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.

Mathematics and Philosophy

Peirce divides mathematics into three areas that correspond roughly to discrete mathematics, mathematics of the infinite, and mathematical or formal logic. We now think of Peirce’s groundbreaking work in mathematical logic as belonging to logic proper rather than being a branch of mathematics. More important though is the role of mathematics as the provider of guiding principles for subsequent sciences, and particularly philosophy. Following his father, Peirce treated mathematics as "the science which draws necessary conclusions." What Peirce means is that mathematics is free from existential concerns about its constructs. In this sense, it is hypothetical and abstract. Peirce, for instance, states that mathematics "makes constructions in the imagination according to abstract precepts, and then observes these imaginary objects, finding in them relations of parts not specified in the precept of construction." What Peirce means is that mathematics creates hypothetical constructions, i.e., constructions which are abstracted and not necessarily actual, and then derives logically necessary connections between them and about them. These "necessary conclusions" about mathematical constructs provide general laws or principles for deriving logically necessary connections between and about all constructs, imaginary or actual. In short, the kinds of reasoning employed in mathematics provide general rules of reasoning, and function as principles to guide our reasoning in subsequent science, particularly philosophy.

For example, we can see the provision of guiding or leading principles from mathematics through the following story about irrational numbers. An irrational number is a number which cannot be expressed as the ratio of two integers (i.e. is a non-terminating, non-repeating decimal whose simplest expression is in the form.