The philosopher Charles Sanders Peirce (1839–1914) did considerable work over a period of years on the classification of sciences (including mathematics).See "Development of Peirce's classification of sciences - three stages: 1889, 1898, 1903" by Tommi Vehkavaara, 2003, and "The outline of Peirce's classification of sciences (1902-1911)" by Tommi Vehkavaara, 2001, His classifications are of interest both as a map for navigating his philosophy and as an accomplished polymath's survey of research in his time. Peirce himself was well grounded and produced work in many research fields, including logic, mathematics, statistics, philosophy, spectroscopy, gravimetry, geodesy, chemistry, and experimental psychology.Burch, Robert (2001, 2009), "Charles Sanders Peirce", Stanford Encyclopedia of Philosophy. Eprint. Also see Peirce's own list published 1906 in the entry "Peirce, C(harles) S" in v. 2, p. 248, American Men of Science, J. McKern Cattell, ed.
Classifications
Philosophers have done little work on classification of the sciences and mathematics since Peirce's time. Noting Peirce's "important" contribution, Denmark's Birger Hjørland commented: "There is not today (2005), to my knowledge, any organized research program about the classification of the sciences in any discipline or in any country".See Hjørland's comments here.Professor in knowledge organization, library science. See faculty profile page As Miksa (1998) writes, the "interest for this question largely died in the beginning of the 20th century". It is not clear whether Hjørland includes the classification of mathematics in that characterization.
Taxa
In 1902 and 1903 Peirce elaborates classifications of the sciences in: "A Detailed Classification of the Sciences" in Minute Logic (Feb.–Apr. 1902), Collected Papers of Charles Sanders Peirce (CP) v. 1, paragraphs 203–283Eprint July 1902 application to the Carnegie institution (MS L75)Arisbe Eprint "An Outline Classification of the Sciences (CP 1.180-202)Eprint and Eprint in his "A Syllabus of Certain Topics in Logic" (1903), wherein his classifications of the sciences take more or less their final form
However, only in the "Detailed Classification" and the Carnegie application does he discuss the taxa which he used, which were inspired by the biological taxa of Louis Agassiz.CP 1.229-230, (from the "Minute Logic", 1902), Eprint
+Taxa of scientific departments Name Characterization ("A Detailed Classification of the Sciences",Collected Papers, v. 1, paragraph 238, 1902, Eprint Feb.-Apr. 1902) Characterization(Carnegie application,Manuscript L75.351-353, (in the "final draft" = actually submitted version of the Carnegie Application) July 1902, Eprint July 1902) Examples(1902 examples are the Carnegie application) Branch of science Branches differ in fundamental purposes. Throughout a branch, there is one same animating motive (though researchers in a branch's different classes seem to live in different worlds). Peirce's three branches (1903): Science of Discovery. Science of Review. Practical Science. Class of science Classes differ radically in observation. Observations in one class (say physical & psychological sciences) cannot yield the kind of information which another class (say pure mathematics) requires of observation. Throughout a class, researchers feel that they inquire into the same great subject (as kinds of inquiry differ but interconnect). Peirce's three classes of discovery science: Pure mathematics. Cenoscopic philosophy. The special sciences. Order of science Two orders within one class or subclass may differ hierarchically, one more general, the other more specialized. Throughout an order, researchers pursue the same general kind of inquiry (but deal with different kinds of conceptions). Peirce's 1902 example of various orders: General Physics. Biology. Geology. Family of science Has special name, journal, society. Studies one group of facts. Researchers understand one another in a general way and naturally associate together. Throughout a family, researchers have the same general conceptions (but differ in skills). Peirce's 1902 example of various families: Astronomy. Geognosy. Genus of science "I can give no such definitions of genera and species, not having carried my classification of the sciences to these minutiae" (of definitions of taxa; he does use the genus taxon). Throughout the genus, researchers have the same skills (but differ in acquaintance with facts in detail). Peirce's 1902 example of various genera: Optics. Electrics. Species of science The species is the narrowest division still having societies and journals, each researcher is thoroughly well qualified in all parts of it. Peirce's 1902 example of various species: Entomology. Ichthyology. Variety of science Researchers devote lives to a variety of science, but not so numerously as to support distinct societies and journals. Peirce's 1902 example of various varieties: Study of Kant. Study of Spinoza.
Sciences
In 1902, he divided science into Theoretical and Practical.Peirce (1902), Manuscript L75.355, Application to the Carnegie Institute, Arisbe Eprint Theoretical Science consisted of Science of Discovery and Science of Review, the latter of which he also called "Synthetic Philosophy", a name taken from the title of the vast work, written over many years, by Herbert Spencer. Then, in 1903, he made it a three-way division: Science of Discovery, Science of Review, and Practical Science.Peirce (1903), CP 1.181, Eprint In 1903 he characterized Science of Review as:Peirce (1903), CP 1.182 Eprint ...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.
Peirce had already for a while divided the Sciences of Discovery into: (1) Mathematics – draws necessary conclusions about hypothetical objects (2) Cenoscopy – philosophy about positive phenomena in general, such as confront a person at every waking moment, rather than special classes, and not settling theoretical issues by special experiences or experimentsPeirce borrows the terms "cenoscopy" and "idioscopy" from Jeremy Bentham. See Peirce: (1902) CP 1.239-241; (1903) CP 1.183-187; (c. 1903) CP 6.6; and (1905) CP 8.199. (3) Idioscopy – the special sciences, about special classes of positive phenomena, and settling theoretical issues by special experiences or experiments Thus Peirce ends up framing two fields each of which is philosophy in a sense: cenoscopic philosophy which precedes the special sciences, and synthetic philosophy (that is to say, science of review), which does take advantage of the results of all the sciences of discovery and develops, for instance, classifications of the sciences.
Peirce opens his 1903 classification (the "Syllabus" classification) with a concise statement of method and purpose:The Collected Peirce (CP), v. 1, paragraph 180 Eprint 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 following table is based mostly on Peirce's 1903 classification, which was more or less the final form. But see after the table for discussion of his later remarks on the divisions of logic.
+ Classification of the Sciences of Discovery Classes Subclasses Orders Other Taxa (suborders, families, etc.) I. Mathematics. The scientific study of hypotheses which it first frames and then traces to their consequences. A. Mathematics of Logic. B. Mathematics of Discrete Series. C. Mathematics of Continua and Pseudocontinua. (Note: By "continuum" Peirce meant, until 1908,See "Peirce's Clarifications of Continuity" by Jérôme Havenel, Transactions Winter 2008 pp. 68-133. From p. 119: "It is on May 26, 1908, that Peirce finally gave up his idea that in every continuum there is room for whatever collection of any multitude. From now on, there are different kinds of continua, which have different properties." a continuum of instants (as he called them) beyond any Cantorian aleph's worth.Peirce, C.S., "Analysis of the Methods of Mathematical Demonstration", Memoir 4, Draft C, Manuscript L75.90-102, see 99-100, Eprint He held that such a continuum was the true subject matter of that which we now call topology, and that the reals, the complex reals, etc., constituted pseudocontinua.) II. Cenoscopy, or Philosophy. (Philosophia Prima, First Philosophy.) About positive phenomena in general, such as are available to every person at every waking moment, and not about special classes of phenomena. Does not resort to special experiences or experiments in order to settle theoretical questions. Epistêmy (1902 classification only).CP 1.278 Eprint A. Phenomenology (or Categorics or Phaneroscopy). (Includes the study of the cenopythagorean categories: Firstness, Secondness, and Thirdness). B. Normative Science. i. Esthetics. (Study of the good, the admirable. Peirce reserved the spelling "aesthetics" for the study of artistic beauty.) ii. Ethics. (Study of right and wrong). iii. Logic (or Semiotic or Formal Semiotic). (Study of true and false.)