csp-frege-revolu

HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? 24
1884]. Thus, by the same token and rationale that we would contest whether Peirce had devised a number theory within
his logical system by 1881, so would also we have to contest whether Frege had done so by 1879.
We have already noted, in consideration of the second criterion for describing the Fregean innovations, that
Frege employed the relations of ancestral and proper ancestral, akin to Peirce’s “father of” and “lover of” relation.
Indeed, the Fregean ancestrals can best be understood as the inverse of the predecessor relation (for an account of the
logical construction by Frege of his ancestral and proper ancestral, see [Anellis 1994, 75–77]). Frege used his definition
of the proper ancestral to define mathematical induction. If Peirce’s treatment of predecessor and induction is inadequate
in “The Logic of Number”, it is necessary to likewise note that Frege’s definition “Nx” of “x is a natural
number” does not yet appear in the Begriffsschrift of 1879 although the definition of proper ancestral is already present
in §26 of the Begriffsschrift. Frege’s definition of “Nx”, as van Heijenoort himself remarked, does not appear until Part
II of the Grundlagen der Arithmetik [Frege 1884].
The only significant differences between axiomatization of number theory by Richard Dedekind (1831–1914)
and Peirce’s was that Dedekind, in his [1888] Was sind und was sollen die Zahlen? started from infinite sets rather
than finite sets in defining natural numbers, and that Dedekind is explicitly and specifically concerned with the real
number continuum, that is, with infinite sets. He questioned Cantor’s (and Dedekind’s) approach to set theory, and in
particular Cantor’s conception of infinity and of the continuum, arguing for example, in favor of Leibnizian infinitesimals
rather than the possibility of the existence of an infinite number of irrationals between the rational numbers, arguing
that there cannot be as many real numbers as there are points on an infinite line, and, more importantly, arguing
that Cantor’s arguments in defining the real continuum as transfinite are mathematical rather than grounded in logic.
Thus, Peirce rejected transfinite sets, maintaining the position that Cantor and Dedekind were unable to logically support
the construction of the actual infinite, and that only the potential infinite could be established logically. (For discussions
of Peirce’s criticisms of Cantor’s and Dedekind’s set theory, see, e.g. [Dauben 1981] and [Moore 2010]).
Affirming his admiration for Cantor and his work nevertheless, from at least 1893 forward, Peirce undertook his own
arguments and construction of the continuum, albeit in a sporadic and unsystematic fashion. Some of his results, including
material dating from circa 1895 and from May 1908, unpublished and incomplete, were only very recently
published, namely “The Logic of Quantity”, and, from 1908, the “Addition” and “Supplement” to his 1897 “The Logic
of Relatives” (esp. [Peirce 1897, 206]; see, e.g. [Peirce 2010, 108–112; 218–219, 221–225]). He speaks, for example
of Cantor’s conception of the continuum as identical with his own pseudo-continuum, writing [Peirce 2010, 209] that
“I define a pseudo-continuum as that which modern writers on the theory of functions call a continuum. But this is
fully represented by, and according to G. Cantor stands in one to one correspondence with the totality [of] real values,
rational and irrational….”
Nevertheless, Dedekind’s set theory and Peirce’s, and consequently their respective axiomatizations of number
theory, are equivalent. The equivalence of Peirce’s axiomatization of the natural numbers to Dedekind’s, and Peano’s
in the Arithmetices principia [Peano 1889], is demonstrated by Paul Bartram Shields in his doctoral thesis Charles S.
Peirce on the Logic of Number [Shields 1981] and his [1997] paper based on the thesis.
These similarities have led Francesco Gana [1985] to examine the claim by Peirce that Dedekind in his Was
sind und was sollen die Zahlen? plagiarized his “Logic of Number”, and to conclude that the charge was unjustified,
that Dedekind was unfamiliar with Peirce’s work at the time.
Peirce did not turn his attention specifically and explicitly to infinite sets until engaging and studying the work of
Dedekind and Georg Cantor (1845–1918), especially Cantor, and did not publish any of his further work in detail, although
he did offer some hints in publications such as his “The Regenerated Logic” of 1896 [Peirce 1896]. (Some of
Peirce’s published writings and manuscripts have recently appeared in [Peirce 2010].)
The technical centerpiece of Dedekind’s mathematical work was in number theory especially algebraic number
theory. His primary motivation was to provide a foundation for mathematics and in particular to find a rigorous definition
of the real numbers and of the number continuum upon which to establish analysis in the style of Weierstrass.
This means that he sought to axiomatize the theory of numbers, based upon a rigorous definition of the real number
system which could be employed in defining the theory of limits of a function for use in the differential and integral
calculus, real analysis, and related areas of function theory. His concern, in short, is with the rigorization and
arithmetization of analysis.
For Peirce, on the other hand, the object behind his axiomatization of the system of natural numbers was stated
in “On the Logic of Number” [Peirce 1881, 85] as establishing that “the elementary propositions concerning number…are
rendered [true] by the usual demonstrations.” He therefore undertakes “to show that they are strictly syllogistic
consequences from a few primary propositions,” and he asserts “the logical origin of these latter, which I here
regard as definitions,” but for the time being takes as given. In short, Peirce here wants to establish that the system of