Here are some extracts from the Introduction:
In 1881 the American philosopher Charles S. Peirce published a remarkable paper in The American Journal of Mathematics called “On the Logic of Number.” Peirce’s paper was a watershed in nineteenth century 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, we will show that Peirce’s axiom system is actually equivalent to these better known systems.
It is not generally known that Peirce’s 1881 paper provided the first abstract formulation of the notions of partial and total linear order, that it introduced recursive definitions for arithmetical operations, nor that it proposed the first general definition of cardinal numbers in terms of ordinals.
Peirce was probably America’s greatest philosopher, and his interest in the foundations of mathematics was closely tied to his main philosophical concerns. Some of his most characteristic philosophical positions – his synechism and his phenomenological categories – bear the direct imprint of his research into the theory of sets and transfinite numbers. Peirce’s 1881 paper, in particular, is important for understanding his view of the nature of mathematics and its relation to deductive logic. It was published concurrently — in the same issue of AJM — with his father’s famous definition of mathematics as the science which draws necessary conclusions.
In the course of tracing out the implications of Peirce’s 1881 paper, we address the problem of locating his mature philosophy of mathematics vis-à-vis the traditional triad of logicism, formalism, and intuitionism. Although we show that Peirce’s view had similarities to and differences from all three, his understanding of mathematics was essentially sui generis. Perhaps the most characteristic aspect of Peirce’s approach is that he did not conceive mathematics to require any sort of epistemological foundation, whether in logic, intuition, or by means of constructive completeness proofs. This is why Peirce, in his scheme of categories, characterized mathematics as a First. “There is no more satisfactory way of assuring ourselves of anything,” Peirce said, “than the mathematical way of assuring ourselves of mathematical theorems.”