by means of relative selection
Using the Weyl and Ricci tensors to prove Darwin’s theories of competition and evolution
and to refute creationism and intelligent design
It is time we acquainted ourselves with the third of our anomalies. We can approach it by remembering that Charles Darwin studied, amongst other things, variations in finch beaks on the Galapagos Islands, using them as evidence of transformations under natural selection. This gives us two different approaches:
The above two would certainly seem a basis for determining the variations Darwin discussed. But if we are to analyse them we need a basis of operations. We have our tensors as a method for studying the transformations involved. We have our 4 × 4 and 3 × 3 grid or tensor. We also understand a little of what it can do. We can use it to measure any biological cycle in any way we want, and gradually accommodate the full panoply of biological diversity.
Although the tensor allows us to describe any cycle in terms of any other, we will use our prototype cell and its ideal cycle. That makes it easier to understand what is happening. It is very well behaved. It gives us the following 4 × 4 “Haeckel tensor” for biology and ecology. It is named after Ernst Haeckel, a German evolutionary biologist who discovered and named thousands of new species, and who reconstructed evolutionary history based on morphology and embryology:
Generation
length : time distribution T’ : τ 
Constraint of constant propagation Wallace pressure : energy distribution P : φ 
Constraint of constant size Mendel pressure : mass distribution M : κ 
Constraint of constant equivalence Work rate : activity distribution S : χ 

Length of time and life style  1:1  
Numbers maintained  1:1  
Body material of organisms  1:1  
Energy and behaviours  1:1 
All the values X :Y in the tensor represent ‘sum : deviation’, ‘multiplier : distribution’ or ‘assimilator : deviator’.
We first look at the second ‘:1’ in the ‘1:1’ that we see in the white row and column headed ‘generation length’ in the above 4 × 4 seeming tensor, and that is therefore some form of distribution or deviation.
In the absolutely ideal case, the amounts of time that our population of prototypical cells will spend with its entry and exit orifices first open, and then closed, is identical. It then spends—or distributes—exactly the same amount of time increasing its internal energy by using mechanical chemical energy, and so acquiring chemical components and resources from the environment, as it does ejecting them back into that same environment, and so losing in its internal energy through its decline in its mechanical chemical energy. The same will hold for the entry and exit apertures. Our population of prototypical cells will distribute exactly as much time increasing its internal energy by taking on nonmechanical chemical energy as it does decreasing internal energy by relinquishing that same form of energy. We shall indicate this temporal distribution of biological activities across both the mechanical and the nonmechanical with the Greek symbol τ.
Since the amounts of time a population of ideal cells distributes across these four activities are the same, then this is what mathematicians call a ‘deviation’ of 1. So wherever we see a deviator of ‘:1’, we straight away know something significant about how the population spends its time and its energy. An even characteristic distribution is always τ = 1.
Now we have a basis for comparison, we can examine some other population. But … we are working with tensors.
No real organism of course spends identical amounts of time taking on and giving off mass and energy. But this is no impediment. We can still take that other population as a basis for τ if we want to. This is the clear advantage of using a tensor. It is simply that the basis we are currently considering for a distribution for time—i.e. our ideal cell—is much easier to work with. But, once again, anything we want can act as the basis.
It is very important to appreciate that we can always use some other, ostensibly uneven, population as our basis for τ. A :1 does not always have to mean an even distribution. It just means, in some cases, that we have compared some second thing to our chosen basis, and that those two things have turned out identical, relative to each other. They distribute themselves in the same way. That :1 again means that we have measured some first population; and then measured another; and got the identcal values both times. Until we have specified some more absolute basis to which we compare them both, we do not know whether or not the distribution τ = 1 is absolutely smooth and even in the sense described above. We just know that the two we are comparing are the same relative to each other.
When, for example, we conducted our own experiment with the plant species Brassica rapa, we found that it spent much more time in its mature phases. This gives it a deviator—relative to the ideal cell basis we have established with our prototype—of τ = 0.625.
We now know that when we compare it to our prototype and reference cell, Brassica rapa has a value of X : 0.625 for this first position in the tensor … where X is an as yet undetermined scaling or absolute or assimilation value that establishes the range for the distribution specified by τ.
A mathematical aside
As a mathematical aside, we should carefully note the other way of looking at this. We can instead take Brassica rapa as our basis. It then defines “normal”, with the prototypical cell then being the one that has the deviation of τ = 0.625. It is important to keep this relative and reversible property of tensors in mind as we proceed.
As for that X in the leading position, it is obviously some form of multiplier, or scaling value, or assimilator … but we again need a basis to act as a standard of measure. We can give ourselves that basis by considering the average singlecelled bacterialike terrestrial organism. This can easily go through a cycle in approximately T = 1,000 seconds (which is just under 20 minutes). We can take that as our basis and set it as T’ = 1. We can then express any other population's cycle in seconds and divide it by 1,000. We will then know how long it is relative to our standard. That will be a multiple and its value for T’.
And that multiplier or assimilator of T’ explains the 1:1 we see in the blank row and column in our tensor for our prototype. That 1:1 is telling us that we are dealing with an organism that has exactly the same generation length as our chosen standard cell (1:); and that it also distributes its time and its energies completely evenly, relatively to the basis, across that given time span (:1). So we now have the length of a generation, plus its temporal distribution. This is T’:τ.
We discovered, in our Brassica rapa experiment, that the plants took T = 36 days to go through their circulation of the generations. When we express that 36 days in seconds so we can get the scaling ratio relative to our standard, we have a multipler for B. rapa of T’= 3,110.4.
The same experiment tells us that Brassica rapa has a temporal distribution of 0.625. This gives us a complete reading, for that species, of: 3,110.4 : 0.625 … all relative to our standard prototype. And since we have a basis, we now know what B. rapa is doing, and how long it is doing it for, as it goes about its cycle of the generations … at least, when compared to the basis provided by our standard cell.
Taken by itself, a reading of 3,110.4 : 0.625 for Brassica rapa does not tell us very much. But suppose we experiment on some other organism and find that it takes T = 5,000 seconds to go through its cycle. Since it takes five times as long to traverse its generation length as our standard cell, we have T’ = 5. If it also has the same deviation as B. rapa (which, as we will eventually see, is impossible), we would have 5 : 0.625.
The value we now have of 5 : 0.625 for our new plant is, however, relative to the basis set by our prototype cell. The advantage of working with tensors is that we can now jettison that prototype. We can use one of these two populations as the basis or reference for measuring the other. They have relative values of 622.08 : 1. The :1 immediately tells us that these two plants share the same overall distribution of activities. These two populations are temporally scaledup versions of each other. They are both behaving exactly like Brassica rapa. The scaling value tells us that one is simply going through the same cycle just over 620 times faster than the other.
We can now compare populations much more easily and directly because we can spot differences and similarities quite easily, both absolutely and relatively. This is the clear advantage of using tensors and comparing everything to an extremely flexible basis regarding amounts and distributions.
Again as a general rule the first figure, X, in any X : Y in our tensor represents some ranging or scaling value as a multiplier or assimilator; while the second, Y, represents some distribution and is our deviator.
It is also a general rule from Einstein's relativity theory that energy does not scale evenly. If we for example keep flinging mass onto a star or planet, it will not just get bigger while staying the same. We instead get such varied planets as Mars, Venus, Jupiter, and Saturn; and we get stars of many different types such as dwarfs and neutrons and reds and whites and so forth, eventually ending up with a black hole.
In the same way, we do not, and cannot, make a scaledup version of an ant. If we keep on adding mass to an ant, we eventually get a creature that cannot support itself on its own legs. They will break. We also never see a mouse as small as an ant.
We shall eventually prove that generation lengths do not simply scale. We shall prove that it is impossible for a first generation to be say ten times longer than a second one without also changing the distributions of activities across time. We shall prove that those distributions gradually change as generation lengths extend or contract.
We should therefore note that we cannot simply change the scaling value without also changing the distribution.
It also works the other way.
We will also prove that when we change the distribution, we automatically change the scale of operations for that property. This is exactly the kind of thing tensors are designed to highlight. This matter of temporal distributions is extremely important. It tells us that the circulation of the generations is marked by the ongoing variations in τ, the sum of which indicate the repetition of set of activities. We shall return to this closer to the end.
We now consider the values in the lightest gray row and column in our tensor. As before, we look first at the second ‘:1’, which represents the distribution or deviator.
The head of the row tells us that this is the number of partitions maintained in internal energy, and so expressed as distinct biological entities. A value of :1 then means that the number distribution for internal energy over time is completely even. The population's minimum and maximum values for partitions and entities are the same, with no changes in numbers. No population members or partitions are ever lost; and so none ever need to be replaced. Or alternatively … the entire population is indifferent to the number of partitionings of internal energy, and to changes in those numbers over the entire generation.
An even distribution in numbers means one of two things. Either (A) we have a population (such as this ideal one) which genuinely never loses any of its members; or else (B) we have a population that is completely indifferent to all changes in its numbers, whether this be absolutely so, or simply relatively to something else. If we measure two populations and their numbers are the same relative to each other at each point, then they have the same distribution and we get a :1 here for the deviator.
Bingo!
Now THAT is something we can measure and test!!
We have found a definite and positive testing method to assess when some property in a biological population is proposed as being free from changes in numbers both absolutely, in terms of no changes at all; and relatively in terms of two things changing in exactly the same way as each other at all times. We have just found a method for assessing what creationism and intelligent design assert. We have a pattern of behaviour for biological entities when they are unaffected by numbers. We proceed … but we keep our eyes wide open!!!
The number distribution in a real population is of course never even. Every real population oscillates between some minimum and some maximum number as its members are first of all lost … and are then replaced in reproduction. Also no two populations or generations can be absolutely identical at all times.
We can always take up a real population, such as we did for Brassica rapa, and measure its minimum and maximum values. We can then express that range as a singlevalued distribution. And if any two populations oscillate between the same minimum and maximum values, then they will have the same value for this distribution and so have :1 relative to each other. They will have exactly the same differentials (i.e. infinitesimal increments dn) at every point. They are first losing numbers to their surroundings, and then reproducing them via their reproductive activities at exactly the same scales and rates. They are the same relative to each other and so act as an identical basis for each other, and so will be :1 relatively.
Our Brassica rapa experiment gave us a number distribution—or size of number differential—of 0.121, again relative to the basis we have established for ourselves with our ideal population. Any other population with that value will be oscillating between the same minimum and maximum number values at the same relative rate. But if we switch our basis and set the two against each other, then that other population will also have a :1 relative to B. rapa.
We then look at the first number—the ‘1:’—in this lightest gray section, which is some assimilator, multiple, or scaling factor. If we now stipulate that all populations should always be expressed so that their average number of partitions in internal energy, over their cycle is n = 1,000, then all populations will always have a 1: at the front. If we adopt this convention then we will only ever have the ranges or differentials to concern ourselves with. This is convenient. It simply makes all comparisons of internal energy between all populations very much easier to handle, because we can then focus our attention on the distributions for we will have a common basis.
A mathematical aside
Expressing the population's average number as 1,000 is simply a way of saying that we are determining a weighted average for that cycle, and then calling that average 1. These weighted numbers will then increase from less than 1 to greater than 1, such as from 0.9 to 1.1. Multiplying such weighted numbers by 1,000 simply allows convenient population sizes to be taken ranging around 1,000 entities, without the potential infelicity of dealing in fractions of entities.
And … since we can link those minimum and maximum number values to the rates, times, generation lengths, and temporal distributions of activities needed to navigate between them, we are beginning to highlight some extremely important features of biological populations. We are linking the above distributions in temporal activities, at each point in the circulation of the generations, to the numbers that traverse them.
We call the value in this column the “constraint of constant propagation” because by the first law of biology that we have already derived, a population must always contain at least one entity—i.e. one discrete partionining of its internal energy—or it is extinct. The population must always maintain a cycle average number, which is to propagate, or to navigate, between necessary minimum and maximum values. The actual numbers must oscillate around some given mean, for that population, as its members die and are replaced.
Every biological population has a characteristic number distribution … its way of thrusting itself and its internal energy into the environment. Mosquitos, for example, reproduce at a much earlier relative age than do, for example, blue whales. They also produce vastly greater numbers at a faster rate. Since the distribution is a part of the constraint of constant propagation and helps determine the energy and so the reproductive behaviour available in this entire column, we give it the Greek symbol φ (most commonly pronounced ‘fie’ to rhyme with ‘my’, but can also be pronounced ‘fee’ to rhyme with ‘me’).
A mathematical aside
We express this constraint of constant propagation rigorously and mathematically as . This simply says that if we sum all the infinitesimal changes that occur in the energy flux or Wallace pressure of nonmechanical energy over an entire generation, then the net sum of all those changes is zero. It is zero because we will first increase from some initial value up to a maximum, and then decrease down to some minimum, and then gradually end up right back where we first started from. The sum of all those changes is oscillating around some definite value. There will be some average over that time span, which is P’. It is characteristic for that generation and cycle. Since both the distributions and the ranges will be different for different species and populations, the values involved will be different, as also the various rates of change. But if we use a common basis, we can compare them all to the common standard formed by that basis and note their mutual differences.
The constraint in the above mathematical aside is a statement about how biological populations partition their internal energy over a generation. We can express in words as:
A mathematical aside
Our number distribution obeys the rules Newton introduced about infinitesimals. So in any infinitesimal time span dt, population numbers and partitions in internal energy will change by dφ. (Since this is the proportionate change centred around unity, we can also, of course, always express the same change as its absolute value, dn)
A mathematical aside
We already of course know that the population uses Mdt kilogrammes of mechanical chemical energy and components, and Pdt joules of nonmechanical energy over the same interval; and that Mdt nm̅ and Pdt np̅. These facts affect the population's net internal energy, and will become relevant very shortly.
We now turn to the numbers in the dim or middling gray row and column.
As we have done above, we first consider the second ‘:1’ which represents the distribution or deviation of mass and components about the mean, and across the population.
Every biological population is composed of organisms that must take in resources from the environment; incorporate them into its internal energy; metabolize them which is to do work upon them; reproduce the relevant number of partitions in its internal energy; and also eliminate wastes as well as die. This is the mechanical chemical energy aspect of its internal energy, but expressed entirely as an interchange of chemical components with the surroundings. Our prototypical cell has its entry and exit orifices to handle this Mendel pressure or mass flux of mechanical chemical energy, M.
The :1 means, in the ideal case, that the mass distribution is absolutely even all across the cycle. Every increase, and rate of increase, in every individual entity is precisely matched by an absolutely equivalent decrease in every distinct entity at every point in time. But it can also, of course, mean that two populations measure each other as identical.
Our experiment with Brassica rapa did not produce an even mass and mechanical chemical energy distribution when compared to our prototype cell. We instead got a value, relative to that, of 0.427.
We also need a scaling value which establishes the absolute amounts concerned, relative to whatever standard.
Since the average terrestrial cell has a mass of one nanogram, we can use that as a basis for our mechanical chemical energy. A scaling or assimilator value of 1:, in this area, then means that the population concerned is manipulating the same amount of mass and resources, over its generation, as our standard cell.
The real benefit of course comes when we compare real organisms to each other. Brassica rapa is considerably bigger than the average bacterium. It has a scaling value, i.e. relative to our prototypical cell, of 9.248 x 1010.
We can now measure some other species in the same genus, and compare both its scaling and its distribution values to the 9.248 x 1010 : 0.427 we have for Brassica rapa, and see how the two measure up to each other. If the two are the same then we should get 1:1 for them relative to each other. Since we will never make such a measure, we can isolate salient properties and differences.
And … when we take the mass distribution together with (1) the time and (2) the number distributions that we already have from above, we are beginning to put together a clear picture of this organism. Our populations and their organisms are varying their numbers, masses, and mechanical chemical energies at very specific rates, and at very specific points in time, over very specific biological activities. They are doing so both absolutely and relatively.
We now call this column, which governs the quantity of mechanical chemical energy in any stock of biological internal energy, the “constraint of constant size”.
A mathematical aside
We can express the constraint of constant size rigorously and mathematically as . This says that the sum of all the infinitesimal changes that occur in the population's mechanical chemical energy, which is its mass of chemical components retained, are ultimately zero. The population's members will acquire mechanical chemical energy, grow, attain a maximum, reproduce, die, enjoy some minimum value for their mass and mechanical chemical energy, and then gradually return to their originals. As with the constraint of constant propagation, there is a characteristic value for the generation or cycle.
We can express the constraint we derived in the above mathematical aside in words as:
The constraint of constant size of course involves our second law of biology, which is the law of equivalence. We are now discussing both the species' genome and the individual entity genes. The constraint of constant size works at the level of individual chemical components and the bonds they form as they exert and express their mechanical chemical energy. Again by the Avogadro constant and the periodic table of elements, two species or populations cannot have the same relative distributions of mechanical chemical energy without having an internal energy composed of exactly the same amount of substance or moles and masses of chemical components point to point … which is to at all times have exactly the same contribution to internal energy arising from exactly the same numbers and types of molecules.
Every population will have its characteristic distribution of mass and chemical components through its mechanical chemical energy. And since that distribution helps determine the mechanical chemical energy available in this entire column, we give it the Greek symbol κ (pronounced ‘kappa’ to rhyme with ‘rapper’).
A mathematical aside
Just like the number distribution above, the mass and mechanical chemical energy distribution follows Newton's infinitesimals procedure. In any infinitesimal time span dt, the quantity of chemical components will change by the proportionate amount, dκ.
But … this is now of great interest. We already know that the absolute amount of mechanical chemical energy or mass, over the same interval, is Mdt. It will certainly be interesting to see if these two infinitesimal quantities dκ and Mdt differ; or are the same; and if they differ, then how and why they differ … especially since we have already been alerted that Mdt nm̅dt.
And, finally, we turn to the values in the slate gray or darkest row and column in our tensor.
First, there is the distribution or deviator of ‘:1’.
Our prototype cell has four ways of communicating with the external world. It has two orifices and two apertures. The orifices handle the mechanical chemical energy expressed in the mass of chemical components; while the apertures handle the nonmechanical chemical energy of heat, light, chemical reactions and so forth that energize those same components. These are the different modes of vibration molecules have at any one given speed or temperature. Each of the orifices and apertures also has its entry and its exit. That gives us four states or combinations. They are of course the source of the temporal distribution, τ.
And that exhausts the possibilities.
These various rates will have different values at different points in the cycle and will establish different distributions, and so values for τ. Since we want to track all these relative changes and variations, we must measure their interaction.
And we can measure the interactions between the population's mass and energy—and therefore track all its changing entropy as well as the distribution of its energy activities over time—simply by setting the energy flux over the mass flux. This gives their structural proportions relative to each other, and is P/M. This measures the population's complete transformations and conformation in both mechanical and nonmechanical chemical energy at any time. It measures the different modes of vibration. We can then call this P/M which measures that complete set of transformations and interactions the ‘work rate’, W, and express it as watts per kilogramme.
A mathematical aside
As with the distributions of time, number, and mass—τ, φ, and κ—two species or populations cannot have the same work rate, W, without having the same genes and genomes that work in the same ways, which is also the same environment and conditions. This is the Gibbs and Helmholtz energies and their declarations of mass, energy, and entropy. This is how they are each defined. They each measure the interaction and transformation capabilities of systems and phenomena with respect to their surroundings.
A mathematical aside
But there will in fact be times when the inverse of the work rate is more convenient to work with. In those cases, we simply set M/P and call it the “visible presence”, V, because it tells us how much mass and mechanical chemical energy a biological organism must build for every joule of nonmechanical chemical energy it wishes to exploit. Visible presence is measured as kilogrammes per joule. The work rate, W, is then the dynamical inverse of the visible presence, V. There is also an individual work rate, w, an individual visible presence, v; and then an average individual work rate, w̅, and an average individual visible presence, v̅, over the populations, along with their distributions.
We should again also carefully note that when this work rate is used in conjunction with the population's mass, it reveals the complete energy and entropy behaviour, which is the scale and nature of all its interactions with its environment, for that is what energy and entropy measure. It is work, energy and entropy which is genes plus genomes plus ecology. We get that complete coordination from the very carefully defined visible presence, and its dynamical inverse of the work rate. It measures all possible ways in which a population uses its various molecular vibrations to do work, which is to change in its internal energy at any given mass.
Since, in the ideal case, the mass and energy fluxes are completely regular, then the work rate will also increase and decrease completely regularly. Since it is their combination, its distribution will follow theirs and also be ‘:1’. This is another way of saying that the mass and entropy combination is always wellbehaved. In this ideal case, work rate only changes when the two apertures are open. It is unaffected when only the orifices are open.
And as is the way with tensors, we can use any other population as a basis, and then establish a pattern of behaviour relative to that.
We can now measure a real case, as we did for Brassica rapa, and express both a distribution and a scale relative either to the ideal prototype, or else to whatever else we want to measure. This give us an X : Y for this section in the tensor, again according to whatever is our basis.
And since both the mass and the energy vary, then this work rate will also vary and reveal information on exactly how the population increases and decreases both its mass and its energy, especially in comparison to any other. It again gives us pertinent information on the population's mass and entropy, which is its entire suite of interactions with its environment. By all the claims of science, any population that returns a 1:1 relative to a basis is behaving in exactly the same way and is using exactly the same chemical components at exactly the same rates and speeds within the same environment.
Since populations must maintain an equilibrium both in their ecologies in space, and down the generations and so through time, we can call this column the “constraint of constant equivalence”.
Granted that we have decided on the convention of expressing population numbers so that their average value, over the generation, is always 1,000, then we must also express this work rate per the thousand. We can then give this work rate but expressed per 1,000 entities the symbol S. It is simply a scaled up version of the population's average individual work rate over whatever time interval we are working with.
A mathematical aside
Using this (normalized value) for work rate, we can express the constraint of constant equivalence rigorously and mathematically as . As above, this simply means that the sum of all the changes over the cycle is zero, because we return to the same value. And as also with the others, there is a characteristic value for the generation or cycle.
We can express the constraint we discovered above in words as:
(Corollary: prodigious savant are always possible; and even “the walls of rude minds are scrawled all over with facts, with thoughts” Ralph Waldo Emerson).
(Corollary: “Bernard of Chartres used to say that we are like dwarfs on the shoulders of giants, so that we can see more than them, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size” John of Salisbury, Metalogicon, 1159).
As with the constraints of constant propagation and constant size, every population will have its characteristic distribution of work rate: i.e. of energy density and intensity, which is of entropy and so of interaction with the surroundings. Granted that the distribution helps determine the energy and the entropy available in this entire column, we give it the Greek symbol χ (traditionally written as 'ch' and pronounced as ‘kie’ to again rhyme with ‘my’).
We can also now explain the significance of the two symbols we place side by side. Since we have three constraints, all of which contribute to a biological equilibrium and the circulation of a generations, then it reflects a simple problem of ordering amongst those constraints.
If, for example, we adjudge the winner of a race to be the one who is fastest to gather six apples and six pears and six bananas, then how do we determine, during the race, which one is ahead? Is the contestant with three of each fruit ahead of the one with say five apples, three pears, and one banana? Or is the contestant with six pears, three bananas, but zero apples ahead? It is not possible to tell, without further information about the course, the relative difficulty of gathering the various fruits, the likely rates of change, and so the likelihoods of which one will reach the end first. In the same way, it is not possible to assess which of any two populations is equivalent to, behind, or ahead of any other upon the generation length. We cannot tell if any particular time period, dt, or change in mass or in energy, dM or dnm̅ or dP or dnp̅, has, or has not, carried that particular population closer to completing its generation over that span of time. We cannot determine this without some additional information.
A mathematical aside
This chemical configuration or energy conformation distribution also obeys infinitesimals. In any such time span dt, a population's energy behaviour will change by the proportionate amount and infinitesimal amount, dχ. But, as above, we already know that the absolute change in energy is Pdt. We must now determine if the changes dχ and Pdt are the same … or different.
And with this tensor, we are measuring both all the totals and all the distributions in any and all time spans and/or generations, and we can use any one as a basis to express any other in terms of any other to determine their behaviours both absolutely and relatively. This is the power of tensors.
An algebraic and geometric topology based proof.
A vector calculus based proof.
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