Merely to say that Peirce was extremely fond of placing things into groups of three, of trichotomies, and of triadic relations, would fail miserably to do justice to the overwhelming obtrusiveness in his philosophy of the number three.
Indeed, he made the most fundamental categories of all “things” of any sort whatsoever the categories of “Firstness,” “Secondness,” and “Thirdness,” and he often described “things” as being “firsts” or “seconds” or “thirds.”
For example, with regard to the trichotomy “possibility,” “actuality,” and “necessity,” possibility he called a first, actuality he called a second, and necessity he called a third. Again: quality was a first, fact was a second, and habit (or rule or law) was a third. Again: entity was a first, relation was a second, and representation was a third. Again: rheme (by which Peirce meant a relation of arbitrary adicity or arity) was a first, proposition was a second, and argument was a third.
The list goes on and on. Let us refer to Peirce’s penchant for describing things in terms of trichotomies and triadic relations as Peirce’s “triadism.”
American mathematician who showed that there are only three algebras with a uniquely defined division: those of the real numbers, complex number, and quaternions.
Charles S. Pierce was a mathematician, astronomer, chemist, geodesist, surveyor, cartographer, metrologist, spectroscopist, engineer, inventor; psychologist, philologist, lexicographer, historian of science, mathematical economist, lifelong student of medicine; book reviewer, dramatist, actor, short story writer; phenomenologist, semiotician, logician, rhetorician and metaphysician.
The most important extension Peirce made of his earliest views on deduction, induction, and abduction involved was to integrate the three argument forms into his view of the scientific method. As so integrated, deduction, induction, and abduction are not simply argument forms any more: they are phases of scientific methodology, as Peirce conceived this methodology. In fact, in Peirce’s most mature philosophy he virtually (perhaps totally and literally) equates the trichotomy with the three phases he discerns in the scientific method.
Scientific method begins with abduction: a conjecture or hypothesis about what actually is going on. Then, by means of deductive inference, conclusions are drawn from the hypothesis about other things that must obtain if the hypothesis is assumed to be true. These other things, it is hoped, can be experimentally tested-for. Finally, hypothesis-testing is performed by seeking experimentally to detect something that has been deduced to obtain from the hypothesis. The entire procedure of hypothesis-testing, and not merely that part of it that consists of arguing from sample to population, is called induction in Peirce’s later philosophy.
The earliest clear statement of Peirce’s pragmatism comes from his 1878 paper "How To Make Our Ideas Clear." In this paper, Peirce introduces a maxim, or principle, which allows us to achieve the highest grade of clarity about the concepts we use. Peirce introduces this principle, which we shall discuss in detail below as the third grade of clarity, as a development of the rationalist notion of "clear and distinct ideas." Combining his pragmatic maxim with notions of clarity from Descartes and Leibniz, Peirce identifies three grades of clarity or understanding.
The subject matter of architectonic is the structure of all human knowledge. The purpose of providing an architectonic scheme is to classify different types of knowledge and explain the relationships that exist between these classifications. Peirce’s own architectonic system divides knowledge according to it status as a "science" and then explains the interrelation of these different scientific disciplines. His belief was that philosophy must be placed within this systematic account of knowledge as science. Peirce adopts his architectonic ambitions of structuring all knowledge, and organizing philosophy within it, from his great philosophical hero, Kant. This systematizing approach became crucial for Peirce in his later work. However, his belief in a structured philosophy related systematically to all other scientific disciplines was important to him throughout his philosophical life.