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Axiomatic Panbiogeography

offers an application of incidence geometry to historical biogeography by defining collection localities as points, tracks as lines and generalized tracks as planes.
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Incidence Geometry
Composite Construction
Quaternion Algebraic Geom
Primate Vicariances
Individual Track Construc
Generalized Tracks
Main Massings
Track Analysis and MetaCo
Martitrack Panbiogeograph
Replies to Criticism
Multimodel Selection
Search Encounter
Track Analysis beyond Pan
Incidence Geometry

If this intuition of distance, of direction, of the straight line, if, in a word, this direct intuition of space does not exist, whence comes it that we imagine we have it? If this is only an illusion, whence comes it that the illusion is so tenacious ? This is what we must examine. There is no direct intuition of magnitude, as we have said, and we can only arrive at the relation of the magnitude to our measuring instruments. Accordingly we could not have constructed space if we had not had an instrument for measuring it. Well, that instrument to which we refer everything, which we use instinctively, is our own body. It is in reference to our own body that we locate exterior objects, and the only special relations of these objects that we can picture to ourselves are their relations with our body. It is our body that serves us, so to speak, as a system of axes of co-ordinate."

The Relativity of SpaceHenri Poincaré (1897)(from Science & Method)

In axiomatic panbiogeography the bodies are not only the psychologically human ones  but the biogeographically different vicariant bodies of evolutionary biology.

Logical Mastery of the Axioms vs. Significant mathematical 
                 insight into the subject thus axiomatized

The incidence axioms and the claims for specific and generalized tracks constitute along with specific reasonings the logical mastery but significant insight is possible only when concepts such as nodes (as described by Heads page 55 MPT and below) and masses and baselines as rejected by Nelson (Sys Zoo) are used counterpuctually.

Gremlin graph traversal language has vertices that "are incident to edges and edges are incident to vertices." This permits the incidence nature of axiomatic panbiogeography to acquire a determinable syntax and semantics.  The key value pair of each edge can acomodate the red and yellow representations (as two kinds of ellipses) above.

What is the body in which this description of distance reveals?

How does incidence geometry "sediment" or ground explanatorially the point at infinity necessarily depicted?

Axiomatic panbiogeography wends a canonical path between these two "doctrines", employing the working mathematician to serialize geographic data through panbiogeographic concepts under a general incidence geometry logically, thus providing the panbiogeographer a robust enough framework to discover patterns (vicariant in moblism or immoblism) in collection data much as the astronomer searched and found patterns in the locations of stars in the heavens organonically.

This is NOT necessarily typological and does and can utilize "population thinking".  The aggregates (represented in sets) simply are not in some statistical mechanics analogy under the apperception of Fisher's difference between blending and particulate inheritance ( on the model of elastic and inelastic gas collisions).

Through a generalization of Poincare's psychology of mathematical discovery to the body of Kant's Opus Postumum the analogy of the formal human experience can really be transferred to the difference between the systematics of nature and system of nature with a logical identity anticipating a perception that may lead to different biogeographic intuitions for the bodies of different logical categories of living things.

This requires a new kind of classification system - indicated by the need to create a panbiogeographic atlas database.

Simply put,  applied incidence geometry provides the general mathematical connectivity within which geographic data can be co-ordinated through tracks to proposed classifications, thus permitting ordinal trends to be realized (no matter the moving forces involved) and taxogenically presented through formations of nodes, masses and baselines.

Progress in axiomatic panbiogeography should go a long way to dissolving the difference of opinion between those who side with Russell or Poincare because theoretical biology will have something other than Woodger's attempt at using Russell's philosophical approach when it comes to deciding how the geography of bauplan differences are to be discerned whether directly in area projections or not. Aggasiz's 1850 remark as Head's reiterated (page 278 Molecular Panbiogeography of the Tropics) becomes workable, "Nothing can be more striking to the observer than the fact that animals, though endowed with the power of locomotion remain within fixed bounds of their geographic distribution."

Vicariance is thus mathematically supported by as many divisions as the data can remand because of the manifold appearence formally identified by phoronomic differences per track in  unity, plurality and totality. Disperal and founder event locomotion caused speciation can affect a correlation between the mathematical logical vs mathematical psychological content per lineage so vicariance based ramifications are considerably simpler and less cumbersome to contain. What fundamentallyl remains to be done is creation of the Panbiogeography Atlas, a datawarehouse where area names are formally resolved into sets of tracks, nodes, masses and baselines, set side-by-side.  Panbiogeography assists the philosophy of math that attempts to search for a relation between arithemetic and geometry.  Algebra can solve the issues relative to taxanomic/systematic organizations. As population sizes become too large for a single species simple increase in numbers  (through form-making) is driectly related to geometry (pattern of population locations) (through translation in space). Non-locomotion explanations are much simpler to mathematize and can utilize the advance of Cantor and Dedekind where progression is arithematized.  This occurs through panbiogeographic concepts.

Basic Principles
"Mathematical axioms (for example, there can be only one straight line between two points) are general a priori cognitions, and are therefore rightly denominated principles, relatively to cases that can be subsumed under them. But I cannot for this reason say that I cognize this property of a straight line from principles. I cognize it only in pure intuition."(Immanual Kant, Critique of Pure Reason 1781 Barnes and Noble 2004 page 199)

 In "Science and Method" Henri Poincare lays into Bertrand Russell's use of probability methodically, because of a philosophical failure in proper identification, of the different versimultude an opposite case, in any concept, can have, on an intuition in some advancement in science. Panbiogeography is not some kind of opposite to Cladistics but the difference in the views of both kinds of practitioners could be seen as such even though their horizons are ostensibly the same.


Russell suggested that Poincare opposed "the quick flashes of insight in mathemtical discovery" to his use of logic and insisted that Poincare "remained faithful to the authority of Kant." But as Russell's use of logic does not accord with Kant's philosophy of the relation between a mediate experience and the intuition it ruled it seems Russell simply sought in the immediate to locate Poincare's intuition in said flashes. Yes, these "flashes" are not at odds with the words of the newer logic. But as we shall see the development of axiomatic panbiogeography at once removes the lack of faith of Russell and the challeges to method, that Poincare used to criticize Russell and others with.


"The English noun incidence comes from the Latin verb incidere, from in meaning in or on, plus cadere, to fall. When we say that a point is incident with a line, we shall mean merely that that point falls on or lies on the line. Likewise, a line will be said to be incident with a point when the line falls on or passes through the point. Here also, we assume that lies on and passes through convey meaning without further discussion."(Harold Dorwart The Geometry of Incidence 1966 Prentice-Hall page 1)


Poincare wanted to have this discussion. This is a subject matter subsumed by axiomatic panbiogeography. We shall. We will get beyond the topological invariant it supervenes.

Poincare put it thus:

 "What strikes us first of all in the new mathematics is its purely formal character.  "Imagine," says Hilbert, "three kinds of things, which we will call points, straight lines and planes; let us agree that a straight line shall be determined by two points, and that, instead of saying that this straight line is determined by these two points, we may say that it passes through these two points, or that these two points are situated on the straight line.”  What these things are, not only do we not know, but we must not seek to know.  It is unnecessary, and any one who had never seen either a point or a straight line or a plane could do geometry just as well as we can. In order that the words pass through or the words be situated on should not call up any image in our minds, the former is merely regarded as the synonym of be determined, and the latter of determine.

            Thus it will be readily understood that, in order to demonstrate a theorem, it si not necessary or even useful to know what it means.  We might replace geometry by the reasoning piano imagined by Stanely Jevons; or, if we prefer, we might imagine a machine where we should put in axioms at one end and take out theorems at the other, like the legendary machine in Chicago where pigs go in alive and come out transformed into hams and sausages.  It is no more necessary for the mathematician than it is for these machines to know what he is doing."

(1908 Science and Method page106 Barnes and Noble 2004)


Axiomatic panbiogeography is this reasoning piano so-called. That is why I independently of the New Zeland panbiogeographers found the relation of Derrida and Croizat compelling. Poincare was mostly concerned to show that logistics as he called it, did not exterminate the Kantian difference with Liebniz nor extinct the "Kantian theory of mathematics" and so as Panbiogeography plays its own tune on these black and white keys we call points and lines and generalized tracks, Poincare can rest assured his argument with Russell was not diminished in the process. It does however seem in th sequence and series of real entities from organic bodies to allow incident lines to be "not straight" with respect to physical (solid bodies) rules.  But I have always wondered why a Salamander need have it's length measured with a ruler?  Would Salamander rather be better measured with French Curves? Straight incident lines in axiomatic panbiogeography are stragiht after both velocities and directions are taken into consideration. Russell had felt that the rise of non-euclidean geometry meant that Kant's ideas of math were not longer valid, but as panbiogeographers get the intuition to describe tracks (working masses into the straightness) that may not exist only on Earth (beyond the simple idea that geodesics need not be necessary to generalized tracks linkages) even Russell's worry will appear too small to worry much about practically. The error there ,lay in the fact that Kant's mathematical notion of right line (both in the sense of one right/straight line for any two points and only one right angle for any division in the plane) had not developed a common metaphysics of morals nor a transition to physics and this really has nothing to do with biology, evolution and biogeography per say but has tangential application only inter say in the legislation that may be constructed for conservation biology. Panbiogeographers will have something to say about this however, and rightly so.




Pituophis and Lampropeltis that may have the same biogeographic homology and thus have a common biogeographic but different geographic model.

 Thus they would have the same compassed "range margin" format as emprically discussed by Burton and Travis .


It is strange that Wright's orthogonal non-adaptive changes associated with isolation by distance has not been pursued to show how the difference of biogeographic and geographic congruence may converge in some cases like this.  Perhaps the more abstract quality of axiomatic panbiogeography was requried to call attention to this possibility.  It may be that Wright's distancing himself from Eimer via Lotka through the notion of "pure line" was not noticed sufficiently enough.



Nelson Papavero, in Comparative Biology and Intuitive Set Theory calls attention to a possible use of logical forms to facts of form in biology but I find that the use of form and formal logic is not as available to the biological intuition than a sense of form simply from the notion of a series which is rife throughout logic itself.  The notion of vicariance can be understood interms of the word “disjunct” of Bertrand Russell in this case. It is not the forms of sets that helps in a discussion of the fact of evolution, nor algebra of trees to some unknown Mendelian thought where the node is usually connoted  but rather the equipollence of point sets in areally homogenous units technically that holds the concept sans explicit shape. Papavero only mentioned panbiogeography in this context but did not say how geographic distributions and biogeographic patterns may be found in the same lingo that then would only leave calculation and experiment as the method. This is possible in authentic panbiogeography.


Phoronomy is helpful because case three

 parallel lines gives historical biogeogeographic monophyly, after a taxon has been added as being biogeographically possible, even if one wishes only hypothetically to consider its geographic space. This is why comparative biogeographers are wrong to say that for collection locality points on Earth, biogeographic space is geographic space. It is not! Kant made clear that as to the quantity, subtractions (of nodes etc) are not the same as additions under velocity and direction!!

Poincare showed/demonstrated that Hilbert's axiomatic geometry depended on prior understood meanings attached to some seemingly random application of mathematical symbols and thus was not as arbitary as Russell's view on logic.  Axiomatic panbiogeography enables one to realize the three step process of (complete induction, geometric properties, specific physical facts).  This follows in the literature of panbiogeography where Croizat discussed (deduction and induction), some geometric consideration of tracks, and Croizat's claim that tracks posses "statistical" generalizations.  The kinds of complete inductions possible with martitracks however does not lead to any(all) kind(s) of statistically validatable physical property because there are specific geometric relations that constrain the kinds of deductions that are creatable within some completed induction of panbiogeography for some particular tracks or nodes etc.

There was much confusion following attempts to introduce Croizat’s world across the continents. I find that only by supporting his method with collection localities as points, tracks as lines and generalized tracks as planes, in an incidence geometry sense, can one hope of resolving or coming to peace with the apparently contradictory proposals about Croizat’s work.  I do not find that this association removes or obliterates any kind of reading of Croizat. Sometimes I have even fancied that this interpretation expanded Croizat’s abbreviated term “panbiog” into Panbiogeography. If this is only about the letter of this law rather than the symbol for the word it may indeed succeed.


It is taught that Hilbert, having relinquished a logical necessity of meaning in the difficulty of defining all the terms of a given specialty subject to proof,  formalistically also returned a result that did not need such meaning. But it is one thing to give terms pure mathematical meaning and another altogether different thing, to find out what applied math the terms may ply empirically,  where proof may nevertheless still remain a part of the praxis.  This rendition of Hilbert’s contribution fails to differentiate fully Poincare’s different logistic appellation given alternatively to Russell and Hilbert within a general psychology of mathematical intuition.




Page183( Mathematical contact therefore is laid at the basis of the physical, but does not alone constitute it; in order that the latter may arise, a dynamical relation must be superadded in thought…) This metadynamical consideration shows up in axiomatic panbiogeography as one attempts to sort the relative relevance of tracks, nodes, masses, and baselines.  The axiomatic or mathematical basis in the incidencde geometry provides as Kant said, “Contact, in a mathematical signification, is a common boundary of two spaces, and is hence neither within the one nor the other space. Straight lines therefore cannot touch one another, but when they have a point in common , it belongs as much within the one as the other of these lines, when  they are produced, that is cut one another.” Thus nodes are neither necessarily within one track or the other and how the generalized track is related to the boundary of individual tracks, nodes, masses, and baselines, is a more complicated expression of Kant’s next sentence, “But circle and straight line, circle and circle, touch each other in a point, surfaces in a line, and bodies in surfaces.” (Kant page 186 Metaphysical Foundations of Natural Science)

The data divisions imagined by Fisher and used by Wright (1933) express the apparent or merely apparent attractive force of Kant.  Genetic segregation and the effect of the environment result from immediate attraction at a distance from the closest linear relation between genotype and phenotype (dominace). Confusion over the explanation via merely apparent attraction and true attraction (not available until after DNA structure was worked out) seems to be the cause of the difference in Fisher’s  and Wright’s numbers which in term has to do with failure to categorize the compressive force effects, leaving Wright to think more physiologically and Fisher more environmentally.




Here is why all of the topology cannot be necessarily and sufficiently located in the nodes.

Under a tree construction of Brownian motion as the area monophyly scales from station to area to realm the chain moves from state K to K+/-(1) with properties of variable dimension



and different relations to other chains that may only intersect at one vertex across this state.


This is necessary if descriptive vs. explanatory dispersal and vicariance are to be kept separate and one intends on finding patterns before modeling mechanisms. The gene trees and the species trees for the same area may have different dimensions due to different states of congruent intersections. The dimensionality depends on the different contributions of Weyl 1-D symmetry.