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| 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."
Henri 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.
| | | | 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. |
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| 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.
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| 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. |
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