The philosopher Charles Sanders Peirce (1839–1914) did considerable work over a period of years on the classification of sciences (including mathematics). 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.
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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". 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:
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.
Sciences
In 1902, he divided science into Theoretical and Practical. 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. In 1903 he characterized Science of Review as:
...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 experiments (3) Idioscopy – the special sciences, about special classes of positive phenomena, and settling theoretical issues by special experiences or experimentsThus 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:
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.
Logic's divisions later
In a piece which the Collected Papers editors called "Phaneroscopy" and dated as 1906, Peirce wrote (CP 4.9):
...I extend logic to embrace all the necessary principles of semeiotic, and I recognize a logic of icons, and a logic of indices, as well as a logic of symbols; and in this last I recognize three divisions: Stecheotic (or stoicheiology), which I formerly called Speculative Grammar; Critic, which I formerly called Logic; and Methodeutic, which I formerly called Speculative Rhetoric
Thus the three main 1903 departments of logic were now sub-departments of the study of the logic of symbols.
In a letter to J. H. Kehler, printed in The New Elements of Mathematics v.3, p. 207 and dated 1911, Peirce wrote:
I have now sketched my doctrine of Logical Critic, skipping a good deal. I recognize two other parts of Logic. One which may be called Analytic examines the nature of thought, not psychologically but simply to define what it is to doubt, to believe, to learn, etc., and then to base critic on these definitions is my real method, though in this letter I have taken the third branch of logic, Methodeutic, which shows how to conduct an inquiry. This is what the greater part of my life has been devoted to, though I base it upon Critic.
There in 1911 Peirce does not mention the 1906 division into logics of icons, indices and symbols. Critic and Methodeutic appear, as in 1902 and 1903, as the second and third main departments of logic. Analytic is now the first department and the word "Stechiology" goes unused. He includes in Analytic the consideration of issues which, back in his 1902 Carnegie Institute application, he had discussed in sections on logic with headings such as "Presuppositions of Logic" and "On the Logical Conception of Mind" that he had placed before the sections on logic's departments (stechiology, critic, and methodeutic).
On the question of the relationship between Stechiology and the Analytic that seems to have replaced it, note that, in Draft D of Memoir 15 in his 1902 Carnegie Institute application, Peirce said that stechiology, also called grammatica speculativa, amounts to an Erkenntnisslehre, a theory of cognition, provided that that theory is stripped of matter irrelevant and inadmissible in philosophical logic, irrelevant matter such as all truths (for example, the association of ideas) established by psychologists, insofar as the special science of psychology depends on logic, not vice versa. In that same Carnegie Institute application as in many other places, Peirce treated belief and doubt as issues of philosophical logic apart from psychology.