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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
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Quaternion Algebraic Geom
Primate Vicariances
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Replies to Criticism

"Cladistic (vicariant) biogeography has declined, primarily because its followers do not recognize the kind of allopatric speciation that takes place when members of a population migrate across a barrier to colonize a new area." Briggs 2008

It is one thing to think that Mayrian allopatry is not as representative as distributions have appeared but it is another thing to imagine what the genetic revolution in such a population was as specific to differences in lineages. In some facts panbvicarance makes it easier to imagine where and when allopatry may be occuring  -- (as soon as one either circumscribes the space by time or by being based in a particular shape of form-making structures).

Waters et al criticism of panbiogeography depends crucially on a designatable and verifiable definition of a difference between what they call “local” dispersal and “long-distance”  dispersal. They may have even missed that Croizat's use in 74 of "dispersal of species" likely referred to Croizat's specific use of the word.  In conversation on Panbiog-L Nelson had attempted to use Croizat’s notion and apparent use of “dispersal” against Robin Craw.  But on that point one is well within a reading of Croizat’s work. It does become a matter of panbiogeography to delimit what and how to use and understand “dispersal” panbiogeographically, say beyond the phrase “the earth and life evolve together”.

Isolating perhaps unfairly to a page or so in STF

“In a previous chapter (see p. 13), I have visualized dispersal as the sum of local form-making and “migration” to agree with a formula: Dispersal = form-making + translation in space, next using this formula sharply to distinguish between dispersal and geographic distribution.  Since, as we have just seen, the factor of translation in space is practically reduced to naught in progressively vicarious form-making, the formula expressive of vicarious, that is, basic dispersal may disregard translation in space altogether, and be written out simply as Dispersal = form-making."

It is perfectly possible to find that Waters et al  criticism substitutes local form-making with local dispersal (locomotion of animals may not be kinematically  the same as ponderability in plants for instance). Waters et al are simply arguing the formula Dispersal = form-making should not inform historical biogeography and thusnot be information useful in panbiogeography anyway.  They are arguing using a part of panbiogeography against and as if  the whole and this not logical.

Once a vicarious process is brought to “naught” or rather near it,  it then in those cases becomes possible,  to really,  investigate form-making that  involves migration (whether mobilism or immobilism etc.) but one must also be in possession of the ESS within that evolution as one does not generally want to get into differences in the presentation of Kimura style pop genetics vs  Fisherian etc etc as  one for any selection the  migration is to NOT indicate,  unless one is prepared to  make statements about the difference between orthoselection and orthogenesis in the network of evolving demes. Othogenesis however I would guess is even more circumspect for Waters etal than vicariance was for them.

If Waters et al are primarily concerned with focus on “ancient vicariances” by post modern authentic panbiogeographers then perhaps they have yet to understand that it is this regard, that should panbiogeography survive its multiple death threats to breath another dust, panbiogeography offers to evolutionary biology what since the late 60s has been abandoned theoretically,  to think that probes back this far in time are possible with data mankind can accumulate and analysize.  My grandfather a dyed in the wool  Darwinian evolutionist was unable to see through the New Jersey glacial geography historically.  Modern Panbiogeography showed me my Grandfathers's vision/insight/intuition to be imprecise and not deep enough.   If we can reliably see through the Miocene ( I am trying to get back into the Permian sensationally )then indeed  authentic panbiogeography is a tool that will be used by others despite premature reports of its having died out locally. It is a non-sequitur to compare panbiogeography to creationism.

Here is  the page-view-flow after the above was posted on Panbiog-L (Thanks!)

The molecular lens that Heads has proficiently applied is proactive, prospective and professional. O'Grady et. al. appears to disagree.

In fact it has spurred me onto thinking what the direct moecular (silent mutations in netural theory by drift vs draft etc. in population genetics) causes are that are associated with vicariance of taxa no matter the level of classification.

I have been led to think of vicariance within the Mendel parent-hybrid and to correlate vicariance and inbreeding.  So in my mind Panbiogeography is actually broader than so called wider  other approaches since none of them have helped me in thinking about the moving forces and forms of energy within form-making. Croizat's method keeps me metaphysically open to a larger range of alternatives.  In truth the lack of physicality (in cases especially with a lack of the earth and life evolve together practiced attitude) in most biogeographic research is more an obstacle to knowledge than to research no matter how broad or how much money is thrown into biodiversity studies. Quotig Vonnegut reminds me of Gould's lack of thinking of the future of evolutionary theory as involving any kind of Kaufmann/D'Arcy Thompson - like order for free. With that I disagree discursively. The use of using fiction to criticize non-fiction is insane when one really thinks about that.  There is a huge difference in heterozygotic advantage (Fisher even recognized) between insects and mammals (modes of selection of viability, fecundity etc etc). I would suggest if vicariance only thought is the intended target that this is the subject under which that debate might continue.  O'Grady et al seem to miss that Heads (unlike me for instance) avoids the dispersal-vicariance confrontation. Heads provided a tide that already came in.  I guess it must have washed well ashore by now. There is likely to be no panbiogeography of the Moon but I am hopeful to see it on Mars before I die. It is striking how far O'Grady et al confound physics and math.  Gould did not do this.

No this criticism is faulty because it confounds the difference of a data driven and knowledge driven programmatics (see). Panbiogeography is not a means of vicariance-only thinking.   Without some knowledge of Panbiogeography it is not sufficient to use institutionally affirmed perspectives to guide logical criticism of horizons of the stoichiology within which (no matter the versimultude) the solidity of a horizon of thinking is sedimented or precipitated.

See that a few people have attempted to read my rather fast and furious response let me slow down and try to explain mhy posisition a little better from within the knoweldge/info that I actually have acquired while attempted to think and do "panbiogeography"

My grandfather Willard F. Stanely was had a Phd in Genetics studying under Sturtevant but was mores so an organismal field biologist mostlly interested in birds.  During his retirement he established a small natural history museum at SUNY Fredonia and I used to spend summers with him collecting specimens for the museum and IDing the large snake collection he had from his time on the Illionis Natural History Survey.  He was the first person to identify the Western Hog Nose snake east of the Mississippi.

I started a 4-H Herpetology club as a teen ager and spent countless hours on a moped collecting and recording localites of Reptiles and Amphibians in Hunterdon County NJ.  The county park system gave me full access to and requested a list of species found in their parks.  Herpetological Associates on Staten Island would contact me for distribution information.  I was motivated by the book The Broken Archipelego and my Grandfather's view of evolution at this time. And I was even asked (at the age of 14) by the Mayor of my township to survey a new construction location for evidence of the Bog Turtle.

I attended Cornell University where I discovered the works of Leon Croizat and a general lack of any other academic means to search for and find patterns in biogeographic data.  I began the first of a kind(for Cornell) independent study for my entire senior year on the HIstorical Biogeography of the Worm Snake after returning from "the Congo" having found a few new species of electric fish with Jud Crawford.  I stopped at Oxford on the way back and Jim Murray invited me to apply for grad school in Mathematical Biology.

That never happened.  I was put on leave of absence because despite clear tasks delineated in writing, Will Provine failed to provide adequate oversight ("mentoring") of the 4 participating profesors and the project was failing to  focus onto a direct path of historical supposoition about how the distribution of Carphophis was formed.  I simply wanted to know if West of the Misissippi was secondary to East of the Appalachians and/or if the middle distribution  area (Indiana -- Mississippi) was where speciation began (as had been proposed in lit.) how did the snake move both East and West  (gene flow etc) and how to relate this analysis proposed to other Colubrids which may have distribtuions that could bias that synthesis.  If this was not possible the thesis would need to state why it was not possible to find out how these snakes moved over space and in time.  I was reading Croizat as part of this project and conducted a grad seminar on the topic that was well received.  Instead I was facing demands that the statistical tehnhique had to be decided (in the Fall not Spring Semester) before even pictures of snakes could be taken.  And that no dissections of snakes could proceed until McDowell at Rutgers approved that there would like be some correlation between scalation and musculature.  I just wanted to become better familiar with internal snake anatomy.  The profs wanted to send me onto grad school.  I was later told that I was in the running for an Oxford Scholarship!!  I just wanted to learn more. I actually was not sure about  Oxford because McMurray wanted me to apply the  zebra reaction diffusion equations to snakes without any reason why snakes and mammals share the same equations.  I also recall a similar conversation with Richard Lewontin about sea snake tail evolution.  He had mentioned in the "Dialectical Biologist" and had his coupled environment - organism interaction equations introduced there and  I was interested so see how these might apply to the flatenning of the tail (compared to the rattle formation say). So when asked he said nothing about math but  rather that one should look to fish first!! ( no comparison to the other snakes!!!).

So for me vicariance is highly relevant right where founder revolution is often supposed and as Hawiian Drosophila are the poster bugs for this (My Grandfather's thesis was on the vestigial wing mutation which Waddington would later write about.   Stan's attention to these invertebrates helped excite the young to be fire-fly expert (James Loyd))  it was not a surprise to see the O'Grady review.  Biogoegraphic space considerations have to include prior spatial relations relative to the biogeography independent of the geography.  Heads accomplishes this by avoiding the older (vicariance -dispersal) debate.  If Nelson can cognize this so should O'Grady et al. recognize.

Just like as a teenager I had a hard time understanding what Ernst Mayr wrote about the physiology of the kindey across species I have equal problems recoginzing how to see the same homologous causation in the "primates" compared to insects which O'Grady et al implicates but have much less of an issue with Head's monkeys and weeds.

Next post I  will include how my knowledge in Panbiogeography has advanced from this questioning level.  I will be attending Evan Cooch's Applied Population Biology  class at Cornell this fall and I am thinking of seeing if the techniques he has devised for within ecological community encounter histories (Linear Design Matrix) might be extended to species biogoegraphic distribution data historicity using graph databases.  I was going to study with Mauro this fall (on-line) but I decided to take advantage of this free class first.

The conflict between a all-vicariance vs all dispersal perspective roughly follows  or could follow a similarity between a purely historical contingency stochasticity vs deterministic attractor permanence for changes across time space and form.

Panbiogeographic investigations under Croizat’s claim of “statistical graphs that tracks are” offer the direction to facilitate concatenations of biogeographic replication across geographic space.  With replicated biogeographies graphed use of vicariance can be applied to extract attractors on which the system dynamics remain.  Head’s book propounds one reading of where and when to look for these islands of common kinetics.  It is not enough to say that the approach is mistaken.  One must show that that times and spaces per forms used are inconsistent or inaccurate.  This can be done by suggesting historical contingencies that obviate the claims made but cannot be done by apriori arm-waving or comparisons to religious like hypotheses.

"Seeing the comments by our dispersalist friends, I wonder then, whether Madagascar was empty, a piece of nude rock, before the "great migration" out of Afr."ca?
>Juerg De Marmels

Croizat thought of a track as divided forkings – like the kinds of divisions that exist between two phylogenetic branches.


He said that the track depends on the start and end not on the series even or odd

“’’’’as the main channel of distribution of both forms, nor will the experienced student be induced to overlook it on account of the records along its course spelling out in one case localities 1,2,…4,,,6…6,,,10, and in another case localities “ 3-----5---7----9, etc the digits are different but the series is the very same 1-10 for both, and it is the series (not the digits that identify  the track in biogeography).       

This can clearly be thought of a the mathematical idea of a bifurcation (Arnold) which “means a forming and is used in broad sense for designating all sorts of qualitative reorganizations or metamorphoses of various entities resulting from a change of the parameters on which they depend.

Here I suggest thinking about this notion of a the track as a blinking fractal thus possessing  several self similarity mechanisms participating in the process of track construction


Cognizing the track as a fractal such as the coast of Great Britain is helpful to understand Heads’ point about centers of origin programs and monphlys relaations between mainland shore and islands.  With a  fractal notion of the area it is possible to avoid those programs with a center of origin but speak about the area rather than only the breaks


This means that the series in the geography can offer a track width as a property of the place.  Further by using the notion of a bifurcation between two different blinking fractal processes it is possible to see how both geology and phylogeny can inform the construction of a given track whether individual or general.


Thus when a start and  end place can be formed tracks can be drawn with varying thicknesses that depend on  grossone computation of the difference between two different series at each point along the track and/or extrapolated beyond.  Geology in the infinitesimal and phylogeny in the infinite.

Thus for a blinking fractal that defined out of the geographic distance a –a. both the thinness of the track Croziat diagrammed left of this pint and thickness east of it could be characterized by different blinking fractals for the two different bifurcations as in a morphometric field on the bifurcations morphodynamically (Ralph Abraham)


Using an example of salamanders in the central us I suggest how phyogenetic/cladisitic nodes can be generated from track separation in panbiogeography through a multi model approach to the bifurcations.

Using these as bifurcations between a  a’



 This uses the principle of structural stability that a small perturbation of any of the geography results in the same catastrophes, That the morphogenetic field controls.


The infinitesimal geology and the infinite biogeography controls the structure of these to relate the tectonics to the phylogenetics. One needs to be able to realize that the arguments against the numerical innovations of grossone are unfounded (as expressed by Moroz  ) through an application of something of this sort. This also offers away around the hierarchical obstacle problem of biogeography and systematics.


The fractal generator is obtained in the orange node as a function from the a a. distance related to he geography for the particular biogeography.

Adrian Paterson wrote:

"We focussed on one of the main tools of panbiogeography, the track. The track connects different parts of a taxon's distribution and has been defined as giving 'shape to or expression to the space and time that necessarily intervenes between disjunct localities' which does not actually tell you how and where to put your lines on a map (which seems to be arbitrary to a particular scenario). Often the tracks are in disagreement to phylogenetic reconstructions for a group and it is not clear then what a track is showing. Finally, as new data is collected, especially fossils outside the current known distribution, track networks will radically change. We conclude that given the lack of utility of panbiogeography, its lack of hard methodology, its lack of quantitativeness and the amount of evidence against the interpretations that it generates, that the only reason that panbiogeographers continue to publish is through some kind of feeling of being fair on the part of editors. However, science is not about giving 'equal time' to all hypotheses otherwise I would have to teach a lot more creation science. If hypothese are repeatedly shown to fail or lack usefulness then they should be put to one side until someone works out how to make the idea useful and successful. We feel that panbiogeography is in this situation and it is currently distracting a number of researchers from using methods that explain the world better."

He thought that it was wrong for panbiogoegraphers to ignore methods based on molecular phylogenetic studies and yet he had some sympathy for the view that life might be thought of a living layer of geology.  The pace and practice of conventional science is no reason to misjudge the objectivity that Croizat utilized. I have often wondered how far his artistic background shaped his perspective and taste but the fact that he could make marks on kmaps to illustrate his points makes it clear that his practice was not the objection to his claims.  If Patterson wanted to criticize panbiogeography for a need to have a hard quantitative methodology then he should have argued from that basis when  he realized how complicated a network of tracks is when it is dissecgted with complex histories and not simply picked as examplars.  I have been interested in panbiogeography over this time  period he narrates and just now am I  beginining to be able to realize panbiogeography as an example of a complex dynamical system.  I may fail as I develop how to automate my own subjective placement of tracks but then I will have an objective reason for dismissing an example method of authentic panbiogeography and have something  based on simply not understanding something.  The fact is that complex patterns of common distributions exist subjectively.  Making an automated means to identify them and hence a hard quantitiavtive method should have been how the  focus from the track as variable width line to network idea proceeded.  It is thekinds of bifurcations that tell one how and where to put the tracks and this needs to be developed from the facts of distribution per say and it can be done with those facts as they are also mappable to phylogenetic reconstrunctions.  The developmenbt of molecular phlylogenetic inference to explain disjunct connections of distributions does not mean that some other quantitative base could not be used to suggest how the split areas were bifurcated.  The form of the bifurcation and the time it referred to were two different things. The "disagreement" must be addressed within the error of line placements vs nodes per mass so if Patterson had been able to seperate the mass from the line then he should have thought back to that before trying to criticize the time where the space was and was by his own admission actually active.  Where Waters et al thought that panbiogeography

was "off" its own tracks it was actually "on" them!!  Editors are not simply feeling they need to be fair - there is sinking feeling that something has been missing in historical biogeography that only Croizat's worked fullfilled.  I was stunned to find no discipline to search for biogeographic patterns. Now that I can think of chaotic determinability as if it was a random process but bounded in space and time it is possible to emprically process Croizat's claims in ways that were only logically possible before.  Page was correct to think that being able to map phylogenies to Earth on Google Earth should be able to enrich panbiogeography but what has been missing is the need to relate vicariance to bifurcation via track width as a macron of nodes with mass where anti-nodes do not exist and develop recursive algorithms in which increasingly nested track relations can be sorted into various phylogenetic tree formats (algebraically with dynkin diagrams perhaps) which can be used THEN as suggestions for taxonomists.  Patterson did what Croizat always said one should not do - to use phlylogenies (molecular) to gudie reasoning in biogeography rather than the other way around.

"So I was well-trained in this field, exposed to all the main players and could explain the methodology and philosophy of panbiogeography when I found myself at conferences in the States. I could tell my tracks from my main massings, my ocean baselines from my minimum spanning tree. I was a panbiogeography cheerleader! So what happened? "  I held a graduate seminar on Panbiogeography at Cornell in the late 80s and was also able to "explain" Croizat to others appreciatively as Paterson remarked at that time as well.  Molecular evolution do not "then" happen. It already existed.  Furthermore Gould was trying to create a whole other some duplicitous notion of evolution itself.  Bifurcation and Vicariance were confused and others like Mayr wanted the species to form somwhere "out" there but not right here where everyone was. So the future formulation is pretty much already here.  In truth Robin Craw said this on Panbiog-L back in 2009!!  Over time I do not feel guilty embracing panbiogeography at all.  Additionally some of my other ideas which were suspect then (statistics that can be done now with multimodels and hypothesis of snake skin to bone via muscle systematic correlation) are now possible and/or are being pursued by others.  Panbiogeography is the only historical biogeographic method that adds what is needed.


If there is some sympatric speciation rather than simple dispersal( Wright immigration for instance) of Ratite ancestors as Zelandia split off this would result in a higher self-similar fractal dimensionality in the generalized track manifested throughout the main massings.  The fact is that this possibility  can be rationally thought out and related (mapped) to geological differences as Heads has done in this case and many others. We know so little about our past that any means to gather possible objective perspectives on the matter ought not be forced out simply because we do not know what the formation forces were.  When Darwin was so interested in the distribution of plants on both sides of the Obi river as observed by Humboldt he was trying to think of the continuum of forces world-wide that could be bring the two different plant worlds (Asian-European) to the same spot. During his time it was the material explanation of the forces that was appearing.  Now since the Evolutionary Synthesis we already accept a material force background and we are working the space vs form temporality with a vicariant timing throughout a given phylogeny (which includes it's space and environment). There is absolutely nothing to do with going back to separate special creations by God with vicariance.  Setting up the biogeography under a bisimulation process already indicated that that is unfounded rhetoric and uncalled for. The ratite case is more difficult than others I have looked into because the track width and basline representation are nearly in the same line and appear superficially as parallels when they are not.