Foundations of Mathematics and Logicism

We have found it necessary to give very full proofs, because otherwise it is scarcely possible to see what hypotheses are really required, or whether our results follow from our explicit premises. (Principia Mathematica, Vol. 1 (1910), vi)

Emphatically I do not mean a neat little set of experiments to illustrate Proposition I and then the proof of Proposition I, a neat little set of experiments to illustrate Proposition II and then the proof of Proposition II, and so on to the end of the book. Nothing could be more boring. (Whitehead, Aims of Education, 5)

1. The Content and Character of Logicism

This paper focuses on Whitehead’s philosophical stance developed and held in the context of his collaboration with Bertrand Russell in the writing of Principia Mathematica between 1902 to around 1911; we will also add some remarks on his subsequent interest in mathematical education, and his 1934 paper on logic. For general references to the theme of this paper see Russell Papers, especially Volumes 1-4, 6 and 8; Mays 1959; Lowe 1985 and 1990; Grattan-Guinness 2000; Code 1985; Gaeta 2002; and Griffin 2003.

The key event for both Whitehead and Russell was the International Congress of Philosophy held in Paris in August 1900, where they learnt of the logical system that had been developed by Giuseppe Peano, and its applications in expressing mathematical theories. Whitehead’s initial reaction was to study the propositional calculus and Cantor’s set theory from various algebraic points of view; in particular, he treated the calculus as a Boolean algebra, and tried out factorising formulae and using group or invariant theory. However, no more general aim or scheme emerged. By contrast, by 1901 Russell converted Peano’s approach into a general philosophy of mathematics (to be explained shortly), which he outlined in The Principles of Mathematics (1903). The task of producing a formalised symbolic version of this position was mammoth, beyond the capacity of one person. Hence, Whitehead’s natural interest in Russell’s progress gradually turned into collaboration.

This collaboration began sometime between 1901 and 1902, and continued until the completion of the manuscript of the first three volumes of Principia Mathematica in the autumn of 1909, and to some extent during the years of its publication (1910-1913) by Cambridge University Press. Although their contacts were close—quite a lot of letters survive in the Bertrand Russell Archives in McMaster University, Canada, and they spend some periods of time together—Russell seems to have taken most or all of the major philosophical decisions, and Whitehead apparently agreed with them.

The main features of the philosophical programme presented in Principia Mathematica are stated in the following seven clauses:

1) They put forward the “logicist” position (to use the normal modern word, which is due to Rudolf Carnap (1929)), that “all” mathematics can be obtained from “logical” objects and devices alone.

2) This logic was mathematical logic, the propositional and multi-order predicate calculi with quantification of variables over individuals, propositional functions and relations, sets, and propositions. Cantorian set theory was subsumed within this logic by defining a set as the collection of objects satisfying some propositional function.

3) They used the theory of definite descriptions as the means especially for defining single-valued mathematical functions in terms of propositional functions contextually within a proposition.

4) The construction of mathematics started out with an elaboration of set theory itself, much within the account of mathematical logic. Then they founded arithmetic on the (contextual) definition of integers as sets of equipollent sets and moved on not only to rational and irrational numbers but also to Cantor’s theory of infinitely large cardinal and ordinal numbers.

Whitehead was due to handle parts of some branches of geometry in a fourth volume of Principia Mathematica. He was to build it upon a logicist theory of both geometry and the construction of space based upon the (non-logical) notion of events. He had outlined this approach in some detail in a long paper (1906a), where he made great use of the logic of relations. In the end, however, the fourth volume never appeared, although by 1917 he had written quite a lot of it (Harrell 1988).

5) Unfortunately the unrestricted use of logic and set theory led to paradoxes of various kinds. The most famous was Russell’s own, concerning the set of sets that do not belong to themselves; but there were others, such as the ancient Greeks’ liar paradox (“I am lying”) and several involving transfinite arithmetic or naming. By way of a solution, they proposed the theory of types, later known as “ramified,” in which the ranges of propositions and propositional functions were stratified into orders and then into types by the criterion of the vicious circle principle (essentially by the kind of variables that were or were not quantified).

6) They followed the contemporary fashion of axiomatizing a (logico-) mathematical theory as fully as possible. But three of the axioms were dubious, as they readily acknowledged. The axiom of choice, found independently by Ernst Zermelo and Russell in 1904, was a non-constructive axiom essential for many mathematical purposes, especially in set theory and mathematical analysis; but its legitimacy was doubted, or at least reluctantly accepted, by many figures. (It posed a special difficulty for logicism, since it posited an infinitude of selections of members from classes whereas the underlying logic was finitary.) The axiom of infinity was needed to furnish the infinite sets upon which much of the mathematics depended. Finally, the axiom of reducibility was an ad hoc reduction assumption adjoined to the theory of types; it was needed to make the mathematics coherent, for without it, for example, four could not be added to seven fifths.

7) Around 1899, under the influence of his friend G.E. Moore, Russell adopted with enthusiasm an empiricist and positivist position in epistemology in general. One of its effects on logicism was that the axiom of infinity had to be an empirical assumption about the infinitude of individuals, because they could hardly be construed as logical objects. For this uncomfortable reason Russell decreed that it be used only when necessary; but Whitehead forgot this and used it liberally in Part III on cardinal arithmetic. He noticed only in 1911, when the second volume of Principia Mathematica was in proof; its publication was delayed for several months while he made difficult modifications.

For a positivist orientation, the status of propositions and propositional functions is also unclear: are they fragments of language, or some things more abstract? Principia Mathematica is not clear on the matter, and the question worried Russell more than Whitehead.

2. Whitehead’s Contributions to the Collaboration

In writing the book, they divided it into Parts and then into Sections. One of them would carry out the initial basic account of a Section, which was then read and discussed with the other, and so on back and forth. Russell seems to have written out the whole final text for the printer; after publication they destroyed the manuscript. The table at the end of this article indicates the range of material covered.

As for Whitehead’s role, he devised the majority of the notations. He also seems to have taken initial responsibility for much of the first chapter of the introduction, which dealt with the basic logic and set theory, though most of the notions described there were due to Russell. He handled Part III on “Cardinal Arithmetic,” several portions of Part V on “Series” and most of Part VI on “Quantity.” This last Part appears to be a rather scrappy ensemble of real-line analysis and vector algebra; much of it was prepared with the needs of the geometry volume in mind. There we would have found a Part on geometry divided into four Sections: on the “projective,” “descriptive” and “metrical” branches, the first two Sections doubtless guided by two of Whitehead’s earlier tracts, (1906b and 1907); and a “Construction of space” presumably elaborating upon the presentation in the paper (1906a).

Whitehead’s apparent agreement with “all” the mathematics of 4) seems surprising, since the range of mathematics actually to be covered is unclear. For example, while much of the apparatus of mathematical analysis was laid out, most of that theory was not attempted at all; in particular, not a word is said on the differential and integral calculus or on complex-variable analysis, although all the preparatory material seems to be there. Again, only parts of geometry were to be presented, but not differential geometry, to mention one important part. Furthermore, no applied mathematics was to be treated. Such topics had received some, albeit rather light, attention in Russell’s Principles; why none at all in Principia Mathematica? And what about algebras of various (abstract, linear, and especially for the author of Universal algebra of 1898, Grassmann algebras), which were not treated in the Principles but lay close to the heart of Whitehead, an algebraist by inclination (as he implicitly confirmed in a late essay of 1939 (1947, 75-86))? One might expect that Whitehead, the better mathematician of the two, would have worried about these silences.

Maybe the answer lies in the implicational form in which Russell cast logicism. For him all mathematical statements took the form “if p, then q,” where p and q were propositions, and where both were required by logicism to contain only logical constants and permitted variables and functions of quantification only if necessary. Unhelpfully he called propositions of that logical form “pure mathematics,” which has no necessary connections with the traditional sense of being free from applications to the physical world, where indeed the same form is often used. While Whitehead did not follow Russell is using “pure” constantly in this sense, he seems to have accepted the implicational form of logicism, perhaps because of its attendant emphasis on variables mentioned in 2). He advocated their importance strongly; they were “the key which unlocks the whole subject” of the philosophy of mathematics as expressed in Russell’s Principles. Further, “the generalised conception of the variable and of its essential presence in all mathematical reasoning” led to a situation “which empties mathematics of everything but its logic” (1911b 234, 237; cf. 239). For in terms of variables propositional functions could be specified, and themselves be the variable arguments of functions of functions, and so on up through the orders and types along with the pertinent quantifications of some variables. The contents of the antecedents or consequents in implications, be they concerned with numbers (say) or with viscous fluids, could be ignored.

When the second edition of Principia Mathematica was mooted in the early 1920s Whitehead was initially enthusiastic, and suggested to Russell some revisions; but in the end he played no role in the writing of the new preface and three new articles. Indeed, he stated his non-involvement publicly and rather pointedly in Mind (1926); maybe he disliked the measure of extensionality that Russell had adopted in the revisions.

However, in the 1930s Whitehead sketched out a new logical system in a difficult paper (1934) that took “indication” as a new basic concept (“Ec!x,” meaning “behold x”) and thereafter imitated Principia Mathematica in its definitions and main results. While ingenious in its constructions, it is philosophically rather incoherent apart from some origins in his process philosophy. It is not a form of logicism, since indication cannot be construed as logical. I shall not consider it further, as it received very little reaction and seems to be limited in its mathematical scope. In particular, his graduate student of that time, W.V.O. Quine, dutifully spoke about the paper to the American Mathematical Society (Quine 1934), but neither then nor later did he adopt either the approach adopted in Whitehead’s paper or the philosophy in general; Quine’s logical work was much more close in philosophical spirit to Russell’s positivism, though with some negative as well as positive influences.

3. On Whitehead’s Later Uses of Logic and Logicism

After Principia Mathematica both men turned to other concerns, and collaboration ceased. Their later careers in and around philosophy were very different. Russell developed a “logical positivism,” in which the Moorean spirit of 7) above was maintained, and devices from mathematical logic were used or adapted (in particular, definite descriptions, the notion of type theory, and the importance of relations). For him Principia Mathematica stated all that needed to be said for the philosophy of mathematics (or would have done if the volume on geometry had been written).

By contrast, for Whitehead logicism was only a part, albeit a central one, of a broader philosophical picture of mathematics. He stated and defended logicism in various writings after 1913, most amply in 1916 when explaining “the organisation of thought.” In addition to stressing the high status given in Principia Mathematica to variables, the use made of relations of various kinds, and the place of set theory, he also divided the prosecution of the logicist programme into four stages: the “arithmetic,” concerning the propositional calculus; the “algebraic,” or predicate calculus with quantification and the theory of types; “general-function theory,” including set theory and definite descriptions; and the “analytic,” where “the investigation of the properties of special logical constructions” is conducted, so that “the whole of mathematics is included here” (1929, 163-170).

However, in this later philosophy Whitehead made little use of the technical devices available from logicism, and also ignored Russell’s positivist rules. The most substantial appearance came from his handling of geometry and space, which from the late 1910s evolved into an extensive account of his own version of the new subject of general relativity theory. Later, especially in his process philosophy he even preferred a version of part-whole theory (sometimes expressed within set theory) to the greater range of relationships available in set theory itself, since “it seems possible that both these conceptions of time-part and space-part are fundamental; that is, are concepts expressing relations which are directly presented to us” (1929, 202).

In Whitehead’s more general writings, logicism also played a modest role. For example, in his book An Introduction to Mathematics (1911a) he treated various basic theories but did not mention logicism at all. In an article (1911c) of the same time on “Mathematics” for the Encyclopaedia Britannica he did drew upon logic and set theory, indeed to a rather excessive extent given the space available; but he also included some remarks on applied mathematics as normally understood (and even commented on the history of mathematics), and his main point was to stress that mathematics involved more than number and quantity, not to claim that all of it was reducible to mathematical logic.

The scope of Whitehead’s vision for mathematics comes out clearly in a rather neglected part of his thought that emerged in the mid 1910s, after his appointment to a professorship at Imperial College in the University of London—mathematical education. He published various articles in and around this topic, and served from 1915 to 1916 as President of the Mathematical Association, the British national society for mathematics teachers. He reprinted some of this material, together with his recent general philosophical papers, as the book The Organisation of Thought (1917); six of these essays re-appeared, along with further educational writings, in the book The Aims of Education (1929).

In an essay of 1912 Whitehead stated that “we have to teach what logic is,” but under the assumption that boys “feel by an acquired instinct what it means to be logical, and to know a precise idea when they see it” and that “logic applies to life,” not that teachers “should indulge in the somewhat futile task of affixing names to elementary logical processes after the manner of primers of formal logic” (1947, 132; girls, it seems, were to look after themselves). More broadly, in his paper on the “organisation of thought” quoted above, he also stressed that “One great use of the study of logical method is not in the region of elaborate deduction, but to guide us in the study of the formation of the main concepts of science,” such as geometry (1929, 175).

To sum up, Whitehead’s philosophy of mathematics (and of science in general) gave a major place not only to the epistemological justification of mathematical knowledge but also to its creative side, both in research and also in some kinds of imitation in education. Logicism was an important necessary condition for fulfilling these aims, but not a sufficient one. The two quotations at the head of this article do not exhibit a contradiction in Whitehead’s thought, but relate to different aspects of his broad philosophy of mathematics.

4. Table: Summary by Sections of Principia Mathematica

The numbers of pages are from the first edition. Volume 2 started at Section IIIA, Volume 3 at Section VD. The titles of the Parts, and numbers of pages (omitting the introductions) were:

I. “Mathematical logic” (251);

II. “Prolegomena to cardinal arithmetic” (322);

III. “Cardinal arithmetic” (296);

IV. “Relation-arithmetic” (210);

V. “Series” (490);

VI. “Quantity” (257).

Section; pages(Short) “Title” or Description: Other included topics
IA: *1-*5; 41“Theory of deduction”: Propositional calculus, axioms
IB: *9-*14; 65“Theory of apparent variables”: Predicate calculus, types, identity, definite descriptions
IC: *20-*25; 48“Classes and relations”: Basic calculi: empty, non-empty and universal
ID: *30-*38; 73“Logic of relations”: Referents and relata, Converse(s)
IE: *40-*43; 26“Products and sums of classes”: Relative product
IIA: *50-*56; 57“Unit classes and couples”: Diversity; cardinal 1 and ordinal 2
IIB: *60-*65; 33“Sub-classes” and “sub-relations”: Membership, marking types
IIC: *70-*73; 63“One-many, many-one, many-many relations”: Similarity of classes
IID: *80-*88; 69“Selections”: Multiplicative axiom, existence of its class
IIE: *90-*97; 98“Inductive relations”: Ancestral, fields, “posterity of a term”
IIIA: *100-*106; 63“Definitions of cardinal numbers”: Finite arithmetic, assignment to types
IIIB: *110-*117; 121“Addition, multiplication and exponentiation” of finite cardinals: Inequalities
IIIC: *118-*126; 112“Finite and infinite”: Inductive and reflexive cardinals, À0, axiom of infinity
IVA: *150-*155; 46“Ordinal similarity”: Small “relation-numbers” assigned to types
IVB: *160-*166; 56“Addition” and “product” of relations: Adding a term to a relation, likeness
IVC: *170-*177; 71“Multiplication and exponentiation of relations”: Relations between sub-classes, laws of relation-arithmetic
IVD: *180-*186; 38“Arithmetic of relation-numbers”: Addition, products and powers
VA: *200-*208; 97“General theory of series”: Generating relations, “correlation of series”
VB: *210-*217; 103“Sections, segments, stretches”: Derived series, Dedekind continuity
VC: *230-*234; 58“Convergence” and “limits of functions”: Continuity, oscillation
VD: *250-*259;107“Well-ordered series”: Ordinals,” their inequalities, well-ordering theorem
VE: *260-*265; 71“Finite and infinite series and ordinals”: “Progressions,” “series of alpehs”
VF: *270-*276; 52Compact, rational and continuous series: Properties of sub-series
VIA: *300-*314; 105“Generalisation of number”: Negative integers, ratios and real numbers
VIB: *330-*337; 58“Vector-families”: “Open families,” vectors as directed magnitudes
VIC: *350-*359; 50“Measurement”: Coordinates, real numbers as measures
VID: *370-*375; 35“Cyclic families”: Non-open families, such as angles

Works Cited and Further Readings

Carnap, R. 1929. Abriss der Logistik, mit besondere Berücksichtigung der Relationstheorie und ihre Anwendungen (Vienna, Springer).

Harrell, M. 1988. “Extension to geometry of Principia mathematica and related systems II,” Russell, new ser. 8, 140-60.

Quine, W.V.O. 1951. “Whitehead and the rise of Modern Logic,” in The Philosophy of Alfred North Whitehead, edited by in P.A. Schilpp (La Salle, Illinois, Open Court), 125-64.

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

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

Whitehead, Alfred North. 1911b. “The Philosophy of Mathematics,” Science Progress in the Twentieth Century: A Quarterly Journal of Scientific Thought and Work, V, July 1910–April 1911 (London, John Murray), 234-39. [Review of H. Berkeley, Mysticism in modern mathematics (1910).]

Whitehead, Alfred North. 1911c. “Mathematics,” in Encyclopedia Britannica, 11th Edition, Vol. 17, 878-83. Also in Whitehead 1947, 195-208.

Whitehead, Alfred North. 1926. “Principia Mathematica. To the editor of Mind,” Mind, new ser. 35, 130.

Whitehead, A.N. 1934. “Indication, classes, number, validation,” Mind, new ser. 43, 281-97, 543 [corrigenda]. Also in Whitehead 1947, 227-40 (without the corrigenda).

Author Information

Ivor Grattan-Guinness
Middlesex University at Enfield
Queensway, Middlesex EN3 4SF, England

How to Cite this Article

Grattan-Guinness, Ivor, “Foundations of Mathematics and Logicism”, last modified 2008, The Whitehead Encyclopedia, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <>.