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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
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Track Analysis beyond Pan
  Composite Construction - when two or three tracks are gathered together...

Banarescu failed to understand the relation of masses to nodes via generalized tracks being developed here and thus did not clearly cognize the difference of units vs members in an order. That is how he could assert that disjunction and vicariism were confounded by Croizat.  They are not.  He did not use Croizat's notion of dispersal formally enough.


This web site is intended as an introduction and study source for information I am calling "axiomatic panbiogeography".  This is a discipline that uses the topology of species' collection localities, starting with an incidence geometry, to deduce biogeographical homologies. It is inspired after the vision that geographic distributions of living things on Earth can be viewed spatially as Kant was inspired by the "starry sky" less the moral law within. It is a modification of recent attempts to mathematize Croizat's Panbiogeographical Method. Thus, it provides content to the "empty form" Poincare criticized Hilbert's program on, enabling a thoroughgoing pasigraphy beyond Woodger answering Poincare against Russell.

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.


These infinite sets may be represented only within the baseline or else are components throughout the track. Regardless this representation enables one to cut individual space and time relations out of physical space-time necessary to match individual evolution as defined as motion in population genetic allele frequency class structure. This permits "x" to belong to P^v.

"Unfortunately, the earliest set-theoretic ideas appearing in Cantor's paper of 1872 had yet to be given any substantial autonomy of their own. Though the necessary elements for the transfinite numbers are recognizable in the sequence of derived sets

P’,P”,…,P(v) ,…,P(∞),P(∞+1),…,

Cantor had no basis for any articulate conceptual differentiation between P(v) and P(∞).  In 1872 he had not yet developed any precise means for defining the first transfinite numberfollowing all finite natural numbers n.  Nor could he begin to make meaningful progress until he had realized that there were necessary distinctions to be made in orders of magnitude between discrete and continuous sets.  Before the end of 1873 he did not even suspect the possibility of such differences.

After such success in applying first species sets to the uniqueness theorem for trigonometric series representations, Cantor must have been intriqued by the reasons for the validity of the result.  The proof had permitted countably infinite sets for which representation was given up, with the points of exception being distributed in a carefully specified way.  Nevertheless, such points could occur infinitely many times, and this fact must have raised an allied question in his mind concerning the relation of infinite sets like the natural numbers, and continuous sets like the reals."


Since fourier series could easily relate to structures in space and time there is no pure math reason to doubt that Cantor's initial insight might have empirical expression in the space and time beyond gene frequency geometry that Darwin's continuum of ordinary generation exists in. It would not matter if this is of a power higher than single continuum.

Morrone has recently recognized baselines within Central America. This requires the sign P(v) of Cantor or P(n) above to expressed without being an empty set(of potential land/water limit points) if the correspondence is to have any meaning.

This page contains a developing introduction to axiomatic incidence panbiogeography and can be found here (.doc) or below





It is because, as these sciences developed, we have recognized more clearly the links which unite them, and at last we have perceived a kind of general design of the map of universal science.There are facts common to several sciences, like the common fountain head of streams diverging in all directions, which may be compared to the nodal points of St. Gothard from which there flow waters that feed four different basins.Then we can make our selection of facts with more discernment than our predecessors, who regarded these basins as distinct and separated by impassible barriers.It is always simple facts that we must select, but among these simple facts we should prefer those that are situated in these kinds of nodal points of which I have just spoken.”Science and Methodby Henri Poincare (page 218 (1908/2004))





This rendition is the effect of trying to synthesize the various attempts to read and work with the voluminous output of Leon Croizat. I attempted to sustain a continuous thought- process from “The Manual of Phytogeography” through “Space, Time and Form” while also reading the commentary on the work of Croizat which largely lay under the rubric of Panbiogeography.  I did not read his “Synthetica/Analytica?” but I did read some of Croizat’s work after 1968. I had come to reading “Croizat” with a background in herpetology and a need to find a way to interpret geographic distributions. I was surprised to find that the discipline did not really exist outside of Leon’s contribution. Under the sponsorship of Amy McCune, I began this study with a 4 credit Independent Study at Cornell University in 1987.  I gave a graduate seminar on Croizat at Cornell.  I also attempted to use his notion of “how does a cell cut” in a thesis as part of the College Scholar Program at Cornell. Later, I spent many hours with Croizat’s texts and found them extremely reflective and useful towards abstracting out the essentials of what seemed to me to be missing in the study of historical biogeography. Most of my work was completed before the 1999 book by Craw, Grehan and Heads.  The most significant motivation towards the development of axiomatic panbiogeography was the finding that there was a lack of discussion about the “width” (in any sense) of a ‘track’. It appeared that Croizat’s orginal notion of a connection among points, gehnoritrea, raised in the Manual remained the basic notion connecting biology and geography.  One of the unique things about Croizat’s corpus is that he provides a biological use/methodical construction of the word “aliorelative” which Russell used from Pierce. It is a notion of asymmetry that contained transitiveness.







There has been some controversy over just what the Croizat ‘method’ is. There has been a certain focus on “Panbiogeography” over “Space, Time and Form.”

 Disagreements arose over what emphasis was to be placed on various concepts that Croizat introduced or used(Nelson on baselines, main masses etc, ancestral areas etc etc).  There were even suggestions that Croizat did not produce any generalizations that were statistically relevant to the evolutionary process itself ( Simberloff). Attempts were made to counter this perspective by utilizing graph theory and showing that Croizat actually drew minimal spanning trees. Nonetheless, the controversy remains with some thinking (Brigss) that a revival of dispersal based studies wrote the last chapter on the involution of Croizat’s method into evolutionary theory. Page (thesis) decided that tracks and cladograms are not to be infused. Morrone in Columbia University Presss , Evolutionary Biogeography: An
Integrative Approach with Case Studies by Juan J. Morrone has the opinion that there are not different “biogeographic” schools but rather the different kinds of practices are components across steps taken during an evolutionary biogeographical analysis. For instance, “Panbiogeography and parsimony analysis of endemicity are useful for
identifying biotic components or areas of endemism,” quotes Ebach.




This is actually NOT the case!  Different kinds of biogeography compete for the same RANGE and focusing on endemisms has hurt as much as it has helped Panbiogeography.  The lack of a track width does not imply that endemisms are objects that Panbiogeographic analysis devolves into. This lack is essential for the construction of a cooridinated concatenation of Croizat’s concepts.


This literature

Seems to miss-develop the difference of biogegographic and geographic congruence.

Example from Snake Distribution under Mauro’s Croizat.



Biogeographic congruence is only in the places OUTSIDE that already achieved by the clade.  Morrone thus failed to appreciate an aspect of Panbiogeography that is not only about biogeographic homology as the goal and can not be reduced to looking for ‘biotic” componenets.  The abotic components as (think of Niche construction for instance) can contribute to this “outside”.


Stoddart has said, “Croizat was undoubtedly right in his basic proposition: that it is distribution pattern and coincidence of pattern that supply the fundamental data of biogeography and indeed which make it geographical at all. Page 300


This map above shows how pattern (yellow and blue) combined “coincide” with purple which yields the notion of “geographical” of Stoddart.  Since it is distribution data of snakes that make up this ‘map’ it is thus biogeographical.  Further the geography that contains the coincidence of pattern is called ‘geographical  congruence’ since the space is the same for the two different kinds (genera) of snakes (yellow (getulus) and blue(triangulum)) vs black/purple (Pituophis) (These two genera hybridize through a third (Elaphe).


Possible origin of Darwin’s view of “Centre” of origin. This center does not seem to be available to Nelson to criticize in comparison Croizat's "Pacific" and "Gondwanna" specifically.

Circles or Ellipse (smooth completed curve) shifted across generations under Malthusian view gave “area” of origin within phylogenetic descent irregardless of whether it was slow or fast. This can NOT be treated as a generic large region but was thought of as a "country" by Darwin.


Alternative view based on increase in fractal dimension during taxogeny of the Genus. Area that structures the species' ranges is outside the clade shape.


This error is apparent in Robinson’s consideration of Etherington.


Dauben wrote,


“Robinson, in fact, had gone into considerable detail on  the subject of “logarithmetic” (or the arithmetic of shapes of nonassociative combinations as Etherington had defined them in a prewar paper, “On non-Associative Combinations”)8.  The basic idea can be illustrated with  a binary noncommutative operation for which the products ((AB)C)D),  ((BA)C)D), and ((AA)A)A) all have the same “shape, whereas D((AB)C) has a different “shape.”  Here Robinson found that an axiomatic approach was of interest.  After first defining the arithmetic system of shapes by a set of axioms, he showed that any realization of the axioms is isomorphic with the synthetic system of shapes.  He preferred to call realizations of the axioms “simple forests,” since “shapes” had also been called “trees” by Arthur Cayley almost a century earlier (although, as he realized, from a different point of view).”


But ((A(triangulum or getulus)B(traingulum or getulus))C)D) has a different biogeographic congruence than ((BA)C)D) even if both shaped, have the same geographic congruence despite the difference in shape of D((AB)C) from ((AA)A)A) (no matter the origin).  This is why Darwin’s ANGLE bounds and binds this difference of the bio IN  the geography. This can not be approached by trying to find biogeographic congruence from TREE SHAPES of CLADES as Cao noticed an extra row is needed.


Not recognizing this Morrone misspeaks when he says, page 52 “any congruence revealed by Panbiogeography will be uncertain because it is possible that it is produced by geographic proximity, and even assuming that the construction…”



The error results from an inadequate notion of range area as seen in Areography minimal spans per continent and Gould’s notion of “fractal”


View of Axiomatic Panbiogeography.


The difficulty of bringing the abotic and biotic components into view is highlighted by Cao et als call for a different/new row of data.


Axiomatic Panbiog takes a step back from this and views collection localities, tracks and generalized tracks as a form of incidence geometry. The biogeographic homology becomes not a goal but an actual construction. This fills in the “empty form” of Poincare.


Hilbert wrote the first chapter of Foundations of Geometry as the Postulates of Connection or those connected with incidence in 1902.  In 1908 Henri Poincare came out with Science and Method wherein he questioned the use of Hilbert’s program.  Stoddart has commented that Croizat was “mechanical” in his methodology and I can not think that this is not somewhat like the notion of “mechanical” that Poincire apologized for Hibert for.  Poincare was concerned rather to show that Kant was not finished with logic and as will be very apparent in the actual use of axiomatic Panbiogeography this concent via and against Russell arises anew but in actual forms.  So in some real sense (where Gould doubted the possible use of Thompson (who refers to Poincare)) axiomatic Panbiogeography supplies a “reasoning piano” that puts pigs in in Chicago coming out transformed into hams. But contra Stoddart THERE is an intuition here and does matter what the authentic panbiogeographer thinks


In light of this background, I thought that the pure math of incidence geometry might help to take Panbiogeography out of these doldrums as the collection locality and the conection of two such  localities (no matter how connected) could be contemplated without necessarily sufficient reference to endemisms which seems to have complicated the discussion of the statistical significance of Croizat’s approach to having the ‘bio’ IN the  bio  GEO graphy.


John Grehan recently wrote, “by comparing the geographic distributions of animals and plant species, Croizat concluded that evolutionary differentiation of biological form (e.g. speciation) results in related taxa (species, genera, families, etc) occupying different geographic sectors without any of these taxa having individually moved to those locations.  This process was made possible by their common ancestor already having a distribution range that encompassed all the descendant locations.”


Axiomatic Panbiogeography is an attempt to abstract the structure of these “ranges” within the co-ordination of Space + Time + Form = Evolution.  The notion of “range” that results needs to be narrow enough to exclude “causal distributions” but at the same time broad enough to permit notions of “parallelism” in form-making no matter whether temporally this is vicariant or not.



The ideal line thus thought is related via the tetration frame to the ideal populations in population genetics. One begins with Wrights notion of a population of 1 relative to the one of tetration. "The extreme case is that of a line propagating by self-fertilization which may be looked upon as self-contained population of one." P106 1931.

This puts a different read of nature than Darwin. Darwin said page 14 “checks determine average number in/of species OR even the existence of the species” (ontology and epistemology of species in Stephen Jay Gould).  Darwin had checks required for “struggle for existence” because he had the struggles “inevitably” follows from high rate  of increase.  An “increase” (or reproductive success of an individual) Darwin decided was conditioned by PHYSICALLY by ‘geometrical’ or multiplicative properties (Malthus rewritten in Nature) Thus considereing that life would always increase (since Nature did not have prudential restraint from marriage) Darwin reasoned that there exist “checks” that prevent the quantity of the increase from reaching very large numbers.  Unlike human activity there would be no way for nature to “artificially” increase food.  Thus concludes Darwin that there is a struggle for existence rather than a law for the duration of a form. This Darwinian view is implict in Morris et al.



Explicitly the competition for the same “biogeographic range” is available by considering Darwin’s diagram under the notion of tetration in the spaghetti diagram of proponents of co-ordinated stais and ecological locking.

Total abundance is associated with morphogentic limits rather than taxogenic ones. There is no possibility of establishing “vicariant time” between ideal populations and ideal lines.



Let us see what we can think, if instead, of unlimited exponentiation between the statistics of populations and the statistics of species, we use the inverse of this, the superlogarithm (properly scaled) to inform the notion of reproduction of forms, form-making,in nature.

Does this offer a different quality ,than the singled out notion of “checks”? Checks rather become integrated into a numerical connectivity as different populations move exponentially from one tetration curve to another no matter for what reason. 



One is not constrained to think with Darwin (Struggle, Check, Indefinite Increase, Individual life).  Thus the average number in/of species is connected directly to the existence of the species and the logical conjunction of “or” Darwin used is not needed where indeed artifical human selections are not the predominant activity.  This is indeed the standard view of populations under “equilibrium” in the forward looking project of population gentics but because of the different curves for even and odd increases back of the ideal population of one propagating, it is possible to frame different tiers of time or vicariant temporalities which must be looking for different genetic mechanics to encompass the trajectories( unlike Gould's claim about "logic"), and are thus proposable WITHOUT Darwin’s notion of origin of a species from a center.

Thus to conclude there there ARE different “schools” with vicariance vs dispersal a principle difference. There is no continuum of “steps” in a common biogeographical synthesis as Morrone had it. 








Vicariance is the easiest mode of adumbrating disjunct distributions , easiest when the opposite is dispersal from a center or region or sector or area.


Other modes of non-Darwinian biogeographic congruence are concieveable.




In a descriptive geometry the fractal dimension across generations reflects some kind of “entropy” increase that might be differentiated mathematically using quaterions or algebraic geometry.

The non-ending upper bound of the superlog provides the junction for this time and it is possible to see NOT a BLOCK of morphogenetic limits as in Morris et al but one that transitions morphogeny into taxongeny as the increase is converted into maturity. Whether stasis is data is acompletely different question.



The relation of projective geometry to affine geometry affords logic for further considerations of how the points are related to the understanding of the bioeograhpic homology.


Incidence Axioms


The opinion of Pascal at the end of his essay “On geometrical demonstration” continues to guide the perspective within axiomatic panbioeography


Pascal wrote, “One man will say the same thing off hand…It is in this fashion that logicians may have borrowed the rules of geometry without understanding their force and thus it does not follow from their happening to include them among the rules proper to logic that they have entered into the spirit of geometry; and unless they give me other indications of it than the passing mention of these rules; I shall be very far from putting them on the level of that science, which teaches the true method for the conduct of reason.  But on the contrary I shall be very much inclined to exclude them, and almost without hope of return.  For to have mentioned these rules in passing without noticing that everything contained in them, and instead of following their light to wander aimlessly in useless investigations, running after what they offer and cannot give, is certainly to show oneself as hardly clear sighted and as having failed to follow the light because one had not seen it.” Page 445 Great Books of the Western World vol33




Hierarchy of Croizatian concepts IN  sentences generatable in axiomatic Panbiogeography.


Finite vs Infinite baseline Representations


Node or Mass as Ellipse Projectively and the aliorelative through Affine geometry.


Kinds of work in Panbiogeography.


3.2.2 GBIF Metadata

Considerations of line and polygon support.

Possible fusion of Wright’s irreversibility and that which structures ranges.


Issue of Orthogeny and happenstance/dominance in Nonequilibrial populations.


David Starr Jordan (published in 1922) set up a way to understand a history of thought on what counts as “orthogenesis”. This can be compared to Wright’s view in 1931 and the seemingly dominant one today of this history among Mayr,Gould and Provine.


Jordan clearly established what he and others like him thought was the “inside” and the “outside”  He says


GS Carter defines “orthogenesis” and “orthoselection” as OG- the theory that the direction of evolution is determined by inherent characters of the organism and not by conditions outside it. OS – natural selection continuing in the same direction over periods of times.


Jordan’s  contrast with Aggasiz expresses this by indicating that non-divergent non-internal changes happened in fish and thus are really orthoselection and not orthogenesis.


Carter by 1959 had expressed the nair current position via a comparison to artifical selection experience. “Still, the very long periods of these trends of change has seemed surprising to many biologists, and the suggestion has been made that they are not soley due to orthoselection, but that we must postulate some inherent character in the animals causing them to evolve in this direction and not in others. Evolution controlled by such inherent tendencies is known as orthogenesis. If such control existed, it would introduce an entirely new principle into our theory of evolution.”


This view however is not univocal with respect to Wright’s 1931 “philosophical” position on orthogeny. Wright discriminated internal orthogenesis requiring “acquired characters” from a statistical  possiblilty via mutations and further explained a position which contained BOTH adaptive and nonadaptive orthogenesis.

Thus Carter’s explanation that “the opinion of biologists is almost universally against the necessity for demanding any inherent orthogenetic tendencies to explain these long-continued trends; the evidence is clearly against the need for postulating time.


Carter uses the heterogeneity of application within groups (or the combined uses of divergence and directum transitiveness) and yet this does not ameliorate Wright’s “irreversible approach” “as some stiffen and move off to the right” etc.


Gould and Provine claim that Wright inappropriately used the notion.


Yet Jordan’s example from Fish shows how the notion may be understood better.

Mayr has gone very far in this line to define orthogenesis as “the refuted hypothesis that rectilinear trends in evolution are caused by a finalistic principle.”


This “definition” takes the trip all the way back to Jordan’s “Agassiz” only now the notion of genitive species can only be against the special creation of species and yet Mayr had gone the distance to separate out teleomatic from teleonomic purposivness and thus in whatever these “species” be it would still be possible for Wright’s notion to withstand a non finality principle. There is no evidence against THIS and thus there is no reason to lump Croizat’s use of orthoselection and recent use by Gray and Grehan of the possibility of orthogenesis from being made “refuted”.


Mayr related orthogenesis back to Kant on adaptation and thus we can find ourselves in Axiomatic Panbiogeography contemplating WHY Poincare resisted Hilbert, namely to find a way to resist, Russell and now we may also find where biophilosophy can begin, where Poincare insisted on an intuition for the synthetic aprirori but this gets us off course a bit as well.


Possible fusion of Wright’s irreversibility and that which structures ranges.



Jordan’s empircis depend on notion of law of growth which may be internal without being by intelligent design.


Mayr’s use of a definition was set up for wrong target


Allmon got Gould on laws of growth wrong.


This is the issue of logic vs geometry in divergence vs speciation.  It is at the level of intuition where Gould thinks Fisher’s complaint against species selection is not valid and works thorugh Poincare via Thompson IN Wright’s non-adaptive branch points under various thermodynamics of isolation by distance.


This position can be made more clear by considering a thought of Charles Zeleny



Was there some rapid change in the environment that doomed Gould to see evolution dieing? NO- I contend it was merely the Croizat’s observation on what he might have called “uniformitarian panbiogeography”  in 1961 that was NOT CHANGING that caused Gould to think otherwise.


It was Gould’s insistence on “retaining clearness of objective” that contributed to him contributing to  keeping “the need” for stimulation by  foreign ideas  from coming into sight.  This clearness Gould sustained by continuing to view species as “stochastic input” to whatever was supposed to be in his thought anagenetic or uniformitarian rates of change.


So when it comes to relating Gould on ontogeny and on evolution, through evolvability and the larger world of punctuation PE actually forms a barrier to any homozygous population of ideas to find its other idea already thought in it.  By relegating winning of Darwins game for a while to the side consequences PE effectively rids progress in evolutionary theory except through his own filtered/sorted/channel. Extension of Zeleny’s analogy provides a remedy to this attempt to find hierarchy IN LEVELS OF SELECTION in AT THE EXPENSE of SOME hierarchy of levels of organization.  Wright had made this clear in 1931 but this was not heeded because Evolutionary Theory generally proceeded without devotion to knitting. There was no common thread at all that even Darwin’s granddad had thought. There was a common relation to the outside culture but that was common to anyone with varying information on what thought was involved in any change.