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 nonadditive variances are closed under
infinity. The principle which enables the link between the countable symmetric
divisions of grossone structures through latent (dseparation(?)) 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 nonadditive 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. Markrecapture 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 formmaking
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 nonadditive
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 nonadditive.
The nonadditve 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
http://researchspace.auckland.ac.nz/handle/2292/1999is available online.
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, “Hostparasite
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 wedgesurface) of speciation than cospeciation in parasitehost
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 coexistence 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
And
“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 nonrelational 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 noneuclidean 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.