Euclid's book, The Elements of Geometry has been published more than any book except The Bible, and there is a similar worship of its contents.However, a satisfactory set of axioms was not available until 1896 in David Hilbert's Grundlagen der Geometrie.
And, even before that, a critical gap in Euclid's axioms was filled in 1880 by Moritz Pasch: an axiom of order. This deficiency is evident in The Elements of Geometry because Euclid had to include an Appendix citing a fallacious proof that all triangles are isosceles. The proof is carried out by "going two ways and finding agreement". Euclid declared this proof to be fallacious because, obviously, any one can see, for example, that a scalene triangle is not isosceles. But Euclid could not explain why, and had no axiomatic tool for doing so.
By attending to order, one sees that "one of the ways" considers vertices of the triangle in a clockwise fashion, whereas "the other way" considers them in a counterclockwise fashion. The "proof" appears to say that "two different sides are congruent", whereas it actually says that a single side is congruent to itself.
I believe that this deficiency in the work which had the greatest influence upon the development of mathematics resulted in neglect of order in other parts of mathematics. For exmple, in the often excellent Mathematical Thought from Ancient to Modern Times, in 3 volumes, by Morris Kline, there is no mention of partial orderings, lattices, etc., and the word "order" does not appear in the Index. This is typical of most historically referent books about mathematics. "No 'Order' within the border."
The important distinction of partially ordered systems did not emerge until the 19th century work of George Boole, Charles S. Peirce, Richard Dedekind, and E. Schröder, thereupon receiving little attention in the literature. Example: at a seminar on lattice theory held at the Naval Academy in the 1980's, it was noted that the supposedly encyclopedic "Bourbaki" had to redo many proofs because limits can be partially ordered (as in a poset or lattice) as well as simply ordered (as in counting).
In 1957, I became aware that several fields of mathematics were concerned only with type (kind), but did not allow distinctions of order (degree):
Later on, I formulated the distinctions of "t-mathematics" (recognizing only type) and "o-mathematics" (recognizing both type and order). The latter long existed in factor theory of arithmetic and in the distributive lattice introduced by Dedekind to provide format for factor theory. My o-sets of 1957 anticipated the "multisets" of the 1960's which had to be introduced to described the algorithms needed in computer science. But my o-logic, o-extension of Venn diagrams and o-extensions of indicator ("truth") tables have apparently never been considered elsewhere.
- set theory;
- statement logic;
- "Boolean" algebra;
- logic circuity;
- combinations;
- etc.
So I believe that ordinology is a frontier waiting to be explored.
How can ordinology be developed? I've a few suggestions in files at this Website. I draw upon the remarkable growth of topology during the 20th century.Similarly, I shall show how
- It was noted that topology could benefit from concepts, procedures, algorithms from combinatorics, a field introduced (along with topology) by the great Swiss mathematician, Leonhard Euler (1707-83), in the 18th century. So combinatorial topology began to develop.
- It was also noted that topology could benefit from group-theoretic ideas of the 19th century, along with other tools from algebra. So algebraic topology began to develop.
- It was further noted that -- whereas ideas from geometry and analysis had initiated topological development -- now set-theoretic concepts of the late 19th and early 20th century could be used to create "a topology". So set-theoretic topology began to develop.
On the other hand, let me quote the eminent American mathematician, Garrett Birkhoff, whose seminal work, Lattice Theory, 1948, did much to advance this subject. On p. 30, Birkhoff writes, "It is well known that the topology of the real continuum can be defined in terms of order; this can be generalized to arbitrary partly ordered sets. The generalization to chains [simple or total ordering] cane first historically; we shall now discuss it." And Birkhoff then gives a definition and theorem for this purpose, later extending this to parordering.
- ideas from combinatorics can contribute to ordinology;
- ideas from algebra can contribute to ordinology;
- ideas from set-theory can contribute to ordinology;
- and ideas from statement logic and indicator tables (indmeasuring) can contribute to ordinology.
However, since what I'm calling "ordinology" then (and until now) consisted of fragments here and there, Birkhoff does not realize the implications of that sentence. IT PLACES TOPOLOGY UNDER ORDINOLOGY. AND, SINCE GEOMETRY IS TOPOLOGY WITH A METRIC -- ALREADY PRESENT IN LATTICES AS RANK -- THIS PLACES GEOMETRY UNDER ORDINOLOGY. This opens up a "frontier" for exploration.
And here's another connection. In the file, "TRANSITIVE CLOSURE", at this Website, I propose adopting as the ordinologically defining transformation a combinatoric which is critical to all of computer science. And in another file, "ORDEXES COMPOSE ORDONALS", I show how to simplify sorting algorithms which transform a parordering into a simple or total ordering. This suggests that Ordinology could become the primary mathematics for computer science.
Given these models and incentives, I call upon some of you who know enough and care enough to be "pioneers" in this "new frontier" of Ordinology.