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
Please note that parts of this chapter use the theory of tensors to comprehensively refute creationism and intelligent design. For that reason, some of what follows is a little more mathematically oriented than is the material in the earlier chapters. Please therefore be sure to read ‘Before We Begin’ before proceeding. You will familiarize yourself with some background, and acquaint yourself with the general strategy we pursue here (e.g. showing that if x + y = x - y then at least one, but more probably both, of x and y will be 0; and that if xy = y⁄x then at least one, but much more probably both, of x and y, will be 1 etc.).
We now have a cast iron method—our first one—for measuring the variations amongst the biological entities that make up any population (see How to Measure Variations). With that in hand, we can soon set about proving what we need.
Our first method for measuring variations rests on the difference between the linear planimeter we see in Figure 19.1, and the polar one we see in Figure 19.2. The linear planimeter goes all around the boundary of whatever area or sector we want to measure. It will start at a given point which will be some distance, x, away from our x-axis, and also some distance y, away from our y-axis. As it travels around the boundary, it will always move some infinitesimal amount in either x, or y, or both. The essence of the linear planimeter is that no matter how circuitous the boundary might be, the sum of all those infinitesimal distances come together to make a rectangular area xy. That is the area we have surrounded.
The polar planimeter works on a slightly different principle. It has one central fixed pin around which it rotates. The tracer arm then goes all around the boundary. By measuring the angles, we will eventually know when we have been all around the circle. We can also measure the radius as we go. That is simply the straight-line distance between the central pin and the tracer. When we have completed the circle, we will have infinitesimally incremented and decremented the radius. It will have varied up and down around some average value. We can now get our bounded area from that circle.
We get both the angle and the radius for the polar planimeter by measuring y and x. We get the angle from y⁄x, which is the tangent. We get the radius from the average differences we measure in x or y, be that as mode, mean, or median. All those infinitesimal changes in x and y give us both the complete 360° for a circle, and the radius.
The linear and polar planimeters may use different methods, but they give us the same value for our surrounded area. There are, however, important differences in how those areas are incremented at each point. One is always working with x + y and x × y, while the other is always working with y - x and y ÷ x. These two planimeters are therefore very unlikely to increment their areas or their boundary lengths in the same ways. We shall eventually prove that those differences—which are the curl—are the source of all variations; of all fitness; of all natural selection; and of all evolution.
We shall start, in this chapter, by refuting creationism and intelligent design. We first prove that the following are inevitabilities for any population proposed as free from all influences of changes in numbers:
We will also derive four equations to make our case. Then in the next chapter, 20: The Proof, we will:
The refutation of creationism and intelligent design is fairly straightforwards. It uses our linear and polar planimeters. Both energy and entropy are defined through very similar processes that also link boundaries to areas. The refutation of creationism and intelligent design therefore rests on the pairs of inevitabilities embedded in the following converse processes conveniently enshrined in the linear and polar planimeters:
We obviously have to separate the mechanical and the nonmechanical forms of the chemical energy that makes up the internal energy of biological entities and populations. We must also contrast the method of fluxes (what the entities in a given population must all do conjointly with their internal energy) with the method of tensors (what they must each do separately, and with the distinct internal energy allocated to each entity or partition). This sets the linear and polar planimeters against each other. Their relative rates of change as they each travel about their boundaries are the different ways in which they each increment:
The difference between the linear and the polar planimeters is that one is highly sensitive to differences in times and totals, while the other is highly sensitive to distributions and to changes in numbers.
A mathematical aside
When we measure across any infinitesimal time interval, dt, the population will certainly use mass and energy. We can calculate that mass and energy in two distinct ways:
We may be primarily interested in reproduction … but that still requires energy. Force and energy, mass and distance, all involve the differences between boundaries and areas. If we want to understand biology, we must properly understand both energy in general, and internal energy in particular.
The modern understanding of energy and entropy—and so of the chemistry and the changes and configurations in internal energy that drive biological organisms—began in earnest in November 1680 with the discovery of Kirch's comet. It was the first to be discovered by telescope, and one of the brightest of the seventeenth century. As in Figure 19.3, we can measure any such planetary body with our linear and polar planimeters. Our linear planimeter goes all around the outside and measures the length of its journey as a boundary, while the polar planimeter sits centrally at the sun which is the centre of its orbit, and measures it as a procession through its areas covered.
Kirch's comet, and the matter of its orbit, immediately raised the issue of energy. This involved both its boundary, which states its velocity; and its area, which tells us the energy it is expending in moving about the boundary. This latter, being the product of an area, always involves the planetary motion we first met in Chapter 9: Energy.
As we learned from reviewing the falling rock in Chapter 9, all energy, whether it be mechanical or nonmechanical; and whether it be biological or nonbiological; is strictly defined, and quantified, as a mass moving through a specified distance in a specified gravitational field … which is the archetypical planetary movement. By the first law of thermodynamics, all energy is equivalent to, and is defined by, that falling rock in conjunction with a heat-resistant or adiabatic wall. Carathéodory showed us how to define energy as the commodity created within a gravitational field, by a falling rock, always behind an adiabatic wall. Wherever energy exists, we can always measure it with a combination of a falling rock and our linear and polar planimeters. That is the conservation of energy. There is no other definition. There is no other meaning. If one exists, then the other does.
Given these definitions, then we cannot understand biology or unravel evolution without a prior understanding of the first great victory of the modern scientific age. Work is a conversion. Work is any conversion of energy from one form to another. Heat is any movement from one temperature to another. Entropy is the reconfiguration and transformation any stock of internal energy necessarily undertakes in providing either heat, or work. There are again no other definitions. Biological populations and transformations are not exempt.
As in Figure 19.3, all planets and galaxies—all of whose movements and dispositions create all time, all space, and all energy—are all orbiting about each other. Therefore, any change in a biological population's energy immediately means it has exploited a change of the kind defined and quantified by the consequences of the laws of planetary motion embedded in the first law of thermodynamics.
Scientists of the era of Kirch's comet had almost universally abandoned Aristotle's views on circular motion and falling bodies in favour of Galileo's concept of linear motion: his ground-breaking realization that the “natural thing” for any object to do if left undisturbed is to continue charging along in its same straight line (or else, if it is at rest, to blithely continue remaining at rest). They also knew that the interactions of comets and planets obeyed Johannes Kepler's three laws of planetary motion, but nobody knew why. Galileo's admittedly superior template for motion did not provide an answer to the riddle of the orbits Kepler had created: i.e. the new rules for the constantly changing motions of the comets and the planets, given Galileo's new insight about the primacy of non-changing or linear motion. It certainly did not provide a basis for energy; and nor did it explain Kepler's laws. And without a cogent explanation for Kepler's laws there is no foundation for the energy and the work a planet does while it orbits, which is force × distance, or W = Fd.
As on the left of Figure 19.4, Kepler's first law, announced in 1605, is his ‘law of orbits’. It involves the linear planimeter—i.e. a set of x + ys and xys—and states a boundary. It states that the arc length or boundary formed by an interacting planet is an ellipse, with the sun standing at one focus.
And as on the right of Figure 19.4, Kepler's second law, announced in 1602 (sic), is his ‘law of areas’. It involves the polar planimeter—i.e. a set of y - xs and y⁄xs—and states an area. If each orbiting planet is connected by the spoke of an imaginary wheel to the sun placed at its focus, then each of its increments covers an equal amount of area in each unit of time.
Kepler's third law, announced in 1619, draws the two planimeters, the boundaries, and the areas of the first two laws together. It is colloquially known as ‘the 3⁄2 rule’. It says that the cube of a planet's average distance from the sun, d3, varies with the square of its orbital period about the sun, t2. There is an intuitively close link between d3:t2 and d2:t with the latter being Newton's famous ‘inverse square law’ based on his discovery of universal gravitation. So although Kepler's 3⁄2 rule made it look tantalizingly as if there was indeed some kind of d2:t or inverse square interaction between a planet's distance from the sun and its period, nobody could initially figure out what that was, or why.
The solution came in late 1680, or early 1681, when the first English Astronomer Royal, John Flamsteed, visited the 37-year old Isaac Newton, then the Lucasian Professor at Cambridge University, and posed the question of the orbit of Kirch's comet to him as a problem. Newton promptly solved it, but did not publish.
Four years later, in August 1684, Edmond Halley also visited Cambridge to discuss Kepler's three laws of planetary motion with Newton. And when Halley asked Newton what paths he thought comets or the planets would describe under an inverse square proposal and/or Kepler's laws, Newton immediately answered that they would follow ellipses for he had already used his infinitesimals method to calculate the exact path of Kirch's comet. He had therefore already calculated the infinitesimals sums of all the distances that a force would push that planet along the boundary which is the work done by the force, and a statement of energy as W = Fd. Newton had also realized it is an area, and had shown that the areas and the boundaries are equal. This is the equality between the linear and polar planimeters, which is the equality between x + y and xy on the one hand, and y - x and y⁄x on the other.
A mathematical aside
If we turn to our linear planimeter and consider the two dimensions of x and y, we can conceive of the force, F, as divided into the two separate orthogonal components F1 and F2. Each acts independently along x and y, together resulting in F, so that F = F1 + F2. The infinitesimals sum for the force is now formed by the variations in distance in both x and y. The resulting boundary is the work done as in . (This is Green's theorem). We see that it has components that (a) require that we integrate; (b) multiply together; and (c) add together before we can determine the value. We have already made sure that our tensor contains all three processes; and also their converse. This is going to have some dramatic consequences, which we shall get to very shortly.
Biological internal energy clearly involves the polar planimeter. A generation is effectively a circular movement. We start with some progeny; they grow; they become progenitors; and they then produce more progeny, which do the same. This is a circulating distribution of biological internal energy.
Since the polar planimeter's radius oscillates in its value as biological populations go through their various transformations, that raises the distinct possibility that numbers will increase and decrease as population members are lost and gained. A loss in numbers in a biological population always means a loss in molecules, which is a loss in energy. The very best that can then happen is that the combined population and cosmic entropies stay the same. In any real situation, however, it will irreversibly increase. And if a population loses in energy in this way, then it must recover that energy or go extinct. The polar planimeter can measure the extent of those variations as its changes in radius all about the circulation.
The linear planimeter differs from the polar one by instead telling us the totals involved. We must properly understand the interactions between these two, and so between force, mass, and distance as they create and drive all energy. These again involve the converse processes of x + y and x - y; xy and y⁄x; and ∫y dx and dy/dx.
If the polar planimeter moves so that its overall change in radius signals a loss in population numbers, then it must eventually move in a converse way and reflect the population making up that loss. The population must recover the energy and resources lost, which is also the total area the linear planimeter is covering. Since the linear planimeter is measuring the total population while the polar one is measuring the individual entities, then the polar planimeter must somehow reflect the changes that each individual entity must go through so the area is recovered. And that curl—which is the difference between the two and so the recovery that the polar planimeter can indicate—also requires energy.
Figures 19.3 and 19.4 confirm that when we track the movements of either a group of molecules or else some cosmic body such as Kirch's comet, we can use x and y as our two dimensions of space. But since we can use either a linear or a polar planimeter to study them, we will measure a boundary and area as well as an angle and a radius. These are equivalent.
Whether the movement in mass that defines any particular quantity of energy happens (a) macroscopically and mechanically, or (b) microscopically and nonmechanically, it immediately establishes both (1) a line or boundary along which, or about which, the objects have moved as they have travelled through space; and (2) an area they have described. The objects, be they macroscopic or microscopic, move through time and space, and so in the x and y dimensions. This is:
These are the converse magnitudes, changes, and rates of change of y with respect to x, and of x with respect to y, as they each go through what eventually becomes the circulation of the generations.
Angles can be measured in a variety of ways. As Figure 19.3 shows, one of those is as a proportion or ratio between sides. An angle's tangent is measured by the ratio y/x. Therefore, whether our linear planimeter is measuring a planet or a molecule, if it either moves in either x or in y, i.e. relative to the other, then it describes an angle.
We can obviously describe our generation using nm̅ and np̅ as totals for the fluxes from the linear planimeter; or else as m̅/n and p̅/n which is a set of angles from the polar one. Whenever the polar planimeter moves through an angle, there is some corresponding linear movement in x and y. That movement immediately describes an area, which is xy. When the changes in x and y have gone through their complete orbit, or set of changes, we have described a circle, or a generation. We have:
We know, from our Brassica rapa experiment, that generation lengths vary as the sizes and the energies of the organisms vary. We can measure the mechanical chemical energy aspect of internal energy, which is the mass flux, M, using the two dimensions of m and n. We can also use the two dimensions, p and n, to measure the nonmechanical energy aspect of the same internal energy as the energy flux, P. Their joint changes describe our circulation of the generations using both boundaries and areas.
But if we set either mass or energy against number as m̅/n or p̅/n, then the resulting angle and tangent stand in for the proportion of the generation length we traverse. That angle states a proportion of the cycle. It is a size of relative sector. And since it is also a temporal distribution, then we are stating a set of biological activities, as well as the periods of time for which the transformations last. This is our temporal deviator, τ. So if m̅, p̅, or n change, we have automatically traversed some proportion of the generation.
The polar planimeter tells us what stage of the circulation we have reached as we are producing that flux. When we have gone through a complete set of changes or as a cycle or an orbit, we have also gone through a complete set of changes in m and n and/or p and n. We have gone through a complete set of configurations in both the mechanical and the nonmechanical energies. The two complementary sets nm̅ and m̅/n, as well as np̅ and p̅/n have gone through their respective cycles. The former is the set of areas, the latter the set of angles, but they have produced the same result.
The location in a biological generation—given by m̅/n and p̅/n—is both an angle and a proportion. But it is also a temporal distribution, τ, through the internal energy stock. It is telling us how long the entities take putting on mass, and/or taking on energy, both of which are changes in internal energy. That is to say, the laws of planetary motion which incorporate Newton's laws of motion are also applicable to the configurations of microscopic objects, for that is the cycle being measured. Since we have m̅/n and p̅/n, they confirm that the time distribution, τ, depends inversely on n.
The fact that we can express the proportion of a generation as a ratio or angle between mass and number has some very important consequences. The size of the relative sectors of a generation depend on m̅/n and p̅/n. If we look at the n, then a decline in n is equivalent to a change from 1⁄3 to 1/2. A decrease in population numbers is an increase in proportion. A decline in numbers is therefore an increase in the overall generation time that the remaining members must expend to restore the full total in internal energy. And if we look at the m̅, then an increase is something like a change from 3⁄5 to 4⁄5. So if mass increases, then the angle again increases, which is an increase in the proportion of the generation time that the population must spend on an activity. The entities spend more time putting on mass or energy. These are both variations. They are both also statements of energy. They automatically involve areas, which are changes in position all around the boundary. Mass and numbers vary inversely, but they both contribute positively to the mass flux. We can measure their relative rates with our partial derivatives.
We now turn to our biological population with its internal energy, which brings together both the mechanical and the nonmechanical chemical aspects. We can straight away notice that if we want to determine the value of both the mechanical and nonmechanical chemical energy fluxes, we need to (a) multiply; (b) add; and (c) use Leibniz's and Newton's infinitesimals.
The mass flux aspect of internal energy immediately depends upon the proportionate and multiplicative changes in both mass and number, which are then added to each other. We can see this from the two components of m and n and p and n respectively. These give the mass flux of: . If creationism and intelligent design are correct and populations are unaffected by numbers, then the multiplicative n dn portion should always have zero effect. There must never be anything to add to, or to affect, changes in mass. We should therefore be able to measure all ongoing changes in mass and show they are independent of all ongoing changes in numbers. Since we have our three dimensional biological space of numbers, mass, and energy, these are certainly claims we can measure and test, for we can easily measure those numbers. It is nothing more than a dimension.
Newton eventually formalized Galileo's original doctrine of straight line motion as his first law of motion: “Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon”. Newton also coined the word ‘centripetal’ to describe the force he described. He took it from the Latin centrum or centre, and petere, to seek out. As he put it, “A centripetal force is that by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre”. Both macroscopic planetary bodies and molecules have masses. They each exert forces upon each other, at their respective scales of operation.
Newton had thus already determined that any first body that exerts a suitable centripetal force upon a second will pull that second body towards it. It will not allow that second body to keep charging straight ahead in the way Galileo had suggested. By Newton's second law of motion, the first body will instead use its centripetal force to compel the second one to change its direction of motion—which is to accelerate—inwards towards it: “The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed”. Anything else would be for that second body to pretend that the first exerted no force upon it so it followed the first law provided by Galileo. Since these two laws hold as much for the first body acting upon the second as they do for the second acting upon the first, then they exert mutual attractive centripetal forces that draw them into an orbit about each other. This is then a temporal distribution of forces and motions which produces the equality of areas and times of Kepler's second law. The area is formed from the movements of the bodies. This interaction is again the origin, and the definition, of all the energy declared in the first law of thermodynamics.
A mathematical aside
We now turn to our polar planimeter. The centripetal or inpulling force, F, acts at an angle to the body, causing it to infinitesimally change in its direction, through being pulled inwards, which is to change in its velocity, dv. Newton gave the force's magnitude as F = Gm1m2/r2 where G is Newton’s gravitational constant, m1 and m2 are the respective masses, and r is the radius or distance between their two centres. This gives the infinitesimal change in velocity d(mv)/dt . But since a change in velocity, dv/dt, is an acceleration, a, this gives Newton's second law: F = d(mv)/dt = m(dv/dt) = ma.
As is well known, Newton originally had no method to calculate all these planetary movements, so he invented one. Although Leibniz had also had the idea, Newton originally invented his method for the express purpose of solving these kinds of problems. As is also well known, Halley was astonished that Newton had a solution to the problem he had posed. He asked to see Newton's calculations … only for Newton to say he could not find them, but would redo them and send them later. Newton's recreation of his work for Halley became his treatise On the motion of bodies in an orbit. Halley realized its importance and Newton was eventually persuaded to expand it into his three volume Philosophiæ Naturalis Principia Mathematica, the Mathematical Principles of Natural Philosophy, which Halley published at his own expense. Generally referred to simply as Principia, it helped make Newton's infinitesimals method, better known as the ‘integral and differential calculus’, available to the world community … such as to Rudolf Clausius, who some two centuries later applied it to internal energy and (without realizing it!) to molecules; explained all nonmechanical energy and its connection by conversion to the mechanical variety; and proved the existence of entropy, which again deals in boundaries and areas. It is thus the foundation of both the second law of thermodynamics and all biological variations.
Energy may be the property of an interaction, but Clausius proved that there are three, and only three, possible kinds of systems and molecular interactions for all brands of internal energy. They allow us to make accurate real-world measurements. Two of them are impossible in real life, but they nevertheless clarify all the issues internal energy raises. They link macroscopic planetary movements to microscopic molecular ones.
Carnot was the first to spot something that now seems extremely obvious, but that no one had spotted before him. As Figure 19.5 shows, we can measure his discovery with our very useful linear and polar planimeters.
Carnot was the first to determine that heat has a favoured direction. It always goes up on the left-hand side, and down on the right-hand side of Figure 19.5. All thermal bodies in other words emulate gravitational ones. Carnot thus discovered that all forms of energy, including the internal energy in our biological populations, behave like the gravitational variety.
The right-hand side of Figure 19.5. shows that hot bodies will spontaneously ‘fall’ down a temperature gradient and so become cooler. However, if we want them to rise up a temperature gradient, as on the left, then we must use a fire or other source of heat. That becoming hotter is not a spontaneous event. Cold bodies will not increase their internal energy by rising in temperature—which is to move up-gradient and become hotter—unless they are provided with an energy source. That energy source must augment their internal energy. It must drag them from the colder temperature all the way up to the hotter one. Unlike the fall in temperature, that is never spontaneous. This fact that some events are spontaneous or happen easily, while others are not and so require effort and work, is critical to the working of biological organisms and populations.
Carnot not only discovered the above fact that hot bodies can spontaneously diminish their internal energy stock and cool down, he highlighted the importance of the additional fact that as a body loses in its internal energy, through these kinds of heat energy transactions, it undertakes a very important change in state. It ever more steadily approaches the temperature of whatever colder body is sucking its heat energy out of it. Just as a falling gravitational body always heads for some specific target, which is the centre of the attracting mass; so also with a body that is cooling down. It is always heading for some specific temperature. It follows a gradient and heads straight for the coldest cold body around. We now know that this is because the molecules decrease their rates of vibration. But this simple fact that hot things always cool down, and which now seems so obvious to us, was not properly known before Carnot discovered it.
Carnot also discovered that when the body undertaking the heat cycle he described enters the other side and interacts so as to move up-gradient and gain in its internal energy, it steadily departs the temperature of some cold body. It approaches the temperature of whatever warmer body is now heating it up. Carnot therefore showed that temperature is akin to gravity. It is an attractive power. It has the power to transform any stock of internal energy. Heat is a form of interaction that internal energy undertakes when driven by temperature.
Carnot also showed that these heat interactions that internal energy will undertake is equally subject to boundaries. And as Newton had done before him for celestial bodies undergoing gravitational changes in state, Carnot gave a method for calculating the relevant thermal changes in state that stocks of internal energy undertake about their boundaries. A given quantity of internal energy acting thermally is declared by its boundary, which is its line of action. We can measure it with our linear planimeter. It is the set of a body's infinitesimal changes in state as it heats up and/or cools down, each multiplied by the temperature at which that change in its internal energy occurs. We simply sum over the indescribably vast collection of microscopic objects composing that internal energy. All the molecules involved are busy obeying Newton's laws of planetary motion upon that microscopic scale.
Carnot had proven, with his calculations, that a steam engine, filled as it is with internal energy in the form of nonmechanical energy, could only, and would only, convert and provide its external and useful mechanical work in that part of the cycle where its temperature fell from a high to a low point. He proved that internal energy did not, and could not, convert itself and provide its capacity for mechanical work and power in the part of the cycle where the temperature was rising. Indeed, some mechanical work would instead have to be done, upon the system, to return the piston to its original position. It was simply that the external mechanical work that would emerge later, in the down stroke, and when the temperature fell, was very much greater than the mechanical work that had to be input to restore it to its original position in the upstroke of the heating cycle. There was very much more energy available for exploitation in the ongoing conversion of internal energy from the nonmechanical to the mechanical as temperature fell than had to be put into the mechanical restorations of pistons that the steam engine required. That ability to mechanically exploit the nonmechanical is what fuelled the James Watt steam engine and the Industrial Revolution … and those conversions of internal energy from one form to another—all of which are work—is also what accounts entirely for biological variations and their circulations of the generations.
Carnot might have showed that it was not possible to get work from the steam engine at any other point than the cooling down stroke, but he did not give what was needed: a reason. In modern language, why would nonmechanical internal energy only covert itself into the mechanical variety across a temperature drop, and not across a temperature rise? It was left to Clausius to state the reasons.
Clausius did not have molecules to assist him, but we now know that the entropy he discovered is an intrinsically molecular interaction. He produced his theory about the unique behaviours of internal energy in 1850. He announced entropy as the reason for the phenomenon Carnot had noted. But his explanation was particularly difficult to grasp because the existence of molecules was still disputed. Without molecules to assist him, his reasoning was very abstract.
Carnot might have discovered that internal energy—and so all energy, both mechanical and nonmechanical—has a boundary; a favoured direction; and a line of action; but it was Clausius who discovered why it had that favoured direction. Clausius also discovered—as had Kepler before him although in a very different arena—that energy had an area and a scope of operations.
Clausius proved his case by dividing all possible interactions into three broad types. They also cover all biological organisms and cycles. Those three are: (A) the ‘isolated’, (B) the ‘closed’, and (C) the ‘open’.
Clausius' three general systems all need the same impossible and unrealizable conditions. They define the scope and scale of operations. His first two help us understand why a population free from Darwinian competition and evolution is impossible.
The first thing Clausius did was define spontaneous, which we can understand as an intrinsic pattern of behaviour. His isolated system achieved this by establishing two impossible conditions. It has both (a) an adiabatic wall that heat cannot breach, and (b) refrains from all exchanges of either mass or energy with the environment. The only thing it is capable of is entirely internal changes in state. This allowed him to establish a pattern for how all systems would behave intrinsically, and entirely because of their own internal natures. Creationism and intelligent design are impossible because they confine biological organisms to these impossible conditions.
Even though Clausius did not have molecules to work with, he proved that an isolated system—which has no connection of any kind with the external world—can still keep changing in its state. As an isolated system it therefore provides a template of behaviour.
Since we now have molecules to assist us, we can express the premise behind Clausius' isolated system by saying that even the molecules of a gas contained in an isolated system, and that comprise its internal energy, will obey Newton's laws of motion. They may be behind an adiabatic wall, so that they are not engaging in heat interactions with the surroundings; but they are still vibrating. They keep colliding with each other. Even a gas in an isolated system will keep colliding its molecules until its average configuration energy, across its extent, is evenly distributed everywhere. All its molecules will then interact equally with each other throughout its available space, in each unit of time, favouring none. Molecules will thus collide until they are evenly distributed across space and time. The search for this state is the transforming power that, as in Figure 9.2, drives a rocket.
An isolated system tells us that if it contains a gas that is initially confined to one corner, that gas will eventually—and spontaneously—interact so as to evenly diffuse throughout the available space. That change in its condition, driven by its intrinsic molecular behaviour, keeps both its internal energy and its mass the same, but redistributes that internal energy evenly across its extent. This spontaneous inevitability of molecules is what entropy measures. It explains energy's behaviour. It is also the source of Darwinian and biological variations.
Clausius thus confirmed that even though an isolated system has no contact or interactions with any other system anywhere in the universe, it can keep its net stock of internal energy constant, yet still somehow interact internally. It can undertake intrinsic changes in its state, behind its adiabatic walls, as its molecules strive to achieve an entirely internal redistribution and equilibrium. His concept of an isolated system therefore tells us how a system would spontaneously behave in any situation. This is again the foundation of the second law of thermodynamics.
The isolated system was the first of the three systems Clausius enumerated. The other two he presented, the closed and the open, differ from his isolated one because the walls are no longer adiabatic. Both can exchange energy with the surroundings. We can nevertheless observe the distinction between them by referring back to our rock-based energy system of Figure 9.1 which had an adiabatic wall and so which presented the isentropic set.
The rock we used in that system successfully delivered mechanical energy. If we drop the rock on it it compresses and gains energy. If we then turn it upside down, the rock will fall off it and we can do some useful work as it loses the internal energy it just gained and cools down. This is the basis of the steam engine that Carnot investigated. We take advantage of gravitational potential to achieve some designated purpose.
Whether its transformations prove useful or not, the system's temperature falls as its volume increases and it returns to its previous state. It loses in its internal energy (and also power) through that temperature fall, which is an internal response to its externally imposed event, and a reflection of the average speed of its molecules. The rock institutes a conversion from mechanical energy to the nonmechanical variety. It thereby augments the system's internal energy.
The rock in this system is entirely external. It acts in and from the surroundings to produce those entirely internal changes in state. This is a closed system for it has a constant number of molecules. The system is closed because the rock's mass, which certainly affects it, does not cross the system's boundary and change its mass. By the same token, no mass or inertia internal to the system leaves. No mass crosses the system boundary. The system affects, and is affected by, the surroundings, but not via any direct changes of material substances. The system can exchange the effects that impinge upon it and its internal energy via that external mass, as (a) work and (b) the nonmechanical energy of heat, which is an exchange between temperatures. These arise because of the rock, but the system itself at all times maintains the integrity of its boundary. It keeps the rock externally to it. The system does not exchange mass itself. It simply exchanges the effects of its internal energy.
A closed system, such as this rock-based one, makes it clear that any system can spontaneously lose in its stock of internal and nonmechanical energy for two reasons. As well as (A) the entirely internal transformations of collisions between molecules open to the above isolated system, it can additionally (B) undertake transformations, such as when we uplift the rock, where it loses in its stock of internal energy and so changes in its temperature. This is a heat and thermal transaction that affects the rate or speed at which its molecules move.
Clausius' isolated system establishes what spontaneous is. It helps us to see that a fall in temperature is a spontaneous event, relative to those surroundings, which are absorbing the heat energy given off via that temperature fall. If we now put the rock back on the closed system, it will both confine itself into a smaller space and rise in its temperature, which is not something any isolated system will ever do. This is not, therefore, spontaneous.
Now we are clear about closed systems, and about their potential to be the source of all Darwinian variations, we turn to examining the third and last of the interactions Clausius elaborated: the open system. Mass and molecules can now breach the system walls and depart. And when they depart, they reduce the system's internal energy while simultaneously causing a net increase in the molecular motions within the surroundings. The system once again loses in energy and power.
Both closed and open systems can change in their entropies, but only the closed one has the option of isentropic changes in state. These are not available to any open system which always has boundaries that are porous not just to molecules but to the heat losses molecules carry with them. No closed system is perfectly closed. Thus an open system can change in its entropy—which is its molecular numbers plus their configuration—because molecular energy is being transferred into or away by molecules. And … this very important distinction between the isolated and the closed systems upon the one hand, and the open one on the other, lies at the heart of Darwin's variations and his natural selection and evolution.
Isolated, closed, and open systems state the complete set of possible interactions and transformations for internal energy, and so for all objects and systems … including the biological. They help us pin down what is, and is not, spontaneous.
Clausius has now given us an alternative and area-based method for measuring nonmechanical energy and its heat-based or thermal changes in state. It partners the boundary-based one of Carnot's. As is the area in Figure 19.4, the area in Figure 19.5 is the total work a system can do with its internal energy as (a) it non-spontaneously undertakes a heat interaction and rises in temperature on the upstroke, so its molecules can increase their net stock of internal energy, and as we input some work and/or heat energy; (b) it non-spontaneously reconfigures itself at its high temperature by adopting different modes of vibration through its changes in area, in configuration, and so in its entropy, and all as we input additional energy and keep it increasing in its stock of internal energy; (c) it spontaneously cools on the down stroke as it loses in its internal energy, converts, and delivers its useful and mechanical external work; and (d) it spontaneously surrenders yet more internal energy to the surroundings by returning to its original mode of vibrations, which is a return to its original configuration.
Both the closed and the open system can go through the cycle that Carnot first discovered. The horizontal line represents the entropy change, which is a change in the number of the available configurations it is exploring, and so in its accessible ‘microstates’. These are also different modes of molecular vibrations, all at the same temperature. That change in its internal configuration is caused by its incoming or outgoing energy, such as when water boils in a steam engine, a rock bears down upon a piston, or steam converts back to water, or water into ice. But only an adiabatic system can do this isentropically. In all real cases, there is an increase in entropy as the system undergoes an irreversible set of changes, spontaneously surrendering some of its energy to the environment as a proportion of its molecules depart. They do not return of their own accord.
Neither isolated nor perfectly closed adiabatic systems exist. All real systems inevitably suffer a loss in their internal energy which is a loss in molecules and energy. This is both (a) a change in configuration and/or modes of vibration; and (b) a loss in the rates of vibrations which is a change in temperature and a heat interaction. Such losses can be computed. They are a specified number of joules. They are also a specified number of molecules transporting a specified amount of heat energy. Molecules in the surroundings gain in energy as the departing molecules are liberated and move outwards at that given temperature. They move into a vastly larger potential set of configurations within the universe at large. An entropy increase always involves a dispersal or spreading out of energy and a loss of molecules to the surroundings. Thus it is a loss in the system's entropy … but a net gain in the universe's. Its value is the system's change in configurative loss at that temperature, and Clausius calculated it as TdS. This simply says: “given how rapidly these molecules are moving at this temperature, here is the change in their internal energy caused by the ongoing change in their configuration”.
Now we understand about boundaries, areas, and types of interactions, we can go back to our primary interest: a biological population trying to reproduce and repeat its generations. We are also—and at last—ready to prove that a population free from Darwinian competition and evolution is impossible. We do this in two broad ways. But it is it is also very important to be clear about two things:
The generation is a cycle or circulation. We will be describing it using tensors, so we can compare any one, reversibly, to any other. If a tensor cannot emulate those cyclical processes no matter which population we compare with which one, then we cannot use it to describe biological entities or their activities.
We are using our tensors to describe biological behaviours. The second law of thermodynamics discusses the combined entropy change caused by the joint interactions between a given system—even a biological population—and the surroundings. This can remain constant at best, but will almost inevitably increase. And once that total has increased, it cannot decrease. No system or sub-system contained within the universe can decrease the combined and shared entropy of the entire universe at large. The best that can happen is for two systems to exchange entropies with each other, leaving everything else the same. This is exactly what our carefully defined prototype biological cell achieves. It can have an invariant and perpetual interaction between itself and its surroundings. The universe at large leaves the cell and its template unchanged as the cell and the environment exchange materials and energy so that the cell and its entire population can reproduce itself indefinitely. It is the very best possible in biology.
Our first broad way of proving that a population free from Darwinian competition and evolution is impossible rests on the way tensors establish the values they attribute to their components, which describe our chosen object(s). If we, for example, use a tensor, or else a component within a tensor, to describe the frequency aspect of a wave-like phenomenon, then that tensor (or those components) must continue to describe that aspect as it changes. If one of those frequencies is 50 hertz and another is 100 hertz, then since one is twice the other we will expect to observe an octave which is the concordance in the phenomena that matches those components. The absolute values might now differ, but we expect to continue measuring a similar concordance. If one frequency grows to 200 hertz and the other to 400 hertz, the one is still twice the other. However, we have shifted from a difference of only 50 hertz between the two in the first case to 200 hertz in the second. One is still twice the other, but the amounts differ. One concordance happens with a one octave difference, the other with two. We have both absolute and relative differences, and rates of change between them, which our tensors must accommodate.
If we now have a first measurement of 200 hertz and another of 201 hertz, then we will expect to measure a beat of 1 hertz. And if we then have a second measurement coupling of 100 hertz and 101 hertz, we will again expect to measure a beat of 1 hertz. However, the proportionate or relative difference is not now the same. It is ½% in one case, and 1% in the other. And if we have a first population of 500 and a second of 5,000, and then increase by one individual in each case, we will see this same situation of absolute and relative differences with population numbers.
If a tensor is going to describe a biological population, it must incorporate (a) additive, (b) multiplicative, and (c) integrative components, because biological populations have all three sets of properties across their various dimensions. Our tensors can record all such sums and differences, which is both their absolute and their relative measures. But this mix of changes is also why creationism and intelligent design are simply impossible.
A tensor is defined through the numbers we associate with it, which are its components. A first task for those components is to indicate the number of different dimensions, or directions, in which we describe the thing.
We should at this point clarify carefully that although we have what looks like a 4 x 4 tensor, the outermost row and column deal with rates. The values at the heads of the various rows and columns are simply the resultants of the inner 3 x 3. Those rows and columns are the separate momentum and kinetic energy values for the mechanical and the nonmechanical chemical energies that we determine as a population thrusts itself both into its ecology, and forwards along its generation through time. Thus the 3 x 3 Owen tensor whose rates we will eventually examine with the 4 x 4 Haeckel one has nine significant components.
Tensors group their components into sets or ‘ranks’ depending on how many components there are, and how they interact. The waves we considered above are a one-component tensor. So if the difference between the objects our tensor is describing can be expressed through the beat between two frequencies, or anything similar, then we need only one number and component-set for them all. The phenomenon we are measuring is a simple “superposition” of linear events. We might have to measure say a frequency of 50 hertz, and also one of 100 hertz, or one of 100 hertz and 101 hertz to get the combined result in a particular case, but it is still a coming together of two one-component tensors, which then produce a third. In our case the rule for bringing them together is fairly simple, but the procedure can be as complex as we like, as long as it is consistent and follows certain rules. The resultant beat we observe will always be a superposition. This holds for waves no matter how many others we might add.
Temperature is another phenomenon that could easily make up a one-component tensor. No matter how many objects we bring together, if we record our final value simply as the resultant of them all; asking no further questions about how the temperature arises; and so not measuring any other factor; then the object or phenomenon is being described using only one component-set, and is technically called a ‘tensor of rank zero’. This is a way of saying that no matter how many of these tensors we have to deal with, they are all of the same kind, and they all follow the same set of rules. The amount of ranking or separation into hierarchies we need to do to determine the final result is zero.
If we now want to know why temperature, for example, might be varying in our results, then we would probably measure some second factor such as the area, or the mass, or the density of our objects. This introduces a second component-set into the discussion. The different ranks that tensors can have thus indicate different significant sets of components and values. In this new situation, we have to group our components into these two separate ranks or hierarchies, according to what they signify. Instead of doing this zero times, we now have to go through our ranking process once for each component to figure out what it is, and so we can put them into one or another of our two interacting groups to determine a final result. This is therefore a ‘tensor of rank one’.
Figure 19.6 shows us what we would expect as we measure a difference within, or between, two such rank one tensors and then associate them with each other. Since we now have two distinct hierarchies of values they interact separately from each other. We have two separate sets of components. It is perfectly possible for two objects to interact so that one components-set changes while the other remains the same. We can for example bring a hot and cold object of the same size together and notice that the final temperature is their mean. If, however, they are of different sizes then the final temperature will have some other value. A rank one tensor thus has two components which must be handled separately—and is in fact a ‘vector’ which is taken from the Latin vehere, because it carries a specific thing in a specific direction.
It is often much easier to diagram a rank one tensor to observe what happens when we bring two of them together. Figure 19.6 shows us trying to find the difference between two vectors, which is to subtract them. Each component must be handled separately.
If we have two objects whose temperatures and densities varied, and we wanted to find their true difference, then the temperatures would separately affect each other; and their areas or densities or whatever else would separately affect each other; both combining to produce a final result. Each would have its distinct effect to produce the final effect we measure. We could represent our object's area with the line's direction or orientation in the diagram; and then represent its temperature with the line's length; … or vice versa. It does not matter which.
We perform a vector subtraction by effectively picking up the second vector, flipping it around, and stretching it out from the end of the other, being careful to maintain the appropriate orientation. We then get the two distinct components values for their difference. The value at the end of the vector marked ‘Vector 1 minus Vector 2’ gives us the values of the two components for the difference between these two rank one tensors. We can now clearly state their exact difference. If we add the subtracted vector—Vector 1 minus Vector 2—to either Vector 1 or Vector 2, we will end up with either Vector 2 or Vector 1, respectively.
The resultant of any comparison, or differencing, between two rank one tensors (or vectors) is another rank one tensor (or vector), which we would then expect to measure across or between the two objects. The diagram tells us how the separately ranked component values interact when we difference or subtract them. Their differently ranked components interact to add and subtract, or whatever other process we want to undertake. We can use the resulting diagram to easily predict the area and temperature differences no matter how many objects we consider. Therefore, when we establish a difference between two or more tensors, we expect to detect or to measure something reflecting that difference. And if there is no difference, then we simply measure the same properties as in the original when we measure the final one. But … that is still something predictable to measure.
So if we now measure a first population at some point in its generation, and then measure it in some succeeding generation at that same time point, we can establish a third tensor that states the difference. We will also have a record of everything that has happened across time, and so in between. And if the proposals of creationism and intelligent design are true, then the tensor that establishes the difference should be the ‘identity tensor’ because we expect the population to be the same in its essentials. We have then added the identity to the original, and kept the original, as in 3 + 0 = 3. That original might not have changed, but it is also not nothing. It is the same thing with zero apparent difference, which is exactly the identity operation with the identity element.
The nice thing about tensors is that we can allocate one to every population, and still relate them to one another. All we are saying is that the identity tensor predicts that we will measure, in the succeeding generation, the same thing we measured in the previous one, and in spite of what might have happened between the one and the other.
In the same way that addition and multiplication each have the identity operations of x + 0 = x, and x × 1 = x respectively, an identity tensor can transform any tensor that it is applied to back into itself, so leaving the original object the same. Thus the identity tensor tells us that the sum of all the interactions at every point, between two generations, has been, and must always be, zero so that the resulting biological entity is the same as the one in the previous generation. If nothing, overall, has happened in the interim so everything is just the same, then x + 0 = x and x × 1 = x, and the identity tensor simply does the same across its entire set of values and operations, and for all its ranks and components.
The identity tensor also tells us what has happened in the surroundings between one measurement and the next. It tells us about the net state of those interposing interactions. It therefore tells us about the measurable universe stretching between those measures.
However … the energy concerned with the identity tensor is again the property, and the statement, of an ecological and environmental interaction. And since the biological entities and populations are interacting with, for example, the sun, then the sun's entropy must be zero. The same goes for the environment.
If creationism and intelligent design are true, then reproduction must be an identity operation for biology, with all organisms being zero or identity elements from generation to generation. They must not, in themselves, add anything to, or take anything away from, each other or themselves. In other words, the entropy of all generations, plus all their surroundings, must not change for the population and generation as a whole, for we are dealing with force and energy, and we cannot just have some “left over”. No entities or their operations must affect averages or distributions no matter how many times they either operate on each other, or are operated upon. These combined entropies must always be zero, so there are no net enduring interactions, and so that both the population and all its interactions can always retain the same values. This preserves the proposed species template … and is the only way to do so, for energy is the property of an interaction with the surroundings through work and heat. Anything else acknowledges the force and power of Darwinian evolution. If creationism and intelligent design are again true, then all biological entities must be identity elements relative to all others, with none having any effect on any other. This is the identity tensor. It is what we must measure as the energy and the interaction for all possible populations.
The theory of tensors is now saying that populations can only follow an abstract creationist and intelligent design template if biological entities are free from all changes, and free from all variations, so that the numbers on all three diagonals for the three constraints of constant propagation, constant size, and constant equivalence, form the :1 deviator of the identity tensor. All off-diagonal elements must be empty. The elements transformed by the identity tensor can then remain identical, with the tensor having unity on the diagonals and zero elsewhere. But this means that there must be no distribution—whatever—across either time or space. All entities in any given population must have exactly the same values at all times, and not just between generations, but also across them. Nothing must happen to biological entities across time or space and between one generation and the next. This is the meaning of the identity tensor.
The tensor we are building up for biology is a ‘tensor of rank two’. This means that we will always associate two components, say a p and a q, to determine every value.
The number of components in a tensor should not now be confused with the number of dimensions. Since ours is a rank two tensor, we will always associate two measurements or comparisons for each component. The dimensions used in each component can vary, but we always take two. So if the three dimensions are x, y, and z, then sometimes we will have to determine an x and a y to produce a component; then sometimes a y and a z; and then sometimes an x and a z and so forth. Each dimension takes it in turns to affect and to be affected by itself, and the other two. If we include the x and x, y and y and z and z couplings, which are direct comparisons, our rank two tensor gives us a complete listing of nine components. They are the nine elements in the 3 × 3 grid. Our 3 × 3 tensor therefore yields the nine components: xx, xy, xz; yx, yy, yz; and zx, zy, and zz. And since we are measuring across time and space, we also of course associate each x, y, and z with itself to see if it has changed, which gives the xx, yy, and zz of the three diagonal positions in the tensor. If we compare a property at a generation's beginning and end, we should have xx, yy, and zz. That diagonal makes the tensor's power to transform equal to its own power to sustain itself; and this is also what makes it so useful to measure across time and space, and between populations.
If a biological identity tensor exists, then every entity must also have its demonstrable and measurable biological inverse. This must be so because every identity operation produces the identity element simply by bringing together some element and its own inverse. So, for addition, we can have 2 - 2 = 0, which is 2 + -2 = 0, or more generally, x + -x = 0. For multiplication we can have 2 ÷ 2 = 1, which is 2 × ½ = 1, or more generally, x × 1/x = 1. The rule is that we must always be able to bring an element and its specific mirror image or converse together to produce the identity element.
If creationism and intelligent design are true, then for every biological entity of mass m1 and energy p1 at time τ1, then there must exist some single specific and corresponding entity with values m2, p2 at time τ2, and so that when they are brought together, they are precise converses. They must together produce exactly the distribution of :1, which is the identity element deviator for this tensor. So for every entity of a given mass and energy at some point, there must be another so that when brought together, they produce exactly the average value and distribution for that population. Every entity is then of equal significance to every other, with none being any more so than any other. Death or loss before reproduction are not permitted, for there would then exist an entity for which there is no converse, and the identity collapses. We would instead have what Darwin proposed: the driving force for evolution. We must preserve the template. That is what the creationist and intelligent design proposals imply. If we cannot do that, then the population contains members with variations, and it is evolving because there is an inherent energy within that population. It is contained in each and every entity that does not have a specific converse and so in that population's ability to evolve through that population's measurably fittest members. We will return to this point shortly.
The theory of tensors tells us that biological entities and their properties must all go through our three inverse processes of (at least) (1) addition-subtraction, (2) multiplication-division and (3) integration-differentiation as they grow and develop over the cycle. The only way to make sure that variations in numbers are of no overall significance, which is what creationism and intelligent design insist is the case, is to make sure that those inverse processes match each other, no matter what may be happening to numbers.
The theory of tensors also tells us that if we ever observe any population in which some entity dies before reproducing, then creationism and intelligent design are immediately refuted. That is the power of the identity tensor … and … observation tells us that things do indeed die before they reproduce. That famous observation is where Charles Darwin begins his thesis. The job of refutation is already done.
But … there is more. The two processes of multiplication and division are x × y and its inverse of y/x. These involve the multiplicative identity which is 1. This takes us back to the matter of areas and angles. The linear planimeter needs boundaries and multiples, while the polar one needs angles and ratios.
We have seen that the work and energy over a generation can be expressed as a combination of the population's mechanical and nonmechanical chemical energies. That is its internal energy, which is a combination of its mass and its energy. That means that if a cycle or generation is to be completed, then all entities must be successfully carried all around the circulation. These each involve a combination of the inverse processes of multiplication and division, addition and subtraction, and integration and differentiation for that internal energy. If numbers—which is the partitioning applied to internal energy—must be constant, as creationism and intelligent design propose, then those inverse processes make a reproductive cycle impossible.
The population's mass and energy fluxes are M = nm̅ and P = np̅. Both these are multiplications. But we have also seen that the population's current location in the cycle is a proportion of the generation length, which is m̅/n and p̅/n. Both these are divisions. We must always both (a) multiply, and (b) divide two values at every point in the cycle, or it is not completed.
The succession of the generations is the constant repetition of nm̅ and np̅ by the linear planimeter. It is also m̅/n and p̅/n by the polar one. The constant repetition is a cycle or wave whose boundary is defined by nm̅ and np̅; but whose proportionate amount is defined by m̅/n and p̅/n. Creationism and intelligent design insist that we remove all dependency on numbers, n. But since the cycle depends on both multiplication and division, then all populations must find some alternative way to create their effects.
The population's internal energy will change by the infinitesimal amounts dn, dm̅ and dp̅ respectively in every infinitesimal interval of time, dt, as its mass, energy, and numbers all change. But if numbers are not allowed to change … yet both the flux total and the angle must be preserved … then we must find some alternative way to ensure the mechanical and nonmechanical energy fluxes, meaning dm̅dn = dm̅⁄dn and dp̅dn = dp̅⁄dn. It is far simpler to represent these as x and y. Both fluxes must therefore satisfy the more general request that x × y = y/x at all times.
Now we have stated the demand more clearly, there is only one way it can be satisfied. We can only have x × y = y/x when both x and y are the multiplicative identity. This then gives 1 × 1 = 1/1.
If x and y are both 1, then dn, dm̅, and dp̅ must all be unity: dn = dm̅ = dp̅ = 1. Every change must now act exactly as the multiplicative identity. Whatever the initial value of m̅, p̅, n might be, they are always multiplied and divided by 1. They must therefore stay exactly the same, all about the generation.
If creationism and intelligent design are true, then the only way any components or variables that depend multiplicatively on numbers can be both multiplied and divided over a complete cycle is to remain exactly the same. There is then neither an area, which is a flux, or an angle, which is a proportion of the generation. Thus the request creationism and intelligent design make that biological populations be free from all changes in numbers is the request that all biological entities maintain their internal energies in a constant and invariant condition … which is simply not what is observed.
There is still more. A circulation of the generations is a constantly repeating cycle, which is an oscillation or wave. If two populations are similar, then their cycles of oscillations in their internal energies—which are their wave functions and their repetitions of values—will also be similar. They will increase and decrease between similar values, in similar ways, and so can immediately superpose. If two populations or generations are brought together, then their values for their various properties will add together to create regular increases and decreases, or crests and troughs, while they add values in some locations, but they must interact so as to leave them invariant at others, so that they share the same mean values. If there are two populations then they will have twice the effect on the universe, but when they have both in the net done nothing, it is as if they have cancelled each other out.
We can determine changes not just by making comparisons, but also by the taking of differences. If things are the same, then there is a difference of zero. So if we measure a property at one point; and measure it again at a different point, after a generation; and if we expect the same value; then we also expect a difference of zero. These are subtractions.
If creationism and intelligent design are true, then we must add masses and energies in some locations; subtract them in others; and still show a zero difference overall so we can leave certain essential properties unchanged from one generation to the next. The additive identity therefore repeats the same problems we had with the multiplicative one. Biological populations must have both x + y = x and its inverse process of x - y = x. Both x and y must be the additive identity, which is zero.
A generation is a set of repeating properties. If two populations are similar, then they have have functions—wave functions, Φ—for their internal energies which must also be similar. Since our biological space has the three dimensions of number, mass, and energy, we can write every population and its internal energy as Φ(n, m̅, p̅). Any Population A will then be ΦA(n, m̅, p̅), with some Population B being ΦB(n, m̅, p̅).
Let Populations A and B have the initial states:
ΦA(ninitial, m̅initial, p̅initial),
and
ΦB(ninitial, m̅initial, p̅initial),
respectively. If we consider them together, they will also have a superposed or joint initial value that is the simple result of the addition of their individual components:
ΦAB((nAinitial + nBinitial), (m̅Ainitial + m̅Binitial), (p̅Ainitial + p̅Binitial)).
We can easily measure all these values to confirm that result.
After a generation, our two populations will end up with final values:
ΦA(nfinal, m̅final, p̅final),
and
ΦB(nfinal, m̅final, p̅final)
individually;
and
ΦAB((nAfinal + nBfinal), (m̅Afinal + m̅Bfinal), (p̅Afinal + p̅Bfinal))
for their superposition.
We can compare any two populations by taking their internal energy differences. We get that difference, in each case, by subtracting initial values from final ones. If creationism and intelligent design are true then this difference in internal energy should be zero, so that any first population always acts exactly as any second one and conversely. This should be true for both populations, as well as for their superposition.
Let us temporarily agree with the creationists and intelligent design advocates that biological populations are indifferent to numbers. All other properties—in this case m̅ and p̅—must still both (a) add by superposition, and yet (b) have their differences taken by superposition to produce zero, so that all necessary values are still the same.
However … we are now asking that x + y = x - y for all those values. But this can only be true when 0 + 0 = 0 - 0. This is to ask that every possible property, and change in such property, throughout all internal energy over the cycle both have, and be, the additive identity, which is zero.
The energy the population needs to go about the circulation is a statement of the net interactions with the surrounding universe. The demand for no change is effectively a demand for no energy, and/or no change in internal energy. Both energy and changes in energy must be zero at all times. Thus the request from creationism and intelligent design that biological entities follow a numbers-free template—i.e. that there be no sensitivity to changes in n, which can only happen if numbers do not change within and between the generations—is immediately the request that the surroundings be constant and invariant … which is simply not what is observed.
We have yet another consideration. Biological entities not only grow and change at each moment, they do so in specified, organized, and coordinated ways. The members of each group or population do so according to their species, which is then the template or code.
Anything that increases must also decrease at some pace that allows a succeeding generation to return to those same values and repeat them. Whatever increases through some ∫y dx, which is an integration, must do so at some given rate, which is a dy/dx. In particular, reproduction involves changes in numbers which are both ∫y dn and dy/dn.
The request from creationism and intelligent design that populations and their generations be indifferent to numbers, and to rates of changes in numbers, is the formal request that every component sets itself equal to its own derivative as in . We must have dy/dx = ∫y dx = e so that all changes in all properties are at all times some power of e.
To divide 5 by 5 creates unity because 5 ÷ 5 = 1 … which is the same as 51 ÷ 51 = 50 = 1. That is just an alternative way of writing exactly the same thing. We begin with 51 because we have used 5 as a base. This is to multiply 5 by itself once, which is the 51. We then divided the 5 by 5. We get 1. But this 1 represents the zero number of times by which we end up multiplying 5 by itself when we divide it by itself, because the powers state that 1 - 1 = 0. Multiplying anything by itself zero times always gets us unity or 1. We can therefore divide a first thing by another by expressing the two things we want to divide as powers to some base; and then subtracting those powers relative to that base. And if we produce a power of zero, then we have divided something by itself to get 1.
To ask that something be equal to its own derivative is to ask, when we express it as a set of powers to some base, that there be no change in those powers at any point. This is again to ask for everything to change by exactly the value of the identity element involved in integration and differentiation. It is to ask that everything at all times go through the process of ex ÷ ex = e0. And since e0 = 1, then that unity must be both multiplied and divided through by everything, so that everything neither increases nor decreases. Therefore, the only way to satisfy the creationist and intelligent design template is for nothing ever to change by any amount or proportion.
Since energy is the property of an interaction; and since it institutes spontaneous changes such as falling objects and temperatures wherever possible; then making any of the above three requests is to ask (a) that systems never change; and (b) that the environment also never changes. Neither a biological population nor the environment may emit energy in either direction, and nor may they do work upon each other, or transmit heat to each other. So a request that biological populations both (A) follow templates, and (B) be free from changes in numbers, is in fact the request that the environment be uniform and constant at all times, which is once again not what is observed. This is the fourth time the idea has failed.
And this is still not all the issues raised. We now turn to the second general category of problems with the creationist and intelligent design proposal. Those problems are all centred on the impossible requests they make of internal energy.
Biological entities and populations contain internal energy, and always interact with the environment. This involves entropy. It means variations and transformations, over time, in modes of molecular vibrations.
We know that if ever we compare two tensors, we will get values to measure, even if the sole difference is the identity tensor. By the second law of thermodynamics, no organized system can maintain itself indefinitely. Therefore, at least some organisms will dissipate in at least some infinitesimal interval dt across the generation length, T. We will get the infinitesimal number loss dn. Since energy is again the property of an interaction then the second law—which is about the interaction between systems and the universe—makes clear that no population can continue indefinitely if it does not institute variations to allow it to cope with the inevitable losses in numbers that will be suffered. If the population does not make a direct response to a loss in numbers, then it will inevitably become extinct. This is the basis of Darwin's proposal.
This gives us yet more problems for the creationist and intelligent design proposal. Since all tensors, and all environments, must match each other to make those notions true, then a request that a population be free from the effects of changes in numbers is the request that each and every population and generation be a perpetual motion machine of the second kind. This is a system that can faultlessly—and without effect either from or on the environment—convert a given stock of internal energy into mechanical work through a given set of heat interactions. Some proportion of internal energy, as molecular vibrations, must then be precisely equivalent, as a stock of thermal energy, and at any given time point, to a specific amount of work done. But the sun must then act as the perfect zero entropy heat reservoir, outer space being the similar cold one. We must measure this in the environment as, at the very least, the difference between any population and itself, over time. And since we must measure this zero difference, then this is also what each and every population must actually be: i.e. ones capable of interacting faultlessly and at all times with zero entropy reservoirs.
Creationism and intelligent design are asking that all biological populations contain our prototype cells which are attempts to establish this exact perpetual motion machine of the second kind. However … we had to establish some rather stringent, and utterly unrealistic, conditions to achieve this. And that it is impossible for such a system to exist in this real world is a basic canon of science, for it requires that the conversion of energy always happen adiabatically and at all times. Observation soon tells us that the sun and the surroundings are not zero entropy devices, which is the only possible way such populations could exist. Our prototype cell is in breach of the second law of thermodynamics for perpetual motion machines of the second kind are impossible.
We then make a yet further observation. Our prototype cell is busy maintaining a specified store of internal energy as mass and energy—permanently—within itself. And since a population is now a system that uses its numbers, its configuration, and its temporal disposition for such purposes, then creationism and intelligent design clearly require that a biological population also be a perpetual motion machine of the third kind. This is a system that suffers no losses or dissipations in its internal energy, and so that can continue indefinitely, without breaking down.
A perpetual motion of the third kind is little different, in its effect, from one of the second kind. It is once again asking that internal energy always be converted adiabatically. Since, as always, energy as both work and heat is the property of an ongoing interaction between a system and its surroundings, then we we need only look at the sun and at free space to note that they are not zero entropy devices. They do not provide the necessary surroundings. All such proposed systems and devices violate the second law of thermodynamics through the character of the surroundings. Our prototype population cannot exist for internal energy does not allow it. It cannot be measurable even as a difference between other populations, for that is still a property of the surroundings. A perpetual motion machine of this third kind is also impossible, and may again not be measured, even as a difference between two populations.
If biological populations are to follow templates, then they cannot be perpetual motion machines of the second or third kinds, leaving only perpetual motions of the first kind as a possibility. These are hypothetical devices that, once started by whatever means, can keep themselves going indefinitely with no discernible dependency on any observable matter or energy source. They suffer no friction or losses of any kind, and also keep themselves going using their entirely internal construction and power. They can affect the environment by producing the mechanical work that all biological populations engage in, but without using the surroundings as that energy source.
Supporters of creationism and intelligent design assert that such perpetual motion machines of the first kind have been created at some specified point in the past. There is one, or so they claim, for every species. Each species and its denizens have been “intelligently designed” to overcome all possible objections. Such perpetual motion machines of the first kind apparently exist simply because biological organisms exist.
The difficulty for this proposal is that every known biological organism expires without external and material sources of both energy and matter to maintain itself. All have ongoing mass and energy exchanges with their surroundings … which also contain others of their kind seeking to do exactly the same thing. They are utterly dependent upon both those others and their external sources and resources, all of which in their turn follow standard scientific laws.
A perpetual motion machine of the first kind is simply impossible. Biological organisms do not meet the definitions and specifications. They manifestly do not demonstrate any capacity whatever to survive without the mass and energy they observably imbibe from the surroundings. They are therefore contingent upon the properties and conditions that those surroundings impose through their not being the zero entropy devices that a perpetual motion machine of even the first kind demands from the surroundings. This creationist and intelligent design template does not, and cannot, exist.
We are still not done. The immediately preceding paragraph makes a well-known difficulty in dealing with these issues clear. Science requires measurement. Without something to measure, entirely theoretical and speculative claims and counter-claims can go on indefinitely. We must have something specific we can measure so we can prove the case either way.
We turn, first, to the two anomalies we have already met: (A) the intersection, within internal energy, of the mechanical and nonmechanical chemical varieties of energy; and (B) the juxtaposition of population totals and individual values.
This immediately raises an issue with tensors. They are very good at stating the relative measures and proportions we want. We can take anything as a base and measure anything else. We can then track all relative changes and proportions without difficulty. However … we eventually need absolute amounts. And if we want to know absolute and actual amounts, rather than just relative and proportional ones, then our tensors must always convert relative and proportional tensor values back to real and absolute ones.
Over any time interval, dt, all populations will change their internal energies by Pdt joules and Mdt kilogrammes. Or alternatively, since P = np̅ and M = nm̅, they will change their internal energies by np̅dt joules and nm̅dt kilogrammes respectively. The former versions are population totals measured by the linear planimeter, the latter are individual totals measured by the polar one. The two taken together allow us to measure the curl: the difference in their rates of change. We need only set one value over the other. The only source for the curl is those differences in numbers. Mass and energy are simple to measure; and entity numbers are just as simple to count.
Our three constraints of constant propagation, φ, constant size, κ, and constant equivalence, χ, can infinitesimally but proportionately increment a population through (a) its numbers as dφ; and/or (b) its mechanical chemical energy as mass through dκ; and/or (c) its nonmechanical chemical energy density through dχ. We have established our biological potential to reflect any such proportional changes in state because μ = dS = dφ + dκ + dχ. We need an expression to isolate dφ, the proportionate or relative change in numbers, dn. This is the actual change at any point. We want to relate the proportional changes any population makes to the real ones both it and others make.
We can now state a first equation that incorporates relative and absolute numbers and amounts. for both populations and their individual entities. (We do not have the time here to derive it, but you can find a more rigorous derivation here). Our biological potential is:
µ = dS = χ(dm̅/m̅) + ΩT’(dV/V).
The first term on the right states all relative changes. We have χ as the constraint of constant equivalence. It has our range stated around our orthonormal central value expressed as unity. It states the proportionate changes of the work rate, and so the entropy, across the circulation of the generations. The second item helps convert relative to absolute values. We therefore have all changes in internal energy, both relative and absolute.
The dm̅/m̅ factor in the first term states the proportionate increase in the mechanical aspect of internal energy at any time. We measure the actual population's average individual mass, m̅; we measure the change over the interval, which is dm̅; and we then state that change as a proportion. And since that proportionate value is applied to χ, which is also a proportion, then we immediately have the total proportionate change in internal energy caused by this mechanical aspect. But since m̅ and dm̅ are real values, we have real and absolute values for this mechanical aspect of the internal energy.
The second term handles the nonmechanical aspect of biological internal energy. It similarly incorporates the proportionate infinitesimal changes in the visible presence, V, which is expressed as dV⁄V. That gives us the proportionate increment in the energy flux, expressed absolutely. But this proportionate increase has the two terms Ω and T’ applied to it. These scale any population's energy relative to our reference.
We have already met the scaling factor T’. Our prototype or standard cell has a generation length of T = 1,000 seconds. If a population has a generation length of T = 5,000 seconds, then it has an eigenvalue or multiplier of T' = 5. All other things being equal, then since the generation length is five times as long as our prototype, we are going to use five times the mass and energy our standard cell uses for any generation. That T’ abutting the dV⁄V is therefore the eigenvalue that scales all changes in visible presence or work rate by the generation length.
Populations also of course vary by numbers. Whatever we choose as a reference or basis, that population will use a very specific quantity of chemical components to reproduce itself at some given numbers. That known and unique collection of both moles and kilogrammes of chemical components is that population's distinctive genomes and genes. If we can directly incorporate that, then our tensor will immediately give us both absolute and relative energy values.
We continue to use our standard cell to create our basis, but we could use any real population and its equilibrium age distribution population of whatever numbers. It does not matter. This is the advantage of tensors.
Our value for Ω is again based on our prototype cell. It sets a standard behaviour. We fix it at some given population number, so we can then scale our internal energies, from that reference, and by proportion. Its sole purpose is to give us a foundation, in numbers of cells, and numbers of entities, for relative scalings and proportion-takings, to and in any other population.
We begin fixing a value for mass and energy for our reference and standard by noting that at temperate latitudes, the earth receives of the order of 100 joules per second. That is then our proposed basis for the Wallace pressure, P. All other values will be scaled against this as some P’. This standard or reference is as effective as using any real population, because it is only ever used for comparison. We scale all populations relative to this and work relatively everywhere, stating everything we want always relatively to everything else. So if we measure some target population as having a Wallace pressure of 200 watts, then we have an eigenvalue of 2, and P’ = 2. We simply put 2P in the tensor everywhere we see P.
We also noted earlier, when developing our prototype, that the average terrestrial eukaryotic cell has a mass of 1 × 10-12 grams. And … that is now the reference or basis mass we use for scaling, and for expressing all other masses and internal energies in relative terms in the same manner. Every time we measure a mass, we also measure an energy density, and so an energy. We saw that, relative to that basis, Brassica rapa has a scaling value of 9.248 x 1010.
We now need to pin down a work rate that links a prototype or basis value for mass and energy to the numbers and moles of both components and entities. We can further specify, therefore, that the stock of thermal energy implied in our standard cell be a multiple of the Boltzmann constant, kB, which is a standard physical constant. This now states the capacity for a definite amount of biological work based on a stock of thermal energy as contained in our internal energy.
We additionally note that at a representative 25° Celsius, the average terrestrial biological organism contains biological matter whose thermal energy is of the order of 0.5 kcal per mole (glycine has an energy content of approximately 979 kilojoules per mole (Haynie, 2001, p. 12)). We can therefore state that our reference cell contains 1,000 kilojoules per mole of components, which now states its energy density, and the scope and scale of its ongoing reactions. We are specifying that genome and those genes, and also stating what they need to reproduce. We could of course measure a real population, if we wanted, but since everything is eventually measured, and expressed, relatively, through our tensor, then these tight definitions are just as effective. There might, after all, be some real entity with these values.
Since most of the entities we are likely to meet in the ordinary course of events are multi-cellular, we can propose a reference entity on similar scale. Estimates for the number of cells in the human body vary from 50 to 75 x 1012 or trillion (Englebert, 1997). We can therefore propose that our reference entity has a cell count of 6.022 136 7 x 1020 cells. If we now take up 1,000 of our entities, we will immediately have 6.022 136 7 x 1023 cells, which is the Avogadro number. This is convenient because it immediately evokes all other physical constants, including the above Boltzmann constant. We thus have a population processing both mass and energy at a measured and known rate, all of which we can use to state all others relatively, and through our tensor. We now have a complete population—numbers, mass, energy, and time to reproduce—scaled and stated to the value represented in Ω which is a multiple of the Boltzmann constant, while simultaneously incorporating the Avogadro constant.
When Ω and T’ are now brought together, they can scale any population to the reference; and, conversely, they scale that reference to any population. They can then be used to express real populations, their sizes, their numbers, their masses of molecules, and their energies in terms of each other. All real populations will differ from our prototype by amounts both relative and absolute. We can express one in terms of the other, and state the mutual transformations they each need to become each other. Values of 1 for all parameters simply mean either (A) that our target population is behaving exactly like the reference in all respects, both relatively and absolutely; or else (B) that two populations being compared are identical. We know how much energy any given population would have by comparison, and we can then produce its precise values for mass and energy, again both relatively and absolutely.
When ΩT’ is now applied to dV/V, we immediately produce the instantaneous—but proportionate—change in the visible presence or energy density for some number and mass of both cells and entities. We have also incorporated changes in numbers. We can then state an absolute value for the Wallace pressure, P, which is the internal energy of record for that population size.
The value for the proportionate change in visible presence, dV/V, now automatically incorporates any change in numbers or partionings of internal energy, because we specified those numbers very closely in defining the ΩT’ that is now applied to them. Given the careful way those have been defined and quantified, P now also gives an exact molecule, cell, entity, configuration, and time count for all stocks of biological internal energy. All these are provided by the tensor and this ΩT’. We now have any change in the Wallace pressure, and in the nonmechanical chemical aspect of internal energy, that is occurring independently of the mechanical chemical energy and mass aspect of that same internal energy. And since we can get the mass and mechanical energy transformation from the previous term—which draws together both the relative in χ and the absolute in dm̅/m̅–—we state the population's change in terms of what is measurable and measured, for dV/V is also a proportionate change that is being applied to an absolute value. We now have the information that we need, and irrespective of the basis. We can now measure absolutely any population relative to any other, and using any basis or standard we want, which is the effect we set out to achieve.
Our above equation confirms to us, with its first term, that if a population is genuinely free from all changes in numbers, then χ = 1 at all times. Any mass change is then entirely responsible for all its population changes in the Wallace pressure, P, because the second term is zero so that there is no ancillary loss or gain from the ΩT’ factor. The rate of change of individual entity values will be equal to the population ones … which is most easy to measure. If we do not measure those two rates as identical, then creationism and intelligent design are false.
Granted that, in the ideal case, we have χ = 1 because mechanical chemical energy is evenly distributed everywhere and at all times, then whenever mass is not changing, all changes are being caused entirely by changes in the configuration, which is dV/V. All changes are therefore coming from the ΩT’ term, which is numbers and configuration. None are again coming from mass. The χ term is then entirely responsible for all changes in the biological potential, µ. Everything is unaffected by changes in numbers. But if χ ≠ 1, then we straight away know we have changes in numbers. Those changes in numbers make a contribution to changes in mass, which we can detect and measure, and population changes should again only be identical to individual ones if the population is ideal. Once again, if we do not measure those two rates as identical, creationism and intelligent design are proven false.
In any real case there will, of course, be changes in numbers, meaning χ ≠ 1. That second ΩT’(dV/V) term then has a value while mass is changing. Neither φ nor κ can now be constantly unity. Since they state proportionate changes, then both mass and energy will now change with changing numbers. It is only in the ideal case that the individual and the total values for mass and Wallace pressure can change at the same rates, both absolutely and relatively, and so that we have dV = 0, to give dV/V = 0, to match the proportionate χ = 1. Thus when mass again changes—in that ideal case—it is entirely responsible for all changes in the biological potential, µ in the equation. But in the non-ideal case when χ ≠ 1, then we will also have φ ≠ 1 and there will be some change in that second term, which immediately signifies some change in the ‘no-go’ or ‘no-entry’ areas of the tensor we highlighted in Figure 15.1. Those are changes due to numbers. And … we can now measure those, exactly, by comparing individual and population rates of change. If the two average individual values of m̅ and p̅ change at different rates from their respective population totals of M and P, then there is mass and energy in those no-go areas, and the population is being affected by those ongoing changes in numbers. We will have refuted creationism and intelligent design.
The above equation therefore gives us a most simple way to test whether or not numbers have an effect. We only need to measure changes in configuration, which is dV/V and a set of absolute values for both mass and energy to see if χ ≠ 1, which is the relative measure, but which can be determined through those absolute values. So if the average individual masses or pressures, m̅ or p̅, change at different rates from their flux totals, M and P, then the method of tensors tells us, through this equation, that population numbers must (a) be changing; and (b) having an effect. These are very simple things we can measure, and as we did in our Brassica rapa experiment. We can use our polar planimeter to measure one, while the linear planimeter measures the other. If they differ, we have a curl. Creationism and intelligent design are immediately refuted. We have achieved our ultimate purpose.
We can state a second, similar, equation (we again do not have the space here to derive it, but you can find that here) as:
µ = dS = κ(dm̅/m̅) + ΩT’(dP/P).
If a population is again free from changes in numbers, then changes in mass must cause equivalent changes in the Wallace pressure through dm̅/m̅. If mass holds constant, then dm̅/m̅ = 0 and since again κ = 1 in the ideal case, all changes in the measured Wallace pressure must be entirely due to changes in configuration. In the ideal case, κ = 1, so the measured absolute change will be equal to the proportionate or relative one. But in the non-ideal case κ ≠ 1 and this will not hold. So if dm̅/m̅ ever changes at a different rate from dP/P, then κ ≠ 1 and the population flux totals are again changing at different rates from the individual ones … meaning population numbers must again be having an effect. We have related the relative to the absolute, and it is now easy to measure any population, and to see whether the individual masses and Wallace pressures do or do not track the population ones. If there is a difference, then numbers are changing and exerting an influence … and we will have again proven our case and utterly refuted and repudiated creationism and intelligent design.
We again do not have the space here to derive it, but we can also describe every biological population using the following Gibbs-Duhem style equation, which once again makes it easy to find out what is happening in any population (you can find the derivation here):
We straight away see that the first two terms are our Mdt and Pdt from the method of fluxes. Those are our two population totals for the linear planimeter.
The final term in the Gibbs-Duhem equation invokes both Newton's infinitesimals and the method of tensors. It is the polar planimeter. It states the effects of any changes in numbers over that interval. It states the amount of mass and energy that every distinct entity will carry into, or out of, the population as it is either inserted or removed, dies or is reproduced. The first two terms are what they must all do, the last one what they must each do.
If creationism and intelligent design are true, then the value for the summation term in this above Gibbs-Duhem equation must always be zero. If an entity is removed, then the averages and distributions over those remaining must remain invariant. Therefore, the one being removed from the population must not affect either the average value or the distribution. The only way this can happen is for it to have exactly the average value at all times. And since any one of them could be removed or inserted, then the same must hold for them all. But we can now measure each distinct entity, compare it to all others, and come up with a precise value for the effect its specific differences are having upon the population, which is its variations.
This Gibbs-Duhem equation simply adds to our armoury of techniques, and again alerts us to properties that are very easy to measure: the population's individual and total rates of change must always be the same if creationism and intelligent design are true. And if they are not the same, then the proposition is not true, and we can state an exact figure for how it fails.
And, finally, we can derive a standard Euler equation. (We again do not have the space here to derive it, but you can find a formal one here). Each term will match an “intensive variable” to its corresponding “extensive variable” … where an intensive variable is some property, like temperature, that does not change with size. This is a coupling between the two planimeters.
We can for example add warm water to a bowl without changing its temperature, which is then confirmed as an intensive variable. An extensive variable, however, measures a sum, or total, and so changes with size. Volume is extensive, and so the amount of water would increase when we added that water even though the temperature remained the same. This is again a population as against a set of individual values.
Our Euler equation matches the average individual Mendel pressure or mass flux, m̅, which is an intensive variable, to the population total Mendel pressure or mass flux, M, which is its extensive one detailing its total population stock of mechanical chemical energy. The Euler equation also matches average individual Wallace pressure or energy flux, p̅, with the Wallace pressure or energy flux, P, its stock of nonmechanical chemical energy. The average individual values then provide the driving force for the extensive one.
The Euler equation then uses partial derivatives to isolate each variable's effects. The change in the population mass or mechanical chemical energy flux, Mdt, is more elegantly expressed as dU, which is the instantaneous change in the quantity of chemical components held in the population, and so that Mdt = dU. It is an internal system change as it interacts with its surroundings.
By the same token, since the visible presence is V = M/P, the change in the mass flux, Pdt, is better expressed as the instantaneous change in V, the visible presence, and so as dV. When that dV is applied to the mass flux it gives us the infinitesimal increase in the Wallace pressure over that interval as a system change internal to the population, and again as it interacts with its surroundings.
We now have an Euler equation of:
The first two terms tell us what happens when numbers hold constant. They are the method of fluxes and the linear planimeter. The first term then additionally holds the visible presence constant to isolate the population mass flux of mechanical chemical energy. It therefore determines the infinitesimal change in the population's stock of biological matter, which is Mdt. In the same way, the second term picks out the change in visible presence, dV, which gives us all changes in the population's nonmechanical chemical energy, Pdt. We thus have our two population totals … which is what they must all do.
It is, however, the latter two terms that are of interest. They are the polar planimeter. They help resolve the anomaly with numbers, and with changes in numbers. That Σ sign shows that we are summing the values and changes of each distinct entity. We therefore trap the ones busy being either inserted or removed. The third term sums the individual masses over the entire but changing population; and the fourth sums the individual Wallace pressures. The effect we want is achieved by the terms j≠i in the two subscripts applied to the summations, which simply mean “apply this summation process to any entities not already incorporated in the population”. This is exactly what we need. Those two latter terms now tell us what they each do in the face of what they must all do and as the population numbers change.
And … once again … if creationism and intelligent design are correct, then these latter two summation terms in the Euler equation should be zero. It is as simple as that.
We have now made all this trivial to determine because all we need to measure is (A) the individual, and (B) the population values all around the circulation of the generations. We measure both sums and averages. We then compare their rates of change. If they change at different rates, the job is done. Creationism and intelligent design are then not merely false, they have been refuted with solid evidence taken right from a population. We will even be able to calculate, from our equations, exactly how much mass and energy any given biological organism expends solely and only because numbers around it are changing, and for no other reason. That is the power of those four equations, and of our biological potential, μ.
Science now tells us that the various terms we highlighted in the above four equations can never be zero or unity (as appropriate). That would be in breach of the Biot-Savart law, the Liouville theorem, the Heisenberg uncertainty principle, and much else besides.
But … since this is science … then only an experiment should be allowed to settle this. We now have our four equations. We also have our identity tensor that allows us to measure their values relative to any population or species. We can very easily measure those various terms in an experiment, both absolutely, and relatively to any population or populations. We simply select any test species, such as the well-known semelparous organism Brassica rapa; and … take some measurements to see if the method of fluxes really does differ from the method of tensors; and if we can truly separate what they must all do from what they must each do. Now suitably armed, we are at last in a position to establish the conditions that will allow us to conduct that very experiment. We turn to that next.
An algebraic and geometric topology based proof.
A vector calculus based proof.
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