Charles S. Peirce On The Logic Of Number
Paul Bartram Shields
Source: DAI, 41, no. 12A, (1981): 5134
The topic of this thesis is a single brief paper
written by Peirce in 1881, called "On the Logic of Number." Despite
its brevity, Peirce's 1881 paper was one of the major achievements of the
nineteenth century in the foundations of mathematics. It contained the first
successful axiom system for the natural numbers. Since scholarship has
traditionally attributed priority in this regard to the axiom systems of
Richard Dedekind, in 1888, and Giuseppe Peano, in 1889, an important result of
this thesis is its demonstration that Peirce's axiom system is actually
equivalent to these latter.
The technical sections of
this thesis provide important historical background on the systems of Peirce,
Dedekind, and Peano. It is pointed out that Peirce's 1881 paper gave the first
abstract formulation of the notions of partial and linear order, and the first
ordinal definition of cardinals. The approaches of all three to such topics as
mathematical induction, use of recursive definitions, cardinality,
categoricity, and the axiom of infinity are discussed. These sections culminate
with the formal demonstration, in Chapter II, that Peirce's axiom system is
equivalent to that of Dedekind. This proof employs onlyl a modest set theory
with no special assumptions. It is more of historical than mathematical value.
The significance of this
proof goes beyond merely giving credit where credit is due. There is a
widespread tendency to view the foundations of mathematics as having sprung
full-grown from the heads of a few individuals, notably Frege, Peano, and
Russell. The work of Peirce, and that of many other nineteenth century
pioneers, from Boole to Schroder, is often neglected, or, at best, consigned to
some vague "prel-history" of foundations. By establishing the
priority of Peirce's axiom system, this thesis takes issue with the
conventional wisdom and attempts to reinforce a more accurate perception of the
gradual and continuous development of foundations in the nineteenth century.
Peirce's interest in the
foundations of mathematics was intimately related to his dominant philosophical
concerns. Some of his most characteristic metaphysical and epistemological
doctrines, e.g., synechism and the theory of the categories, bear the direct
imprint of his work on sets and transfinite numbers. In particular, his 1881
paper is important for understanding Peirce's classification of the sciences.
Although it has apparently escaped the attention of most Peirce scholarship,
his 1881 paper was published simultaneously with his father's famous definition
of mathematics as "the science which draws necessary conclusions."
This historical connection is important because Peirce eventually adopted his
father's definition as the centerpiece of his own philosophy of mathematics.
Hence it indicates how Peirce himself understood the philosophical implications
of his axiom system, and sheds light on how he viewed the relation between
mathematics and logic, and thus the classification of the sciences as a whole.
In its final section, this
thesis addresses the problem of locating Peirce's mature philosophy of
mathematics vis-a-vis the traditional positions of logicism, intuitionism, and
formalism. Similarities and differences with all three are found, but the
diffrences are emphasized. Perhaps the most important,
and contemporary, single feature of Peirce's conception of mathematics is that
he does not conceive it to require any foundation at all, whether in logic, in
intuition, or by means of constructive completeness proofs. Mathematics, for
Peirce, is essentially independent and self-sufficient.