Order and Change: The Memoir “On Mathematical Concepts of the Material World”

Le cose tutte quante
hanno ordine tra loro, e questo è forma
che l’universo a Dio fa simigliante
(Dante, Paradiso I, 103-105).

In the middle of a difficult decade of work, during which the monumental achievement of Principia Mathematica was completed, Whitehead delivered before the Royal Society a valuable memoir dealing with “the mathematical investigation of various possible ways of conceiving the nature of the material world” (MCMW, 65). The memoir was received in September 1905, and read later in December. Whitehead had not previously published on the topic, nor would he within the next ten years. In despite of this, MCMW was tightly connected to his research programme as sketched in 1912:

During the last twenty-two years I have been engaged in a large scheme of work, involving the logical scrutiny of mathematical symbolism and mathematical ideas. This work had its origin in the study of the mathematical theory of Electromagnetism, and has always had as its ultimate aim the general scrutiny of the relations of matter and space.[1]

As we shall see, the problem of the action between distant bodies is the source of Whitehead’s suspicion of Cartesian physical views, as they were developed according to Newtonian mechanism. Once nature had been divided into separate classes of points, instants and particles, there was no way, Whitehead thought, to restore the relatedness of the material world that we perceive. What needs clearing up is the definite import of the conceptual partitions between space, time, and matter that were introduced during centuries of scientific advancement. Though the use of some scientific terms had become established, and almost unavoidable, the meanings attached to them needed to be revised in order to take new phenomena into account.

Such a task is beyond the abilities of any single person. Nevertheless, since the beginning of his career, Whitehead held that mathematics allows the substitution of human thought by the manipulation of conventional signs which, in turn, serve as substitutes for things and their properties.[2] Mathematics functions as a mirror of the world, at least of the ordered part of it. The manipulation of signs, according to fixed rules, shadows the course of natural events. On the other hand, mathematicians proceed inductively, seeking the simplest generalizations that will both make individual theories more coherent and provide means of connecting previously disparate theories. It was the power of mathematical abstraction and simplification that Whitehead sought to harness in his own work.[3]

MCMW can be read as the opening of a stream of thought that leads to Whiteheadian philosophy of science. However, the actual purpose of the memoir was to show how geometry is supposed to work when applied to the material world. An improvement of an abstract model of correspondence between mathematical language and observed nature should, in Whitehead’s view, pave the way for the subsequent unification of both geometry and physics. Therefore, the theoretical model of MCMW is not really concerned with the choice between different concepts of the material world. The variety of concepts is of the essence of the hypothetical reasoning as extended to mathematical physics. The final choice must be left to experimental verification, unless the logical scrutiny has revealed some inconsistency affecting the internal structure of the concepts.

The following account will stress the epistemological import of the memoir, as well as the variety of intellectual streams that Whitehead synthesized. In addition, the multiple purposes enclosed in this dense masterpiece, variously stated by the author, need to be disentangled. Concerning the historical circumstances of the memoir, I will refer often to the thorough work of Ivor Grattan-Guinness.[4]

1. The Scientific Context of the Memoir

MCMW drew upon at least three lines of research, all of which sought to demonstrate the fundamental cohesion of scientific knowledge. These are, roughly speaking: the subordination of mathematics to formal logic, the axiomatization of geometry, and the unification of physical laws.

1.1. Russell’s Logicism and the Theory of Relations

The first of these, in the words of Bertrand Russell, maintained that:

all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and […] all its propositions are deducible from a very small number of fundamental logical principles (Russell 1903, xv).

In fact, it is doubtful that Whitehead completely subscribed to logicism. The idea that mathematics could be nothing but a branch of logic is in a sharp contrast with Whitehead’s previous statement of mathematics as being “the development of all types of formal, necessary, deductive reasoning” (UA, vi).[5] However, Whitehead’s projected second volume of UA was set aside, and the fruitful co-operation between him and Russell began. Whitehead had perhaps realized that his second volume “did not have the mathematical or philosophical weight carried by Russell’s ambition, whose fulfilment would require too much effort for even that young man” (Grattan-Guinness 2002, 433). On the other hand, it is no surprise that the logical scrutiny of mathematics as a whole could appear to Whitehead as an end in itself. For many years, following Grassmann’s Ausdehnungslehre, he had investigated the general properties of algebra. His own results showed that, for instance, by assigning suitable spatial meanings to the symbols, a great number of geometrical propositions could be translated into algebraic language and fully elaborated through it.

At the outset of the twentieth century, mathematicians became increasingly aware of the advancements of predicate logic. The symbolism adopted by Giuseppe Peano helped to bring greater attention to the work of Gottlob Frege. Russell was greatly excited when he discovered Peano’s mathematical logic at a congress held in Paris.[6] Within the same year, he wrote the bulk of The Principles of Mathematics. Apart from its zealous advocacy of logicism, Russell’s book was a striking catalogue of what could be done with few logical concepts and principles. From Whitehead’s point of view, logic may have well appeared as the valuable formal language through which a higher level of mathematical generalization could be attained. In a sense, logic appeared to be more universal than algebra.

Russell’s focus on the logic of relations is perhaps his most brilliant contribution to Whitehead’s intellectual development. It may be true that Whitehead “was born a relativist,” as Russell recalled a long time afterwards (Lowe 1985, 299), yet the logical treatment of relations gave to his natural bent a definite impulse whose relevance cannot be denied. Beginning from MCMW, relation became the fundamental category of Whitehead’s thinking, both mathematical and philosophical, until it was absorbed by the broader notion of process.[7] Moreover, the analytical approach of his former pupil to highly controversial matters may have encouraged Whitehead to adopt it also, though more cautiously than Russell. This is perhaps the case for the primacy of atomic entities, which is essential to Russell’s theory of classes, and which dominates the first half of Whitehead’s memoir. Even the absoluteness of space, as opposed to Lotze’s theory,[8] is defended by Russell on purely logical arguments.

1.2. Veblen’s System of Axioms for Geometry

The axiomatization of geometry was undertaken in its modern form by Moritz Pasch. His Vorlesungen über neuere Geometrie (1882) proposed a system of nuclear propositions about undefined concepts in terms of which all other geometrical concepts must be defined, and from which all other geometrical propositions must be deduced. Though Pasch’s nuclear propositions were still supposed to be empirical, he had developed “the prototype of an abstract science, which ignores the origin of its principles and does not care about the applicability of its conclusions” (Torretti 1978, 211).

Pasch’s methods were soon adopted by Peano, who wrote the axioms of projective geometry in a formal language.[9] His followers, namely Gino Fano, Mario Pieri, and Alessandro Padoa, stressed that the validity of a system of axioms ought not to depend on the particular meanings assigned to primitive terms such as “point” or “line.” David Hilbert’s Grundlagen der Geometrie appeared in 1899, and was a milestone of the scientific development: it made possible higher levels of comparison between the structures of different geometrical theories, and, against former intuitionalist theories, argued the hypothetical status of the axioms.

In April 1903, Oswald Veblen presented to the American Mathematical Society a system of twelve axioms for Euclidean geometry. Veblen’s group of axioms presupposes “only the validity of the operations of logic and of counting” and is stated “in terms of a class of elements called ‘points’ and a relation among points called ‘order’” (Veblen 1904, 344). All remaining geometrical concepts are defined in terms of point and order. Whitehead was impressed by the idea of deriving geometry from a single triadic relation. The procedure could be easily translated into the logic he was elaborating together with Russell. Indeed, Veblen’s set of axioms is applied throughout MCMW. Each mathematical concept of the material world is endowed with a set of axioms, according to the varying properties of the fundamental relation of order. Euclidean geometry is then derived from each set of axioms.

Instead of providing for the treatment of both projective and metric geometry, as Veblen did, Whitehead dealt exclusively with the former. Metrical ideas are not found in MCMW. The omission may be due to the primacy Whitehead accorded to the notion of order, as opposed to measure, for conceiving of the material world. The preface of his contemporary tract on projective geometry includes among “the triumphs of modern mathematical thought” (APG, v) the theorems by which it is proved that “numerical coordinates, with the usual properties, can be defined without the introduction of distance as a fundamental idea” (APG, v). In the same book geometry is defined as “the science of cross classification,” whose system “dominates all external existence” (APG, 5).

From Poncelet to von Staudt, projective geometry had lost many of the original links to perspective view. The character of new entities such as the point at infinity eludes even graphical representation. Accordingly, spaces which are based on projective elements are almost impossible to grasp intuitively. Moreover, Whitehead knew that points and lines are interchangeable, inasmuch as the theorems of plane projective geometry are true of the plane whether we regard it as a set of points grouped in lines or as a set of lines grouped in points. MCMW may be viewed as the extension of the so-called “principle of duality” to three-dimensional space.

1.3. Maxwell’s Theory of Electromagnetism

Whitehead was educated at Trinity College, Cambridge, since the autumn of 1880. During this period, James Clerk Maxwell was reaching the height of his fame and he was teaching at Cambridge[10] when the Treatise on Electricity and Magnetism appeared in 1873. However, “the impact of the Treatise was at first muted, and at the time Maxwell’s reputation rested largely on his work on molecular physics and gases” (Harman 1998, 1). It was only after his death in 1879 that the theory of the electromagnetic field received experimental confirmation and “came to be regarded as one of the most fundamental of all physical theories” (Harman 1998, 1). The successful unification of electric, magnetic and luminous phenomena had contributed significantly to the goal of unifying all natural knowledge.

The young Whitehead, who had attended lectures on both pure and applied mathematics, wrote his fellowship dissertation (now lost) on the theory of electromagnetism.[11] Three years later, he published a paper on the motion of viscous incompressible fluids (Whitehead 1888). As a matter of fact, Maxwell’s first paper about the theory of the electromagnetic field had developed the physical analogy between magnetic lines of force and the flow of an incompressible fluid in tubes of varying section (Maxwell 1856). The analogy showed that the already known hydrodynamics equations could suitably represent the intensity of the electromagnetic force. Maxwell also assumed the primacy of Faraday’s lines of force, though in his subsequent works he insisted that a mechanical model of the ether was required in order to explain electromagnetic induction and the generation of electric currents. The man who had turned Faraday’s physical ideas into elegant mathematical equations never got rid of the mechanical mentality he had inherited.

Whitehead, on the other hand, grew up without such a mentality. Just as the mathematics was no longer being based on quantitative ideas, so the mechanical paradigm was questioned by the rise of field as a basic concept. In addition, the development of vector physics, to which Grassmann and Hamilton had contributed ante litteram,[12] had great formative influence on Whitehead. It is not too surprising, from our modern point of view, that both MCMW and Einstein’s first account of the theory of relativity (Einstein 1905) were written in the same year. Physics has always been the horizon of Whitehead’s mathematical research.[13] Is the notion of field, as applied to the logic of relations, really extraneous to the same notion as referred to contiguous elements of the space in the neighbourhood of the electric or magnetic bodies? The answer to such question is intimately bound up with philosophical interpretations of MCMW.

2. The Contents of MCMW

2.1. An Essay of Mathematical Cosmology

It is not an easy task to point out the purpose of the memoir. Whitehead himself gives different statements in the introductory section. On the other hand, different disciplines are intertwined, or else amalgamated into a new one. From an overall viewpoint, the memoir turns on the connection between the material world and its mathematical concepts. The material world is defined as “a set of relations and of entities which occur as forming the ‘fields’ of these relations” (MCMW, 466). It is plain that “material world” is the name given to an implicit cosmological theory dealing with both relations and entities. That is, the phrase does not refer to the physical world itself. On the other hand, a mathematical concept of the material world is a “complete set of axioms, together with the appropriate definitions and the resulting propositions” (MCMW, 466). The whole frame is therefore expressive of a definite mathematical theory about a vaguely stated cosmological theory.

Mirroring the intuitive description of the cosmological concepts, five mathematical concepts are then developed by assuming the essential relation of order between points as the argument of the various sets of axioms. The mathematical concepts also possess the property that the theorems of Euclidean geometry can be deduced from each complete set of axioms and of definitions. As a consequence, the connection between the cosmological theory and the corresponding mathematical theory is such that different interpretations of the primitive terms do not affect the validity of Euclidean geometry as expressing the fundamental properties of space. Since the axioms settle the formal relation of order holding between points, the notion of order is supposed to be the connecting link between cosmological thought and mathematical reasoning. At any rate, such epistemological ideas are left unquestioned. The pattern of correspondence is purely hypothetical.[14]

Along with mathematical concepts of the material world, two alternative theories of space are considered. The absolute theory of space, tailored to the requirements of pure mechanics, dominates both Concepts I and II. According to Whitehead, the theory means at least that points without extension are the real entities constituting space. That is, points exist apart from both particles of matter and instants of time. The relative theory of space is termed “Leibnizian.” Euclidean points are dropped, while the name “point” is referred to a bundle of linear entities which are neither spatial nor material in nature. Rather than absolute essences, space and matter are modes of relating the same primitive entities. Far from being timeless, relative space is altered from instant to instant. In a Leibnizian[15] concept of the material world, the essential relation always includes linear entities as well as an instant of time. It is a momentary relation of order, resulting in “a protest against exempting any part of the universe from change” (MCMW, 467).

Whitehead proves Euclidean geometry to be sound for each of the Leibnizian Concepts III, IV and V. Hence space can be Euclidean without the further postulation of either the atomicity of points or their absolute separation from time.[16] This task was accomplished by suitably chosen geometrical means, namely the theory of intersection points (or interpoints) and the theory of dimensions. The development of such theories, which is proposed as “the main object of the memoir” (MCMW, 466),[17] occupies several pages, even though both theorems and proofs are written in the concise language of Principia Mathematica.

2.2. Spaces, Relations, Entities

The concepts of the material world can be grouped differently depending on the type of space, the class of fundamental relations, and the kind of entities which are considered. Space is either absolute or relative in the above sense. The class of fundamental relations always includes (i) a polyadic relation R which is called the “essential relation;” (ii) the time-relation holding between instants; and (iii) a set of additional relations containing at least one member. The entities form the fields of the fundamental relations, so that the cosmological theory referred to in the previous section at least means that every entity of the material world bears some kind of relation to other entities. However, no hint can be found that relatedness precedes individual existence as far as the primitive entities are concerned.

The whole class of the entities is named the class of the “ultimate existents.” The class includes all the instants of time together with the sub-class of the “objective reals.” Both points and particles are objective reals, yet Whitehead adds a third kind of entity, “the lines of force of the modern physicist” (MCMW, 482). The so-called “linear objective reals” are never found together with points, though they coexist with particles in Concept IVA.

(See Table 1 attached infra.)

Concept I is a rough account of Newtonian cosmology in which the influence of Russell’s analytical manner is clearly present. The concept is “punctual,” since the class of the objective reals includes points instead of linear entities. It is also “dualistic” as far as the same class includes both points and particles. The essential relation R holds between points, whereas an indefinite number of “extraneous” triadic relations are required holding between each material particle, a point, and an instant of time. Such triadic relations, determining the position of particles, are “extraneous,” as are the points and particles between them: “relations between strangers” could be a good paraphrase of the term. Indeed, in the “classical” concept space is absolute. It lies apart completely independent of time, matter, change and motion, perfect in its beauty as revealed in geometry. “Unfortunately,” Whitehead writes ruefully, “it is a changing world to which the complete concept must apply” (MCMW, 479). That is why, he supposed, a separate class of atomistic particles had to be introduced as a necessary substratum for change and perception. Whitehead seems to consider both particles and their extraneous relations to points as awkward remedies for the embarrassing disagreement between the competing claims of logic and experience.

From a physical point of view atomicity is best suited to mechanical explanation, though the evidence of actions between distant bodies forces the supposition of an ether filling all space. Instead of multiplying entities in order to keep the absoluteness of space, Whitehead aimed at a more streamlined account of the entities needed for any physical theory: ether is no longer divorced from space, and a different way of conceiving the entities may result in a monistic concept of the material world, with no ultimate ontological barriers between space, time and matter.

Concept II gives monistic form to the classical concept in which the particles are abolished. Nothing is left to be perceived but dyadic relations holding between points and instants. This is hardly a concept of the material world. It is in fact an ironic remark that absolute space is much more rooted than matter in the classical concept. As Russell boldly wrote: “it is plain that the only relevant function of a material point is to establish a correlation between all moments of time and some points of space” (Russell 1903, 468). If a choice is necessary between space and matter, so much the worse for matter.

The remaining concepts increasingly alter the “classical” assumptions on absolute space, punctual entities, and indefinite extraneous relations without questioning the validity of Euclidean geometry. Indeed, Concept III abandons the idea of absolute spatial position. Since the instants of time are members of the field of the essential relation, “R;(abct) may be read as stating the objective reals a, b, c are in the R-order abc at the instant t” (MCMW 480). Though still composed of punctual entities, which can be either particles of ether or moving points, space is no longer exempt from change. The pattern of the entities which constitute space can vary from instant to instant.

As a consequence, the notion of matter as a substratum for change becomes irrelevant. Bodies are simply volumes, and different volumes are defined by some peculiarity of the motion of the ether/points. The rule of change to which a set of punctual entities is subject identifies the latter as a body. In short, bodies resemble waves. What we recognize when looking at a single wave on the sea, for instance, is the persistence of a certain type of motion in a fluid which is continuously changing. However, a slight variant of this concept results if “the persistence is that of the same objective reals in the same special type of motion” (MCMW, 482).

The whole class of the objective reals is composed of moving points or ether particles. The specific advance of Concept III, compared to Concept I, is that space and matter become indistinguishable. Instead of many extraneous relations determining the position of each particle of matter, a single tetradic relation S of orthogonal intersection is required which holds between three straight lines at any instant of time. In so doing, a system of reference is provided in order to define both velocity and acceleration. According to Whitehead’s early notion of time, “corresponding to any instant t in the fourth term, there is one and only one line for each of the other terms respectively” (MCMW, 481). This is to say that an absolute system of kinetic axes is associated to each instantaneous space of Concept III. As relations between points change in time, so the axes are subject to translation. The continuity of motion, in spite of the atomicity of time, rests on the assumption that different spaces at different times bear similar structures of Cartesian axes.

Concepts IVA and IVB enlarge the relativity of space by repealing the idea of points as primitive entities. Here the notion of a projective point becomes Whitehead’s model. The whole bundle of linear entities concurrent “at a point” is to be taken as the point proper. Points are “knots of relatedness,” according to Enzo Paci’s beautiful description (1965, 57). Networks of linear entities appear and disappear ceaselessly. As a consequence, “the points of one instant are, in general, different from the points of another instant” (MCMW 483). Linear objective reals, on the other hand, are enduring entities in a temporal sense as well as indivisible in a spatial sense.

Concept IVA is dualistic. A class of particles is associated with both points and instants by means of triadic extraneous relations. Particles may be seen as plunged in a network of linear entities acting as lines of force. Accordingly, “the motion of the particles may be conceived to be influenced by that of the linear objective reals, and vice versa” (MCMW 491). Since any two points (or bundles) have in common a linear entity, the particles of matter which occupy those points cannot be quite separated in a spatial sense, as long as space is the pattern of the linear entities. The problem of action at a distance is avoided, since, in a sense, there is no distance. Absolute separation becomes impossible. The unbroken relatedness of the linear entities is such that any point of space is bound up with every other point. The same spatial structure is found in Concept IVB, which is a monistic variant similar to Concept II.

The notion of point as a complex is not without defect. Indeed, the definition of point as a bundle of lines concurrent at a point is circular. To overcome this tricky problem, the geometrical theory of interpoints is worked out. Though the use of terms such as “point” or “intersection” is avoided in its formal statement, the theory can be explained less rigorously as follow. The linear objective real a intersects the objective reals b, c, d in the order bcd at the instant t. The linear objective reals b, c, d are members of three different bundles or interpoints centred on a. That is, the three different interpoints have a single linear objective real in common which is a. The intersection order bcd holds whatever the members of the interpoints which are considered apart from a. In other words, the pentadic relation R;(abcdt) can be read as the statement that every member of any three interpoints upon a has a “similarity of position” with respect to the intersecting member a at the instant t. Accordingly, the intersection of four linear objective reals, as expressed by R;(abcdt), assigns a linear order to the interpoints. This relation of linear order is the essential relation R of both Concepts IV and V. In the respective systems of axioms it takes the place of the triadic relation between points, by the help of few additional definitions.

Concept V is the simplest from a logical point of view, though its mathematical treatment is by far the most esoteric. The class of the extraneous relations is reduced to one member only, while the class of the ultimate existents includes only the instants of time together with the linear objective reals. Despite the logical simplicity of the scheme, Concept V is completely remote from intuition. As for Concept III, the corpuscles are to be taken as volumes defined by some peculiarity of motion of the objective reals. However, since the objective reals are linear instead of punctual entities, they pass the volumes and proceed at infinity. The distinction between a corpuscle and the surrounding field of force is no longer due to the nature of the respective ingredients. It is rather derivative from different patterns of relation between the same physical entities. In any case, Whitehead is faithful to the hypothetical approach of the memoir. The laws of motion of the linear objective reals are not actually sought, even if the hint is given that “[t]he endeavour to state such laws appears to reduce itself to rewriting with appropriate changes a chapter of any modern treatise of electricity and magnetism” (MCMW, 491-92).

The points of Concept V are much more complicated than interpoints. From the outset Whitehead has in mind the physical analogy which is sketched in the final section of the memoir, where different types of points are associated with both negative and positive electrons as the component parts of a corpuscle. The harmonization of the most advanced concept of the material world with modern physical ideas required an increasing number of subatomic particles to be assumed without abandoning the primacy of the linear objective reals. Roughly speaking, different types of points were needed for physical reasons.[18]

The brilliant achievement of Concept V rests on the “theory of dimensions.” An abstract property of flatness is defined that applies to classes of straight lines in a three-dimensional space. A special case of flatness, which is termed “homaloty,” is then referred to classes of linear objective reals and the notion of “cogredience” is developed as expressing the idea of parallelism. By the aid of three axioms introduced on purpose, the relation of interpoints to points is stated. According to the axioms, a point can contain both a secant and a nonsecant part. The secant part of a point is made up either of one or of several interpoints, while the nonsecant part of it, if present, is a single flat stream of linear objective reals concurrent at infinity. Such baroque architecture is the point of Concept V. As a further step, a point-ordering relation is defined holding between three points and an instant of time. In fact, the relation is the projection of the linear order between three interpoints upon the objective real x (as defined through the theory of interpoints) on three points upon the objective real a, not intersecting x, according to the “universal preservation of order by ranges in perspective on a pair of lines” (MCMW, 509). Finally, Whitehead proves that the point-ordering relation of Concept V has the same properties as the essential relation R of Concept I. Hence, the ordinary Euclidean geometry holds of points, lines, and figures which, at the same time, may represent particles, trajectories and atoms in a physical sense.

3. The Unfortunate Fate of MCMW

Thirty years later, Whitehead still considered MCMW to be “the most original thing he had done.”[19] Yet despite the remarkable synthesis it represents, the memoir was destined to oblivion. Not even the fame of the Principia Mathematica, written in the same formal language, could draw attention to it. In 1941, when Victor Lowe published his fundamental work on the development of Whitehead’s philosophy, Morris Cohen alone had referred in print to MCMW (1932). It was not until 1953 that the memoir was reprinted.[20]

There is no doubt that Lowe’s penetrating account paved the way for all subsequent interpretations of the memoir. His clear statement of the subject is coupled with suggestive links to the scientific advancements of the time. Also, the anachronistic attitude of tracing Whitehead’s metaphysics back to 1905 is wisely avoided. Though MCMW unquestionably presents the first Whiteheadian criticism of scientific materialism, the point is that “[t]he criticism is logical, not physical or philosophical.”[21]

In 1961 Wolfe Mays wrote the first article about MCMW. Mays considers both mathematical and physical issues at length, though the accent is placed on seeming adumbrations of Whitehead’s mature philosophy. It may be true that the axiomatic method of the memoir bears “a family resemblance” (Mays 1961, 258) to the philosophical method of PR. However, such recognition depends on the reader’s familiarity with PR: the memoir should be evaluated philosophically on its own terms, as much as possible, without reference to Whitehead’s later work.[22] Lowe has rightly criticized Northrop’s statement that Concept V is likely to be “the metaphysical system which was stated in [Whitehead’s] technically modified English prose in Process and Reality.”[23] On the contrary, he concurs with Bendall’s statement that “it is just as harmful to overestimate the significance of [MCMW] to the development of Whitehead’s philosophy as it is detrimental to neglect it altogether.”[24]

Robert Andrew Ariel’s examination of MCMW points out the influence of the formal mathematical approach on Whitehead’s later thought.[25] Following the above mentioned similarities with PR, he argues that Whitehead was always a philosopher, even in his early writings. In her 1975 article, Margaret O’Rourke Boyle applied the notion of “similarity of position,” on which the theory of interpoints rests, to her model of the spatial relationship between God and the universe (Boyle, 1975). During the last fifteen years, there has been renewed interest, from a mathematical perspective, in Whitehead’s early writings.[26] However, MCMW seems to be generally neglected by scholars. The exception here is Ivor Grattan-Guinness whose work provides a detailed historical background of the memoir, together with a versed exposition of the geometrical issues (Grattan-Guinness 2002). He also discusses some of the work’s limitations, with regard to mathematical axiomatization, mechanics and physics.

4. Is there a Philosophical Strategy in View?

If a philosophical strategy was hidden behind the cryptic symbols of MCMW, Whitehead did not divulge it. The mathematical investigation of the subject, he wrote, “has an indirect bearing on philosophy by disentangling the essentials of the idea of a material world from the accidents of one particular concept” (MCMW, 465). The accidents of the classical concept were summed up by dualism. As Lowe maintains: “Whitehead looks upon the dualism of points and particles as a challenge to theoretical thought. He was never happy with final, unrelieved dualisms” (Lowe 1985, 297). Accordingly, a philosophical strategy may be supposed which was aimed at, so to speak, “including space in matter” through the notion of bodies as permeable volumes.

Concerning Whitehead’s possible philosophical strategy, let us make an analogy with Plato’s Timaeus. In order to provide a likely explanation of the relationship between the receptacle and the physical elements (Timaeus 48A-56C), Plato resorted to the most advanced geometrical methods of his time. The exhaustion of plane surfaces by means of triangles was due to Eudoxus of Cnidus, while Theætetus had discovered that regular polyhedrons were finite in number. Plato’s wonderful vision of the elements as different types of polyhedrons begins with the attribution of depth to bodies. But all bodies with depth are bound by plane surfaces, and plane surfaces are made of triangles. Two types of right triangles are selected as the ultimate entities which compose the five polyhedrons. In a quite analogous way, Whitehead summoned up the resources of both modern axiomatic and projective geometry to strengthen the physical idea of linear entities, neither spatial nor material in the usual meaning, as being the ultimate elements of the universe.

This hypothetical strategy involves some kind of correspondence between mathematical theories and physical ideas. The relation of order makes the correspondence feasible. Indeed, the essential relation holds between the symbols occurring in the axioms of the mathematical concepts as well as between the ultimate existents of the physical concepts. In a sense it is the same relation, but what is then a relation within the theoretical frame of the memoir? Does a relation exist apart from its field? It was definitely beyond Whitehead’s resources to answer such questions in 1905. My assumption is that relations belong to unexplored meta-theoretical levels forming the vestibule of Whitehead’s epistemology.[27] The direct contribution of MCMW to his intellectual development was due to the practice of axiomatization as a method for connecting different kind of theories. Thus, hypothetical reasoning became the characteristic mode of thought of the future philosopher.

Table 1


“All things that are have order/Among themselves, and it is this their form/That makes the universe a mirror of God.”

[1] Letter to the dean of the University College London, in Lowe 1985, 155-56.

[2] See UA, Book I, Chapter I, §§ 1-2, 5-6.

[3] “By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race” (IM, 59).

[4] See Grattan-Guinness 2002, § 3.

[5] It is noteworthy that, while announcing his own treatment of Boolean logic in the second book, Whitehead adds “the results of this book are not required in any of the succeeding books” (UA, ix).

[6] The first international congress of philosophy was held in Paris in August 1900, under the auspices of the Revue de métaphysique et de moral. See Grattan-Guinness 2002, 431.

[7] As Whitehead wrote in a endnote of PNK, “the book is dominated by the idea that the relation of extension has a unique pre-eminence and that everything can be got out of it. During the development of the theme, it gradually became evident that this is not the case, and cogredience had to be introduced. But the true doctrine, that ‘process’ is the fundamental idea, was not in my mind with sufficient emphasis. Extension is derivative from process, and is required by it” (PNK, 202).

[8] See Russell 1903, Chapter LI.

[9] See Kennedy 1980.

[10] In 1871 Maxwell had accepted an offer from Cambridge to be the first Cavendish Professor of Physics.

[11] See Russell 1959, 43.

[12] See Crowe 1967.

[13] Cf. Athearn 1997.

[14] Whitehead still dress mathematical clothes, and “[t]he business of mathematics is simply to follow the rule” (UA, p. vi). Any investigation of the philosophical import of order would exceed the legitimate scope of mathematics. Indeed, the memoir shows how the primitive notion of order could be applied for connecting geometry to physics, whatever order may be apart from axiomatic definitions.

[15] Whitehead knew the Leibnizian theory of space only through Russell 1900 and Couturat 1901.

[16] It is worth noting that Whitehead never ceased to prefer Euclidean space in the subsequent years. At Harvard he tried to persuade Albert Einstein that the theory of general relativity should assume space to be Euclidean for philosophical reasons (see R, Part I, Chapter II).

[17] The other parts of the memoir are “explanatory and preparatory […], though it is hoped that they will be found to have some independent value” (MCMW, 466).

[18] Introducing Concept V, Mays noted: “it looks as if Whitehead first arrived at it by reference to electromagnetic theory rather than as a result of purely logical deliberations” (1961, 248).

[19] The remark was made in a conversation with Lowe on December 2, 1936 (Lowe 1985, 296).

[20] This is in the Anthology edited by Northrop and Gross (1953).

[21] Lowe 1951, 35. The opinion is confirmed in Lowe 1962, 158 and Lowe 1985, 297.

[22] See Gaeta 2002, 9.

[23] Northrop 1961, xxii, in Lowe 1985, 332, note 16.

[24] Bendall 1973, 3, quoted in Lowe 1985, 332 note 16.

[25] Ariel 1974; cf. Gaeta 2002, 72-75.

[26] See for example Henry and Valenza 1993.

[27] Compare the relevance accorded to the inclusion relation in “La théorie relationniste de l’espace,” Revue de métaphysique et de morale, 23, 1916, 423-454.

Works Cited and Further Readings

Ariel, Robert Andrew. 1974. “A Mathematical Root of Whitehead’s Cosmological Thought,” Process Studies, 4, 107-13.

Athearn, Daniel. 1997. “Whitehead as Natural Philosopher: Anachronism or Visionary?” Process Studies, 26, 3-4, 293-307.

Bendall, R. Douglas. 1973. On Mathematical Concepts of the Material World and the Development of Whitehead’s Philosophy of Organism (Berkeley, Master’s thesis for the Graduate Theological Union—forthcoming with Éditions Chromatika, Louvain-la-Neuve).

Boyle, Margaret O’Rourke. 1975. “Interpoints: A Model for Divine Spacetime,” Process Studies, 5, 191-94.

Couturat, Louis. 1901. La logique de Leibniz d’après des documents inédits (Paris, Félix Alcan). Reprinted 1985, Hildesheim, Georg Olms Verlag.

Einstein, Albert. 1905. “Zur Elektrodynamik bewegter Körper,” Annalen der Physik, 17, 891-921.

Gaeta, Luca. 2002. Segni del cosmo. Logica e geometria in Whitehead (Milano, LED).

Grattan-Guinness, I. 2002. “Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A.N. Whitehead,” Historia Mathematica, 29, 427-62.

Henry, Granville C. and Robert J. Valenza. 1993a. “The dichotomy of idempotency in Whitehead’s mathematics,” Philosophia Mathematica 3, 1, 157-72.

Henry, Granville C. and Valenza, Robert J. 1993b. “Whitehead’s Early Philosophy of Mathematics,” Process Studies, 22, 1, 21-36.

Harman, P. M. 1998. The Natural Philosophy of James Clerk Maxwell (Cambridge, Cambridge University Press).

Kennedy, Hubert C. 1980. Life and Works of Giuseppe Peano (Dordrecht, Reidel).

Lowe, V. 1985. Alfred North Whitehead: The Man and his Work. Vol. 1: 1861-1910 (Baltimore, John Hopkins University Press).

Maxwell, James Clerk. 1856. “On Faraday’s Lines of Force,” Transactions of the Cambridge Philosophical Society, X.

Mays, Wolfe. 1961. “The Relevance of ‘On Mathematical Concepts of the Material World’ to Whiteheads’ Philosophy,” in The Relevance of Whitehead, edited by I. Leclerc (London, George Allen & Unwin).

Northrop, F.S.C. and Gross, M.W. (eds.). 1953. An Anthology (Cambridge, Cambridge University Press).

Northrop, Filmer Stuart Cuckow. 1961. “Foreword,” in Donald W. Sherburne, A Whiteheadian Aesthetic (New Haven, Yale University Press).

Plato. 1929. Timaeus (Harvard University Press).

Paci, Enzo. 1965. “Logica e filosofia in Whitehead,” Relazioni e significati, Vol. I, (Milano, Lampugnani Nigri).

Russell, Bertrand. 1900. A Critical Exposition of the Philosophy of Leibniz. (London, George Allen & Unwin).

Russell, Bertrand. 1959. My Philosophical Development (London, Allen & Unwin).

Russell, Bertrand. 1996 [1903]. The Principles of Mathematics, 2nd ed. (New York, W. W. Norton; Reprinted with new introduction, 1937, London, Allen & Unwin).

Torretti, Roberto. 1978. Philosophy of Geometry from Riemann to Poincaré (Dordrecht, Reidel Publishing).

Veblen, Oswald. 1904. “A System of Axioms for Geometry,” Transactions of the American Mathematical Society, V, 343-384.

Whitehead, Alfred North. 1888a. “On the Motion of Viscous Incompressible Fluids. A Method of Approximation,” Quarterly Journal of Pure and Applied Mathematics, 23, 78-93.

Whitehead, A. N. 1888b. “Second approximations to viscous fluid motion: a sphere moving steadily in a straight line,” Quart. Jour. Of Pure and Applied Math., 23, 143-152.

Whitehead, A. N. 1906. “On Mathematical Concepts of the Material World,” Philosophical Transactions: Royal Society of London, series A, Vol. 205, 465-525. Reprinted in Alfred North Whitehead: An Anthology, edited by F.S.C. Northrop and M.W. Gross (Cambridge, Cambridge University Press, 1953), 11-82. Whitehead, A.N. 1911a. An Introduction to Mathematics (London, Williams and Norgate).

Author Information

Luca Gaeta
Politecnico di Milano, Dipartimento di Architettura e Pianificazione
Via Bonardi 3, 20133 Milano (Italy)

How to Cite this Article

Gaeta, Luca, “Order and Change: The Memoir ‘On Mathematical Concepts of the Material World'”, last modified 2008, The Whitehead Encyclopedia, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <http://encyclopedia.whiteheadresearch.org/entries/thematic/mathematics-and-logic/order-and-change/>.