part among infinite others, which together form event A, the same holds for B' with B. The Cantor Theorem, in its turn, shows that, since X is a set, then the set of parts of X is larger than the set X (#X < #P(X)). In this way, the relation between events A and B, thought as mathematical sets, is in fact more complex than the relation between any of its parts, A' with B', but cannot be more complex than the relation between all its parts, P(A) with P(B). That is, the relation between A and B is simpler than the relation between the set of parts of A, P(A), with the set of parts of B, P(B). This leads us to conclude that the greatest complexity between events is given horizontally, that is, in the relation of two events at the same depth level in the aggregate cluster of events delimited for analysis.


Whitehead’s Notion of Creativity in new Approaches to the Living: Inspiration and Differences

Roland Gazalis (Belgium) lecturer and researcher in the Department of ‘Sciences, Philosophies, Societies’ of the University of Namur (Belgium), and lecturer at the Jesuit faculties of Paris , France

The category of the ultimate refers to the first concepts that are decisive for the series of notions and principles that derive from them. This is the case for creativity in Process and Reality. Creativity is the operator for the unification of new existences from the already made up ones (PR, 73). Since Whitehead distinguishes living nature from lifeless nature, scholars, Whitehead's readers, have proposed alternative approaches to life. In this study, we mention two models, which show the pervasive nature of the notion of creativity. Various categories are used to echo this notion, while keeping their own particularities.

The first approach is Akkerhuis's Operator hierarchy. Broadly speaking, it is a ladder that ranks all types of physical particles and types of organisms, according to a discrete transition into the complexity of their organization. It provides with a basis for a definition of life. The second approach follows Francisco Valera's work. Miguel Benasayag proposes this model, which he names Mamotreto. It is an organogenesis that helps to envision the particularity of life in a hierarchical model with three levels: the aggregates, the biological field, and the mixed. More generally, this model results from Benasayag’s dialogue with mathematicians Giuseppe Longo and Francis Bailly, about the physical singularity of life. Both models respectively approach living organisms from the evolutionary biology perspective, and from an individual living organism situation.

We show that these models indirectly help to understand the notion of creativity, and what it generates. In each case, in light of a physical analogy, the geometry of an attractor can describe its trajectory. In the operator theory paradigm, the emergence of new entities highlights that in the space of all processes, only a few sets of entities subsist. Thus, the evolutionary trajectory landscape suggests networks with attractors’ basins. In Benasayag’s paradigm, we understand the living as an "extended critical situation". As such, the authors compare the living to the critical state, in physics, described as a singularity in a process, i.e., a configuration briefly accessible during that process, before the observed overall state changes. A system cannot remain in a critical state indefinitely. However, the biolon, or the unit of life, as these authors call it, which includes its constituents, regulations, and an environment, remains an "extended physical singularity". In this case, this phenomenon is similar to a more complex mathematical figure, in which the trajectory seems to loop on itself in a singularity. In short, we show that these models allow us to evaluate in a pragmatic way the notion of creativity as an invariant of the organic process, by the way it holds the process together, while guiding it and providing its impulse.


The Idea of Creativity in Whitehead’s Writings on Mathematics

Vesselin Petrov, Bulgarian Academy of Sciences

The question that will interested me in the present exposition is, first, when the idea of creativity appears in Whitehead’s writings on mathematics and, second, how this idea works in mathematics itself and in philosophy of mathematics? So, I shall follow chronologically the appearance of the idea of creativity in Whitehead’s writings on mathematics and the evolution of his intellectual development with regard to the function and role of creativity in mathematics. The key role of creativity in Whitehead’s metaphysics is emphasized in his last publication Immortality (1941). The connection of creativity with mathematics is emphasized by Whitehead in his last publication on mathematics Mathematics and The Good (1941). the conclusion of my exposition is that Whitehead’s views concerning the idea of creativity in mathematics was formed in his last mature period of intellectual development after 1920ties and he has supported firmly these views till the end of his life.


Nothing in Whitehead's philosophy and in modern physics makes sense except in the light of the relatedness of things

Christian Thomas Kohl, University of Education Freiburg

In my contribution I will exemplify this sentence by [10-30] key sentences of Whitehead‘s contributions and by [3] key concepts of quantum physics (entanglement, interaction, complementarity) and by Einstein‘s sentence: „A courageous scientific imagination was needed to realize fully that not the behaviour of bodies, but the behaviour of something between them, that is, the field, may be essential for ordering and understanding events“.