On the value of axiomatic panbiogeography

Abstract

Although criticism of panbiogeography exists to the point of describing its putative decline, the past few years have witnessed new uses of computer programs to bring the discipline forward (Calvacanti, EE). As more attempts are made to use computers to advance the discipline beyond some of the early attempts of Page more scrutiny is being paid to the basic concepts that pertain in the study of Croizat’s work. Without having to make completely transparent all differences in approach it is possible to show that the view of tracks as derived from the Martitracks program is inadequate as one attempts to sort out how to evaluate the multitude of suggestions being put forward by phylogeographers. Axiomatic panbiogeography provides a large mathematical structure in which these evaluations can more adequately take place.

Introduction

Axiomatic panbiogeography offers authentic panbiogeographers a larger mathematical/logical unity from which to extend constructions of hierarchical relationships within the domain of historical biogeography than that described by Daniel Rafael Miranda-Esquivel and Susy Echeverria. Although qualified as to “approach” the authors wrote:“The Panbiogeographic method involves basically three main steps (Morrone, 2004): Firstly, construction of two or more taxon individual tracks (minimum spanning tree from distributional localities). Secondly, delimitation of generalized tracks through geographic congruence between individual tracks. Finally, determination of nodes within the intersection areas between generalized tracks.”

In axiomatic panbiogeography the notion of the generalized track is more general than this. One begins with the notion of the point or collection locality, the line or individual track and the generalized track or incidence geometrical equivalent of the plane. The intersection “areas” between generalized tracks can better be cognized as a 3-dimensional entity if the hierarchical relationships among the distribution patterns becomes complex. How complex this needs to be for the benefit of axiomatic panbiogeography to apply is likely to be found to be in those cases where simple catastrophes fail to topologize the mathematical range of the points in the domain.

The three-fold method of: individual track construction, composite track interpolation and node junctioning, results in hierarchies only so deep but is useful in investigations where multiple baselines are not needed or where there is little relationship between nodes and masses conceptually. This methodological procedure does not divide out the third phoronomic (angle) case from the first two amongst the points.

Body

This position can be displayed by investigating speciation in salamanders in the Eastern US under potential construction of a generalized track that if real is one common to other taxa (trees, snakes and lizards etc). In particular the difference of node shape as understood from quaternion tension and

torsion yields hierarchical tree structures of more complexity than is possible from tracks generalized and connected by nodes as developed in Martitracks. This is observed in graphing the distributions of three salamander genera (Eurycea, Plethodon and Desmognathus) while attempting to derive a areagram as suggested by comparartive biogeographers.

Maps of the three genera under discussion look like this:

This will become more important as we expand the clades considered in the track.

The question simply put is, is there a generalized track for the Genus Desmognathus, perhaps as indicated

That is part of the same track as that suggested below for many species which may form part of a biotic area homology

This generalized track as constructed with a Pantrack martitracks parameterization could look like this –

This depends on how homology and analogy scales with geography

but the method of interpolating generalized tracks from individual tracks and only using nodes to join so generalized tracks (in Martitracks) fails to provide a deep enough tree structure if the relation of salamander genera Plethodon and Eurycea are representable within the same track as suggested by their geography below:

Especially if this nodal shape to distribution can be strictly temporalized with the vicariant time of the following catastrophe model

This model interrelating time and space through a specific form-making can be used to explicate Darwin’s notion of‘sides’ - “produced on one side alone” vs. “have not been able to migrate to the other side” for a spot (that Darwin argued may be against independent separate production and thus) continuous as to area (vs. larger sized area homology) in some sense but potentially discontinuous on a full station by station basis. Genes can move to the other side even while “parents” may not have.

This conclusion should likely be verifiable on extension to the genus Ambystoma or else the general argument needs be made via Cladistic Proposition 2.6 “Given two proposed tracks been parallel by higher order empirical statements before tested individually (i.e. whether baseline finite or infinite), not Proposition 2.3, one of the three lines of Proposition 2.5 is a track or a track part/segment and crosses some sea or ocean between said parallel (when only).”

The generalization of tracks as conducted with Martitracks cannot retain complex unified topologies .

What we will find is that Desmognathus has an extremely fast velocity across the space of the generalized track and that Plethodon is faster again than Eurycea when it comes to the plurality of directions in this same line. This is possible to discover in axiomatic panbiogeography but not with pantracks no matter how martitracks is parameterized. This expresses itself in the shallowness of the tree structures of martirack suggested areas of similarity which are then useable to find area

homologies. The difficulty is that direction with respect to body vs relative space is not computable by apurely geometric function as in martitracks but is available where the incident axioms become things of space between population genetics and historical biogeography via congruity.

The topological complexity that Pantracks via martiracks appears incapable of enclosing can be theorecti cally investigated by making arbitrary path trajectories through scaleable spaces of composite homologuos and analgous areas as conceived in the project of comparative biogeographers.

One of the early figures about the complexity of areas available to biogeography was from the book Areography

By making a database of fractal self-similar spaces of differing places and densities of homology and analogy overlaid onto longitudes and latitudes using a quaternion calculus in quaternionic analysis

The set H of all quaternions is a vector space over the real numbers with dimension 4. (In comparison, the real numbers have dimension 1, the complex numbers have dimension 2, and the octonions have dimension 8.) The quaternions have a multiplication that is associative and that distributes over vector addition, but which is not commutative. Therefore the quaternions H are a non-commutative associative algebra over the real numbers. Even though H contains copies of the complex numbers, it is not an associative algebra over the complex numbers.

it is possible to show how node junctioning of generalized tracks Such as this

can miss a large proportion of the possible spaces of potential biotic area homology. Transcendental numbers can be used to catalog collection localities in this modeled place space so as to retain the more complex algebraic relationships that become geometrically reduced in the Martitrack implementation of the pantrack idea.

"...the thing about a Quaternion 'is' is that we're obliged to encounter it in more than one guise. As a vector quotient. As a way of plotting complex numbers along three axes instead of two. As a list of instructions for turning one vector into another..... And considered subjectively, as an act of becoming longer or shorter, while at the same time turning, among axes whose unit vector is not the familiar and comforting 'one' but the altogether disquieting square root of minus one. If you were a vector, mademoiselle, you would begin in the 'real' world, change your length, enter an 'imaginary' reference system, rotate up to three different ways, and return to 'reality' a new person. Or vector..." — Thomas Pynchon, Against the Day, 2006

Conclusion

The pantrack ideal of martitrack implementation gives reality, to things that Poincare distrusted could be attached practically to the incidence axioms of Hilbert, as one decides when taxonomically (increasing the clade extent) the mosaic structure of connected generalized tracks cannot be adequately a function of the concurrent area monophyly areagram.

But when attempting to scale e-w and n-s (lat/long) differences into an area monophyly from a very particular place Oichita/ Arkansas mountains in the sections of pantracked martitacks segments of a generalized salamander track the form is only within the constructability available in axiomatic panbiogeography There is no possibility for this from the generalization of tracks in a martictracks structure of any simple topology we currently posses an intuition for. Tension and torsion of quaternions at nodes does not alter the area homologies that may be induced in the case but clearly assist in creating a mechanically larger hierarchical nesting structures (both in terms of depth and width) than can be conceived sans data from that via the concept available from martitracks. Axiomatic panbiogeography is thus the more proper mathematical system from within which future computer implementations should be cognized.