*Principia Mathematica*

*First published Tue May 21, 1996; substantive revision Tue Mar 30, 2010*

*Principia Mathematica*, the landmark work in formal logic written by
Alfred North Whitehead and
Bertrand Russell,
was first published in three volumes in 1910, 1912 and 1913. Written as a
defense of logicism (the view that mathematics is in some
significant sense reducible to logic) the book was instrumental in
developing and popularizing modern mathematical logic. It also served
as a major impetus for research in the foundations of mathematics
throughout the twentieth century. Along with the *Organon* written by
Aristotle and the *Grundgesetze der Arithmetik*
written by
Gottlob Frege,
it remains one of the most influential books on logic ever written.

- 1. History of
*Principia Mathematica* - 2. Significance of
*Principia Mathematica* - 3. Contents of
*Principia Mathematica* - Bibliography
- Other Internet Resources
- Related Entries

Interested readers may wish to view the

- Title page of the first edition of
*Principia Mathematica,*Volume 1 (1910) - Cover of the first edition of
*Principia Mathematica to *56*(1962).

## 1. History of *Principia Mathematica*

Logicism is the view that (some or all of) mathematics can be reduced to (formal) logic. It is often explained as a two-part thesis. First, it consists of the claim that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of the vocabulary of logic. Second, it consists of the claim that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of the theorems of logic. In Bertrand Russell's words, it is the logicist's goal “to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms” (1959, 74).

In its essentials, logicism was first advocated in the late seventeenth century by Gottfried Leibniz. Later, the idea was defended in greater detail by Gottlob Frege. During the critical movement initiated in the 1820s, mathematicians such as Bernard Bolzano, Niels Abel, Louis Cauchy and Karl Weierstrass succeeded in eliminating much of the vagueness and many of the contradictions present in the mathematical theories of their day. By the late 1800s, William Hamilton had also introduced ordered couples of reals as the first step in supplying a logical basis for the complex numbers. In much the same spirit, Karl Weierstrass, Richard Dedekind and Georg Cantor had also all developed methods for founding the irrationals in terms of the rationals. Using work by H.G. Grassmann and Richard Dedekind, Guiseppe Peano had then gone on to develop a theory of the rationals based on his now famous axioms for the natural numbers. Thus, by Frege's day, it was generally recognized that a large portion of mathematics could be derived from a relatively small set of primitive notions.

Even so, it was not until 1879, when Frege developed the necessary logical apparatus, that the project of logicism could be said to have become technically plausible. Following another five years' work, Frege arrived at the definitions necessary for logicising arithmetic and, during the 1890s, he worked on many of the essential derivations. However, with the discovery of paradoxes such as Russell's paradox at the turn of the century, it appeared that additional resources would need to be postulated if logicism were to succeed.

By 1903, both Whitehead and Russell had reached this same conclusion.
By this time, both men were in the initial stages of preparing second
volumes to their earlier books on related topics: Whitehead's 1898 *A
Treatise on Universal Algebra* and Russell's 1903 *The
Principles of Mathematics.* Since their research overlapped
considerably, they began collaborating on what would eventually become
*Principia Mathematica*. By agreement, Russell worked primarily
on the philosophical parts of the project (including the book's
philosophically rich Introduction, the theory of descriptions, and the
no-class theory), while the two men collaborated on the technical
derivations. As Russell tells us,

As for the mathematical problems, Whitehead invented most of the notation, except in so far as it was taken over from Peano; I did most of the work concerned with series and Whitehead did most of the rest. But this only applies to first drafts. Every part was done three times over. When one of us had produced a first draft, he would send it to the other, who would usually modify it considerably. After which, the one who had made the first draft would put it into final form. There is hardly a line in all the three volumes which is not a joint product. (1959, 74)

Initially, it was thought that the project might take a
year to complete.
Unfortunately, after almost a decade of difficult work on the part of
both men, Cambridge University Press concluded that publishing *
Principia* would result in an estimated loss of approximately 600
pounds. Although the press agreed to assume half this amount and the
Royal Society agreed to donate another 200 pounds, that still left a
100-pound deficit. Only by each contributing 50 pounds were the
authors able to see their work through to publication. Today there is
not a major academic library anywhere in the world that does not
possess a copy of this landmark publication.

## 2. Significance of *Principia Mathematica*

Achieving *Principia's* main goal proved to be
controversial. Primarily at issue were the kinds of assumptions that
Whitehead and Russell needed to complete their project. Although
*Principia* succeeded in providing detailed derivations of many
major theorems in set theory, finite and transfinite arithmetic, and
elementary measure theory, two axioms in particular were arguably
non-logical in character: the axiom of infinity and the axiom of
reducibility. The axiom of infinity in effect stated that there exists
an infinite number of objects. Thus, it made the kind of assumption
that is generally thought to be empirical rather than logical in
nature. The axiom of reducibility was introduced as a means of
overcoming the not completely satisfactory effects of the
theory of types,
the theory that Russell and Whitehead used to restrict the
notion of a well-formed expression, thereby avoiding paradoxes such as
Russell's paradox.
Although technically feasible, many critics concluded that the axiom
of reducibility was simply too ad hoc to be justified
philosophically. As a result, the question of whether mathematics
could be reduced to logic, or whether it could be reduced only to set
theory, remained open.

Despite these criticisms, *Principia Mathematica* proved to be
remarkably influential in at least three other ways. First, it
popularized modern mathematical logic to an extent undreamt of by its
authors. By using a notation superior in many ways to that of Frege,
Whitehead and Russell managed to convey the remarkable expressive
power of modern predicate logic in a way that previous writers had
been unable to achieve. Second, by exhibiting so clearly the deductive
power of the new logic, Whitehead and Russell were able to show how
powerful the modern idea of a formal system could be, thus opening up
new work in what was soon to be called metalogic. Third, *Principia
Mathematica* reaffirmed clear and interesting connections between
logicism and two of the main branches of traditional philosophy, namely
metaphysics and epistemology, thus initiating new and interesting work
in both of these areas.

Thus, not only did *Principia* introduce a wide range of
philosophically rich notions (such as propositional function,
logical construction,
and
type theory),
it also set the stage for the
discovery of classical metatheoretic results (such as those of
Kurt Gödel,
Alonzo Church,
Alan Turing
and others) and initiated a tradition of common technical
work in fields as diverse as philosophy, mathematics, linguistics,
economics and computer science.

Today there remains controversy over the ultimate substantive
contribution of *Principia*, with some authors holding that,
with the appropriate modifications, logicism remains a feasible
project. Others hold that the philosophical and technical
underpinnings of the Whitehead/Russell project simply remain too weak
or confused to be of great use to the logicist. Interested readers are
encouraged to consult Quine (1966a), Quine (1966b), Landini (1998),
Linsky (1999), Hale and Wright (2001), and Hintikka (2009).

## 3. Contents of *Principia Mathematica*

*Principia Mathematica* originally appeared in three
volumes. Together these three volumes are divided into six
parts. Volume 1 begins with a lengthy Introduction containing sections
entitled “Preliminary Explanations of Ideas and
Notations,” “The Theory of Logical Types,” and
“Incomplete Symbols.” It also contains Part I, entitled
“Mathematical Logic,” which contains sections on
“The Theory of Deduction,” “Theory of Apparent
Variables,” “Classes and Relations,” “Logic of
Relations,” and “Products and Sums of Classes”; and
Part II, entitled “Prolegomena to Cardinal Arithmetic,”
which contains sections on “Unit Classes and Couples,”
“Sub-Classes, Sub-Relations, and Relative Types,”
“One-Many, Many-One and One-One Relations,”
“Selections,” and “Inductive Relations.”

Volume 2 begins with a “Prefatory Statement of Symbolic Conventions.” It then continues with Part III, entitled “Cardinal Arithmetic,” which itself contains sections on “Definition and Logical Properties of Cardinal Numbers,” “Addition, Multiplication and Exponentiation,” and “Finite and Infinite”; Part IV, entitled “Relation-Arithmetic,” which contains sections on “Ordinal Similarity and Relation-Numbers,” “Addition of Relations, and the Product of Two Relations,” “The Principle of First Differences, and the Multiplication and Exponentiation of Relations,” and “Arithmetic of Relation-Numbers”; and the first half of Part V, entitled “Series,” which contains sections on “General Theory of Series,” “On Sections, Segments, Stretches, and Derivatives,” and “On Convergence, and the Limits of Functions.”

Volume 3 continues Part V with sections on “Well-Ordered Series,” “Finite and Infinite Series and Ordinals,” and “Compact Series, Rational Series, and Continuous Series.” It also contains Part VI, entitled “Quantity,” which itself contains sections on “Generalization of Number,” “Vector-Families,” “Measurement,” and “Cyclic Families.”

A fourth volume on geometry was planned but never completed (1959, 99).

Contemporary readers (i.e., those who have learned logic in the last few decades of the twentieth century or later) will find the book's notation somewhat antiquated. Even so, the book remains one of the great scientific documents of the twentieth century.

## Bibliography

- Chihara, Charles (1973).
*Ontology and the Vicious Circle Principle*, Ithaca: Cornell University Press. - Church, Alonzo (1974). “Russellian Simple Type Theory,”
*Proceedings and Addresses of the American Philosophical Association*, 47: 21–33. - Church, Alonzo (1978). “A Comparison of Russell's Resolution
of the Semantical Antinomies with that of Tarski,”
*Journal of Symbolic Logic*, 41: 747–760. Repr. in A.D. Irvine (ed.),*Bertrand Russell: Critical Assessments*, vol. 2, New York and London: Routledge, 1999, 96–112. - Copi, Irving (1971).
*The Theory of Logical Types*, London: Routledge and Kegan Paul. - Frege, Gottlob (1893, 1903).
*Grundgesetze der Arithmetik*, Band I (1893), Band II (1903), Jena: Verlag Hermann Pohle. Ed. and trans. by M. Furth in part as*The Basic Laws of Arithmetic*, Berkeley: University of California Press, 1964. - Gabbay, Dov M., and John Woods (eds) (2009).
*Handbook of the History of Logic: Volume 5 — Logic From Russell to Church*, Amsterdam: Elsevier/North Holland. - Hale, Bob, and Crispin Wright (2001).
*The Reason's Proper Study*, Oxford: Clarendon Press. - Hintikka, Jaakko (2009). “Logicism,” in Irvine,
A.D. (ed.),
*Philosophy of Mathematics*, Amsterdam: Elsevier/North Holland, 271–290. - Irvine, A.D. (ed.) (2009).
*Philosophy of Mathematics*, Amsterdam: Elsevier/North Holland. - Landini, Gregory (1998).
*Russell's Hidden Substitutional Theory*, New York and Oxford: Oxford University Press. - Link, Godehard (2004).
*One Hundred Years of Russell's Paradox*, Berlin and New York: Walter de Gruyter. - Linsky, Bernard (1999).
*Russell's Metaphysical Logic*, Stanford: CSLI Publications. - Proops, Ian (2006). “Russell's Reasons for Logicism,” Journal of the History of Philosophy, 44: 267–292.
- Quine, W.V (1960).
*Word and Object*, Cambridge: MIT Press. - Quine, W.V (1966a).
*Selected Logic Papers*, New York: Random House. - Quine, W.V (1966b).
*Ways of Paradox*, New York: Random House. - Ramsey, Frank P. (1931).
*The Foundations of Mathematics*, London: Kegan, Paul, Trench, Trubner. - Rodriguez-Consuegra, Francisco (1991).
*The Mathematical Philosophy of Bertrand Russell*, Boston: Birkhäuser Press. - Russell, Bertrand (1903).
*Principles of Mathematics*, Cambridge: Cambridge University Press. - Russell, Bertrand (1919).
*Introduction to Mathematical Philosophy*, London: George Allen and Unwin. - Russell, Bertrand (1948). “Whitehead and
*Principia Mathematica*,”*Mind*, 57: 137–138. - Russell, Bertrand (1959). My Philosophical Development, London: George Allen and Unwin, and New York: Simon and Schuster.
- Shapiro, Stewart (ed.) (2005).
*The Oxford Handbook of Philosophy of Mathematics and Logic*, Oxford: Oxford University Press. - Urquhart, Alasdair (1988). “Russell's Zig-Zag Path to the Ramified
Theory of Types,”
*Russell*, 8: 82–91. - Whitehead, Alfred North (1898).
*A Treatise on Universal Algebra*, Cambridge: Cambridge University Press. - Whitehead, Alfred North (1906).
*On Mathematical Concepts of the Material World*, London: Dulau. - Whitehead, Alfred North, and Bertrand Russell (1910, 1912, 1913).
*Principia Mathematica*, 3 vols, Cambridge: Cambridge University Press. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as*Principia Mathematica to *56*, Cambridge: Cambridge University Press, 1962. - Wright, Crispin (1983).
*Frege's Conception of Numbers as Objects*, Aberdeen: Aberdeen University Press.

## Other Internet Resources

*Principia Mathematica*, reproduced in the University of Michigan Historical Math Collection.- Principia Mathematica: Whitehead and Russell, by Stanley Burris (Mathematics, U. Waterloo)

## Related Entries

Frege, Gottlob |
Frege, Gottlob: logic, theorem, and foundations for arithmetic |
Leibniz, Gottfried Wilhelm |
logic: classical |
logicism and neologicism |
*Principia Mathematica*: notation in |
propositional function |
Russell, Bertrand |
Russell's paradox |
type theory |
Whitehead, Alfred North