Tuesday, September 13, 2011

First Forms:
The Enary Plenary Parenthetics

In 1969, George Spencer Brown published Laws of Form (abbreviated among the cognoscenti as “LoF”), the classic and exhaustive study of the simplest possible analysis, involving two indexes or indices and transition between those indices, providing an elegant and powerful calculus for such analysis; extending it to the corresponding binary arithmetic and algebra; treating questions both fundamental and advanced about such analysis, calculus, arithmetic and algebra; and applying that algebra in Appendix 2 to the binary resolution or analysis of propositional logical arguments and to set analysis.

The book has gone through many editions since, and deservedly so.

Since the publication of LoF, there has been no comparable study of the next least complex or most simple analysis, three-index or ternary analysis, still less four-index or quaternary, five- or quinary, and least of all the generalized N-ary or enary (nor should one discount the validity and eventual elucidation of the transfinary).

The companion volume to this one, Enaries (now, July ’11, in progress), begins the study of enarics or enary analysis, by examining the enary plenaries, the fundamental forms that become available as order of analysis increases, from unary to binary, binary to ternary, ternary to quaternary, and so on, up to the septenary (N = 7), each new plenary form using each and every index available at its order once and only once.

The companion data CD to this volume and that, The Enaries CD (also available as a download) (also in progress), contains various digital forms of both volumes, the separate figures, graphics and tables, extended graphics and tables, and the software used to generate all of the above (generally more or less simple original programs written in the very simple, ancient, beautiful and powerful C programming language).

This volume simply presents the enary plenaries, as parenthetics, from the unary (N = 1) to the terdecenaries (N = 13), simply enumerated and tabulated, for their beauty alone, certainly, but also for examination of their fundamental characteristics and properties and consequent taxonomizings.

Note that each abstract plenary form here represents N! equiforms; for example, the ternary plenaries “(())” and “()()” each represents 3! = 3*2*1 = 6 possible forms, eg "A(B(C))", "A(C(B))", "B(A(C))", "B(C(A))", "C(A(B))" and "C(B(A))" in the first case, and "A(B)(C)", "A(C)(B)", "B(A)(C)", "B(C)(A)", "C(A)(B)" and "C(B)(A)" in the second.

The Enary Plenary Parenthetics

(As usual, this research could use some financial support.)

[20110927 Jim Snyder-Grant of the Yahoo group lawsofform has found several missing enary plenaries in the nonaries, eg "(()()())((()()))", so that means the posted list is incomplete.

[The list will be expanded as soon as possible.]

keywords: enarics, enaries, enary analysis, foundations of analysis, George Spencer Brown, laws of form, LoF, mathematics, philosophy, science