+++++++++++++++++++++++++++++++++++++++++++++I was actually surprised by a comment by John Baez over at the n-Category Cafe about his surprise that Ernst Schröder, a mathematician of whom he had known through Schröder's work on mathematical logic (ML), had published (in 1870) on the number sequences now often called the little (A001003, Wikipedia) and large (A006318, Wiki) Schröder numbers. I was surprised because I had been aware of Schröder only through these somewhat important number sequences, not his later work on ML, on which he published three-volumes between 1890-1905 after studying the research of Peirce and De Morgan, among others.
Wikipedia on the little Schröder, or Hipparchus-Schröder, numbers says,
Plutarch's dialogue Table Talk contains the line:
"Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952."
This statement went unexplained until 1994, when David Hough, a graduate student at George Washington University, observed that there are 103049 ways of inserting parentheses into a sequence of ten items. The historian of mathematics Fabio Acerbi (CNRS) has shown that a similar explanation can be provided for the other number: it is very close to the average of the tenth and eleventh Schröder–Hipparchus numbers, 310954, and counts bracketings of ten terms together with a negative particle.
The problem of counting parenthesizations was introduced to modern mathematics by Schröder (1870).
Plutarch's recounting of Hipparchus's two numbers records a disagreement between Hipparchus and the earlier Stoic philosopher Chrysippus, who had claimed that the number of compound propositions that can be made from 10 simple propositions exceeds a million. Contemporary philosopher Susanne Bobzien (2011) has argued that Chrysippus's calculation was the correct one, based on her analysis of Stoic logic. Bobzien reconstructs the calculations of both Chrysippus and Hipparchus, and says that showing how Hipparchus got his mathematics correct but his Stoic logic wrong can cast new light on the Stoic notions of conjunctions and assertibles.
Peirce is known for contributions (as early as 1882) to graphical logic (see the Wiki logical graphs and Anellis and Abeles below).
Given the potential connections among Peirce's work on graphical logic and Schröder's earlier interest in combinatorics (a somewhat compulsive addiction according to one biographer) and later interest in ML, I have some questions:
Did Schröder develop any diagrammatics directly related by him to mathematical logic?
(I'm not sure how related Schröder's interest in parenthesizations/bracketings was to any interest he may have had at that time in ML. Most likely his interest was purely combinatorial at that time and related to functional iteration, but see my footnote.)
If he made somewhat significant/influential contributions, did at least some of these independently precede and then influence the work of Peirce on graphical logic, was it the other way around, or rather was it a matter of concurrent interplay/synergy between the two?
(The pernicious effects of compartmentalization)
Schröder was not always a logician. He published his work (and began work in earnest) on ML well after his work in combinatorics and functional iteration. See the book "A History of Complex Dynamics: From Schröder to Fatou to Julia" (1994) by Alexander for some history on Schröder's contributions to these latter two fields circa 1870. On page 10 are even the compositional inversion polynomials of OEIS A134685, related to integer partitions of $2n$, that Schröder naturally encountered in some computations on iteration, which reduce to the Ward polynomials, famous for their connections to phylogenetic tree space and, therefore, associahedra and bracketings. Schröder most certainly was not aware of all these connections, but he probably was aware of similar work by his contemporary Cayley. Only in 1877 does he begin to publish on logic, and Dipert in "The Life and Work of Ernst Schröder" (p. 12) claims that Schröder's interest in logic was first aroused probably no earlier than 1872-1873. The point is his early career in combinatorics and complex analysis, specifically functional iteration, although dovetailing nicely with his later interest in ML, was most likely distinct from ML and any general diagrammatics developed specifically for ML.
Cayley, influenced by Boole, published in the 1840s on invariant theory and; in the 1850s, on tree graphs in characterizing iterated derivatives; in 1871, a two page "Note on the calculus of logic", referring to Boole's "The calculus of logic", about the time Dipert thinks Schröder first became interested in logic; and in 1879, a modest paper on functional iteration related to Newton's method for finding zeros. (See "The Historical Sources of Tree Graphs and the Tree Method in the Work of Peirce and Gentzen" by Anellis and Abeles for an account of Cayley's influence on Peirce.)
However, Schröder's and Cayley's research interests overlapped in the theory of invariants (Cayley--1840s-60s, Schröder--1870-71), combinatorics (Cayley and trees and bracketings--1857, Cayley and dissected polygons--1891, Schröder and bracketings--1871), functional iteration (Cayley--1879, Schröder--1871, Schroeder had a more ambitious approach than Cayley), and then later in algebraic logic (Cayley--1871, Schröder--1877 onward}. The famous Kirkman-Cayley numbers enumerate dissections of polygons, and, Schröder in his doctoral thesis in 1862 even defines $p/q$-polygons, polygons with a fractional number of sides. i.e., fractional polygons. The two definitely crossed paths several times.
Both Cayley and Schröder were influenced by Boole's and Hermann Grassmann"s works on invariant theory. Interestingly, "A formalization of Grassmann-Cayley ALgebra in COQ and its application in theorem proving in projective geometry" by Fuchs and Thery (2010) introduces binary trees. Schröder's teacher Hesse was keen on algebraic geometry and projective spaces. In "A Tutorial on Grassmann-Cayley Algebra" White states, "The Grassmann-Cayley algebra is first and foremost a means of translating synthetic projective geometric statements into invariant algebraic statements in the bracket ring, which is the ring of projective invariants." In "On the exterior calculus and invariant theory," Barnabei, Brini, and Rota wrote, "To the best of our knowledge of published work, the first mathematicians to understand, albeit imperfectly, the program of <H. Grassmann's> Ausdehnungslehre were Clifford and Schröder ..." and "It was Schröder, in an appendix to his “Algebra der Logik,” who first stressed the analogy between the algebra of progressive and regressive products, and the algebra of sets with union and intersection."
Schröder's work is usually approached by three different camps--analysts (functional iteration), combinatorialists (Schroeder's numbers), or logicians (ML)--all somewhat on average insular in their approach, but as put in the MacTutor bio:
Schröder "never considered himself to be a logician, as Peckhaus points out [...]:
His very own object of research was absolute algebra in respect to its basic problems and fundamental assumptions. What was the connection between logic and algebra in Schröder's research? ... one could assume that these fields belong to two separate fields of research, but this is not the case. They were intertwined in the framework of his heuristic idea of a general science.
In fact Schröder started out being interested in mathematical physics, and his move towards logic was simply an attempt to deepen its foundations."
Schröder studied mathematical physics under Kirchhoff (cohomology and homology have their roots in the rules for electrical circuits formulated by Kirchhoff in 1847--algebra and diagrams) and chemistry under Bunsen (Cayley introduced diagrams for chemical combinations in 1875, see Anellis and Abeles above for more discussion on the synergy between diagrammatics in chemistry and logic).
