Selection from C.S. Peirce, “A Syllabus of Certain Topics of Logic” (1903)
An Outline Classification of the Sciences
180. This classification, which aims to base itself on the principal affinities of the objects classified, is concerned not with all possible sciences, nor with so many branches of knowledge, but with sciences in their present condition, as so many businesses of groups of living men. It borrows its idea from Comte’s classification; namely, the idea that one science depends upon another for fundamental principles, but does not furnish such principles to that other. It turns out that in most cases the divisions are trichotomic; the First of the three members relating to universal elements or laws, the Second arranging classes of forms and seeking to bring them under universal laws, the Third going into the utmost detail, describing individual phenomena and endeavoring to explain them. But not all the divisions are of this character.
The classification has been carried into great detail; but only its broader divisions are here given.
181. All science is either,
- A. Science of Discovery;
- B. Science of Review; or
- C. Practical Science.
182. By “science of review” is meant the business of those who occupy themselves with arranging the results of discovery, beginning with digests, and going on to endeavor to form a philosophy of science. Such is the nature of Humboldt’s Cosmos, of Comte’s Philosophie positive, and of Spencer’s Synthetic Philosophy. The classification of the sciences belongs to this department.
183. Science of Discovery is either,
- I. Mathematics;
- II. Philosophy; or
- III. Idioscopy.
184. Mathematics studies what is and what is not logically possible, without making itself responsible for its actual existence. Philosophy is positive science, in the sense of discovering what really is true; but it limits itself to so much of truth as can be inferred from common experience. Idioscopy embraces all of the special sciences, which are principally occupied with the accumulation of new facts.
185. Mathematics may be divided into
- a. the Mathematics of Logic;
- b. the Mathematics of Discrete Series;
- c. the Mathematics of Continua and Pseudo-continua.
I shall not carry this division further. Branch b has recourse to branch a, and branch c to branch b.
186. Philosophy is divided into
- a. Phenomenology;
- b. Normative Science;
- c. Metaphysics.
Phenomenology ascertains and studies the kinds of elements universally present in the phenomenon; meaning by the phenomenon, whatever is present at any time to the mind in any way.
Normative science distinguishes what ought to be from what ought not to be, and makes many other divisions and arrangements subservient to its primary dualistic distinction.
Metaphysics seeks to give an account of the universe of mind and matter.
Normative science rests largely on phenomenology and on mathematics; metaphysics on phenomenology and on normative science.
(Peirce, CP 1.180–186, EP 2.258–259, Online)
to integers, either non-negatives 
or positives 
I was especially interested in codings that were also surjective, thereby revealing something about the target domain of arithmetic.
and finite partial functions from 
All of this comes straight out of the primes factorizations. That type of bijection may remind some people of Dana Scott’s 
Corresponding to the positive integers there arose two species of graphical structures, which I dubbed “riffs” and “rotes”. See these links for more info:
Inquiry Into Inquiry 