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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
Work A
Work E
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Multimodel Selection
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Track Analysis beyond Pan



  The work of (BioGeoInteroperatiblity) the potential TDWG substandard task group

no doubt will continue to involve the discussion of standards as well as how they are used. The need towards  a continued discussion of standards in a vision of a future BIODIVERSITY GEOSPATIAL DATA INFRASTRUCTURE



Unfortunately, Barry Cox has already decided (1998) against Panbiogeography after Croizat but completely missed that Croizat said, a track could be read upside down or not.  The difficulty of orienting a track remains within the difference of semantics and syntax but my analysis in terms of GVIS needs revision.
Journal of Biogeography, 25, 813-828
Panbiogeography: method and metaphor in the new biogeography

I had had the idea that there was an isomorphism between two graphs (Pituophis (black/purple- one species per graph) and Lampropeltis (yellow&blue -two species).  This is what I meant as a "generalization."  The isomorphic appearence is closer geographically in the above than spatially below. That was why I stressed geographic congruence as distinct from biogeographic congruence.

By creating a mathematical structure to bind down/constrain the notion of Carter’s divergent and successional evolution one is in a better position to adjudicate claims for internally channeled operations vs. trial and error externalism while one attempts to sort out patterns of change over time. This construction enables internal cleavages to be evaluated in the population genetics of species as a whole in such a way that one may be able to evaluate the relative frequency of centered dumbbell speciation and vicariance  vs peripherally originated form-making.

Carter defined these different evolutions (One Hundred Years of Evolution page 152)  thus ,

 “It may often happen in the course of evolution of a speices divided into demes that, as a result of some change in the environment, a deme or a group of demes becomes completely isolated from the rest of the species. Land may become separated as an island, an area of forest separated from other forest areas by the development of desert between them, an arm of the sea may be cut off from the open sea.  If the isolation is complete, the isolated part of the species will evolve independently.  It will begin to diverge as soon as the isolation becomes complete and its divergence will continue so as the isolation is maintained.

We must distinguish this divergent type of evolution (which on a small scale occurs in the demes of all species that are divided into demes) from evolution of a population living without isolated parts and evolving by closer adaptation to the conditions of the environment and to any changes that occur in it ,  this latter type of evolution may be called successional.”


One difficulty in bringing the forward bearing approach of population genetics forward is that there is no uniform way to consider data cleavages as indicated by development or behavior for instance that may be on accelerative time scales divergent but in terms of overall velocity only indicate a slope relative to some topology be successional. In the creation of a quaterionic calculus a common figure can be described which may bind cleavages across various combinations of successional and divergent periods of varying durations.  This has application to the claim of Gould that major demonstrated guppy and anole population changes are merely the boulder in Mandelbrot’s fractal by providing a population genetic measure or fractal dimension for the entire monophly of populations undergoing the change.  The model will assist in finding regularities of acceleration, deceleration and velocity within a common catastrophic topology and permit various degrees of external trial and error and internal channeling.  The notion that the evolutionary synthesis suffered from “adaptive” hardening is merely the consequent failure to locate in space biogeographically the functionality described below. There was simply a lack of consideration of how isolation was to be contained geometrically. This is not bound to the particular geography of Earth but is by the topological connections in catastrophe theory. Isolation must be considered in terms of space, time and form jointly and singly.


When applied to vicariance we find that tachytellic population structures’ reality are divergent while hortotellic populations are orthogenetically successional and bradytellic populations are static/ecclesiastical while vicariance is going on in the same metric that the cleavages exist. This provides a structure in the function between fitness landscapes and this surface on which a given population “moves” where track width is dependent on both internal sequences and series of contingent external environmental syncing.  The continuum thus describable quaterionically enables the application of macrothermodyanmic differential equations to actual geography  and the possibility of finding a means that change can be “locked up” (to use Gould’s neologism) or isolated within various agglomerations of supramolecular configurations carried across generations in various sets of cleavages.  Biquaterionic and Octonionic representations provide other tools to explore how track width may be organically distributed onto population genetic variances.


Grossone Statistics- Extensions to Path Analysis

Differentiating the statistical from the causal obstacles in the way of applying infinity to empirical approximations to reality is not going to be easy.  Frege always thought it would never be possible. Russell ignored Cantor and set up his own subjectivity.  Jourdain was on track when he was insisting on getting from Cantor his thoughts on the possible ideational heritage of enumerability from closure of the P series. Cantor never capitulated and it was left to Sergeyev to instantiate the question into the design of a computer with infinite radix.  Von Neumann memory is long past but the future remains before us.

(the last equality in particular makes the extension possible (combined with the fact that 2^2 == 2 tetrrated to itself twice between gametes and zygotes)

Here we show how the technological versioning of grossone numeral structures permit generalizations of a particular use case of Wrightian Path Analysis within symmetric multivariate statistics.  A research programme that fuses panbiogeography to expressions of real populational variable homogeneity from latent individual heterogeneity is causally approached as the grossone relative enumerations sets of additive and non-additive variances are closed under infinity. The principle which enables the link between the countable symmetric divisions of grossone structures through latent (d-separation(?)) is the causally supported observation that continual inbreeding decreases genetic variance (and renders homozygous formats full) to zero (never zero exactly, but to some infinitesimal amount). This permanent state transition is assumed analogous to one in which maximal vicariance in the lineage under consideration is realized.  Grossone computers are used to simulate the individual heterogentiy that results in populational homogenity as differences in the division of additive and non-additive variance is distributed onto the grossone collective representative of the correlation between relatives.  Explict models of Fisher’s difference between elastic and nonelastic gas theory heritability are developed with Wright’s orthogenesis as cases of grossone computer manifestations at Cantorian enumerablitiy of realized technology of orthogonal axis multivariate statistics.  Mark-recapture is suggested as a means to verify this new empirical discipline that connects the generation of hypotheses, to new tool formation to test for significance of the effects suggested, to better physical presentations of the organon instructing the relation of math to the physics in the biology (stoichiology)  reproducing continuously in a discontinuous space statistical abstractions (method). Social evolution of offspring infrastructure selection as a replacement for sexual selection of the same organization of cooperative vs. competitive trajectories are dynamizable in this new means to relate theory and experiment in population biology’s expansion of population genetics. A particular view of Lewontin’s hope that analytical population biologists may one day be able to solve the problems they have posed to themselves is no longer in question. The history of the Mendelian- Biometric debate is commented on.

The concept of mobilism/immoblism implies that there are genetic changes that correlate with what otherwise would simply be conceived of speciationally as a “barrier”.

The use of Ecosystem Engineering to prob the geographic distribution boundaries as to survival, provides a means to investigate if geneotypes can move either around or subspeciationally through barriers which may mask former changes in phases of mobilism/immoblism. This provides a discipline that can investigate collection localites of organisms differently than one would the distribution of minerals as metamorphosis of rocks does not have a consistent genetic component and organisms posses biogeographic dispersal abilities which may change function between periods of mobilism and immobilism.
In the future, it may be possible to distinguish effects of rotation vs revolution on vicariant form-making in situ. This will require a better use of Wright's distinction of short range and long range dispersal under ecosystem engineering.

Thus when we put in the interaction and elastic energy exchanges in the virial we wil have a different way of thinking about the evolution of dominance.  In particular the dominance may be partial and so can the recessive and not purely symmetrical but it will depend on the interaction (and here in it is in the non critical points and ridges but rather in the surfaces that connect to these and these can transfer blend energy across generations not only distribute them by Gladyshev substance stability latent heat affect.

Fisher did this combination of biometric and men delian by getting rid of Darwin’s blending. But the blending is still possible in social selection.

“The normal distribution and the correlations between relatives followed from these assumptions. “

We get similarly the normal distribution in the fold catastrophe but we also approach the cause rather than the correlation of the relatives. We do not use Fisher’s gas but what is essential to gases the virial elasticity.  The impediment Provine speaks of is a vield need for such a theoretical development.  This is more so needed with ESSs social selection and latenet vs relaizied inidivdual to population survivial.

“Fisher thus showed how a very attenuated sense of Mendelaism”


We bring back the fuller Mendelian vision with the catastrophe version.

“was compatible with the results of biometry.”… many mathematical models in evoltuoianry biology are nonemprical mathematical truths – not so!!

“Fisher’s demonstration answered two “how possibly” questions simultaneously: How is it possible that the known correlations between relatives could be determined by discrete genetic factors? And, how is Darwinism compatible with Mendelism?”

It seems that it may be possible to sort out different proportions of additive and non-additive variance to a mean by different grossone numeral systems. W, w+1,W+2, 2W, 3W being additive and the number of infinities  Gross^ 2 etc, added being non-additive.  The non-additve are out of the realized connections but bind the latent to real experienced. Elastic energy transfers are empirical bindings of + and X to a particular infinity and thus contingent but become beyond versimultudes to probabilities!!

The relation of nodes, masses and baselines to tracks provided in axiomatic panbiogeography works in the process of establishing the best partial linear space for a phylogeny. Nodes for instance need not simply be connecting areas between two tracks that otherwise approximate the existence of a partial linear space (of cross classification or area monophyly value) but rather specify conditions under which every two distinct points in a clade are incident with at most one generalized track line.  Masses and baselines assist in helping to decide which point(s), otherwise apparently so (area similarity vs. area homology) are distinct. A node thus is related to a track both in determining what collection localities are to be considered distinct and as well as aggregating geographic points into linear spaces (lines that have at least two points incident). It is thus dual. The different relations of masses and baselines sort out the difference of the partial linear space and the linear space of the phylogeny with respect to the generalized track that may be common across phyla.  This incidence geometric quality in axiomatic panbiogeography is not algorithmitized in martitrack panbiogeography geometry nor is the  mass/baseline differentiation properly cognized in comparative biogeography.

that the Roderick Page PhD thesis
is available on-line.

The thesis opens with a need of Stoddart about putting the “geo” back into biogeography, that this science reached the stage it is at today because theory was left in the form of Hutchinson’s student legacy or museum specialists' work and shortly thereafter, after neglecting the relation to MacArthur, Page, following this intellectual thrust (of seeming to need to answer only one side), decided to say that, “Host-parasite cospeciation presents the same analytical problems as biogeography,”
although he did indicate that a larger scope is possible. This statement is false on its face and precisely so between the kind of theory derivable from the Hutchinsonian line (infinite niche) and that from those who sought computer programs (infinite phenotype) to do a better job than the naked eye (and legs).

Simon Levin had told me  in the 80s to see Hutchinson before he passed but I had already been well down that road by then. The difficulty lay in getting BACK into the organism after space is addressed and so the biogeographical analysis is more complicated and has a greater obtusness (when related to a common wedge-surface) of speciation than co-speciation in parasite-host systems.
Proposition 2.1 - If l and m are any two distinct generalized tracks that are not parallel, then l and m have a unique point in common -
                                  - if this point is a node then there is a derivative at the node
                                  - if this point is not a node then the coninuousness in the area is a plane boundary cutting colinear points.
Proposition 2.2 - For every individual track or geographic variation there is at least one node outside the area, in the vicinity of which where,catastrophe theory (Thom 1975) may be applied
Proposition 2.3 - For every collection there is at leat one individual track or geographic variation not passing through it's area (principle of terminal taxons)(the attempt to distinguish Taxon Area Cladograms and Areagrams are involved in this proposition)
Proposition 2.4 -For every collection locality there exist two distinct tracks oriented from some baseline that pass through the geographic coordinates of the locality (Let MrN be three points reconstructing the baseline, Prop 2.4, therefore an indivdiual baseline exists{baseline, collection localities} or co-existence was extant)
Comparative Biogeographers fail to apply this Proposition and miss the opportunity to combine the area similarity and area homology in the channels Croizat considered (regardless of how endemism is defined or Darwin's motion to the other side is used).
Proposition 2.5 There exist 3 distinct lines such that form SPACE+TIME+FORM, either the two distinct lines of Proposition 2.4 are not time (temporal) or time is correctly inferred (unresolved nature of cladogram construction that plaques area homology classification systems as well) and is Craw (1983 (SYS ZOO 32(4)428 reduction of the information content of the cladogram.
Craw's work thus obviated the attempts of Ebach and Parenti
Proposition 2.6 Given two proposed tracks, been parallel by higer order empirical statements before tested individually (i.e. whether baseline is actually represented finitely  or infinitely), not Prop 2.3, one of the three lines of Proposition 2.5 is a track or track part (segment) and crosses some sea or ocean between said parallel (when only).
This proposition enables one to utilize panbiogeography OFF EARTH.  This is a point that Robin Craw(personal correspondence) commented he agrees with me on.

 I will discuss Page's thesis below. I do not have a problem combining "clades" and "tracks".

Hierarchy Representation Function as a Database Key

It appears that failure to appreciate an affirmative judgment in line with Wright’s notion of parallels is cause to not combine tracks and clades.



“A continuous and essentially irreversible evolutionary process thus seems inevitable even under completely uniform conditions.  The direction is largely random over short periods but adaptive in the long run.  The less the variation of gene frequency about its mean value, the closer the approach to an adaptive orthogenesis.  Complete separation of the species into large subspecies should be followed by rather slow more or less closely parallel evolutions, if the conditions are similar, or by adaptive radiation, under diverse conditions, while isolation of smaller groups would be followed by a relatively rapid but more largely nonadaptive radiation.”page150




“Finally in a large population, divided and subdivided into partially isolated local races of small size, there is a continually shifting differentiation among the latter (intensified by local differences in selection but under uniform and static conditions) which inevitably brings about an indefinitely continuing, irreversible, adaptive, and much more rapid evolution of the species.  Complete isolation in this case, and more slowly in the preceding, originates new species differing for the most part in nonadaptive respects but is capable of initiating an adaptive radiation as well as of parallel orthogenetic lines, in accordance with the conditions.”page 158


Evolution in Mendelian Populations GENETICS 16:97 Mar 1931


The generalized track thus is likely to be an empirical measure of orthogenetic lines across phyla. It seems possible to construct a database key ((((baseline),mass),node),track) that in a nonrelational system to  access lat/long and species information for generalized tracks as well geographic distributions (large population subdivided) for individual tracks. Croizat’s observation seems to accord well with the relational nature in logic and Wright’s irreversible parallels provides the full figure to pin the track to the clade itself. Different subdivisions changing under similar conditions create vicariant data.  Page seemed unwilling to go this far. It is an empirical claim that Croizat validated a part of the process to a significant frequency that is doubted for general reasons of single line radiation. It still remains to show how the gene trees and species trees are bound within the Croizat method and how this specifically (rather than geometrically) is accomplished in parallel per generalized tracks. Through the construction of a non-relational database this should be able to be made clear as  table structures are likely to confound the relation between the tracks and baselines.


I am working on writing up an expansion to the notion of orthogenesis. This to involve non-euclidean geometry to link relations amongst panbiogeographic concepts.


I start with Jordan’s distinction of orthogenesis as directed vs divergence both of which are clearly NOT creationist.  Then we see that trial and error externalism might indeed contain orthogenesis if paths of divergence are comprehended without the possibility of side consequences dimensionally.


I (will)then suggest that similarities in Nelson’s diagrams from Candolle to Haeckel through Darlington is due to an inversion of insides and outsides as described by Poincare on a continuum resulting from a smoothed history no matter the intricacy of the original conditions which he explains in terms of relative space and a wrong small size by chance.


If this works it will explain why modern biologists can only find teleology with orthogenesis and why they insist on using Kellog as the historical guide. There is a better way to understand the history of biology if one investigates how Kant understood the work of Charles Darwin's grandfather Erasmus' use of Linneaus which is different than Charles with respect to space at the notion of sexual selection. Furthermore the irreversibility in Wright’s 1931 paper provides the means to dismiss as miss designed by Gould, the idea that orthogenesis expires on Fisher’s initital argument for particulate inheritance (punctuation generally vs hierarchy) which was empirically decided through the rates of mutation but  because the externalism  in a case can be internal as to divergence but orthoselected as to the trends in internalism externally which is why (being internal) Gould felt the case was closed on orthogenesis.



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.



It has been recognized increasingly since at least (On the Foundations of Incidence Geometry by Francis Buekenhout and Dominique Buset 1988) that it is possible to produce geometry in various contexts. Incidence geometry has found application with respect to how it is conceived in pure math as well as in physics. This is an attempt to situate it in biology. (? Dorpat(1960) )had suggested that at its foundations incidence geometry is between topology and projective geometry. While it is obvious that incidence geometry as math is independent of the  diverging applications being made with it when certain objects are assumed constitutive conceptually of the basic elements it is possible to resolve some of Poincare’s apprehension about the horizon of “empty forms” with these sets. This in turn with decided use of the notion of parallels permits the math itself to be developed in a more organized and on-going way materially.

It is hoped that this use of math will help panbiogeographic thought to differentiate itself from the more popular association of panbiogeography with vicariance biogeography and in the meantime to help panbiogeography become a mainstream addition to developments in historical biogeography. It will be shown that panbiogeographic classifications of distribution patterns provide flexible empirical labeling capable of advancing the field.



  This website will help assimilate the criticism of

 The critic  (in the above link) wrote, "there may be something of heuristic value in a panbiogeographic depiction of patterns" and indeed there is. It is more than simply "of heuristic" value.


Stephen J. Gould published this paragraph in "The Structure of Evolutionary Theory" on page 244. He developed his conclusion without recognition of Croizat's distinction of wing distribution patterns within a systematic whole of biogeographic similarities across diverse taxa. He rather concluded that Darwin missed the opportunity to gainsay "species selection". Darwin's attempt to explain diversity via natural selection WITHIN "fans" fails in the observation of Croizat that biogeographic patterns form a "network" rather than fill a volume. A defense of the differential value of extreme "stations" only works if there is no such thing as biogeographic homology. The deductive sytem of axiomatic panbiogeography sustains the proposition that Darwin was mistaken in attributing a quantity to a quality instead. Darwin thus destroyed a modality that never existed for the combination of geographic and named neighbors. The "products" of the fan edges may be better explained by alterations of character recombination under periods of alternations of mobilism and immoblism.


8. Croizat and PostModernism
“Its decline became noticeable after the turn of the century as the dispersal counterrevolution began to have its effect.”
Panbiogeography: Its origin, metamorphosis and decline
Briggs simply fails to affect a proper appearance of a spatial evolution.

Craw was going in the correct necker cubism when he wrote,
Robin Craw*
Panbiogeography and Structuralist Biology
Riv. Biol. - Biol. Forum 80 (1987), pp. 226-229

“Darwin developed his
biogeographic ideas as a refutation of the views of idealist biolo-
gists who postulated independent multiple creations as an explana-
tion for discontinuities in the distribution patterns of species.
Having rejected species fixity and multiple creations Darwin at-
tempted to demonstrate that the distribution patterns of life could
be explained by species origin in single centres of origin, descent
with modification, subsequent migration out of the centre across
barriers and colonization. The Darwinian biogeographic method
consists of determining the centre of origin of a taxon and then
interpreting its geographic distribution in terms of this centre and
dispersal from it by the species means of dispersal. ”

but did not take this far enough post-modern wise.

I asked Derrida while he was alive about how he related his work to science and he told me that I would have to talk to the scientists themselves so, the field is wide open to understandings on the relation of deconstruction of separate creations rather than idealistic biology as started in
Rivista di Biologia/Biology Forum(81/4 - 1988).

I feel that teaching Panbiogeography can be worked through some version of deconstruction through Derrida’s “white mythology” under the topic of “spatial evolution” as I begin to present here.
/Documents/Teaching Panbiogeography-a postmodern directum.ppt
Thus it does not seem to me to be the “hegemony of the ‘modern synthesis’”(Colicino and Grehan) that continues to eclipse Panbiogeography but simply a lack of Darwin scholarship which shows that Darwin’s conception was actually not able to separate out religious connotations across levels which are separable with quaternions which possesses two operations (rotation and tension) this however has to be read out of Croizat’s ½, 1/3, etc. and this does not seem to me to have been done. The modern synthesis (evolution in mendelian populations) can incorporate the sour and forlorn Croizatist if the work is actually being done to a conclusion.


Gareth Nelson was trying to decide just what “insights” might be ascribed to panbiogeography and what to other biogeography last spring on Panbiog-L.

In my last post I was suggesting how panbiogeography is different than comparative biogeography but after reading Libermann’s review

of “Comparative Biogeography” (see my comments here)

 and doing ‘some’ “serious reading” of West Indies Croizat (again – last time was about 6 years ago) I can present the following “map”(attached) wherein the blue curve diagrams for me my first thought that links geometrically some kind of use of Croizat’s mass, track and channel.  This is clearly panbiogeographic and not, say, simply depending on a difference between area homology and analogy.  It may not be true (using a lizard distribution to inform a salamder’s distribution) but it gave me an opening via Croizat’s discussion of the West Indies geographically to authentic panbiogeographic concepts.  Still it did not decide fully if CHANCE “ dispersal” or “vicariance” was to go but it was based on distributions rather than dispersal as Gary mentioned it last year.


There was nothing hard or execeptional in all this. I simply was reading Parenti and Ebach’s book and Croizat’s Chapter 7 after having attended Darwin Days in Ithaca and hearing  Kelly Zumudio


 about a proposed history (phylogeographically) for Pseudoeurycea  lebrosa. A question from the audience attempted to interpret her graph of the species distribution geologically (volcanically) but I got the same result/graph panbiogeographically (without the reciprocal geological illumination as inherent in comparative biogeography). 

The unexplained (as presented by the Kelly) Western  (two western most mountains in the distribution) biodirectionality of gene flow in this salamander was explained by combining the geographic expressions of tracks, masses and channels! Incidence geometry contra Poincare and differentiating the history of Math beyond Descartes only is able to have this foreground. Cladistic relationships in “trees” (no matter the mathematical maturity) where the node carries all of the topology is unable. So while the cladistic aspect in systematic biogeography has light to shine beyond Kansas anymore it dims if one reads Croizat for half a dozen more hours.