by Christoph Wassermann

Christoph Wassermann (Breslauerstr. 7, 7858 Weil am Rhein, West Germany) is Research Associate at the Faculty of Theology of the University of Geneva. He is working to extend Whitehead’s theories to quantum theory, especially by coordinating the work of C. F. von Weiszäcker with Whitehead. The following article appeared in Process Studies, pp.181-192, Vol. 17, Number 3, Fall, 1988. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.

Most of Whitehead’ s publications prior to 1911 were intended exclusively for the world of professional mathematicians. This was especially the case in his Treatise on Universal Algebra (1898) and in Principia Mathematica (published together with B. Russell in 1910 ff.). In some minor publications he in addition had mathematical physicists in mind.¹ This, however, changed in 1911 with the publication of An Introduction to Mathematics. Here we find him for the first time writing for a larger public.

This had consequences for both the contents and manner of presentation in this new work. He no longer could assume such a profound mathematical knowledge as with his previous readers. The title of the book already reflects this circumstance. It ‘only’ proposes to be an introduction to mathematics -- however, a very profound one, as we shall see. Regarding the contents of the book, this meant that all the subjects of higher mathematics (such as mathematical logic, group theory, analytic, non-euclidian, and projective geometries, and integral calculus) could not be dealt with.² The simpler manner of presentation was conditioned by the fact that neither the necessary mathematical symbols (such as the symbolism of mathematical logic used in PM or that of analytical geometry used in UA) nor the rigorous mathematical methods (such as axioms, definitions, and proofs) could be utilized. Thus the insights into the remaining areas of mathematics that were dealt with (the theory of numbers, algebra, geometry, the differential calculus, and topology) had to be reduced to the most basic points of these various branches.³

However, these external conditions did not make Whitehead’s presentation of mathematics superficial. We rather find him in IM dealing anew and in depth with three areas: philosophical, historical, and applied mathematics. He touches these questions anew, insofar as they had already delivered important problems in his earlier works on pure mathematics (philosophical problems in UA, MC, and PM; historical matters in MC; and applied mathematics in his earliest scientific publications).

The significant interest in philosophical mathematics follows from the amount of space allotted in IM to the treatment of the three basic problems that had occupied Whitehead in his work up to that point. They are the question about the nature of mathematics, its unity and internal structure, and its applicability to nature. This is also indicated by the aim of IM as enunciated in chapter one: "The object of the following chapters is not so much to teach mathematics, but to enable the students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena" (IM 2).

Whitehead’s statements on the origin of mathematics, the criteria and characteristics of its historical development, and the present conceptions about the interrelation of its basic disciplines belong to the field of historical mathematics. In all three phases of its development the language of mathematics played an important role. Whitehead’s selection of and emphasis on individual moments in the development of mathematics is primarily guided by his philosophical interest in understanding the growth in unity and interconnection of mathematics as a whole. He ignores areas of mathematics which are not especially suitable in clarifying these philosophical aims, and suppresses periods in which various subfields were developed independently of each other.⁴

His special interest in applied mathematics can be seen in the large passages dealing with the mutual influence of physics and mathematics. Here also the topics are chosen so as to elucidate philosophical points, especially the question of how and why mathematics applies to nature at all. One of the areas that is not covered in IM is the relationship of mathematics to the humanities or, more generally, to the subjective side of reality. However, one can already find hints intimating the forthcoming emergence of that question.

The aim of the following analysis of IM is to trace Whitehead’ s keen interest in mathematics and its philosophical foundations, and to show how on the one side the basic insights of his previous works on pure mathematics are here condensed into a few highly ingenious and self-evident concepts, and how on the other side these concepts can be regarded as pre-figurations of basic concepts in his later philosophy of organism.⁵ The historical and physical passages from IM will only be used to illustrate the basic structure of Whitehead’s philosophy of mathematics.

Whitehead sums up the nature of mathematics in the following statement: "the leading characteristic of mathematics [is] that it deals with properties and ideas which are applicable to things just because they are things, and apart from any feelings, or emotions, or sensations in any way connected with them. This is what is meant by calling mathematics an abstract science" (IM 2f).

A problem with this conception of mathematics seems to arise when the two sciences of algebra and geometry are compared with each other. While the numbers of algebra can be universally applied, even to things that can never be perceived by the senses, the space of geometry seems to be much less abstract (see IM 179). The reason is that spatial conceptions cannot be applied to all things as numbers can. Whitehead argues: "This, however, is a mistake; the truth being that the ‘spaceness’ of space does not enter into our geometrical reasoning at all . . . . [The] space-intuition which is so essential an aid to the study of geometry is logically irrelevant . . . . It has the practical importance of an example, which is essential for the stimulation of our thoughts" (LU 180f). Thus even in geometry the leading characteristic is its abstractness (see IM 182).

The abstract nature of mathematics had occupied Whitehead more than two decades prior to IM, and had found an exact establishment in UA and PM. This was done by showing that the concepts hitherto regarded as basic to mathematics, like numbers and space-points, were not that basic at all, nor bound up with our intuition of nature, but could be deduced from the axioms of mathematical logic. Even on this new level of foundation for mathematics, the axioms and definitions of formal logic could be formulated without any direct introduction of the contents of assertions, thereby assuring a complete independence from any particular occasion in which these ideas were applied.

This analysis of mathematics seems to be the reason for Whitehead to attach e attribute of a "particular individuality" (SMW 229) to eternal objects in his later philosophy. By this he meant that "the [eternal] object in all modes of ingression is just its identical self" (SMW 229). The reasoning behind this statement is that, if the ideas of mathematics and their respective properties were dependent on diverse cases of application, they would not be able to maintain the same identity in every mode of application.

Thus the abstract nature of things, as first uncovered by the analysis of mathematical ideas, if extended to pertain to nonmathematical ideas as well, is the condition for membership in the realm of eternal objects.

The comparison between algebra and geometry led Whitehead to another conclusion, in a sense complementary to the arguments above. He states: "Space-perception accompanies our sensations, perhaps all of them, certainly many; but it does not seem to be a necessary quality of things that they should all exist in one space or in any space (IM 182).⁶ Here he does not only stress the independence of abstract mathematical ideas from any special application to nature, as described above, but extends their independence to the point where these abstract ideas are no longer bound to find application in nature, as perceived by our senses.

This aspect of the abstract nature of mathematics, that not all of its results necessarily can be applied to reality, plays an important role in UA and PM. Already in UA the problem of an "uninterpreted calculus" (UA 5) or a "calculus only partially interpretable" (UA 10) has become prominent in Whitehead’s mathematics.

In his metaphysical cosmology these thoughts are further developed, and result in statements such as: "the metaphysical status of an eternal object is that of a possibility for an actuality" (SMW229). For we could not think of pure possibilities, if all eternal objects, of which the ideas of mathematics only constitute a subgroup, necessarily had to find an application in nature. The partial detachment of the realm of eternal objects from the actual world, which is important in Whitehead’s later thought, seems to originate in his appreciation that mathematical ideas need not find application in the spatial world of matter.

On this side, the abstract nature of mathematics, again extended to include non-mathematical things, provides the necessity of assuming the existence of an independent realm of eternal objects. Though sufficiently separated from the actual world, this realm is not completely detached from it, as we shall see below.

Altogether then, we have two lines of thought ensuing from the abstract nature of mathematics. One fixes on the independence of mathematical ideas from any special instance of their application to nature and thus focuses on the area of contact between mathematics and reality. This line of thought controls the necessary condition for membership in the realm of eternal objects. The other fixes on those areas of mathematics that are partially inapplicable. As can be shown, Whitehead reaches these conclusions through an analysis of the internal structure of mathematics. The result was a sufficient condition for postulating an independent existence of the realm of eternal objects. In the following, we will further expound these two separate lines of thought. We first take up the second aspect, and analyze Whitehead’s statements on the internal structure and historical development of mathematics. Only then will we come back to the first line of thought, and systematize Whitehead’s analysis of the area of contact between mathematics and natural science.

In connection with the abstract nature of mathematics Whitehead specifies three concepts that underlie all mathematical disciplines. He says: ‘These three notions of the variable, of form, and of generality, compose a sort of mathematical trinity which preside over the whole subject. They all really spring from the same root, namely from the abstract nature of the science" (IM 57).

Of these three, the notion of the variable is the most fundamental. In UA this concept is called a "substitutive sign," and its universal importance in mathematics follows from the assertion: "The signs of [any] Mathematical Calculus are substitutive signs" (UA 3). In PM the notion of the variable is carried into the realm of mathematical logic in order to emphasize and establish its profound significance in the overall structure of mathematics. Its foundation is laid in the "Theory of Apparent Variables’ (PM I 127ff). The variables are defined in close connection with the predicates and quantifiers of the calculus of propositional functions. Here the quantifiers define the range of the variables (some =x, and any = (x)), and the predicates represent the propositions assigned to the individual variables. This system of definitions is still in use today.

In IM Whitehead condenses this notion of the variable in the two concepts any and some." Whitehead can say: "Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things" (IM 7). It is clear that this pair of concepts simply represents a non-technical rendering of the rigorous formalism of the theory of propositional functions (see IM 8).

As the following assertions show, the two mutually exclusive terms "any" and "some" play a fundamental role in all basic disciplines of mathematics. "The ideas of any and of some are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic" (IM 7). "As in algebra we are concerned with variable numbers . . . so in geometry we are concerned with variable points" (IM 88; cf. 176). Even in laying the foundation for the differential calculus one cannot do without this pair of concepts (see IM 175).

Similar things could be said of the notion of form. In UA, for example, the idea of mathematical form is founded rigorously on the concepts of "Equivalence" and "Operations" (UA 5ff and 7ff). And in PM it is the general notion of a relation (PM I 187ff) that represents the logical analysis of form. Just like the variable, this notion also plays a fundamental role in algebra and geometry (see the summary IM 88).

Using the notion of generality, Whitehead wants to point out that mathematics always seeks expressions which, taking up the notions of the variable and of form, are able to unite as great a subdivision of mathematics as possible, using only one uniform formalism. It was Russell’s and Whitehead’s aim in utilizing the formalism of mathematical logic to unite the whole of mathematics in this way. Thus the notion of generality is the condensation of the main results of PM as enunciated in the preface: "(1) . . . what were formerly taken tacitly or explicitly, as axioms, are either unnecessary or demonstrable⁷; (2) . . . the same methods by which supposed axioms are demonstrated will give valuable results in regions, such as infinite numbers, which had formerly been regarded as inaccessible to human knowledge. Hence the scope of mathematics is enlarged both by the addition of new subjects and by a backward extension into provinces hitherto abandoned to philosophy. (PM I.v). All this is accomplished by using only one and the same formalism. This generalization of a mathematical formalism⁸ results on the one hand in the progressive extension of the range of applicability of a single mathematical idea,⁹ and on the other hand in the unification of parts of basic mathematical disciplines on a higher level.¹⁰ Whitehead himself states: "As they [the branches of mathematics] become generalized, they coalesce" (IM 84).

It is especially these two last notions of form and of generality, and the underlying correlations between variables and mathematical formalisms, that play an important role in Whitehead’s later philosophy. Just as in mathematics the quantities represented by the variables are not treated independently, but in a complex network of interrelations, so Whitehead later postulates: "An eternal object, considered as an abstract entity, cannot be divorced from its reference to other eternal objects" (SMW 229f). Here lies the mathematical pre-figuration of concepts such as "realm of eternal objects," "abstractive hierarchy," and "nexus" (cf. SMW 232f, 241f; PR 22/32).

Of the two results that issue from the generalization of a mathematical formalism mentioned above, we will here discuss the possibility of unifying subordinate mathematical disciplines in a new formalism. The emergence of coordinate geometry from the unification of algebra and geometry will serve as an example. (For the following see figure I)

In chapter 9 of IM Whitehead shows the correspondence between algebra and geometry regarding their main abstractive processes. Just as in algebra the variables are an abstraction from specific numbers, so in geometry variable points are generalized from points. The same can be said of the algebraic transition from special equations to general algebraic forms, and the geometrical extension of figures to general geometrical loci. Only on this more abstract or general level of the two branches was a unification possible. Variable points and variable numbers are united in the idea of coordinates making it possible to identify algebraic correlations with geometrical loci.

In connection with this analysis of coordinate geometry Whitehead makes some far-reaching statements on the philosophical implications of the ideas involved. He starts by pointing to the prerequisites for being able to use the formalism of coordinate geometry, namely the arbitrary choice of an origin and coordinate axes. From the point of view of abstract mathematics both choices seem "artificial and clumsy . . . . But in relation to the application of mathematics to the event of the Universe we are here symbolizing with direct simplicity the most fundamental fact respecting the outlook on the world afforded to us by our sense" (IM 91). Every one of us has a restricted and finite outlook on the universe. The origin we choose and to which we refer our sensible perceptions of things "we call here: our location in a particular part of space round which we group the whole Universe is the essential fact of our bodily existence" (IM 91).

This simple philosophical interpretation of coordinate geometry also has its forerunner in Whitehead’s mathematics. It can be derived from certain earlier ideas (see MC). His definition of an interpoint (cf. definition 1.21, MC 485) has two characteristics relevant in this connection. First, the underlying pentadic relation of linear objective reals is based on the idea of an intersection of two such objective reals without the prior conception of a point in space. Such a meeting of two lines with no location in space comes very near to Whitehead’ s later concept of feeling (see PR 23/35 and 219ff/334ff). Secondly, as with the origin of coordinate axes, round which the whole universe is grouped, the definition and existence of an interpoint also involves all of space, insofar as an infinite number of linear objective reals, covering all of space, go to make one interpoint.

The same can be said of the definition of a - (or homaloty-) point (cf. definition 3.42, MC 495 and 506). Here the -point is defined as the -concurrence of the whole -region (i.e., all of space) with a three-membered -axial class. This means that all of space is conceived as being arranged around one point. The difference from the interpoint is that the -point involves a -axial class, i.e., the nonpunctual analogon to the axes of a coordinate system (cf. definition 3.22, MC 494).

Here again we have the pre-figuration of an important concept of the philosophy of organism. It is the concept of "prehension into unity of the patterned aspects of the universe of events" (SMW 213f). Just as we, when applying coordinate geometry to our physical existence, choose an origin and axes round which the whole universe is grouped, so later Whitehead conceives every event as prehending the whole universe of events into one new unity.

Before concluding this section, we need to make some remarks about his treatment of mathematical symbols. Contrary to the popular idea of mathematical symbolism as comprising the language of mathematics or more generally the language of the exact sciences, we find mathematical symbols treated here in a significantly different way. They are not the elements of a new language, distinct from our ordinary language, but simply represent a shorthand for our everyday speech, as pertaining to mathematics. Two statements underscore this understanding of symbolism.

First, Whitehead emphasizes that the signs for numerals, letters, and mathematical operations are not the outward side of a language essentially different from, and more mysterious than everyday language, but that they are introduced to relieve the brain (see IM 39) and "to make things easy" (IM 40). For "by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which other-wise would call into play the higher faculties of the brain" (IM 41). Thus the function of mathematical symbolism is to help perform "important operations . . . without thinking about them" (IM 42).

Secondly, Whitehead shows that every mathematical symbol must be interpretable. "A symbol which has not been defined [i.e., interpreted in ordinary language] is not a symbol at all. It is merely a blot of ink on paper which has an easily recognized shape" (IM 64). These properties of mathematical signs, connected with the main body of mathematics by the reciprocal relation of symbolism and interpretation (see figure 2), were essential for the advance of mathematics. Without their help the basic concepts governing mathematical structure and development, i.e., variables, form, and generality, could hardly have been used in actual mathematical research.

The question about the relation of mathematics to physics must have occupied Whitehead since the beginning of his scientific career. For his fellowship dissertation on Maxwell’s electromagnetic field theory,¹¹ as well as his first two scientific publications on special problems of the hydrodynamics of incompressible fluids¹² testify to an at least open relationship to problems of mathematical physics. This must be one of the reasons why Whitehead devotes so much space to questions concerning the methods and principles of applying mathematical ideas to the phenomena of nature, and why he sees himself obliged to write that "all science as it grows to perfection becomes mathematical in its ideas" (IM 6).

Whitehead’ s statements on the relation of mathematics to the experience of nature can be summarized in three pairs of opposing concepts: events/things, things/mathematical ideas, and mathematical ideas/variables. To these correspond the interrelational conceptions of relations between things, correlations of mathematical ideas, and form. (For the following see figure 2.)

With a similar aim as in MC, Whitehead conceives the world as "one connected set of things which underlies all the perceptions of all people" (IM 41). He also calls this world "the general course of events" (IM 14). The first step in the direction of a mathematical handling of nature is "to recognize [amid the general course of events] a definite set of occurrences as forming a particular instance" (IM 14) of what is to be mathematically grasped. These things, isolated for their mathematical treatment, have relations to each other (see IM 4). We thus must distinguish between the "general course of events" and the "things" with their respective relations to each other. The process of differentiation consists of two opposite movements: the isolation or discovery of these things in nature (see IM 15) and their re-cognition or verification in "the general course of events." For this it is necessary "to have clear ideas and a correct estimate of their relevance to the phenomena under observation" (IM 18).

The second step crosses the actual region of contact between mathematics and the experience of nature, and thus represents the science of mathematical physics or applied mathematics. It distinguishes between the actual mathematical ideas involved, such as numbers or points, and the things of nature attached to them. At the same time it separates the mathematical correlations from the prevailing relations between things in nature (cf. IM 2, 32). This process of differentiation again consists of two opposing movements: of abstraction and of application. As mentioned above, abstraction is that direction which determines the nature of mathematics and insures the existence and independence of the mathematical ideas as against the things of nature.

Application goes in the opposite direction, as Whitehead postulates: the "correlations between variable numbers . . . are supposed to represent the correlations which exist in nature . . ." (IM 32). While abstraction stresses the distinct existence of mathematics, application emphasizes the connection between mathematics and nature. This treatment of the reciprocal relationship between the things of nature and the ideas or concepts of mathematics, together with the respective correlations, is of great significance for understanding Whitehead’s philosophy of nature and his subsequent metaphysical cosmology. For here we have the germination point Out of which later grew his concept of the ingression of eternal objects in the world of actual events. For every eternal object Whitehead later demands "the general principle which expresses its ingression in particular occasions . . . . An eternal object, considered as an abstract entity, cannot be divorced from its reference to other eternal objects and from its reference to actuality generally; though it is disconnected from its actual modes of ingression into definite actual occasions" (SMW 229f).

The final step has already been treated above (in section 2) and consists of distinguishing between special and general mathematical ideas (e.g., the difference between numbers and variables), as well as between special mathematical correlations and their generalization in the concept of form. In this part we also have to distinguish two opposing directions. By successive steps of generalization the scope of the individual branches of mathematics is enlarged, and these branches are brought to a position where they can be united. By substitution special ideas and correlations are produced which can possibly find application in nature.

Now each of these three steps contributes in an individual manner to the clarification of the tasks in the bordering area between mathematics and natural science. The actual laws of nature originate in the reciprocal process of abstraction and application of step two. For Whitehead writes:

The task of mathematics in this complex network of relations is by no means to prove the truth of the laws of nature. It can merely provide mathematical certainty about the properties of correlations used in these laws of nature. In Whitehead’s own words:

Thus the truth of a law of nature can only be established within the context of step one, namely under the control of the reciprocal concepts of discovery and verification. Here, however, Whitehead points to an essential problem:

Whitehead’s analysis of the applicability of mathematics to nature, as set forth above, is basically formal in nature. The influence on his later thinking is apparent. This influence is so great that we must see in Whitehead’s mathematical publications not only pre-figurations of later thoughts, but also part of the coercive force that caused him to develop his metaphysics the way he did. For if mathematics holds in any way, and if it is at all applicable to nature, then it is, by its very existence, a guarantee for the reality and significance of eternal objects and their ingression into nature. This is a clear indication of the continuity and overall consistency of Whitehead’s philosophy in its various stages. All of this, however, resulted from the formal side of the interaction between mathematics and science. But, as can already be seen in MC, Whitehead is in addition very much concerned with the material side of this interaction. In IM Whitehead also touches the material questions regarding space, time, measurement, and the dynamic conception of nature. But in his treatment of these topics (IM chapters 4, 16, 17) there are marked differences with his later thinking. Neither his relational conception of space, which is basic for understanding his concept of extensive abstraction and which gave his theory of relativity its unique character, nor the problem of the bifurcation of nature, with its differentiation between the materialistic and personalistic outlook on the world, seem to be clearly in Whitehead’s mind at this time. This seeming indication of a clear discontinuity, however, is relative. For even amidst these apparent shifts in the overall conceptions of Whitehead’s thinking there are strong points that can be made in favor of a continuous and organic development of his philosophy. ¹³ A detailed discussion of this material side of the relationship of mathematics to nature is beyond the scope of this paper, for it involves not only Whitehead’ s philosophical analysis of geometry but also his philosophy of nature.

IM -- Alfred North Whitehead. An Introduction to Mathematics. London: Oxford University Press, 1982.

MC -- Alfred North Whitehead. "On Mathematical Concepts of the Material World." Philosophical Transactions of the Royal Society of London Series A, 205 (1906): 465-525.

MVI -- Alfred North Whitehead. "On the Motion of Viscous Incompressible Fluids: A Method of Approximation." Quarterly Journal of Pure and Applied Mathematics 23 (1889); 78-93.

PM -- Alfred North Whitehead and Bertrand Russell. Principia Mathematica. Vol. I (to *56). Cambridge: Cambridge University Press, 1980. All other material from PM is quoted according to the respective first edition: Vol. I, 1910; Vol. II, 1912; Vol. III, 1912. All published Cambridge: Cambridge University Press.

UA -- Alfred North Whitehead. A Treatise on Universal Algebra. With Applications. Cambridge: Cambridge University Press, 1898.

WPR -- Granville C. Henry. "Whitehead’s Philosophical Response to the New Mathematics." Southern Journal of Philosophy 7 (1969): 341-349.

1Cf. MVI and MC.

2This can be seen from a direct comparison of the contents of IM with the article "Mathematics" in the 1911 Encyclopedia Britannica, now printed in ESP, 282-284.

3All this already results from a very external comparison of the layout of the text in IM with PM or UA.

4 E.g., Islamic and medieval mathematics and the development of mechanics in the eighteenth century. A detailed analysis of the general philosophical implications that ensue from the way Whitehead discussed the history of mathematics and its relation to physical science in IM can be found in C. A, Clark, "Intimations of Philosophy in Whitehead’s Introduction to Mathematics," Adds del Segundo Congreso Extraordinario Interamericano de Filosofia, Julio 1961 (San Jose: 1962, 157-161).

5G. C. Henry in WPR (reprinted in Explorations in Whitehead’s Philosophy, ed. Lewis S. Ford and George L. Kline [New York: Fordham University Press, 1983]: 14-28) has also pointed to the fact that Whitehead’s later philosophy (especially his doctrine of eternal objects) is rooted in his earlier basic mathematical concerns. Henry’s very instructive treatment of the subject is, however, only based on an analysis of UA, MC, and PM, with no mention of IM.

6As Whitehead remarks at IM 185f, the same can be said about the mathematical properties of time.

7 E.g., the purely logical derivation of the natural numbers. Cf. PM I 331ff and II 3-26. Later on Whitehead realized the impossibility of a complete derivation of number theory from logic. Regarding the influence of this insight on the development of his philosophy see WPR, especially 345ff.

8 In UA this is comprised in the concept of "substitutive signs." See UA 8.

9 E.g., of numbers in the generalizations of number theory. See IM, chs. 6-8.

10 E.g., coordinate geometry as the unification of algebra and geometry. See IM. ch. 9.

11 See The New Encyclopedia Britannica 1983. Macropedia 19, 816.

12 Cf. Whitehead’s MVI and "Second Approximations to Viscous Fluid Motion: A Sphere Moving in a Straight Line, Quarterly Journal of Pure and Applied Mathematics 23 (1889): 143-152.

13 See Michael Welker. Universalität Gottes und Relativität der Welt. Theologische Kosmologie im Dialog mit dem amerikanischen Prozessdenken nach Whitehead. Neukirchen-Vluyn: Neukirchner Verlag. 1981, especially ch. 2, part A.