Peirce and Ordinal Numbers

The main categories of Peirce's System are 'Firstness', 'Secondness', and 'Thirdness'. He gives no precise definition of them, but some triadic examples include, respectively, Possibility-Actuality-Necessity, Quality-Fact-Law, and Sign-Object-Interpretant. He acknowledges that his triadic structure is somewhat influenced by the Kantian and Hegelian Dialectic of Thesis-Antithesis-Synthesis, but while Thirdness similarly links the other two components, it is not as their sublation, but as their mediation. A clue as to the nature of this mediation is his description of the categories as 'cenopythagorean', presumably meaning 'neo-Pythagorean'. On that basis, a geometric construal suggests a point-line-triangle progression, e. g. in which the vertex of the triangle mediates between the endpoints of the opposing side. But the invoking of Pythagoras also suggests why 'Firstness', 'Secondness', and 'Thirdness' are inappropriate names for the categories. For, the Pythagorean characteristics that Peirce attributes to these categories are Numerically more accurately Cardinal, not Ordinal, i. e. are more accurately 'Oneness', 'Twoness', and 'Threeness'. Most obviously, the infinitude of the becoming-diverse of the Ordinals does not accommodate, as does Peirce, the closure of the sequence at Thirdness. Furthermore, Thirdness, unlike Pythagorean Threeness, merely succeeds Firstness and Secondness, and does not mediate between them. Likewise, Peirce's concept of Interpretant, which mediates between Sign and Object, is more accurately Threeness, not Thirdness.