21: Conclusion

We conclude by examining Darwin's ideas on variations. We show how a form of relative selection, based on them, can create a new species. We shall:

  1. examine the criteria for defining a species;
  2. discuss Darwin's proposals regarding the Origin of Species; and
  3. outline an experiment—based on our theoretical and experimental proofs of Darwin's theory we outlined in previous chapters—that we hope to conduct in the near future to demonstrate Darwinian principles of selection through variations in action by creating a completely new species.

The key is variations. Nevertheless, Darwin's ideas on variations and his associated theory of evolution are indelibly associated with Herbert Spencer's phrase “survival of the fittest”. Although there are important differences between Darwin's original theory and Spencer's appellation, such is the popular mythology surrounding the two that the one is now almost synonymous with the other. This is unfortunate for it makes it unnecessarily difficult to grasp the essence of Darwin's theory. His theory is primarily about variations, and the results of variations, and not about the fittest, howsoever they might be conceived.

When Darwin first presented his ideas on variations, and selections through variations, claiming that they were the sole cause of the origin of species, he also proposed an engine:

I have called this principle, by which each slight variation, if useful, is preserved, by the term Natural Selection (Darwin, 1869, p. 76).

Unfortunately, even though “slight variations” and “natural selection” are central to his theory—the latter is the force that both produces and results from the former—Darwin did not properly define either. Both can be, and have been, interpreted in a multitude of ways. He therefore left the field wide open for his theory's most well-known characterization, which is Spencer's ideas on fitness:

Alfred Russel Wallace, the cofounder of evolutionary theory, was struck by “the utter inability of many intelligent persons” to understand what he and Darwin meant by natural selection and suggested they substitute Spencer’s phrase. When Darwin obliged him by using “survival of the fittest” in later editions of the Origin, readers were still confused; everyone seemed to have his own interpretation of what was meant by “the fittest”.

However, the phrase caught the public’s imagination and became completely associated with Darwin. Critics said it was a meaningless tautology—a proposition that simply repeats itself. Since the fit are the individuals who survive, they argued, wasn’t it another way of saying “survival of the survivors”? (Milner, 1990, p. 424).

Nothing could make the seemingly tautological nature of natural selection clearer than its proposed definition in the Collins Dictionary of Biology where we find ourselves referred right back to the beginning of the same definition!:

natural selection, n. the mechanism, proposed by Charles Darwin, by which gradual evolutionary change takes place. Organisms which are better adapted to the environment in which they live produce more viable young, so increasing their proportion in the population and thus being ‘selected’. Such a mechanism depends on the variability of individuals within the population. Such variability arises through mutation, the beneficial mutants being preserved by natural selection (Hale & Margham, 1988).

Figure 21.1
0-sphere and 1-ball

There is a problem with the terminology. Just as gravitational attraction can occur between any two masses, with gravity being the resulting force, Figure 21.1 shows the clearer understanding we can now give not just to Spencer's admittedly evocative phrase survival of the fittest, which is more akin to the force; but also to the slight variations and natural selection that Darwin proposed as driving it. It is the slight variations that lead to the attractive force that then produces what we call the fittest. The figure shows a group of biological entities congregating together, as the fittest, through sharing (a) the same values, and (b) the same rates of change. That congregation, across a range of specific variations, is the essence of Darwin's theory.

It would be most helpful to have a word to describe this process of gathering together around a common set of values. It would be even better if it sounded like Spencer's ‘fittest’. We therefore now define “plessist”.

Plessist is taken from the Greek ple̅sios meaning ‘like’, ‘near to’, or ‘neighbour’. A plesiosaur, for example, is near to, or like, a reptile. “Survival of the plessists” then refers to the joint survival, in each others' neighbourhoods, of a group of biological entities that specifically abide by our four laws of biology which are Law 1 of existence, Law 2 of equivalence, Law 3 of diversity, and Law 4 of reproduction; our four maxims of ecology, which are Maxim 1 of dissipation, Maxim 2 of number, Maxim 3 of succession, and Maxim 4 of apportionment; and our three constraints which are Constraint 1 of constant propagation, φ, Constraint 2 of constant size, κ, and Constraint 3 of constant equivalence, χ. Those biological entities that abide by all these—jointly surviving in that measurable way—are then neighbours. They are of the same kind through constantly sharing the same average values, while also always exhibiting the same average rates of change, over the same periods of time, and therefore being on the same path as the set of equivalents defined by Law 2.

‘Average’ as we use it here—i.e. in association with survival of the plessists—means something more than just the mean. It carefully includes all associated rates of change, and therefore all associated quantities, all of them measurable.

The association of average with both (A) quantities, and (B) rates, is important in this context. It also harks back to average's original meaning.

There is some evidence that average originated from the Arabic awār and/or 'arwāriya, meaning ‘damaged merchandise’. There is much clearer evidence that its etymons date to the twelfth century Italian avaria and the thirteenth century French avaries. Those referred to the taxes and duties levied on goods shipped from one port and/or country to another. It was certainly being used by the fifteenth century to refer to the loss and damage that goods invariably suffered in transit. The convention grew that the most just and equitable remedy for all parties concerned was to distribute the financial liabilities implied by those losses out evenly amongst the owners of the ships, the owners of the goods, and other interested parties. The amount levied from each was then the awarage amount that each party should bear. From this usage as a fair and equitable apportionment of quantities and values, average gradually came to signify the more general arithmetical mean. But we must add to that concept.

Figure 21.2
0-sphere and 1-ball

As we see in Figure 21.2, we liken the biological generation itself to a three-dimensional helix or helicoid that winds itself around a set of average quantities and values. It is a repetition of core values over time. Those core values set it the centre, between the extremes, and are therefore propagated over time. But since that upward pointing arrow of time is indicating a carriage of average values and ranges that stretch from the past, as one pole, towards the future, as another pole, then those averages are being poloidally maintained over time, and always at a set rate. The population of plessists winding about that helicoid is then the set of biological neighbours that congregate about a complete set of maximum and minimum values, which is meridionally. They thus share a time rate of oscillation about that helicoid toroidally, for they move about on a track between their shared inner and outer boundaries, which form a torus. Those entities that do not share those minimum and maximum values, plus rates of traversing them, will gradually cease being plessists or neighbours. They do not share the same torque and will fly out, off-track, and so beyond the shared surfaces and boundaries. Plessists therefore stand constantly near to each other in volumes of shared quantities and rates.

The conceptual problem that Darwin and his theory faces is simply that the boundaries around plessists, and so around their averages and their rates, are not fixed in stone. For one possible boundary around Homo sapiens, then in May 2010 a woman named Stacey Herald, from Dry Ridge, Kentucky, USA, became the world’s smallest mother. She was only 71 centimetres, 2’ 4”, tall—no taller than the average three year old—when she gave birth to her daughter. She was so small, during her pregnancy, that her belly protruded so much that she was almost as wide as she was tall. Her developing foetus was also so heavy, compared to her own body weight, that she could not stand up. She was confined to a wheelchair for most of her nine month pregnancy.

Herald suffers from a rare disease called osteogenisis imperfecta which stunted her own growth and development. Her doctors discouraged her from having children at all because they thought she would not survive the pressures placed on her lungs and brittle bones. Her first and third children have the same disorder, while her middle child is normal. The smallest of her children was only five inches long at birth, and weighed a mere 964 grams, 2lb 1oz at birth. Those are therefore values for one boundary. They are not fixed in stone, but are nevertheless a boundary.

In complete contrast to Herald, and for another possible Homo sapiens boundary, the 40-year-old British woman Maxine Marin gave birth in Hospital Marina Salud, in the Mediterranean city of Denia, in Spain, to a daughter who was then the biggest baby ever born using natural methods. This particular baby weighed in at 6.2 kilogrammes, 13.6 lbs, and measured 57 centimetres, 22 inches long. She was almost as large at her birth, as was the adult and full-grown childbearing mother Herald. This second baby's values are therefore another boundary. Those values are also not fixed in stone, but are also nevertheless a boundary.

Herald and Marin and their babies thus define, between them, the current inner and outer bounds for our species as an associated group of plessists. All of us fit within the bounds these mothers and their babies define. The rest of us congregate around the averages we conjointly establish in our survival as plessists … and … nothing else does. But since these boundaries are not set in stone, then a different smallest and/or largest set of defining values is always possible. The average is itself maintained with each of our contributions, and so varies with every one of us. That is the essence of Darwin's theory.

Figures 21.1 and 21.2 suggest that plessists tend to move away from such edges, and their boundary values, and inwards towards the mean as a body volume. Those biological entities whose values deviate to become less than average, in their initial quantities, will also tend to have a set of greater than average rates of change to keep them on the helicoid and behind a surface; while those that deviate to become greater than the average, again in their initial quantities, will instead tend to have lesser rates of change and create a different surface. The former strive to fly away towards an inner boundary and surface, while the latter strive to fly away towards an outer one. Those entities that stay close to each other share the body volume in between. They define the mean and are the plessists.

When Carl Friedrich Gauss was fifteen years old, which was the year before he went to study at the University of Göttingen, he discovered his ‘method of least squares’. This is a way of determining an average, or a most likely, value for a set of properties. He had realized that the more values we have in a set, then the more likely they are to be distributing themselves around what we now call the mean. That mean is their summarizing value. But no distinct value ever has to have, or to be, that mean. They are all free to show their variations. That mean is instead what they all hold in common.

Gauss had realized that each individual measurement is just a sampling of the whole. Each distinct value contributes to the whole with both (a) its own value, and (b) its weighting or error. The whole is a combination of these. It is then a question of determining how accurately each value is describing the true, shared, or mean value. Gauss gradually realized that he could minimize these differences or variations using what is now called a bell-shaped curve, or a Gaussian distribution. We then know not just any given value, but also how close to the mean each one is. Or expressed differently, we also know its slight variations. He did not, however, publish his discovery, nor let anyone know about it.

On January 1st, 1801, Giuseppe Piazzi, director of the observatory in Palermo, Italy, was looking in the night sky when, quite by accident, he caught sight of something new and unexpected. He observed it for a while—and then excitedly announced to the rest of the world that he had seen a completely new star of magnitude eight moving through the skies. Although he at first thought it was a star, on further observation he decided that it might be a planet instead. But he could only make three observations, about 1/40th of its entire projected orbit. It was not particularly bright, so it disappeared when it got closer to the sun. Since there had not been enough time to determine either an accurate mass or distance, Newton’s gravitational laws could not be applied to predict its trajectory. However … Gauss's way of looking at the situation was not merely a study of planets. It was a study of the methods to summarize behaviours, and thus to predict them from knowing their slight variations in values.

When Gauss heard about Piazzi's discovery, and about the desperate search the collective astronomers of Europe were making to rediscover it, he used his least squares method to calculate its most likely trajectory. He could not predict its exact trajectory, because nobody knew its mass. However, Piazzi's observations of its slight variations were enough for Gauss to compute a most likely trajectory, along with a margin of error. He then announced that the ‘clod of dirt’, as he later came to call it, would show up again before the end of the year, and he predicted its most likely location. Ceres turned up where and when he said it would. He was still only 24.

A year later, in 1802, Gauss' friend Heinrich Olbers discovered the asteroid Pallas. Located between the two massive bodies Jupiter and Saturn, it had a highly eccentric orbit and was particularly prone to perturbations. Gauss again used its slight variations to predict both the most likely position, and the most likely approximations for Olbers' discovery.

Gauss' masterwork in astronomy, Theoria Motus Corporum Coelestium or Theory of the Movements of Celestial Objects described these techniques. His much later memoir Theoria combinationis observationum erroribus minimis obnoxiae or Theory of the Combinations of Observations Least Subject to Errors carefully discussed variations in measurements, and their effects. He had recognized that in spite of variations due to instrumental variations; and in spite of random variations, systematic variations, observational variations, and “variations due to the personal equation”; all properties in any set of such slight variations are characterized by a shared true value, along with a characteristic distribution. A perfect data set is not possible. But a large number of samples-plus-observations will always (A) exhibit a true value, and (B) be accompanied by a distribution that states the underlying behaviour as the variations.

The average is a location parameter. It tells us where a shared property is most likely located, given a collection of attributes. We then know not just a given average value, but also a set of behaviours regarding where all other similar properties and components are also most likely to be located, and how they are distributed. The further away we get from the mean, then the less likely we are to find properties and components … and also the more likely those are to be exhibiting the behaviours that will restore the population mean.

Our concept of average is somewhat more extensive and explicitly incorporates a set of similarly average rates of change where one thing changes relative to another. Those rates of change again of one thing relative to another are a part of what creates a set of plessists that survive together:

  1. The plessists share the same associations, conjoinings, and distributions which are the first, second, and third integrals. They form coordinated line-like, surface and area-like, and volume-like collections of attributes. And since integrals are a way of simultaneously ‘adding to’ and ‘multiplying with’ what is already possessed, then plessists build, and constantly rebuild, a shared morphology and physiology.
  2. They also share, as plessists, the same transforms, directives, and aberrancies, which are the first, second, and third derivatives. They form a coordinated set of velocity-like, acceleration-like, and jerk-like transformations of entities and their properties. And since derivatives are a way of simultaneously ‘subtracting from’ and ‘dividing amongst’ or ‘proportionately allocating to’ what is currently possessed, then these particular biological entities have a constantly shared set of instantaneous interactions and responses to their surroundings.
  3. Plessists have the same average quantities and average rates of change, and a shared distribution across the same sets of:
    1. poloidal or temporal intervals;
    2. meridional or resource-based quantitative ranges between minima and maxima; and so therefore
    3. toroidal distances, which are complete time periods and circulating generations of shared behaviours, dispositions and lengths.

Plessists therefore survive close to each other all about the circulation of the generations through having (a) the same quantities, and (b) the same rates of change which are the averages they describe with and around each other. They produce similar average progeny that act similarly, complete with their own similar sets of neighbours, and that can then do the same. This is the survival of the plessists.

Plessists survive near to each other because they again use very similar masses and energies through sharing the same rates of change. They procure those masses and energies from the surroundings using their similar, and current, physical and ecological propensities. They then metabolize what they acquire into themselves at very similar rates, and for very similar biological purposes. They thus evidence the same quantities, which they use at the same rates of change, both metabolically and physiologically.

Figure 21.3
0-sphere and 1-ball

We measure the quantities and the values that biological entities use with both the rows and columns of the tensor we see in Figure 21.3. We confirm their similarities and their averages by, firstly, putting all their absolute values and their clock times along the rows; and then, secondly, by putting all their relative values and so their transformations and generational times in the columns. Plessists then share the same absolute and relative values. Those are the coordinated values highlighted by the magnifying glass in Figure 21.3. In the other components in any column we have the same relative values, but different absolute ones; and in the other components in any row we have the same absolute values, but different relative ones. The absolute and relative are only the same along the diagonal. In all other locations there is either the one or the other, but not both. Plessists then always seem normal and accessible to each other, throughout their joint transformations, and even though they might become abnormal and inaccessible to others near to the edges of their shared inner and outer boundaries. The tension between the absolute and the relative is the source of all variations and the driving force of natural selection and Darwinian fitness, competition, and evolution.

The rows in our tensor measure properties at each discrete moment in clock time t. The present moment is t0, the immediate past one is t-1, and the future one is t1. The distance between moments is dt. So when Stacey Herald's baby daughter weighs in with her 964 grams and five inches of length, we record those values.

And … we already know, from comparing her absolute values relatively to those of other newborns, that Herald's infant daughter is extremely small indeed, relatively. We are not at all surprised to learn that because of that very small relative size, she spent the first part of her existence being closely monitored in an intensive care unit. And if she puts on 10 grams, then we record that; as well as how long it took to acquire them.

And when Maxine Marin produces her daughter, we record that 6.2 kilogrammes birth weight, along with the 57 centimetres of length. We again know from those absolute values, and by comparing to others of this kind, that she is very large, relatively. And when this second baby also puts on 10 grams, we record that, as well as how long she takes to do so, and also again relatively.

We measure all these mass and energy values relatively, both to themselves, and to the generation of length T. So when we record Stacey Herald's baby's 10 grams of increase in weight, we also record that it is is just over 10% of her original birth weight. We can then turn to Maxine Marin's baby and find out how long she takes to put on the equivalent 10% of her initial body weight. We record these in terms of how long each needs to live to do it, so we can express the transformations in terms of how long each has lived thus far. One baby will have to live proportionately longer than the other to do the same mix of absolute and relative things. Even if both, for example, attain an adult height that is twice that each one had at two years old, they are still likely to spend different amounts of times and resources achieving both those sets of heights. We place all those relative sizes and timings in the columns. The present set is τ0, the immediate past set is τ-1, while the immediately upcoming one is τ1. These columns give us the biological actions, sequences, and intervals dτ.

Plessists, or near neighbours, are again those biological entities that always have shared absolute and relative values on our tensor diagonals. They do the same things at the same times using the same absolute and proportionate resources and energies for the same general purposes defined by their common generation length, which is always the same absolute-relative timings and patterns. Every time interval dt along an absolute clock, and row, is also always some specific biological and action-distance dτ that carries that population about its circulation of the generations. Their absolute and their relative values must always match all the way around the cycle. Plessists then share the same average values of mass, of energy, of processing, and of resources in both these absolute and relative respects. That is the same sequences of biological activities sharing the same descripta of t-1τ-1t0τ0–t1τ1 all about each of their equivalent circulations. They build the same surfaces about the helicoid and share a volume of activity stretching between those surfaces defined as the commonality of their relatives and absolutes.

Only plessists can share all such moments, masses, and energies continuously through sharing the same rates of change, as well as the same values. Only plessists have the same absolute and relative values. The absolute values govern their surface relations to the surroundings, while the relative ones govern their internal and volume relations to each other. Their integrals govern their constructions and their histories, while their derivatives govern their current behaviours and their futures.

All those biological entities that can continue to fit their times and their activities into the same volume, the same “tunnel of reproductive accessibility”, and the the same neighbourhood event-sets will be members of the same breeding community. As our two babies from Herald and Marin suggest, those that go either too slowly or too quickly; and/or are too large or too small; and/or are configured too intricately or not intricately enough; they will not have a sufficient number of neighbours on their volume track on the helicoid. Those that do not share the same sets of absolute and relative averages will eventually start showing the variations that gradually move them away from their neighbours. They will not contribute to the survival of the plessists.

But the plessist concept is not quite enough for our purposes. In 1909 the Danish scientist Wilhelm Ludvig Johannsen coined the word “gene”, which he took from the Greek genea meaning generation or race, to apply to the fundamental physical, functional, units of heredity described by Mendel's factors of inheritance. And in 1989 Richard Dawkins coined the word “meme” to refer to fundamental and transmissible ‘units of culture’. But those are still also not enough.

We need to refer, explicitly, to these patterns of behaviour–plus–morphology that one group of organisms can have, as plessists, but which can also directly or indirectly affect the behaviour–plus–morphology of another group … and so that at least one of them changes either its genes, or its memes, or both, in response. Examples are when hunters run down their prey, or viruses successfully infect their hosts. We therefore coin the word “pleme”.

A set of plemes is the collected set of genes–plus–memes that we can allocate to a group of plessists, where we explicitly refer to a discrete set of organisms, all of which are sufficiently near to each other to be sharing the same sets of average values and average sets of rates of change in mass and energy over a determinate and measurable period of time. Dogs and human beings on the one hand, or human beings and wheat on the other, would both serve as examples of plessists, along with interacting plemes.

Dogs and humans, and humans and wheat, do not breed with each other. But they each, separately, meet our criteria for being plessists through each demonstrating average values and rates for survivals of the fittest. They each have their separate sets of both genes and memes, each of which is discernible, and each of which is transmissible and heritable within each distinct group. They can also reproductively and heritably affect each other and their surroundings. There are actions and circumstances they can impose on each other that have trackable consequences on each group of genes and/or memes.

We can now talk about plemes because dogs and humans, and humans and wheat, influence each other as distinct travelling bands of plessists. Humans have not directly interbred with either dogs or wheat, but they have greatly influenced each others' breeding habits all down the generations … and they have been reciprocally influenced by those that have responded to those efforts. Thus each of these sets of plessists have travelled back and forth and influenced each other, with specific and determinable behaviours, so that those behaviours and characters have each themselves changed over time. The hunters affect the hunted by catching them, but the hunted have in their turn back-affected their hunters by being caught; just as those that succeed in not being caught affect those that have failed to hunt them down. Those are their joint plemes.

Humans have benefitted from the plemetic responses of dogs, just as dogs have benefitted from the plemetic influences of humans in selecting them through both their genes and their response behaviours and memes. The same goes for wheat. Plemes, therefore, are the memes and behaviours we can allocate to specific groups of genetic neighbours, each living as plessists. That coordinated set of memes-plus-genes is influenced by, and is influencing, other similar sets … and that is a plemic or plemetic interaction.

Although our concept of plessists is clear in the sense that it is defined with a demonstrable set of average values and average sets of changes in terms of shared values plus first, second, and third integrals and derivatives, it still needs handling with care. The plessist concept is obviously skirting around the far more well-known “species” one, but it is considerably better defined. It also allows us to discuss variations and the forces that cause evolution.

The evocative German term Rassenkreis or ‘ring of races’ highlights these issues. The more formal scientific term is ‘polytypic’. It refers to geographically distributed species where it becomes increasingly unclear where boundaries between entity-types should be drawn. So salamanders of various types, including the species Ensatina eschscholtzii and its equally close cousin E. eschscholtzii klauberi, can be found ranging from British Columbia, in Canada, southwards through Washington, Oregon, and California in the USA, on down into Mexico; and seagulls including the herring gull, Larus argentatus, and its close cousin the lesser-black-backed gull, L. fuscus and a variety of others can be found around Great Britain and the eastern North Atlantic, and the northern polar regions stretching from the Aral and the Caspian Seas to the Mediterranean, Siberia, Alaska, and into northern America.

California has its famous Great Central Valley where at least seven species or subspecies of salamanders skirt the valley, some of which cannot interbreed. They live at both the northern and southern ends, as well as on the eastern and western sides. So we can begin at the south end of the valley, and pick breeding salamanders heading northwards on the eastern side. We can pick interbreeding types all the way up and then return southwards over on the western side, still picking interbreeding types all the way back down. But by the time we return to where we first started in the south, there will have been so many changes amongst the various salamander subspecies that we will find ourselves with (at least) two distinct types that cannot interbreed … even though there is an unbroken chain of successfully breeding cousins and types all the way round.

There is a similar pattern with Larus argentatus and L. fuscus. In or around the Pleistocene epoch, some 2-½ million years ago, various sub-groups of an original L. argentatus population seem to have branched out and settled all around the northern polar regions. The original radiation found L. argentatus living in a huge geographic circle. Local subpopulations then reproduced preferentially with each other. They formed distinct reproductive enclaves. Thus the Caspian gull, L. cacchinans seems, from its DNA, to be an offshoot of an original L. argentatus population that stayed near the Aral and Caspian seas. It now has the subspecies the Steppe or Baraba gull, L. cachinnans barabensis. Then there is L. glaucoides, the Icelandic gull, which lives in the Arctic Ocean, near Baffin Island. It has its own subspecies, Kumlien’s Gull, L. glaucoides kumlieni, dwelling around Canada’s Arctic coasts. It seems to have been L. cacchinans which gave rise to L. fuscus, the lesser-black-backed gull, which then expanded outwards, leaving L. cacchinans in its original territories. But the original L. argentatus continued surviving both on the eastern coasts of Siberia, and in Alaska. From there, it gradually spread into North America, approaching the L. fuscus territories. But differences had accumulated so interbreeding is now largely not possible. Another genetically distinct variety is L. grucoides, a North American form of the original L. argentatus, and which also met and bred with L. fuscus.

The Association of European Rarities Committees recognizes six Larus types: Larus argentatus, the European herring gull; L. smithsonianus, the American herring gull; L. cacchinans, the Caspian gull; L. michahellis, the yellow-legged gull; L. vegae, the Vega gull; and L. armenicus, the Armenian gull. But in 2010 the British Ornithologists’ Union Records Committee stopped recognizing the Caspian gull, insisting it was not truly distinct. There is also debate about the Mongolian gull and whether it should be titled L. vegae mongolicus, or else L. cachinnans mongolicus, and so questioning whether it is or is not a subspecies of either the Caspian or the East Siberian gulls. Some argue, however, that it is a distinct species, in its own right, and that it should rightly be called L. mongolicus.

There are several well-known examples of Rassenkreis. Whether or not such rings should be classified as a single species, or should be divided into separate ones, is a matter of continuing debate.

There is also a problem with what are called eusocial species such as ants and bees. The only vertebrate species known to engage in true eusocial behaviour are Heterocephalus glaber, the naked mole rat, and Cryptomys damarensis, the Damaraland mole rat.

A true eusocial species is any animal species living in colonies, and that also lives in multigenerational family groups. They also have the distinguishing feature that the vast majority are infertile and cooperate for the benefit of the very few reproductively capable group members. There is often a very high degree of specialization in tasks, which maximizes efficiency for the group, but at the apparent cost of the fitness and the efforts of the many, who seem to derive little benefit in strict fitness terms. They do not pass on genes. The workers never reproduce, yet they spend a great deal of time and energy aiding and abetting the reproductive efforts of the queen, who is typically their mother. Those are her genes, and she is their provider.

Although eusociality is very specific, there are broadly comparable behaviour patterns in many other species and orders, such as amongst wolves, when only the high-ranking or alpha male and female wolves breed. The alpha pair is most usually the only pair to mate. This is an expression of their dominance. They are generally the pack progenitors … the pack patriarch and matriarch. When the female is in oestrus the dominant pair can sometimes even move temporarily out of the pack to restrict her availability, and to prevent interruptions in their breeding. Dominance can sometimes switch from the male to the female, informing other pack members of whom to serve.

Wolves are not eusocial. Non-alpha pairs can still potentially mate. They are not permanently infertile. Some studies have in fact shown that when conditions are more favourable enough, then in as many as twenty or thirty per cent of cases there can be pups from at least two litters: i.e. from a non-alpha female or male. Nevertheless, the dominant males tend to chase other males away, and the females strongly discourage others and maintain the pack hierarchy, thus preventing subordinate females from breeding through both physical and chemical methods.

The question then remains as to why so many individual biological entities, across so many species or collections of plessists, can make such efforts to benefit others and others' genes when they themselves are either incapable of, or else are are prevented from, transmitting their own genes through to their own distinct progeny. Worker ants and bees are not even channels for genes. They pass nothing on. Fitness, as generally conceived, is the survival of one's own genes manifest through one's own reproductive channel of transmission, and not merely the coincidental or willing custodianship of some others'. Darwin himself mentioned this “one special difficulty, which at first appeared to me insuperable, and actually fatal to my theory”. The issue is still outstanding. We can now try to resolve it.

The concept of plessists and plemes makes the issue far easier to address. We need only to distinguish a curved line from a straight one, and to identify a minimizing tendency. We need only a set a average values and a range, such as we measured in our Brassica rapa experiment.

Figure 21.4
0-sphere and 1-ball

Figure 21.4 shows us constructing a square that is a “Levi-Civita parallelogramoid” … or more exactly, a “Levi-Civita squaroid”. These are named after the Italian physicist and mathematician Tullio Levi-Civita, renowned for his work in tensor calculus and differential geometry. They allow us to much more carefully define our groups of plessists and their plemes. We can also identify the straight lines and the minimizing tendency we need.

We begin with two highly trained commandos. In this first and simplest case, they are clones and identical. Whatever one would do, in a given situation, the other would do exactly the same. Each has a helicopter to provide him with all ropes, pitons, crampons, speedboats, motorbicycles, all-terrain vehicles, and anything else needed to navigate the terrain. They each also have a compass, and each keep in constant radio communication both with each other and with each of their helicopters.

Our two commandos start off from the same location. Commando 1 goes due south, Commando 2 goes due east, each for one day. Whatever terrain each one meets—be it oceans, deserts, marshes, mountains, open lakes, thick woods or jungles—each selects the appropriate transport and goes just as far and as fast as he can.

At the one day point, they switch directions. Commando 1 now goes due east instead of due south, while Commando 2 goes due south instead of due east. And … if one goes first south then east, while the other goes first east then south, then each one's times and directions of travel are each parallel to each of their previous ones. They are also zeroing in on each other to form a square. They use their radios, on the last stretch, to make sure they adjust their speeds so they meet up exactly when that second one day combination stretch has elapsed. We now have a square made up of matching one day stretches of time, of terrain types, and of their various rates of traversal. Those lines linking them are straight, because each always chose the most direct path, at the fastest possible speed. There was no other path possible, and they would each have done the identical thing in those conditions.

Although distances as we ordinarily think of them are not the determining factors in the squareness of this Levi-Civita parallelogramoid or squaroid, all such basic concepts still make sense. It is still very easy to understand that our two commandos can each travel at 90° to each other, which is orthogonally. Independently of the terrain, and independently of physical distances, we have built a square based on shared times and rates, and carefully defined neighbourhoods. Given those careful descriptions, then this square's sides are guaranteed even and parallel, no matter what the actual physical terrain and physical distances might look like.

We can also now propose, as in Figure 21.4, that our two commandos have a piece of string attached to both their heads and their feet. The lines stretching between them will always form a gradient, even if it is flat. Whenever it is flat, we know they are at the same height. If one starts climbing a mountain, both we and the commandos will know from that line's gradient. That line is also always direct. No matter how curved a path they might travel on, to get around obstacles, we will always know exactly how far apart they are, for we have our direct line. We can even hang an electronic digital plumb bob that always positions itself exactly in the middle of the line, and that always hangs straight down under gravity. It always tell us what is going on. So we now only ever need to measure from one commando to the middle of the line, and we will know exactly where the other one is from the length we have already measured, and from the gradient. (We are now building a Frenet-Serrat trihedron).

Since these two commandos are the same height, we can also measure the distance each one travels in “height-lengths”. So a commando will move at both (a) so many height-lengths-per-minute, so that we are expressing all distances relatively and proportionately; and also (b) at so many kilometres per hour, so that we additionally express all their movements absolutely. And since they are identical, they always both have exactly the average height value, both relatively and absolutely.

And since each commando has a piece of rope attached to their feet, we can always record the surface and the area they have covered, since they started. This is a good way of recording the resources they have covered and used. We know that they need so many calories an hour, and that they must forage for so much food and nutrition. This is also an integral.

And since each of our commandos has a height, along with another piece of rope attached to their heads, then they also create a box and a volume between them as they run. We can easily record the total volume of atmosphere they have jointly pushed their way through. It is a good measure of the total work they have done and the energy they have used. This is also an integral. We can measure those areas and volumes both relatively and absolutely.

It is also very clear that if we swap in a different commando, who is a different height, then the descriptions of what is happening will change. If the height is different, then that mix of absolute and relative units we are constructing cannot be the same. The average also changes. We can nevertheless always express each commando as being so many percent of the average. And if one of them is 10% bigger than their shared average, then we know straight away that the other one is correspondingly smaller. We also know each one's contribution to our lines, surfaces, and volumes.

If our new commando now runs at the same height-length-per hour as the old one, then we will have a different absolute value for kilometres per hour; and if the new commando runs at the same absolute kilometres per hour, then we will have a different relative value for height-lengths-per hour. And, of course, our lines, our gradients, our areas in our imaginary resource squares, and our volumes in our imaginary atmospheric cuboids, all these will also change immediately a commando changes. But thanks to our mix of absolute and relative measures, we can spot all variations.

If we send out lots of commandos we will gradually build up a “most expected” picture for how the terrain is covered. We will get a picture both in absolute terms, and in terms of how the average one performs. We will get a picture for the route that the average commando takes, and the modes of transport he selects. The occasional commando might select to swim a particular section where others have used a rowboat, but the greater the number of commandos that go about the course, the better defined that average becomes, in the sense of being the most likely to be chosen. We also build up a picture of the terrain itself.

We can gradually find a point in the middle of the terrain from which we can describe the entire behaviour of these commandos in both relative and absolute terms. That will be where the plumb bob gravitates the most often. We can follow that plumb bob and assign all averages to that spot, its movements, and its behaviours. We can use it to refer everything to, and to measure everything from, both relatively and absolutely. We will know such things as how far away any commando is from either the beginning or the end; how long it will take to get from one to the other; and we can assess all rates of movement everywhere. If our electronic plumb bob is lower or slower than expected, it tells us about the commandos, and we can soon find out who is responsible, relative to the others.

We will gradually observe the most expected journey over this terrain, which is the surroundings, in terms of both kilometres per hour, which is absolutely, and in average height length covered, which is relatively. We will additionally get figures for expected areas and changes in areas, and volumes and changes in volumes, again both relatively and absolutely.

And if we send out lots of commandos simultaneously, we simply increase the rate at which the areas and volumes build up, while gradually getting more accurate mean values, both relative and absolute. Therefore: recording both the absolute and the relative values is a sure-fire way of telling us what is happening, and finding out whether or not anything is changing from one commando to the next.

We can now transfer over these easy-to-understand concepts to biological entities; to their circulations; and to our plessists and plemes. So when, for example, a human being is ill, then we can equally easily understand that he or she is likely to spend a few days in bed fighting say a fever. His or her mass does not change, but his or her energy level does. We can think of one of these properties as changing without the other. The energy level or density is therefore one complete dimension. It can change quite independently of mass. And, by the same token, mass can change independently of energy level. Those two are as orthogonal as are north and east. We can imagine one changing as much as we like, without the other changing, just as we can imagine a commando going as far east or west as we want, without moving north or south in any way. We can now use that idea of their independence, in conjunction with their relative and absolute measures, to build biological versions of Levi-Civita squaroids and parallelogramoids.

Figure 21.5
0-sphere and 1-ball

Figure 21.5 shows us building such a biological Levi-Civita parallelogramoid. Instead of commandos, we use a human being developing from infancy to adulthood. Either of Childhoods A or B is possible. So are infinitely many others. But we can look on all those proposed developments as two-dimensional transformations. For one dimension, we can think of each person as, firstly, growing exclusively in mass, all the way from infant to adult size. This is done without developing, and so without changing any other features whatever. We thus end up with a baby of adult size. And when our infant has at last reached full adult size, he or she can then turn and undertake all needed development, all at that adult size.

For every Homo sapiens that undertakes this growth, we get a more accurate set of average and relative values we can use to describe all others. And if our two time intervals are the same; and if the rates of transformation and energy intake are also the same; then we have a form of biological Levi-Civita squaroid.

We have also made sense of the 90° concept, but within biology. It just means that one thing can change when another does not. And even if these rates and times are not the same, so that this is not a square, then this is certainly some form of rectangle, for those are clear and parallel gradients linking all the various points in the Levi-Civita parallelogramoid. There is a definite maximum size, and there is a definite configuration energy for the maximum development point. The times, rates, and quantities involved are a resultant of the biological terrain that our human beings pass over in their varied infant-to-adult transformations.

The alternative is, of course, for the infant to first undertake all possible development, while staying constantly at that infant size; and then once development is completed, to gradually grow, with all features intact, to the adult size. It does not matter which way we look on it. Just as travelling first south then east is the same as travelling first east then south, the end result here is the same. We can describe any biological transformation in terms of these factors. We can also do so both relatively and absolutely. We can describe any journeys of growth and development from infant to adult entirely in terms of these absolute and relative measures, in these two dimensions.

We can describe both Herald and Marin's babies, and all other human beings, in exactly these terms. Seat belt and child restraint manufacturers, for example, are especially concerned with the niceties of human development, and the interplay of absolute and relative patterns. Human children show a steady gain in weight from the tenth day after birth onwards, putting on approximately 0.9 kilogrammes or two pounds per month for the first three months. That might be the average, but it is also the entire birth weight of Herald's very small newborn. Birth weight should have tripled by the end of the first year, and quadrupled by the end of the second. The gain then slows so that by age five, body weight is approximately six times birth weight; and by ten years it should be about ten times. Body length should have doubled by age five, and tripled by age thirteen. Subcutaneous tissue therefore increases rapidly in thickness for the first nine months, bodily growth then slowing. Although the average five-year old child has about half the subcutaneous layer that he or she did as a nine month old infant, the inherent chubbiness makes it difficult to create appropriate seat restraints which must be strongest around the limited skeletal components, yet allow for the inevitable slippage induced by the the larger amounts of soft tissue.

We can now describe all biological entities in terms of (a) their changes in mass relative to whatever might be their initial value; and (b) through whatever might be their changes in energy density, again in terms relative to their initial value. We can do this all across the generation, both absolutely and relatively in mass and energy. And just as two commandos are identical if their absolute and relative descriptions are the same, so also with two biological entities. They can only have identical absolute and relative descriptions if they are indeed identical; and so that they build the identical Levi-Civita parallelogramoids; also in identical terrains and conditions. They are plessists if they share close enough values as their means.

We then have to classify these plessists, their plemes, and the masses and energies they use. The German philosopher Immanuel Kant is widely regarded as one of the three greatest philosophers in history (the other two being Aristotle and Wittgenstein). He proposed three ways of classifying all human knowledge. Firstly, facts and knowledge can be classified according to the type of objects they study and describe. So numismatics studies coins, genetics studies genes, and herpetology studies snakes. Secondly, facts and knowledge can be classified temporally, which is an essentially historical perspective. It reveals the interconnections between things. So Lennon and McCartney make little sense without Chuck Berry and Little Richard; Debussy makes little sense without Chopin; and quantum physics is extremely hard to understand without first grasping its roots in classical physics. And then thirdly, facts and knowledge can be classified spatially, which leads to such subjects as geography and anthropology. So Kant said:

I treat geography not with the completeness and philosophical exactitude in each part, which is a matter for physics and natural history, but with the rational curiosity of a traveller who collates his collection of observations and reflects on its design (Quoted in Casirer, 1981, p. 52).

In Kant's view, geography, as a discipline, draws various other subjects together, synthesizing them with concerns of area and space. It tells us how humans regard and differentiate the world around them. It in particular reveals the human in action as a synthesizer. Geography thus has various sub-disciplines as moral, physical, political, commercial, mathematical, and theological.

Kant pointed out that the geographer's essential objects of study are regions. But regions are not set in stone. They show considerable variations. Since they are spatial, they cannot be moved. Each remains clear and distinctive, yet no region has a set boundary. But any region can seem different when approached in a different way, or from a different perspective.

Each region is characterized by a natural environment with its landform and climate. It can be as small as the square inch of ground occupied by some ants under study. But no matter how small, it always elicits a different investigation from say chemistry or physics. Or, a region can be as large as the entire earth. But it is still then different from say a planetary-sized cosmological or astronomical investigation.

Each region has a distinct relationship between its physical and its elemental attributes, and how humans and/or other biological entities relate to it in the present, and have done in the past. Each region has its sociocultural context, which is a record of some group's way of interacting with it. That fairly stable culture and context—that collection of memes—awaits new arrivals or immigrants into that region, who can then both adapt to it, and contribute to it.

A region's size and nature depends at least as much upon the geographer’s scope of investigation as it does upon the region itself. So a lush tropical jungle can be just as valid a region as the South American country that contains it. The two could even be physically co-extensive … but they are still different regions to the geographer.

All biological entities depend on regions for their habitats. And … just like regions … the proposed boundaries that classify them into species are not set in stone. They depend entirely on the arena of investigation. Thus the snowshoe hare, Lepus americanus, its immediate predator, Lynx canadensis, and the woody browse and vegetation in which they live, together formed a valid region or field of study for L. B. Keith. The entire set of snowshoe hares, lynx, and woody browse meets our criteria for plessists because they all abide by our four laws, our four maxims, and our three constraints. Their plemes also affect each other in a plastic and interactive way. Those plemic interactions create a discernible ten-year cycle as first one set of plessists flourishes until another is stressed; which then has consequences on the third; which then re-affects the first. There are no hard and fast boundaries around any of these plessists, but they each nevertheless describe an average and a set of means that come together to create a discernible set of interactions which we can easily divide, in this case, into three distinct groups based on three distinct sets of reproductive behaviours that produce three distinct sets of entities.

Figure 21.6
0-sphere and 1-ball

Figure 21.6.A suggests a suitable biological space, while 21.6.B suggests the interactions between some of the plessists within that space. The two spheres are the inner and outer bounds. That entire contained volume, in Figure 21.6.A, is created with the strings we attach to the entities, and as we did with our commandos. The area on the proposed surface is the area that the strings on the entities create as they interact with their resource base. The various lines on the surface are their action lines defining the various shapings and possibilities of traversing the terrain. The dots are the values and the positions established by the juxtaposition of absolute and real values.

None of the values that define any such proposed biological and plessist energy sphere is fixed in stone. If, for example, it is raining when our commandos approach a terrain type, they will tend to cross it one way; while if it is windy they will probably use a different form of transport. The average then becomes just as much a function of what the weather is like as it does of the approaching commandos. Nevertheless, for every viable collection of plessists and surroundings, there is a minimum shared set of lines. There is also a set of maximum possible areas and volumes that is a statement of their means and their associated rates of change. These together define them and their distributions. Every set of plessists creates such a set of means, minima, and maxima.

All biological entities are parts of plessist collections and regions, along with their associated sizes, boundaries, and ecologies and habitats. And … just like regions … plessist congregations have no set boundaries. Those regional classifications depend both upon the plessists themselves, and the investigator's scope of inquiry. So if an Ursus arctos horribililis or grizzly bear, for example, tries to breed with an Ursus maritimus or polar bear, then a biologist might feel that that defines a new region, along with a new average and a new set of rates of change. And if those two do not breed then no such averages and rates of change would seem to be created, even though both types of bears continue to exist. But whether they do or do not breed, the progeny they each produce is certainly an ursid or member of the genus Ursus. They will continue to produce something that is an expression of their averages and rates of change. They are still both plessists, and have simply become extremes relative to each other, that still seek a mean and set of common values relative to both.

In the same way, Homo sapiens is currently defined by the two extremes of the two contrasting babies Herald and Marin produced. All plessists have definite and measurable averages and rates of change, yet all are also without clear boundaries, which are never hard and fast. They are simply arenas of interaction defined by neighbours that have similar averages and rates of change.

There is a variety of lines of action that can go from any one point to any other upon the spheres in Figure 21.6.A and that represent a set of plessist and plemic interactions. The precise line chosen under any conditions depends upon how the sphere is flexing and changing in the prevailing conditions, which is the congregation of all associated plessists and their interaction with their surroundings. There is no one fixed line of action. There is instead a group of plessists existing together. Sometimes we get an Ursus maritimus, sometimes an Ursus arctos horribililis and sometimes something with elements of both. The species designation may become indeterminate, but we will not produce anything of the genus Homo from any two in Ursus. If they are both in Ursus then the result will be in Ursus. The resulting mean and its rates of change are always a function of the interacting and associated extremes, which define the possibilities.

A species, as commonly understood, is simply a set of plessists with a minimal average set of values that together define a path, along with a unimodal distribution. Or as our Law 4 of biology expresses it, “In the allowed set is at least one path such that mass is surrendered, and such that a further entity possessing the required set, and satisfying these four laws”. A species is simply a group of plessists with only the one clear path, and a clear and single set of actions for their reproduction with its minimum and its maximum set of values surrounding an average, which then serves as a range of variations.

Plessists can accumulate in many different types of groupings and conglomerations, some of which will have boundaries smaller than a proposed species, others of which are larger. Pekingese, for example, do not generally breed with bull mastiffs. If we are more interested in discussing breeds, which is the province of animal husbandry, then we place a boundary in one location. But if we are more interested in Canis familiaris in general, then we place the boundary elsewhere. The biological norm is to indicate such discrete reproductive isolates with the term species … although a tight definition is impossible. No matter where the dividing line, we always have plessists and their plemes which abide by the given rules.

Since plessist groupings represent only averages, and average rates of change, then it is very easy for proposed species ranges to overlap. This is our Species 1, 2, and 3 in Figure 21.6.B. Those overlappings are distinct groups of biological organisms that can interact (a) with each other and so within their most usual distributions; and/or (b) with the indicated plessists either side of them; while also perhaps (c) entities in those separate groups either do not, or cannot, interact reproductively. That entire set is possible. Pekingese do not generally breed with bull mastiffs, and grizzly bears do not generally breed with polars. The only difference is that there is a host of entities ranging across the Canis familiaris landscape so creating a genetic contiguity, whereas we do not currently have a host of ursids ranging across the apparent boundaries defined by contemporary examples of Ursus maritimus and Ursus arctos horribililis. We can, however, use this idea of a spread of variations, around a mean, to understand Darwin's proposal for how to create a new species, using the relative selections close to boundaries.

Figure 21.6
0-sphere and 1-ball

Figure 21.6 is another group of very familiar looking plessists that depart their home meadow first thing in the morning. They will all return at nightfall.

Group 1, the recumbent group, barely moves. The ones in that group amble straight over to the nearest tree and lie down, remaining there the rest of the day. Group 2 walks a little further away, to a field slightly behind, in the near distance, and then spends the rest of the day there. Group 3 goes a little further away yet again, to the middle distance, so spending yet more time walking, and a little less grazing. And the final Group 4 is very peripatetic. It goes up and down and round and about, travelling to the grazing a considerable distance away on the furthest hills.

At some point before nightfall, those furthest away in Group 4 return to the home meadow. Those cows are therefore the first back.

We now trace the path the Group 4 cows took. We are interested in the areas their strings form. So we infinitesimally increment the displacement or distance, d, they walked, in each equally infinitesimal increment of time, t, that they were absent from the home meadow. That produces Group 4's absement, A, given by A = ∫ d dt. This is its time-distance combination that states its total awayness value, from the home meadow, in kilometre-hours.

Group 3, in the middle distance, waits a little longer before returning to the home meadow. The extra time those plessists spend away is just enough to compensate for the extra distance that Group 4 walked. They therefore end up with the same absement value.

And then a little later yet again, the near distance Group 2 cows return to the home meadow. And once again, the extra time they spend away is just enough to create the same value for absement, A, as for Groups 3 and 4.

And then finally, the recumbent Group 1 sits in its field until it has accumulated the same value for absement as the others. It then ambles back to the home meadow.

As is always the case, the cows that travelled small distances were close to the inside of the helicoid or spiral staircase, and have large time values for each increase in absement. Those, on the other hand, that travelled large distances were close to the outside of the helicoid or spiral staircase and have small time values for each increase in absement. They may increment in different ways and for different reasons, but these absements—each of which is an area and a multiplicative phenomenon—nevertheless end up the same. We can therefore determine the sources and consequences of the variations in these plessists. These are two-dimensional or surface integrals. We can easily create volume and distribution ones.

Although our four groups of plessists have the same absements, they do not have the same jerks, accelerations, velocities, or actual distances walked. They may all have the same absement, but Groups 2 and 3 are the closest to the mean for all those other values. Groups 2 and 3 walked the distances that are the closest to being the same, and therefore the speeds and the accelerations they deployed to cover those much more equal distances, in that same amount of time, were closest to being the same. So also … four different rectangles can all have the same area, but two can be closer to each others’ shapes by both being more square-like, while the other two differ with one being much longer than it is tall, the other being much taller than it is long. Groups 1 and 4 therefore define the inner and outer boundaries, with Groups 2 and 3 defining their similarities. But since we were careful to adjust the timings so their absements were the same, then these four groups are the same in their surface or area attributes. They will therefore be even more alike in all their volume or three-dimensional ones. They nevertheless show differences in the Levi-Civita properties used to create them. But wherever and however we compare them, Groups 1 and 4 will always show the largest, relative variations of attributes, and will define the boundaries, while Groups 2 and 3 are always closest to the mean.

We can now propose the source of speciation. It is defined by the range that the first, second and third integrals establish as quantities either side of any plessists' joint means; and that the rates of change that the first, second and third derivatives establish either side of those same quantities and means. Speciation is the difference between the ensuing absolute-relative values maintained at one boundary, as compared to the absolute-relative values maintained at the other. Speciation results from the differences between those absolute and relative values either side of some definite mean established by the set of integrals, and across a range established by that mean; which range then leads to changes through derivatives or differences in the rates of changes and exchanges of masses, energies, and timings, both with each other across that range, and between those boundaries and with the surroundings.

Those biological entities that, for example, wish to define themselves as Homo sapiens must have brains that grow rapidly in the period before birth, slowing considerably during per-school years. The newly born entities must have a body weight that is 5% of their adult weight, but with each brain still having 25% of its adult weight. About half the brain volume must accrue in the first year, and it must be at about 75% by end of second year … although their genital organs must develop considerably more slowly, only reaching adult size in the second decade of life. They must also evidence those relative changes at suitable absolute rates and magnitudes. Nevertheless, those boundaries are not set in stone. They have no absolutely definite value. But any organism seeking to be a member of H. sapiens must meet those criteria, within acceptable limits of chemistry, metabolism, and physiology.

The plessists across any proposed range and set of variations must have neighbours whose values and rates of change are closer to their own than are those beyond their boundaries. But those at the inner or the outer boundaries, for any group of plessists, are less likely to directly share values with each other. Our Group 1 cows are much more likely to interbreed through spending more time closer to each other, and similar for the Group 4 cows. This does not destroy a basic reproductive compatibility and communication across all cows or other entities. They simply preferentially establish distinct breeding communities. There will always be near neighbours and plessists.

Darwin's proposal is simply that the plessist boundary entities have an infrequency of contact so that the range and dissimilarity gradually increases until the means and rates of change they need for reproductive success can no longer be met by any viable genetic or reproductive process. Those differentials and ranges between such groups lead to speciation as boundary entities draw away from each other, until the inner and outer boundaries that at one time surrounded a mean acquire further boundaries between them that coalesce around different means, initially located within their boundaries, and so with different inner and outer boundaries that each arise across and within that prior range. There are then completely separate, reproductively isolating means and rates of change established through a reproductive discontinuity.

We can now use our inner and outer boundary concept to demonstrate how to create a new species. Our initial Brassica rapa experiment has already proven:

  1. that population numbers are inversely correlated with generation length;
  2. that every group of plessists is distinguished by the three averages of:
    1. generation length, T,
    2. average individual mass, , and
    3. average individual energy, ; and
  3. that the amounts that every population expends on its fitness and competition, and therefore on maintaining itself at its set of average values, can be precisely quantified.

We now intend to follow our original experiment with one that will demonstrate Darwin's assertion that variations are the proximate cause of speciation.

We found, in our experiment, that when we planted Brassica rapa at a density of four seeds per pot, it returned a generation length of 44 days, but was inclined to shorten. That is therefore one boundary. And when we planted at fourteen seeds per pot, B. rapa returned a generation length of 28 days and was inclined to lengthen. That is therefore the opposite boundary.

In its 44-day situation, Brassica rapa tended to increase its numbers by producing more seeds. It also tended to decrease its mass and increase its energy density and intensity. In its 28-day situation, by contrast, it tended to decrease its numbers by producing less seeds, while increasing its mass and decreasing its energy density and intensity. We can now use these two boundary behaviours to drag those sub-populations away, and to increase the variation-range so as to create … not just one new species, but two. We shall create one new species either side of the demonstrated B. rapa helicoid boundaries.

Since our experiment showed that Brassica rapa tended to expand out to one boundary but to reduce away from another, we shall plant a Tranche 1 and a Trance 2 of seeds to capitalize on these disparate trends.

  • We shall plant Tranche 1 at a density of three seeds per pot. We shall also place them in extremely favourable conditions for light and water. This tranche of seeds therefore has every possible advantage. We shall maintain this set at that advantage by consistently thinning out each following generation to this three plants per pot level, so that irrespective of its seed production, and its efforts to increase, it maintains that low average pot density. The average mass should then grow consistenly as its generation length increases, so that what is currently regarded by these plants as favourable should gradually become the norm, and so that a new mean, and a new shared and expected average rate of increase results.
  • We shall plant Tranche 2 at a density of fourteen seeds per pot. We shall place them in extremely unfavourable conditions for light and water. This tranche is therefore forced into continuing high-stress situations. We shall consistently cram seeds fourteen seeds into these pots to maintain that high number density survival tendency throughout succeeding generations, and again so that a new mean, and a new shared and expected average rate of increase results.

We then frame these two hypotheses that we test in this proposed experiment:

  1. Hypothesis 1:
    That Brassica rapa's first port of call will be a display of Rassenkreis. We hypothesize that as the number of generations mounts, the two extremes that are boundary populations will at some point fail to breed, directly, with each other … but that they will still be able to breed with the original B. rapa population. There is a fairly large difference between a 28-day and a 44-day cycle, but both are still sufficiently close to the shared original 36-day one.
  2. Hypothesis 2:
    That the boundary populations will eventually fail to breed not just with each other, but with the original B. rapa population from which they are extracted. We will then have our two new populations, one produced by extending out either side of the two extremes and boundaries that surrounded the original population, using them as sources for variations.

The only proviso, here, is that B. rapa has an average generation length of T = 36 days. This species is therefore likely to take an inordinately long time to demonstrate its proclivity to use variations to create a new species. It would therefore seem advisable to find some alternative species, such as a bacterium or other prokaryote … i.e. some fast growing, one-celled, micro-species whose generation length is much closer to the one-hour mark. We can then find its carrying capacity and determine two analogous subpopulations that define its inner and outer bounds through their maximum variations in mass, , energy, , numbers, n, and generation length, T. We can then place the inner boundary tranche in unfavourable conditions, and the outer bound tranche in favourable ones … and then observe Darwin's thesis enacted as they create two new species through the force of natural selection. This will be … the creation of species by means of relative selection.

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