**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 *Forms of Concrescence: Alfred North Whitehead’s Philosophy and Computer Programming Structures *(Bucknell University Press, 1993).

Robert J. Valenza is W. M. Keck Professor of Mathematics and Computer Science at Claremont McKenna College. He is the author of *Linear Algebra: An Introduction to Abstract Mathematics *(Springer-Verlag, 1993).

The following article appeared in *Process Studies*, pp. 21-36, Vol. 22, Number 1, Spring, 1993. *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.

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These authors, both mathematicians, are amazed that Whitehead saw a complete system of relationships in process thought but could not see this fuctionality in mathematics.

We examine Whitehead’s *early *philosophy of mathematics in this article because it was his *only *explicit philosophy of mathematics. After *Principia Mathematica, *Whitehead let major new mathematical developments pass him by, and he never returned seriously to a philosophy that considered those new directions in mathematics.^{1}

In looking for a work of Whitehead that singularly and most accurately describes his early mathematical philosophy, we should not choose *Universal Algebra *(UA) and the variety of formalism espoused there. For Whitehead explicitly states in the only review of another’s book he ever made: "I think that the formalist position adopted in that chapter [Introduction to *Universal Algebra*],* *whilst it has the merit of recognizing an important problem, does not give the true solution. . ." (SPTC5:239). Whitehead was not a formalist.

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 *Principia Mathematica *and its philosophy of logicism interpreted and restricted by the theory of types. For his original enthusiasm for the theory of types, given in the statement "All the contradictions can be avoided," (MAT 293) gave way to mild revulsion when he realized that "our only way of understanding the rule is nonsense" (MG 111). Whitehead did not remain a logicist.

Formalism, set theory, logicism, and intuitionism are the four major recognized contemporary schools in the philosophy of mathematics.^{2 }If Whitehead did not advocate any of these, including intuitionism (which he never engaged probably because of its Kantian roots), what was his position? We believe that Whitehead viewed mathematics as consisting primarily of ideal objects radically abstracted from human experience. In the simplest of terms, Whitehead was an empiricist -- an empiricist with a romantic streak of Platonism. He was not, however, a pure Platonist. Plato accepted his forms as ontologically primary. Whitehead always accepted experience as more fundamental than ideal objects abstracted from it.^{3}

Ours is a simple thesis with respect to powerfully general but unfortunately vague philosophical words, such as *empiricism, formalism, intuitionism, Platonism. *They obscure many important and subtle distinctions in mathematics and its philosophy. We have an obligation to speak carefully about mathematics and philosophy in order to present our position for consideration and criticism. But where should we start? With what work or works of Whitehead should we begin? Not *Universal Algebra *or *Principia Mathematica, *or for that matter, any of his professional mathematical or philosophical works. We best begin with a work written for lay folk, first published by the Home University Library in 1911, called *An Introduction to Mathematics *(IM). We think that Whitehead spoke more fundamentally in this work about mathematics than he did to professionals in philosophy or mathematics. At least we see a basic continuity in the book between Whitehead’s earliest mathematics and his final philosophy. In addition, the mathematics covered is what an undergraduate today would have in her first courses in calculus. We intend to use this subject matter of mathematics to begin to explain Whitehead’s early philosophy of mathematics, including that implicit in *Universal Algebra *and *Principia Mathematica, *as well as to introduce contemporary issues in mathematics that affect an interpretation of his mature philosophy.

**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 *Process and Reality *a metaphysical explanation of how we may objectify precisely an individual thing from vague feelings by his description of indicative feelings (PR 260).

Abstraction *is *objectification; that is, the activity of abstraction from our experiences produces ideal objects. In the process we "put aside our immediate sensations" and recognize that "what is left is composed of our general ideas of the abstract formal properties of things; . . . the abstract mathematical ideas" (IM 5). Mathematics applies to the physical world *because of *its abstraction. By abstraction we get to mere things. The configuration of abstract things in abstract space at different (abstract) times is the mathematical science of mechanics, "the great basal idea of modern science" (IM 31). "The laws of motion . . . are the ultimate laws of physical science" (IM 32). Mechanics is the foundation of science. How strange to hear these words from the philosophical anti-mechanist of *Process and Reality.*

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 *variable, *which is a letter that can refer to general things of the world. It can also stand for ideal things like numbers and even for other variables, which, of course, may refer to things ideal or physical of any sort. Later in chapter five, statements about variables and numbers, such as algebraic equations, are called *algebraic forms, *which Whitehead does not define because "the conception of form is so general that it is difficult to characterize it in abstract terms" (TM 45). Finally, in Chapter 6 after discussing generalizations of number, Whitehead introduces the notion of *generality, *which with the ideas of variable and form "compose a sort of mathematical trinity which preside over the whole subject" (IM 57). In commenting on Whitehead’s notion of generality expressed in *An Introduction to Mathematics, *Christoph Wassermann states, "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" (PS17:184).

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 *Principia Mathematica, *the first volume of which was published in 1910, a year before the publication of *An* *Introduction to Mathematics. *However, he does give a most prominent place to logic by tying it to the importance of variables ("The ideas of *any *and of *some *are introduced into algebra by the use of letters" [IM 7]) and proceeds to discuss the quantifiers *any *and *some *in a way that clearly indicates a reference to the logic of *Principia Mathematica. *Whitehead is more forthright about the relationship of logic to the idea of a variable in his review published also in 1910. "This discovery [the generalized concept of a variable] empties mathematics of everything but its logic. For the future mathematics is logic . . ." (SPTCS:237).

The mathematical content of *An Introduction to Mathematics *begins with standard generalizations of number: from natural numbers to integers, rational numbers, real numbers. In this context Whitehead mentions Cantor’s proof that the real numbers cannot be arranged as countable and comments that this discovery "is of the utmost importance in the philosophy of mathematical ideas" (IM 55). Complex numbers -- Whitehead calls them imaginary numbers -- are presented in a "new" guise as ordered pairs of real numbers. Their addition and subtraction as ordered pairs illustrates the parallelogram law, which Whitehead had shown to be of great practical merit, "It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications" (IM 71). Coordinate geometry is also made practical. The origin of a vector. "the root idea of physical science," illustrates our location in relation to the world as "nearly here" (IM 92). Analytic geometry and conic sections are discussed as illustrations of the principle of generality. In the chapter on functions, Whitehead celebrates the clarity of Weierstrass’s definitions of limit and continuity. After a *neighborhood *definition of continuity, Whitehead states "If we understand the preceding ideas, we understand the foundations of modem mathematics" (IM 19). Trigonometry is shown in terms of periodic functions. An introduction to series and then differential calculus is given. Finally some brief geometry is portrayed in which Whitehead states that "the fundamental ideas of geometry are exactly the same as those of algebra; except that algebra deals with numbers and geometry with lines, angles, areas, and other geometrical entities" (IM 178).

*An Introduction to Mathematics, *surprisingly, seems completely uninformed by *Universal Algebra *or *Principia Mathematica, *at least in the sense of what might be new or creative in these two major works. It gives no hint of any of the new algebra examined in *Universal Algebra, *and does not mention formal logic at all. The entry *logic *is not even in the index. Whitehead seems to be describing the comfortable orthodox analysis of the late nineteenth century *as mathematics, *with a few nods to the creative work of Cantor and Weierstrass. Furthermore, he sees this mathematical analysis to be an abstraction from the objective physical world and as such constitutes the mathematical basis for science. There is nothing strange or wonderful or even bothersome in the staid mathematics of Whitehead’s work for lay people. He is backing off from the adventuresome spirit in mathematics, never again to be really creative there.

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 *Introduction to Mathematics, *formal logic was an example of the passion of mathematicians to establish connections within mathematics and to attempt to unify the whole of mathematics.

In *Universal Algebra *Whitehead sought to achieve what he calls *generality *by trying to unify by a common interpretation apparently disparate algebraic systems that to many did not appear to be mathematics at all. In *Principia Mathematica *he sought to unify mathematics by logic. Both attempts failed. The supposed common interpretation of generalized spaces in *Universal Algebra *was not satisfactory. When his system of logic with its assumption of the theory of types was objectified and compared with other mathematical systems, it was shown to be paradoxical. Further, Gödel showed that it was incomplete for arithmetic. That Whitehead’s early attempts at a philosophy of mathematics were inadequate, does not mean that his empiricist position was wrong. We believe that his mature philosophical position, an extension and modification of his earlier empiricism, is an adequate and satisfactory foundation for a contemporary philosophy of mathematics.^{4 }Whitehead, however, never re-examined mathematics from his later philosophical position. This is new and fertile ground. In order to cultivate it adequately we have to examine Whitehead’s mathematics and philosophy of mathematics in *Universal Algebra *and *Principia Mathematica.*

**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 *An Introduction to Mathematics; *his philosophy of mathematics was probably also a version only implicitly contained therein. We do not know exactly when he encountered Hermann Grassmann’s *Ausdehnungslehre, *published in 1844, or Hamilton’s *Quaternions, *1853, or Boole’s *Symbolic Logic *of 1859. He did, however, recognize that the subject matter in these works was quite different from conventional mathematics. He also had the conviction that it was good mathematics. At the age of thirty he began *A Treatise on Universal Algebra, *which was published seven years later in 1898.^{5} His goal was to present both the old established and the new unconventional mathematics as part of a unified and, using his term, *universal *algebra.

What were some of the characteristics of the new algebras that challenged the old mathematical analysis? In a review of *Universal Algebra, *G.* *B. Mathews gives an admittedly tongue in cheek caricature of this challenge. We present it here not because it is mathematically precise, but because it addresses in simple terms the mathematics of *An Introduction to Mathematics, *which as we have said is that kind of mathematics contained in contemporary college calculus courses. Even in its misleading clarity, we think that it was this kind of provocation that also motivated Whitehead. (Our readers who did not take mathematics beyond calculus may find it especially engaging.) One can also see the challenge to typically secondary school algebra and geometry.

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 [*Universal Algebra*]*. . *. .which appears to set every rule and principle of algebra and geometry at defiance. Sometimes *ba *is the same thing as *ab, *sometimes it isn’t; *a ***+ ***a *may be *2a *or *a *according to circumstances; straight lines in a plane may be produced to an infinite distance without meeting, yet not be parallel: and the sum of the angles of a triangle appears to be capable of assuming any value that suits the author’s convenience (N58:385-6).

How did Whitehead attempt to rectify these apparently paradoxical assertions? By insisting that there are no inconsistencies within an *individual *algebra. This means there is no longer just one algebra or one geometry. There are many self-consistent structures that can lay claim to being algebras or geometries, which may, however, differ from each other. In some of these *a ***+ ***a= a *and in others *a *+ *a 2a*. Whitehead called each of these algebraic structures an *algebraic manifold, *which in his definition is a set with a commutative and associative operation.

In modern terminology Whitehead’s algebraic manifold is a *commutative semi-group. *We mention this fact to point out that at this stage in his development Whitehead did not accept, or apparently understand, that a group (or semi-group) structure could be a means of relating his different algebras, which were themselves semi-groups. In "Sets of Operations in Relation to Groups of Finite Order," he chose explicitly to "abandon the idea of a group of . . . operations . . . on some unspecified object, as being an idea which . . . appertains to a special interpretation of the symbols" (PRSL64:319-20). He affirms that the *operations *must be considered as *objects. *Whitehead was in a severely objectifying mood, not in a relational one, even though his primary task was to relate disparate algebras. That he did not lay claim to the work of Cayley on the abstract and relational nature of groups published in 1849 and 1854 was a crucial failure of oversight on his part that essentially separated him from the future direction of mathematics.

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 *vice versa, *Whitehead saw the new algebras as interpretable in terms of generalized mathematical spaces. This position was never satisfactory, as Whitehead eventually recognized, because of his attempt to interpret *objectified *algebraic systems in terms of other generalized *objectified *algebraic or geometric spaces. We now know that there is no one objectified algebra, geometry or other general content that forms a ground for all of mathematics. We must go to a relational route which we shall examine later in much more detail.

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** **is the business of experience or philosophy. The business of mathematics is simply to follow the rule. In this sense all mathematical reasoning is necessary, namely, it has followed the rule" (UA vi).

In contrast to Mathews’s strongly supportive review of *Universal Algebra, *Alexander Macfarlane took Whitehead to task for his arbitrary, formal approach to mathematics:

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 *Universal Algebra *would be an important book for the generality of its formalism. Macfarlane felt that the work suffered by virtue of that same aspect. Macfarlane was right: *Universal Algebra *has been largely ignored, although not for its alleged empty formalism.^{6}

In *Universal Algebra *Whitehead attempted a synthesis of mathematical experience. As Mathews pointed out, to some extent he succeeded. Such a synthesis should be important, whether written by a formalist, a mathematical realist, or the most ardent, bean-averse Pythagorean mystic. Why, then, has *Universal Algebra *been of such little consequence? Is it mathematically flawed? To some extent, yes. For example, the classification of algebras into two genera by the law of idempotency *(a ***+ ***a *= *a) *ultimately proves inept, and Whitehead’s discussion of positional manifolds repeatedly confuses the distinct notions of what we now call affine and projective spaces. But we do not feel that these technical mistakes are really at issue, except insofar as they perhaps suggest a false perspective. Newton, after all, did not get the foundations of calculus right, but he suffered neither mathematical irrelevance nor obscurity for his oversights. The failure of *Universal Algebra *is more subtle.

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),* *Peano (1858-1932). With the exception of Grassmann, Whitehead was most affected by the work these men did in connection with the foundations of mathematics. Issues of continuity, cardinality, set theory and logic, and the foundations of arithmetic dominate. But more remarkable is the following list of mathematicians seldom or ever mentioned by Whitehead: Dirichlet (1805-59), Kummer (1810-93), Galois (1811-32), Cayley (1821-95), Riemann (1826-66), Dedekind (1831-1916), Poincaré (1852-1912), Hilbert (1862-1943). The work of these men led directly to the key mathematical structures, methods, and programs that have persisted through this century: groups, rings, modules, and field extensions; algebraic and analytic number theory; algebraic geometry; algebraic topology and qualitative analysis of dynamical systems; Hilbert’s twenty-three problems. These domains -- the principal legacy of nineteenth century mathematics -- play no role in *Universal Algebra; *in this light, it is no surprise that *Universal Algebra *plays no role in twentieth century mathematics.

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,"^{7 }and even less sense of passion for mathematical adventure. Later, he would declare that mathematical form does not even admit emotional subjective form for its feeling (AI 251). For Whitehead, during this time of transition between the nineteenth and twentieth centuries, abstraction is foremost a tool of science, and *Universal Algebra *takes this view to the limit. Hence, when he surveys the field with a unifying eye, he sees on the one hand, symbolic logic *(a ***+ ***a= a) *and, on the other hand, real or complex linear algebra *(a ***+ ***a ***_ ***a) *and its extensions. The vast middle ground (including number theory and algebraic geometry, for instance) is lost in the deep shadows cast by rational, empirical science. The resulting formalism is too enfeebled to support the objects and methods of twentieth century mainstream mathematics, and the great irony of Macfarlane’s criticism becomes this: the failure of *Universal Algebra *lies not in relentless, arbitrary abstraction and formalization but in the narrowness of its extensive base.

*Principia Mathematica*

We have already remarked on the anomaly of Whitehead’s giving a general description of mathematics in *An Introduction so Mathematics *(1911) without considering any of the results of his work in *Universal Algebra *(1898) or the first volume *of Principia Mathematica *(1910). Wassermann attributes this to White-head’s reluctance to presume a technical knowledge of mathematics among lay people. As confirmation, he specifies that Whitehead did include more contemporary mathematical content in the article "Mathematics" from the 1911 *Encyclopedia Britannica *(PS17:192, Footnote 2) where he was addressing both lay and professional audiences. It is true that the article "Mathematics," in contrast with *An Introduction to Mathematics, *participated fully in the spirit of *Principia Mathematica. *For example, after trying a number of definitions of mathematics, Whitehead settled in that article on Russell’s definition of mathematics as the "science concerned with the logical deduction of consequences from the general premises of all reasoning" (MAT 291), In fact, the article "Mathematics" is the most accessible, most approving and best summary of *Principia Mathematica *ever done by Whitehead.

On examination, however, the *mathematical *content of *An Introduction to Mathematics *and "Mathematics" seem quite similar. Remember that *An Introduction to Mathematics *primarily discusses natural numbers, integers, rational numbers, real numbers, complex numbers, as well as coordinate geometry, periodic functions, series, differential calculus, and geometry. The mathematical content of "Mathematics," as indicated by its chapter headings, consists of *Cardinal numbers, Ordinal numbers, Cantor’s Infinite Numbers, The Data of Analysis *(in which the rational, real and complex numbers are defined), one paragraph headed *Geometry, *and *Classes and Relations. *Whitehead had said in a discussion of the definition of mathematics that "the traditional field of mathematics can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line" (MAT 291). The definitions of number and geometries depend on the theory of classes and relations (MAT 292). The reason for including classes and relations as part of the content of mathematics in "Mathematics" is that the theory of classes and relations, like all mathematics under the thesis of *Principia Mathematica, *is supposed to be deducible from the "ultimate logical premises" (MAT 292). However, when we compare traditional mathematical content in *An Introduction to Mathematics *and "Mathematics," we see little difference. Aside from a very brief discussion of geometry, both begin with numbers, distinguish between cardinal and ordinal ones, and then develop rational, real and complex numbers.

One wonders, then, what was the mathematical content of *Principia Mathematica? *No less a mathematical authority than Alonzo Church, in his review of the second edition of volumes II and III, claims that in the whole of volume I (over 700 pages of closely argued mathematical logic introductory to the theory of cardinal numbers) and together with volumes II and III (themselves enormous tomes) one gets "cardinal numbers, relations and relation-numbers, series, well ordered series and ordinal numbers, and finally the continuum of real numbers" (BAMS34:237). Not surprisingly then -- given that the rationale for the work was foundational -- there is no significant new mathematics developed in *Principia Mathematica. *This was probably one of the reasons Whitehead made no reference to *Principia Mathematica *in his book on mathematical content for lay people. Another reason may have been that Whitehead was already concerned about the paradoxical assumptions in the theory of types through which an attempt was made to develop the real numbers by Dedekind cuts. The foundations for real numbers, which physicists as well as mathematicians must have in order to do their work, were insecure under the thesis of *Principia Mathematica.*

Whitehead was of two minds in 1910 and 1911, one expressed in *Principia Mathematica *(1910) and also in "Mathematics" (1911); the other in *An Introduction to Mathematics *(1911). It is interesting to us that in the book for common people, Whitehead paused and chose the route of caution, prudence, clarity and, if we may say so, integrity. In the article "Mathematics," reflecting *Principia Mathematica, *Whitehead was so caught up with Bertrand Russell in the professional development of his scholarship that he affirmed positions that later came crashing down around his feet. After some time, Whitehead came back professionally to his empirical roots and began a brilliant philosophical career.

Although *An Introduction to Mathematics *and *Principia Mathematica *are similar in mathematical content, these two works differ considerably in their approach to mathematics. We can see this best by contrasting the idea of *number *as it appears in both works.

In *An Introduction to Mathematics *numbers apply to everything – "to tastes, to sounds, to apples and to angels, to the ideas of the mind and the bones of the body" (IM 2) -- because the idea of numbers, as well as the idea of mere things, is abstracted from *actual things. *A cardinal number, say two, in this empirical view is an abstraction from, and therefore a property of, sets of things that have two members, for example a set consisting of a cow and a rock. We say that the set of cow and rock has the numerical property of *twoness. *From an empirical perspective, it makes little sense to speak of *the *definition of number; there are many interpretations of number, most of which coalesce to a common understanding through communal experience. To become mathematically precise, however, one has to become systematic, that is, work within some formal system. Within a system a unique definition of number becomes appropriate. This idea of *twoness *above is not as vague as it sounds, for we can agree on a certain arbitrary model set containing what we call two things, our cow and rock if we wish, got by counting or other means, and declare that any other set has two things if it can be put in one-to-one correspondence (in modern terminology, *bijective *correspondence) with our model set.

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 {}* *and 1 is the set containing the null set {}*. *The model set for two, built up from the definitions of 0 and 1 would be {}*, *{}and contains what we can determine by counting to be two items. Notice that 0 contains no items and 1 contains one item. This method of determining model sets motivates a definition of the successor of a number as the set containing the number and its members. (The number 2 is the successor of 1 because it contains 1, which is {}*, *and its member, the null set {}.)* *What we have done here is (a) accept that mathematics arises from experience, (b) recognize that we can get a general idea of *twoness *from our experience, (c) accept constraints on our experience -- what we can assert as existing and what we can construct -- by accepting some formal system, in this case a system defining set theory, and (d) acknowledge that we can define precisely within that system what we mean by *number, successor *of a number and in the process *twoness. *Even though the definition of *twoness *within the System is radically abstract, it arises out of a common understanding of *twoness *in our experience. We should point out that even when we construct mathematical definitions that may have no apparent reference to any items of our experience, we are doing so in terms of our activity, a kind of experience, often subject to the constraints of some formal system.

In contrast, the definition of number in *Principia Mathematica *has an entirely different feel than that outlined above. First, Whitehead and Russell are looking for *the *definition of cardinal number. There is little sense of multiple systems with differing definitions of number within *Principia Mathematica,* because its goal was to unify mathematics by deducing all of it from an ostensibly common logic. Second, the definition of number becomes radically extensive, Thus the cardinal number two becomes a huge set, the set of all sets of doublets. In his review of Volume II of *Principia Mathematica, *C. I. Lewis clarifies the situation:

The cardinal number of a given class is ordinarily thought of as a *property *of the class, but the attempt so to define cardinal number would rock the "Principia" to its foundations. Throughout the work, the procedure is to determine such properties *in extension, *by logically exhibiting the class of all entities *having *the property (JP 11:498).

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 *can be related *so that its members are one on one with the model set. In Whitehead’s definition, a set has a certain cardinal number if it *exists *as a member of the set defining that cardinal number. It is interesting to contrast Whitehead’s extreme objectifying and abstractive position reflected here and stated concisely in *Science and the Modern World *that "Mathematics is thought moving in the sphere of complete abstraction" (SMW 21) with his later statement in *Modes of Thought: *"Hence the absolute generality of logic and mathematics vanish" (MT 98). We see here an example of the transition from Whitehead’s romantic Platonism (following Russell) to the reclaiming of his empirical roots in his philosophy of process.

To develop cardinal numbers further in *Principia Mathematica *and avoid inconsistencies required the theory of types, largely due to Russell. (For exampie, the notion of cardinality sketched above leads at once to the anomaly of a set belonging to itself -- a hazard to which no one could be more sensitive than Russell.) Lewis comments on the theory of types: "This theory can not be made clear in a brief space, -- almost one is persuaded it can not be made clear in any space... (JPI1:498). Here is what Whitehead said in 1911 about the theory of types: "All the contradictions can be avoided, and yet the use of classes and relations can be preserved as required by mathematics, and indeed by common sense, by a theory which denies to a class -- or relation -- existence or being in any sense in which the entities composing it -- or related by it -- exist" (MAT 293). But thirty years later Whitehead wrote:

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 *Principia Mathematica *and by the inelegant but obligatory theory of types as well, but he did not want to express these concerns professionally in 1911. Instead he wrote a book for lay people in which he was much more relaxed and, without criticizing (or even mentioning) his work in *Principia Mathematica, *laid an empirical foundation for his monumental metaphysical work of *Process and Reality.*

A final comment on the times of the first quarter of the twentieth century. In the Introduction to the Second Edition of *Principia Mathematica *published in 1925, Whitehead and Russell, in trying to repair the theory of types by the axiom of reducibility, mentioned a new suggestion proposed by Ludwig Wittgenstein for philosophical reasons (PM xiv). This suggestion "that functions of propositions are always truth-functions" (PM xiv) was made in his *Tractatus Logico Philosophicus *(TLP), a non-Platonic, tight-knit, precise logical system that had a tangential but critical influence on the logical positivist movement. Wittgenstein, who wrote *Tractatus Logico-Philosophicus *partly to address problems in *Principia Mathematica, *was so pleased with his book that he gave up philosophy, because he thought that he had solved all the philosophical problems that could be solved. F. P. Ramsey, in his review of the second edition of *Principia Mathematica. *proposes that "the whole trouble" with the theory of types "really arises from defective philosophical analysis" (N116:128) and gently chides Whitehead and Russell for not taking more seriously the suggestion of Wittgenstein.

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.^{8 }His *Logical Investigations *in which he established a new -- how shall we say it --* relational *philosophy based on simple language games has become the primary reference of the contemporary philosophical position called language analysis and was a massive attack on *Tractatus Logico-Philosophicus. *Like Whitehead, Wittgenstein was "born again" philosophically, and also, like Whitehead, repudiated the fundamental thesis of *Principia Mathematica.*

It should not be surprising that *Principia Mathematica *has had no significant lasting influence on twentieth century mathematics. We can see early evidences of its failure to engage contemporary mathematicians in a review of the Second Edition by B. A. Bernstein in 1926. "When one considers the caliber of our authors and the fact that the *Principia *has occupied a prominent place on mathematical shelves for fourteen years, one wonders that the book has influenced mathematics so little" (BAMS32:711). Bernstein gives a number of examples of the source of this failure, as explanations of his general belief "that the authors have admitted into the book concepts and principles based on considerations not sufficiently convincing -- concepts and principles based on views opposed to those forced on mathematicians by the work of Peano, Pieri, Hilbert, Veblen, Huntington" (BAMS32:712).

There is one major mathematical legacy of *Principia Mathematica *in which it** **is referenced in the title of perhaps the most significant paper that affects mathematics and its foundations of the twentieth century, "Ûber formal unentscheidbare Säze der *Principia Mathematica *und verwandter Systeme I" ("On formally undecidable propositions of *Principia Mathematica *and related systems I") (UUPM). It was written by Kurt Gödel in 1931. In the article he proved, not just suggested or forcibly argued, that the thesis of *Principia Mathematica *is false. One cannot deduce arithmetic, much less mathematics, from logic. There is no set, finite or infinite, of well defined axioms from which all the true theorems of arithmetic follow. Two years prior to 1931, Whitehead published *Process and Reality, *in which the thesis of *Principia Mathematica *cannot hold, although he never mentions it. Two years later in 1931 Whitehead claims "We cannot produce that final adjustment of well-defined generalities which constitute a complete metaphysics" (AI 145). At that time he probably also believed this statement with the word *mathematics *substituted for *metaphysics.*

**Whiteheadian Mathematics and Process Thought**

So far we have explored the technical shortcomings of Whitehead’s most significant mathematical works, *Universal Algebra *and *Principia Mathematica, *and their connections with and implications for his philosophy of mathematics. We conclude with some remarks on what -- with nearly a century of hindsight -- might be considered methodological shortcomings and their surprising relation to his mature metaphysical thought.

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 *commutative ring, *which is therefore an appropriate generalization of both objects. The formalism entifies, at least linguistically -- no ontological commitment is implied here -- and algebraists speak of and study rings. Whatever their abstract properties might be, they are shared by the integers and continuous real-valued functions.

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 *magma: *a set together with an operation, subject to no restrictions (e.g., associativity) whatsoever. Magmas are certainly more general than groups, but are they correspondingly more central to mathematics? Of course not; they are so general as to be jejune. One could similarly relax the axioms for a topological space to achieve more generality at the expense of losing the key features of spatiality.

*Universal Algebra, *in precisely this sense, is a poor framework for mathematics insofar as it unites spatial manifolds and symbolic logic by introducing the common notion of an *algebraic manifold *(Whitehead’s terminology) or a *semi-group *(current standard terminology), an object with very little structure or intrinsic interest.^{9 }In this case, generalization comes at the expense of abstract sterility. While *Universal Algebra *does have its moments, it is rich mathematically only insofar as Whitehead transcends the generality of his algebraic manifolds and deals in the specifics of Boolean algebra or Grassmannian manifolds.

(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 *Principia Mathematica *(or, equivalently, from those of Zermelo-Fraenkel set theory). We face at once the heuristic paradox, *if fields as diverse as, say, geometry and number theory might be flattened by logicism into the same essential activity, how is it that we sense them as diverse in the first place? Moreover, how is it that we so easily distinguish the big theorem from the throw-away lemma and the throw-away lemma from the empty inference -- valid, but with no interest whatsoever? *We set this paradox aside, however, to focus on the deeper and more decisive issue: the approach leads to failed levels of description.

Just as a pixel-by-pixel account of Seurat’s *Sunday Afternoon on the Island of La Grande Jatte *would be unappreciated as painting, just as a bit-by-bit digitized readout of Ravel’s *Chansons Madécasses *would be unrecognizable as song, and just as a physician would find a quantum mechanical description of his or her patient irrelevant to a diagnosis, *the view of mathematics set forth in *Principia Mathematica *is irrelevant to the working mathematician. *It is simply the wrong level of description for the activity in question. In our opinion, this, and not Gödel’s Incompleteness Theorem, is the basic functional failure of logicism. We might wonder idly from time to time whether Fermat’s Last Theorem has slipped through the net of the formally decidable, but this is of no consequence to anyone seriously engaged in number theory. As a matter of practice, one does not eschew logicism for its incompleteness but for its ineffectiveness. And. as a matter of esthetics, one does not ask the artist to leave the searing colors and throbbing life of a tropical paradise for a gray, lifeless plain,

(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.^{10 }We shall give one brief elementary technical illustration, but we emphasize that our interest here is only with broad concepts.

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 *embedded *in the group of reals. The point is that the structure of the former fits precisely into the structure of the latter. Now finite groups abound also (the permutations of finite sets, for example) and a structural relationship that one might consider between two finite groups G and His the possibility that *H *can be embedded in *G. *One can show that if this is so, then the number of elements that constitute *H *must divide the number of elements that constitute *G. *This, then, is an example of functoriality: the relationship of embeddability for groups corresponds directly with the relationship of divisibility for integers. Now witness an example of the power latent in this correspondence: Consider a group G consisting of 128 elements and a group *H *consisting of 120 elements. There are many possible structures for both *G *and *H, *but no matter, we can in full generality assert that *H *is not embeddable in *0 *(to rephrase, *H *cannot be structurally a part of G) because of the functorial relationship with integer arithmetic: 120 does not divide 128.

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.

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**References**

BAMS32 -- B. A. Bernstein. "Whitehead and Russell’s Principia Mathematica" *Bulletin of the American Mathematical Society *32 (Nov. Dec., 1926): 711-13.

BAMS34 -- Alonzo Church. "Principia: Volumes Hand Ill" *Bulletin of the American Mathematical Society *34 (1928): 237-40.

ESS -- *Essays in Science and Philosophy. *New York: Philosophical Library, 1948.

EWM -- Lewis S. Ford. *The Emergence of Whitehead’s Metaphysics, 1925-1929. *Albany: State University of New York Press, 1984.

IM -- *An Introduction to Mathematics. *(Number 15 in the Home University Library of Modern Knowledge.) London: Williams and Norgate, New York: Henry Holt and Company, 1911. London: Oxford University Press, Inc., 1948, 1958, 1969.

JP11 -- C. I. Lewis. *The Journal of Philosophy *11(1914): 497-502.

MAT -- "Mathematics" in ESS. Published originally in *Encyclopedia Britannica, *11^{th} ed., 1911,17, 878-83.

MFF -- Saunders Mac Lane. *Mathematics: Form and Function. *New York: Springer-Verlag, 1986, pp. 455-56.

MG -- "Mathematics and the Good" in ESS. Published originally in *The Philosophy of Alfred North Whitehead. *Edited by Paul Arthur Schilpp. Evanston and Chicago: Northwestern University Press, 1941, pp. 666-81.

N116 -- E P. Ramsey. "The New Principia." *Nature *116, 2908 (July 25, 1925): 127-28.

N58 -- G. B. Mathews. "Comparative Algebra." *Nature *58 (1898): 385-86.

OO -- Murray Code, *Order and Organism: Steps to a Whiteheadian Philosophy of Mathematics and the Natural Sciences, *Albany: State University of New York Press. 1985.

PRSL64 -- "Sets of Operations in Relation to Groups of Finite Order." Abstract Only. *Proceedings of the Royal Society of London *64 (1898-99): 319-20.

PS 17 -- Christoph Wassermann. "The Relevance of *An Introduction to Mathematics *to Whitehead’s Philosophy." *Process Studies *17/3 (Fall, 1988),

QJPAM23 – "On the Motion of Viscous Incompressible Fluids. A Method of Approximation." *Quarterly Journal of Pure and Applied Mathematics *23 (1888): 143-52*. *"Second Approximations to Viscous Fluid Motion. A Sphere Moving Steadily in a Straight Line." *Quarterly Journal of Pure and Applied Mathematics *23(1888): 143-52.

S9 -- Alexander Macfarlane. *Science 9 *(1899): 324-28. SPTC5 -- "The Philosophy of Mathematics." *Science Progress in the Twentieth Century *5 (October, 1910): 234-39.

TLP -- Ludwig Wittgenstein. *Tractatus Logico-Philosophicus. *Translated from the German by D. F Pears & B. F McGuinness. First English edition, 1922. London: Routledge & Kegan Paul, 1961.

UA -- *A Treatise on Universal Algebra. *Cambridge: Cambridge University Press, 1898.

UUPM -- Kurt Gödel. "Ûber formal unentscheidbare Sätze der *Principia Mathematica *und verwandter Systeme I." *Monatshefte für Mathematik und Physik *38 (Leipzig: 1931D): 173-98.

WPRM -- Granville C. Henry. "Whitehead’s Philosophical Response to the New Mathematics," *Explorations in Whitehead’s Philosophy. *Edited by Lewis S. Ford and George L. Kline. New York: Fordham University Press, 1983, pp. 14-28. An earlier version appeared in *The Southern Journal of Philosophy *7 (1969-70): 341-49.

** **

**Notes**

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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.

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2. Mac Lane in his analysis of schools in the philosophy of mathematics accepts two others, Platonism and Empiricism (MFF 455-456).

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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.

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4. We share this opinion with Murray Code who has expressed it in OO.

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5. See "Autobiographical Notes" (ESS 16).

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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.

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7. Personal conversation.

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8. See Norman Malcolm and G. H. Von Wright. *Ludwig Wittgenstein: A Memoir *London: Oxford University Press, 1958. 12-13.

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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.

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10. Categories and functors were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. See MFF for a technical introduction.