On May 14, 1867, the 27–year-old Charles Sanders Peirce, who eventually founded pragmatism, presented a paper entitled "On a New List of Categories" to the American Academy of Arts and Sciences.
Among other things, this paper outlined a theory of predication involving three universal categories that Peirce continued to apply in philosophy and elsewhere for the rest of his life.
[1][2] The categories demonstrate and concentrate the pattern seen in "How to Make Our Ideas Clear" (1878, the foundational paper for pragmatism), and other three-way distinctions in Peirce's work.
So Peirce's claim that three categories are sufficient amounts to an assertion that all manifolds of meaning can be unified in just three steps.
That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think through, into being subjects thought of.
Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into Realms for the different Predicaments.
Part of the justification for Peirce's claim that three categories are both necessary and sufficient appears to arise from mathematical ideas about the reducibility of n-adic relations.
Others, notably Robert Burch (1991), Joachim Hereth Correia and Reinhard Pöschel (2006), have offered proofs of the Reduction Thesis.