Whitehead’s Early Philosophy of Mathematics by Granville C. Henry and Robert J. Valenza Granville C. Henry is Professor of Mathematics and Philosophy at Claremont McKenna College, Claremont, CA, 91711. He is the author of Robert J. Valenza We examine Whitehead’s In looking for a work of Whitehead that singularly and most accurately describes his early mathematical philosophy, we should not choose Neither should we choose any of the numerous works in which Whitehead establishes mathematics as derivative from the abstract theory of classes or intuitive set theory, because in these works he acknowledges the paradoxes in set theory that drove him to affirm for a time Russell’s logistic thesis that mathematics is the "science concerned with the logical deduction of consequences from the general premises of all reasoning" (MAT 291). Whitehead did not ground mathematics in set theory. Nor should we center Whitehead’s philosophy of mathematics in the monumental Formalism, set theory, logicism, and intuitionism are the four major recognized contemporary schools in the philosophy of mathematics. Ours is a simple thesis with respect to powerfully general but unfortunately vague philosophical words, such as An Introduction to Mathematics Whitehead’s theme, begun in the first chapter and maintained throughout the book and, in our judgment, for the rest of his philosophy, is that mathematics begins in experience and as abstracted becomes separated from experience to become utterly general. "We see, and hear, and taste, and smell, and feel hot and cold, and push, and rub, and ache, and tingle" (IM 4). These feelings belong to us individually. "My toothache cannot be your toothache" (IM 4). Yet we can objectify the tooth from the toothache and so can a dentist who "extracts not the toothache but the tooth," (IM 4) which is the same tooth for both dentist and patient. Whitehead would give later in Abstraction
Because we can objectify things as things individually and communally we have a common world of things, which is not only the abstract domain of mechanics but becomes, as extended, the subject matter of arithmetic. Arithmetic, therefore, "applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body. The nature of the things is perfectly indifferent, of all things it is true that two and two make four" (IM 2). Whitehead then identifies the leading characteristic of mathematics, not just of arithmetic, as that subject which "deals with properties and ideas which are applicable to things just because they are things, and apart from any particular feelings, or emotions, or sensations, in any way connected with them" (IM 2-3). An abstract or ideal thing that has no reference to "particular feelings, or emotions, or sensations" is what Whitehead later would define as an eternal object (see PR 44). Eternal objects form a realm -- a Platonic realm? Not quite. Whitehead remains an empiricist, but shows early this romantic streak of Platonism that is given expression in his doctrine of the realm of eternal objects. In the second chapter Whitehead introduces the idea of a It is curious that Whitehead does not mention explicitly in this context the formalism that he and Russell had been developing for a decade to unify mathematics, namely the symbolic logic of The mathematical content of
In a summary of Whitehead’s position, mathematics is abstracted from human experience to become ideal objects which initially represent general things that are symbolized in classes by variables. The variables can then become ideal objects as parts of forms, which themselves may become objects in more general systems. Whitehead asserts that mathematicians seek to extend their systems so that operations and relations are defined most generally, e. g., the natural numbers extended to the integers so that subtraction always has meaning, as well as desiring to show relationships between general systems. These general systems and their perceived interrelationships are examined for consistency and completeness by means of logic, which Whitehead believed was a universal language for the presentation of all mathematics. At least for him, at the time immediately prior to the publication of An In
Universal Algebra In the next to last decade of the nineteenth century, Whitehead was in his twenties and was working on the applied problem of the motion of viscous incompressible fluids (QJPAM23). His mathematics was at most a sophisticated extension of that outlined above in What were some of the characteristics of the new algebras that challenged the old mathematical analysis? In a review of In the good old times two and two were four, and two straight lines in a plane would meet if produced, or, if not, they were parallel. . . .Here is a large treatise [ How did Whitehead attempt to rectify these apparently paradoxical assertions? By insisting that there are no inconsistencies within an In modern terminology Whitehead’s algebraic manifold is a To show the relationship between algebras, each must be objectified clearly. At least Whitehead did that and created a work that as reviewer Mathews said "ought to be full of interest, not only to specialists, but to the considerable number of people who, with a fair knowledge of mathematics, have never dreamt of the existence of any algebra save one, or any geometry that is not Euclidean" (PRSL64:385-6). We wish that we could have asked Whitehead in his later years about his earlier passion to objectify mathematics to the detriment of its relational aspects. His mature philosophy was so thoroughly relational. How did Whitehead attempt to relate his disparate algebraic manifolds? He did so in two ways: by interpreting them in terms of the general abstract mathematical properties of space and by asserting a formalist posture on the nature of mathematics. The former is much less interesting than the latter, but we shall say a few words about it. Just as Euclidean geometry can be interpreted in terms of algebra and Whitehead’s formalist position is stated by him in plain terms: Mathematics is the development of all types of formal, necessary, deductive reasoning. The reasoning is formal in the sense that the meaning of propositions forms no part of the investigation. The sole concern of mathematics is the inference of proposition from proposition. The justification of the rules of inference in any branch of mathematics is not properly part of mathematics; it In contrast to Mathews’s strongly supportive review of Is geometry a part of pure mathematics? Its definitions have a very existential import; its terms are not conventions, but denote true ideas; its propositions are more than self-consistent -- they are true or false; and the axioms in accordance with which the reasoning is conducted correspond to universal properties of space. But suppose that we confine our attention to algebraic analysis -- to what the treatise before us includes under the terms ordinary algebra and universal algebra. Are the definitions of ordinary algebra merely self-consistent conventions? Are its propositions merely formal without an objective truth? Are the rules according to which it proceeds arbitrary selections of the mind? If the definitions and rules are arbitrary, what is the chance of their applying to anything useful? (S9 325-6). Where Mathews thought that In The list of mathematicians who most influenced Whitehead is remarkable: Grassmann (1809-77), Boole (1815-64), Weierstrass (1815-97), Cantor (1845-1918), Frege (1848-1925), The lists above and other evidence suggest not merely that Whitehead backed the wrong horses, but that his horse sense was somewhat eccentric. His mathematical research tended to two extremes: applications and foundations. The mainstream mathematical culture, which, regardless of ontological commitment, is driven as much by esthetics as by science, seems to have had little meaning for him. In spite of his great interests in esthetics generally, he had only a narrow sense of mathematics as, in the words of C. H. Clemens, "an esoteric art form," Principia Mathematica We have already remarked on the anomaly of Whitehead’s giving a general description of mathematics in On examination, however, the One wonders, then, what was the mathematical content of
Whitehead was of two minds in 1910 and 1911, one expressed in Although In If pushed to be more accurate, we can claim, as is often done, that our model set of two is the set containing 0 and 1, where 0 is the null or empty set {} In contrast, the definition of number in The cardinal number of a given class is ordinarily thought of as a Defining a cardinal number as the set of all sets having a certain numerical property is an example of Whitehead’s radically objectifying tendency during this period, as contrasted with a relational one. We have offered a definition of number that is significantly more relational, and certainly less ostentatious. A set I has a certain cardinal number if it is bijective on some model set, that is, if it To develop cardinal numbers further in Russell was perfectly correct. By confining numerical reasoning within one type, all the difficulties are avoided. He had discovered a rule of safety. But unfortunately this mle cannot be expressed apart from the presupposition that the notion of number applies beyond the limitations of the rule. For the number "three" in each type, itself belongs to different types. Also each type is itself of a distinct type from other types. Thus, according to the rule, the conception of two different types is nonsense, and the conception of two different meanings of the number three is nonsense. It follows that our only way of understanding the rule is nonsense (MG 111). This statement was written some sixteen years after Whitehead had discovered temporal atomicity and developed a thoroughly relational process philosophy on this discovery. We can not help but believe that Whitehead was troubled by the odd mix of formalism and near Platonism expressed in
A final comment on the times of the first quarter of the twentieth century. In the Introduction to the Second Edition of Ironically, it was a visit by Ramsey and his attendance of a lecture by the great intuitionist mathematician Brouwer that set Wittgenstein again to the task of philosophy.
It should not be surprising that There is one major mathematical legacy of
Whiteheadian Mathematics and Process Thought So far we have explored the technical shortcomings of Whitehead’s most significant mathematical works, Recall the Whiteheadian mathematical trinity: generality, variable, and form. How does one achieve generality in mathematics? We discuss three approaches, admittedly related, but with distinct flavors. (1) Perhaps the most naive approach is through the generality of objects or forms. To illustrate, consider the set of integers and the set of continuous real-valued functions defined on the real numbers. If we posit these as concrete objects in our metaphysics, what form do they share? It is not difficult to show that both admit addition and multiplication subject to some very familiar laws, upon which we need not digress. The point is that both are subsumed under the modern mathematical structure of a Axiomatic systems such as rings, groups, fields, and topological spaces distill gradually out of mathematical experience. One sees that by the latter half of the nineteenth century the method of generalized forms is beginning to blossom, both as a means to unify mathematics and as a means to isolate the key properties of well-studied objects. But neither in the arts nor in mathematics is mere methodological awareness to be confused with genuine creativity, and the capacity to identify viable forms is a quintessential mathematical talent. The notoriously austere axioms for an abstract group or a topological space resemble cosmetically any number of simple axiomatic systems that one might construct. Their particular richness derives from two mutually contentious attributes: (i) They are sufficiently general to encompass a wide spectrum of mathematical phenomena. (ii) They are sufficiently restrictive to capture essential features of some part of the mathematical landscape. Point (i) alone is insufficient. Should we enlarge the definition of a group, we might reach a structure -- a non-structure really -- called a
(2) A second approach to generalization may be framed in terms of activities rather than Objects. The premier example is logicism, the reduction of mathematics to formal logic. Under this program, geometry and number theory are unified insofar as they are part of the same activity: deriving consequences from the axioms of Just as a pixel-by-pixel account of Seurat’s (3) The last approach to generalization that we consider is the path not taken by Whitehead, at least not in his philosophy of mathematics. This is a mode of organization stressing structural relationships across distinct classes. We designate this with a word borrowed from the technical lexicon of twentieth century mathematics:functoriality. Consider the rational numbers, a set in which we can add and subtract subject to an associative law and thus constituting a mathematical group. The real numbers likewise constitute a group with respect to addition, and clearly the reals contain the rationals. We say that the group of rationals is While the previous example is trivial (most undergraduate mathematics majors will have seen it), functoriality is a key feature in some of the deepest mathematics of this century. By stressing relationships across classes, it neatly sidesteps the contention between generality and richness discussed above. Functorial relationships allow one to bring to bear the full knowledge of one class to the analysis of another. They bring about unification without retreat to insipid common objects or inept common methods. Whitehead, a mathematician of note to his contemporaries but of small consequence to his successors, never scented a relational approach to mathematics. Perhaps functoriality had to await the further maturation of cross-disciplinary fields such as algebraic topology and algebraic geometry, but in light of Whitehead’s eccentric tastes, we doubt that fifty years would have made much difference. He seems implicitly to have accepted a condition of ontological stasis for the mathematical world. All the more remarkable, then, that Whiteheadian metaphysics explicitly countenances the occasions of actual entities through the dynamic, relational process of concrescence, a process remarkably similar to the dynamic evolution of mathematical forms. The holism of functoriality is the holism of process thought. We stand in amazement that Whitehead saw this so clearly in his adopted field of philosophy but not in his native field of mathematics.
References BAMS32 -- B. A. Bernstein. "Whitehead and Russell’s Principia Mathematica" BAMS34 -- Alonzo Church. "Principia: Volumes Hand Ill" ESS -- EWM -- Lewis S. Ford. IM -- JP11 -- C. I. Lewis. MAT -- "Mathematics" in ESS. Published originally in MFF -- Saunders Mac Lane. MG -- "Mathematics and the Good" in ESS. Published originally in N116 -- E P. Ramsey. "The New Principia." N58 -- G. B. Mathews. "Comparative Algebra." OO -- Murray Code, PRSL64 -- "Sets of Operations in Relation to Groups of Finite Order." Abstract Only. PS 17 -- Christoph Wassermann. "The Relevance of QJPAM23 – "On the Motion of Viscous Incompressible Fluids. A Method of Approximation." S9 -- Alexander Macfarlane. TLP -- Ludwig Wittgenstein. UA -- UUPM -- Kurt Gödel. "Ûber formal unentscheidbare Sätze der WPRM -- Granville C. Henry. "Whitehead’s Philosophical Response to the New Mathematics,"
Notes ^{1. Murray Code has written a good introduction to Whitehead’s philosophy of mathematics in his book (OO) based on Whitehead’s later works. It is not, however, an exposition of Whitehead’s mature position as it could be made relevant to contemporary mathematics. Co-author Henry of this article examined the philosophical development of Whitehead in terms of his reaction to mathematics in an article (WPRM) written over twenty years ago. This present article, in contrast to the older one, seeks to evaluate Whitehead’s early philosophy of mathematics in terms of Whitehead’s mature philosophy and contemporary mathematics.
2. Mac Lane in his analysis of schools in the philosophy of mathematics accepts two others, Platonism and Empiricism (MFF 455-456).
3. Whitehead saw Plato to be of two moods, one in which he thought of mathematics as "a changeless world of form.. contrasted...with the mere imitation in the world of transition," and the other in which he "called for life and motion to rescue forms from a meaningless void" (MT 97). Whitehead was a Platonist in this Second sense.
4. We share this opinion with Murray Code who has expressed it in OO.
5. See "Autobiographical Notes" (ESS 16).
6. In retrospect, Macfarlane’s criticism was not fair. Whitehead understood well that abstraction does not operate under unlimited license, but once a formal system has coalesced, it may develop independently of its extensive base.
7. Personal conversation.
8. See Norman Malcolm and G. H. Von Wright. Ludwig Wittgenstein: A Memoir London: Oxford University Press, 1958. 12-13.
9. Journals devoted to semigroups do exist and manage to fill their pages with interesting mathematics, but only through examination of special subclasses. In contrast, both groups and topological spaces are interesting for their bare-bones abstract structure as well as their special subclasses. Consider, for instance, the immense treasure-trove of mathematics engendered by the problem of classification of finite simple groups.
10. Categories and functors were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. See MFF for a technical introduction.
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