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

We have nearly created the structure that will allow us to prove that a population free from Darwinian competition and evolution is impossible. The key lies in using our 4 × 4 Haeckel tensor to juxtapose multiples and distributions, and totals and averages, both quantitatively and temporally, and then to apply basic scientific laws to a population's internal energy.

The key also lies in remembering that when we measure a population's energy, we also measure its total interaction with the surroundings. That is how energy is defined. That interaction is thus its environment and its ecology.

Darwin's thesis is that variations lead to natural selection, which then leads to evolutionary change. He claims that individual transformations endure. Natural selection is then a consistent difference extending over time, both absolutely and relatively. The claim is that a relative difference at one moment gradually extends to become absolute. And … we now know how to measure all differences, both absolutely and relatively.

Figure 17.1

There is a very simple way to look at this situation. Figure 17.1 shows both the outer 4 × 4 Haeckel tensor, and the inner 3 × 3 ‘Owen tensor’, named after Richard Owen. Since we can measure all populations relative to each other; and since we can reversibly and invariantly substitute their measurements for each other; then the properties we have indicated with the no entry signs state those for the ideal population. We can interpret creationism and intelligent design as the declaration that the four coloured squares creating a much smaller 2 × 2 grid can change their values and transmit them directly to the 4 × 4 Haeckel one, but without affecting any of the values in the no entry signs in the 3 × 3 Owen one located between them. In other words, creationism and intelligent design assert that any and all changes in quantities and rates can occur without any necessity for changes in numbers.

The no entry signs in the 3 × 3 Owen tensor that surround the 2 × 2 one represent all situations in which a population is unaffected by numbers. Those no entry signs show that changes in numbers affect nothing in those locations. They similarly show that nothing can reciprocally affect either those numbers, or the population. These invariant properties in those locations are necessary if population numbers have no effect on their surroundings, and the surroundings have no effect on population numbers. There will then be no difference between (a) what the members must all do jointly as a population to establish totals, and (b) what they must each do, as individuals, to make their distinct contributions to that total.

We can prove what we need because we have two different ways of measuring the same thing. We have (a) the two fluxes; and (b) the versatility of our tensor. It is a trivial exercise to make measurements from any population or generation's vantage point, and to measure both whole populations and individual members in each. Thanks to our tensor we can easily change our basis and our standards of measure, and thus state each population in units and quantities relative to the other.

We can tie the different ways of measuring populations together very easily, because the method of tensors gives us the collection of distinct entities and properties within the population. This is what they must each do. But at the same time, the method of fluxes gives us the entire population. This is what they must all do. We will get 1:1 in every position. But we have already defined this 1:1 as our ideal population.

Thanks to the tensor, then if two populations or generations are proposed as identical, we should get 1:1 as their relative measure. The relative measure for the difference between any two identical populations is our ideal population. We measure our common notion: “things that are equal to the same thing are also equal to each other”. This involves their relative temporal distribution, τ.

Time is the fourth dimension. Biological activities involve modest amounts of mass, and so do not have the spacetime bending properties of black holes. They do, nevertheless, induce spacetime differences in activities at different points in the generation. That is a possible driving force for evolution. This involves the temporal distribution, τ. We can assess it with the outermost 4 × 4 Haeckel tensor.

For now, we direct ourselves to the internal 3 × 3 Owen tensor. We therefore eliminate the outermost row and column in the Haeckel one from contention. We have coloured that outermost row and column in the 4 × 4 Haeckel tensor white.

Eliminating the outermost row and column just leaves the three constraints of constant propagation, size, and equivalence—along with their distributions of φ, κ, and χ that are the three columns in the Owen tensor. So the question now is whether those no entry signs really do have no effect, which is what creationism and intelligent design insist is the case.

We should now consider that you, as the reader, are at this very moment reading these words either on some form of electronic device with a monitor, or else on a piece of paper. Either way, it almost certainly has the two orthogonal dimensions of up-down and left-right in front of you. The third dimension, again mutually orthogonal to the previous two, will be into and out of the monitor or paper, spearing directly into and out of your chest and body. These three are traditionally known as the *x*, *y*, and *z* dimensions of space.

We can now look on our three constraints and their distributions in biological space as three separate and “orthogonal” dimensions (i.e. mutually at 90°) whose totals and distributions affect biological populations. Those three dimensions are then:

- Numbers,
*n*, at any given time. - Mechanical chemical energy, measured as
*m*per each individual entity; and as*m̅*as their average along with its distribution. - Nonmechanical chemical energy, measured as
*p*per each individual entity; and as*p̅*as their average plus its distribution.

Those three dimensions will make up any population and will fully describe its internal energy:

- Combining (1) and (2) produces the mass flux of mechanical chemical energy as
*M = nm̅*. This is the number and type of chemical components. - Combining (1) and (3) produces the energy flux of nonmechanical energy as
*P = np̅*. This is the way in which the components are currently vibrating. - Combining (2) and (3) produces either the dynamical work rate
*as W*=*P*/*M*, or else the visible presence as*V*=*M/P*. It states the work being done by each component or set of components.

We can now put our three proposed biological dimensions to good use. They also bring two of our anomalies together: the distinction between

- the mechanical and the nonmechanical chemical energies'; and
- the individual/average values and the totals.

The three dimensions we have proposed for biology are orthogonal. We also measure each one by finding its average, and using that as a basis for measure. That average therefore acts 'one' or unity for each dimension. Everything greater than one is greater than average, and everything less than one is less than average. This means that our three dimensions also have an “orthonormal basis”. They follow all pertinent rules. They behave exactly as do the three dimensions of ordinary physical space.

Using the three dimensions of space as our basis for argumentation, then creationism and intelligent design are claiming that biological populations can move freely in the two dimensions that keep them flat on a monitor screen or piece of paper, and so similarly on the *xy* plane you are reading this on. The supporters of that idea insist that biological populations never engage in the motions that would lift them up out of, or push them into, the *z* dimensions of the monitor or piece of paper, which would be a change in numbers. So … all we now have to do is find some way to take some measurements to see if that two-dimensional proposal of independence from the third dimension of number has any merit.

Since we now know what we are looking at, all of this is becoming so easy to measure. We have the constraint of constant propagation.

The constraint of constant propagation and its distribution of φ help highlight population numbers by determining the relative number of partitions for internal energy. They establish the distribution of biological energy in the third dimension of biological space. They tell us all changes in numbers over the entire generation both relative to a population itself, and to any other we care to compare it to. If a population's overall behaviour is never affected by these changes in numbers, then when compared to itself it will behave exactly like our ideal population, whose numbers also never change. We have a base line for comparison, and also know the result that we must find. We must find the ideal population.

The constraint of constant size and its distribution of κ together determine the relative scale and entropy of all resource-based interactions with the surrounding biotic and non-biotic environment. They help govern the uptake and elimination of all components and so the size of the mass flux, *M*, which is the mechanical chemical energy. They also, as we have seen, play a vital part in determining a population's entropy through its genome and its genes. It is one of our biological dimensions.

In the same way, the amount of energy the population absorbs and/or emits at any time depends upon the chemical conformation, or overall energy density, which is the constraint of constant equivalence, along with its relatively stated distribution, χ. These are the behaviours of nonmechanical chemical energy which in their turn help govern the quality and quantity of the Wallace pressure or energy flux, *P*, the complete chemical behaviour of those same genomes and genes. It is another of our biological dimensions.

If a population is going to be free from the effect of variations in numbers, then as again in the no entry zones in Figure 17.1, no population may divert any of the energy it uses for either mass or chemical configuration into the third dimension of numbers. This must not happen either absolutely or relatively. It must not happen when any population is compared to any other. If creationism and intelligent design are true, then no biological entity may compete by growing a little faster simply because resources have been made available due to losses in numbers, and so due to decreasing competition. To compete is to show a variation, which is then a value in an off-diagonal element in the tensor. It is a change that has only happened because of numbers.

As in the Figure, the constraints of constant size and equivalence may freely exchange and interchange with each other, but neither may interact with the constraint of constant propagation. Those are no-entry and no-go phenomena. They are both absolute and relative. We can compare populations across space, and generations across time to ensure that this is so.

We are surely in business.

We borrow a leaf from Sir Isaac Newton. We know from him that we can measure things by considering an infinitesimal increment of time, *dt*.

We are now moving into the territory in which science reigns supreme. Measuring exactly these kinds of infinitesimals into and out of the surroundings—which is energy and entropy—is a triviality. Our tensors can measure the rates and amounts at which biological populations thrust themselves into their surroundings. These biological issues are becoming no more problematic than finding out how far east, and north, and vertically something has moved, which are the *x*, *y*, and *z* dimensions of physical space.

So let us now do as Newton suggests. Let us see what our population does within its surroundings over an infinitesimal period of time, *dt*, and as it impacts its ecology as measured both as a unit of time, and a span of the generation length.

As Figure 17.1 indicates, each of our constraints, which are our dimensions, will now change across that infinitesimal time span *dt* by the infinitesimal and proportionate amounts *d*φ, *d*κ, and *d*χ respectively, where these are the infinitesimal changes in numbers, mass, and energy, but always expressed as proportions of the current population values.

We are well-used to making measurements in three-dimensional physical space, and measuring and determining the infinitesimal increments *dx*, *dy*, and *dz*. We are now determining the infinitesimal biological transformations of *d*φ, *d*κ, and *d*χ in our equally three-dimensional biological space. These will be both relative and absolute, because we will always have some basis on which our relative values are being stated, even if that basis is busy changing along with the values we are measuring. An organism's mass, energy, and numbers are singularly easy to track for they are all interactions with the environment. And that environment—those surroundings—is exactly where we measure.

We simply have to do what science requires. We simply have to take a few measurements, in the surroundings, and see if any infinitesimal change in numbers, *d*φ, that we might measure always has a zero effect, again in those surroundings, and as creationism and intelligent design assert.

It is obviously going to be a mouthful to write out *d*φ, *d*κ, and *d*χ every time. It is much too inconvenient to keep saying ‘the proportionately infinitesimal increments *d*φ, *d*κ, and *d*χ’. Since those incrementals are any change whatever in our biological population and tensor, we are going to call their combination the **biological potential** and give it the Greek symbol μ. The biological potential is simply an entire population's infinitesimal and proportionate change in its energy over some infinitesimal period of time, which is itself some proportion of a generation length. We record that change as a part of the biological potential, no matter what its source: numbers, mass, or energy.

All we are now saying is that we can think of all those distributions of φ + κ + χ together. The whole then infinitesimally increments, so that μ = *d*φ + *d*κ + *d*χ. So any change in a biological population now happens because of its biological potential, μ. We can then track down exactly what has contributed to any specific change by measuring things before and after, for we have both absolute and relative measures of:

- numbers;
- the mechanical; and
- the nonmechanical energies.

We will be able to track down what is being caused by what anomaly because we can measure in any and all dimensions.

The process is very simple. We shall run an experiment. We shall measure a population. We shall measure its values of *d*φ, *d*κ and *d*χ at every point. That is our biological potential, μ. We can then look at the results and see which theory is correct: Darwin's, or the creationists' and their confrères who instead favour intelligent design. If the latter are correct, then we will measure *d*φ = 0 at all times. The values in the rows and columns with no-entry signs will be either constant or zero. And if creationism and intelligent design are incorrect, we will have *d*φ ≠ 0 overall. This is a nonzero value maintained for at least some part of every cycle, meaning all tensor rows and columns will get a value.

And now we have used our tensor to master the first way of analysing changes in our population, we can turn to the second, which is directly with the Mendel and Wallace pressures, or the mass and energy fluxes.

A mathematical aside

We have in fact already met the conjoined commodity φ + κ + χ. It is the sum of the mass and energy fluxes for a specified number. It is the average individual work rate, *w*, scaled proportionately to 1,000, and to which we have given the symbol *S*. Its infinitesimal increment, or rate of change, is therefore the biological potential, so that μ = *dS*.

- CONTENTS

- Introduction
- 1: Setting the Scene
- 2: The Requirements
- 3: Reproducing the Cycle
- 4: The Sun and the Cosmos
- 5: The Earth
- 6: The Ecology
- 7: The Biological Entity
- 8: The Genome
- 9: The Energy
- 10: Existence and Equivalence
- 11: Physiology and Metabolism
- 12: Death and Dissipation
- 13: Reproduction
- 14: The Anomalies
- 15: Measuring the Anomalies
- 16: The Basis
- 17: The Biological Potential
- 18: How to Measure Variations
- 19: The Refutation
- 20: The Proof
- 21: Conclusion
- Biography

An algebraic and geometric topology based proof.

A vector calculus based proof.

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