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# 48. The definition of a species

No matter how it is expressed verbally, the opposition to Darwin is grounded in numbers. Our two proposals—one Darwinian and one Aristotelian—may disagree markedly about fundamental biological affairs, but they do not disagree on the properties and behaviour of numbers. And since the disagreement between them rests on number, we can speak with complete clarity about their differences and similarities.

We should perhaps first remind ourselves that for each moment in a generation from t to T, then every variable X in a given population of size N, a relevant mean and distribution of x̅ = X/N exists for that t and that is immediately a property of all the N. Both an x̅’ and an X’ which are each their respective generational weighted averages of all the x̅sand Xs over all the N for every moment t over the entire cycle of length T also exist, along with their sums or totals, and which are each again a property of all the N—and etc. Their respective y̅ and Ys also exist where Y =dX/dt and etc. Those sums, means, and their dynamical and stationary complements are then distributed over the three constraints, and over the population and the generation length, so as to affect system behaviour. Whether it is now Darwin’s principle of natural selection or the contrary proposal of the Aristotelian template that is correct, then since X = Nx̅, the fact that if X changes then either N or x̅ or both must also change and conversely, and for all such values and variables, both dynamic and static, is a far broader and more general reality that has dominion over both these proposals.

Figure 56 shows Green’s and Stokes’ theorems—two of the four fundamental theorems of the vector calculus—being evaluated in the plane. Limit point-elements—which could easily be, or else be within, a biological entity—are busy swirling some arbitrary field flux—which could easily be energy, or mass, or both—about themselves. Green’s theorem again uses a double integral to relate the line integral of the curve forming such a boundary to the area it contains:

If R is a simply connected plane region whose boundary, ∂R, is a simple and positively oriented closed curve, then if there exists a vector field F(x, y) = (F1(x, y), F2(x, y))T which is smooth, and where T is the unit tangent vector, then:

∮∂R F • ds = ∬R(∂F2/∂x - ∂F1/∂y) dxdy.

Stokes’ theorem now relates that same line integral to a surface integral, thus directly incorporating the behaviours of any elements that are constituents of that surface, and as are found inside that boundary, to that perimeter or boundary:

The circulation of a vector field over a closed path C is equal to the integral of the normal component of the curl of that field over a surface S for which C is a boundary (Fleisch, 2008, p. 116).

As earlier stated, Stokes’ theorem is most usually considered in three dimensions, as in electromagnetism, where a signal is radiated out in a third and orthogonal dimension, through free space, as the relevant Maxwell vortex seemingly spins about itself on two orthogonal axes, and according to Fleming’s right-hand rule. Stokes’ theorem is a more general statement of Green’s (Stewart, 2007, p. 1129). However, it also operates in two dimensions and as we consider it here.

In Figure 56 we see the net circulation about the boundary being evaluated through the integral and differential calculus. The surface is being considered, in the usual integration fashion, as a selection of infinitesimally small squares that tile the entire boundary, and within each of which is found the limit point—which could again be a biological entity—about which the swirling or curl occurs. As is clear, every interior boundary is counted twice, once in each direction. These interiors therefore cancel out. In the limit, therefore, the circulation can be reckoned simply by tracing the entire boundary, which then sums all the curls about them all, and so states both the total area and the circulation of the micro- or other constituents through Green’s and Stokes’ theorems respectively, and simply by counting the point-entities, and in the manner we have already determined.

As upon the left in Figure 56, we have an initial population of twelve given squares through which our flux is passing as it swirls around through, and so around, the constituent ingredients. Each square has a side length of one unit. Each area is one square unit. The boundary length is therefore 14 units. The total area is 12 square units. We thus have two totals. And since both the population count and the total area are 12, then the divergence or flux density is 12/12 or unity. And further since the curl is the circulation per unit area, we therefore have 14/12 or 1.167. We thus also have two averages.

Now suppose, as on the right of Figure 56, that one of the limit-points is lost. A square is therefore lost. Since this is a closed and simple path that does not cross itself and that is also simply connected across the entire region, the gradient theorem of the line integral, the third of the four fundamental theorems of the vector calculus, applies imediately, and we have established a gradient across our surface, and which is also a potential. We now consider four of the possibilities that can occur in response to that gradient:

- As depicted, and for the first option, let the side length per each square remain at one unit. Each individual area thus remains at one square unit. This tells us that the divergence, an average, does not change. The boundary length, or ciculation, also remains at 14 units. Thus the circulation, a total, also does not change. But since an entire unit has been lost, the total area now changes to 11 square units. This tells us that the flux, a total, has changed. Since the curl is the circulation per unit area and an average, then it must change to 14/11 or 1.272. Thus simply by losing one unit from the population, the curl has immediately increased while the individual unit area, which is the divergence, has remained the same. The circulation has also remained the same while the total area over the population, which is the flux, has decreased. We are more formally expressing this in vector calculus terms by saying that since the numbers have changed, then there has been a change in gradient over the surface; and since there has been a change in gradient then there has also been a change in flux, a total; in curl, an average; but there has been no accompanying change in either the divergence, an average, or the circulation, a total.
- A second alternative is to keep both the curl and the individual unit area or divergence—both averages—at their initial values of 14/12 and 1 respectively. But although the curl or circulation per unit area remains the same at its initial value of 1.167, the total population area still reduces to eleven square units as the unit is lost. The flux in other words still decreases. Therefore, the circulation around the entire population must change—it must also decrease—to 12.833 units. Thus if the curl is to remain the same when a unit is lost then the total area or flux must decrease and the circulation must decrease. There has again been a change in gradient across our surface, but no change in curl or divergence, our two averages, although the circulation and the flux, our two totals, have both changed.
- In both the above scenarios, we allowed the total area, which is the flux and a population total, to decrease when a unit was lost. The third scenario is, of course, for that population value of the total surface area to remain the same at 12 square units. But if the area remains the same even as a unit is lost, then the remaining squares must increase their individual areas to 1.091 square units each. The divergence must in other words increase. This demands that the side lengths of those squares also increase to 1.044 each … which immediately increases the circulation about them all to 14.622, while the curl increases to 1.218. So as the gradient again changed across our surface on this occasion, and the flux, a total, remained the same, the circulation, also a total, and the divergence and the curl, both averages, all changed.
- The final possibility we shall consider involves leaving the curl, an average, at its initial value of 1.167. The circulation, a total, then decreases to 12.833 units. But we now seek to also leave the flux, a total, at its initial value of 12 square units, which of course means increasing the divergence or flux per unit area and an average to 1.091 square units.

And … no matter whether the population of record is a Darwinian or an Aristotelian one, these analyses exhaust the possibilities.

Since the curl, the flux, the circulation and the divergence jointly depend upon an interplay of both averages and totals, then it is impossible for all other values and interrelations to remain the same if even one value changes. There must be other accompanying changes.

We now turn to Gauss’ theorem, the divergence theorem, and the fourth and last of the fundamental theorems of the vector calculus. It now links our surfaces to the volumes that our surfaces create between them, and through which the flux therefore passes:

The flux of a vector field through a closed surface S is equal to the integral of the divergence of that field over a volume V for which S is a boundary (Fleisch, 2008, p. 114).

We can now bring all our newly acquired information together to stipulate—in a completely rigorous manner—how a population seeking to follow the proposal of the Aristotelian template must behave. It is required, for such a population, that when n changes:

- the gradient across the surface must remain flat or else show no change due to that change in n;
- the population must remain incompressible or solenoidal and show no changes in its divergence again because of n;
- the population must be irrotational and show no changes in its curl as are due only to n.

All other changes in values are of course permitted … it is simply that the above three—i.e. due entirely to changes in n—are not. Thus gradients, fluxes, divergences and curls are still free to change for other reasons, even in populations purporting to be those of the proposal of the Aristotelian template.

Now that we have properly described the template proposal, we are ready to pick up our planimeter and use it to closely examine Darwin’s theory, using the vector calculus. Our variables arise from our function, f(n, q, w), and from our three constraints of constant propagation, constant size, and constant equivalence. Our variables are, simply: mass, energy and number. As we have already proven earlier, that is all that we require.

We now turn to our line segment or tunnel of molecules as depicted in Figure 29, and as arranged across a generation of length T. We use our planimeter to measure that line segment all around that boundary. Since we have both (a) a mass; and (b) an energy flux; we shall use our planimeter twice. But by Green’s theorem, we also know that the force F acting all around that boundary is divided into the two orthogonal component forces F1 and F2, which we must now identify.

The first occasion we use our planimeter is as in Figure 57. We measure around the entire boundary for the generation which is the time period, T, relevant to our chosen vector commodity. We count the number of entities, n, at each time point t about that population. We also use the standard methods of chemical analysis to determine the number of moles, q, of each chemical substance and chemical element of which each and every entity is at that moment composed, as each one moves energy and mass through our Gaussian surface. By the Avogadro number this tells us the atomic and molecular mass, mA, of each such component. The sum of all those components at each t over T will tell us the mass flux, M, at that t and for that population. We can calculate a q̅ as the average number of components and molecules held per entity and per second. Again through the Avogadro constant, this q̅ will immediately tell us the average individual mass at that t … with this m̅ then also being the divergence in the mass flux for that population for each t, and therefore also over the entire generation length T, and as it undertakes all activities necessary to produce its progeny through a given cycle of biological potential, µ.

Once we have gone all around the boundary, our first exercise in measurement is complete. We now have the values for (a) the divergence in mass at every point which is m̅; and also (b) the total mass flux, M, both at each point and for the generation. The divergence in mass again measures the number and type of chemical components held per entity per second over the entire population. But in measuring the mass flux, M, we have also measured the constant pressure work undertaken by the population, which is the total mechanical chemical energy used in acquiring all needed components from the environment. This is the first component, F1, of the vector force F acting all around the boundary and so over the entire generation, as the population at large does work and produces its progeny.

As in Figure 58, we now again pick up our planimeter. And on this occasion … at and for every t over T, we measure the total energy in the population in joules. And since we have already—on our previous excursion—measured the masses and numbers of components at every one of those points, and over the same population count, then we are also and automatically computing the energy density at every point … and which is therefore all the changes in the nonmechanical energy over the population, which is the constant volume work. This then allows us to compute both the values for, and the ratios pertaining between, the mechanical and the nonmechanical energies over the entire population, as well as the totals of energy used at each and every point per entity and per unit mass. We are now measuring and comparing all possible changes in DNA and in configuration over an entire population as it grows, develops, and produces progeny. As we measure the energy at each point, which is the Wallace pressure, P, we can also apply the population count at that same point to produce p̅ which is the divergence in the energy flux. And when we apply P to M or p̅ to m̅, we derive the visible presence, V, which is the energy density and the Gibbs energy per unit mass, also at each and every point. We therefore now have the second component, F2, for the vector force F acting all around the boundary and forming the generation for our population.

We can now turn to the Liouville theorem. This tells us, as Gibbs also proved, that the phase density, the distribution of ρ over time, is conserved. The phase space volumes and the accompanying energy totals that this population exhibits define it no matter what changes and transformations it may undertake.We have just used our planimeter to measure the total energy by measuring about the equilibrium age distribution's boundary. Again by the Liouville theorem, this is a defining property for that population, and over that given generation time. And since the time period is the span of a generation as given ratios and properties are repeated, then both the time span T and the energy used over that T are again definitive for that population, for T is simply the distance over which the force, F, has acted through its two components of the mechanical and the nonmechanical energies. And, furthermore, by the first law of thermodynamics, every manifestation of energy must occur through some material medium, which we have also measured as the mass flux, M, for that same generation and at the requisite and relevant points t over T. Thus there must be—and there is—some energy density, V, at every point determinative of the bonds and configurations adopted and so governing the changes and the rates of change between the potential and the actual energies for that population in so far as these are the quantities of mass and the specific configurations of chemical components used. And so therefore, the quantity of chemical components, q, used at every point t over T, and for that generation, is also definitive of that population for it must be uniquely configured for that population to produce the energy and the Wallace pressure being used at that point. Therefore, and through the Liouville theorem, the generation length T and the energy and mass fluxes—P and M—are together a part of the unique specification for any population. Therefore—and again by the Liouville theorem—we have here two fluxes that together uniquely define any population working over any given T and thus stating the boundary conditions. That T and those two fluxes work together with the three constraints of constant propagation, constant size, and constant equivalence to provide the boundary, phase volume, and density distributions.

We now turn to the Helmholtz theorem. This tells us that the curls and the divergences in mass and energy, m̅ and p̅ are in their turn defining properties for any population. Molecule by molecule and chemical bond by chemical bond, these two values between them establish (a) the numbers and types of chemical components acquired from the environment; and (b) how they are configured and are then made available per each unit of time, per each entity, and per unit of mass over that population. The biological entities then use those components, as distributed to them, to form themselves and produce progeny. Therefore, and by the Helmholtz theorem, we have two averages which establish divergences and curls and which also uniquely define any population.

By Proviso 3 of Maxim 1 of ecology, the maxim of dissipation, we at all times have the relation M = nm̅. The mass flux taken over the entire generation is the complete cycle of operations in kilogrammes per second for each t over T, and as establishes the complete work done against constant pressure within that environment and by that population. This is again the complete mechanical work done about that boundary. It is the force and energy relevant to the Liouville theorem as it establishes the work, impulse, and energy for the entire neighbourhood of points through its phase volume. But by Proviso 1 of Maxim 2, the maxim of number, we also and at all times have P = np̅ where P is the sum of both the mechanical and the nonmechanical work done about that same boundary, and for all those values of energy. The value for P, the Wallace pressure, now incorporates all the work done and heat emitted—δW and δQ—over the population … and which are inexact differentials.

We should now also notice that we have described our population with the vector relation P(n, q) insisting it is a vector. We are now describing the force about the population by F(F1, F2), and which is also as a vector, and where n, q, F1 and F2 are scalars. These are both indeed vectors because P and F both obey the relevant rules such that if a, b and c are specified vectors within each of those fields; and if c and d are scalars applicable to both; then:

(1) a + b = b + a;
(2) a + (b + c) = (a + b) + c;
(3) a + 0 = a;
(4) a + (-a) = 0;
(5) c(a + b) = ca + c ;

(6) (c + d)a = ca + da;
(7) (cd)a = c(da);
(8) 1a = a (Stewart, 2007, p. 810).

Those are the complete rules for well-behaved vectors, and P and F abide by them all. Therefore, both biological populations and the forces that act upon them are vectors. We have already demonstrated that they abide by all standard theorems within the vector calculus, these being:

- the gradient theorem of line integrals;
- Green’s theorem;
- Stokes’ theorem; and
- Gauss’ theorem or the divergence theorem

which together link points to lines to areas to surfaces, and so in zero, one, two, and three dimensions and as again as points, then curves, then surfaces, then volumes.

Each of our vector fields P(n, q) and F(F1, F2) also has its standard basis vectors or unit normals … and each of them again follows all the rules for vector spaces and for their bases:

basis of a vector space. A set of linearly independent vectors such that every vector of the space is equal to some finite linear combination of vectors or the basis (James and James, 1992, pp. 27, 194).

Each is therefore independently a separable two-dimensional vector space, with orthogonal components. As thus defined their dot products can vanish, and each of their two components is therefore linearly independent. They are orthogonal in the Euclidean sense. They therefore form axes that define the entire plane of interaction between them. In the case of F, the two components F1 and F2 therefore completely define the circulation in mass that F produces under the influence of biological potential, µ, and as n and q—or whatever else may be our chosen variables—loop about the circuit of the generations under the influence of this force and energy, as it does the work associated with F1. We now identify F1 as the source of the constant pressure mechanical chemical work done by biological entities. It therefore supervises the removal of mass from the environment in growth and also for reproduction; or else returns mass to that environment in dissipation and degradation and always in a measurable mass flux. We then identify F2 as the constant volume nonmechanical work as chemical bonds are configured and reconfigured. And since F acts directly on M, the mass flux, then it is that mass flux that is being distributed both about the boundary and over the surface as it therefore curls about the entities composing that surface, and so enables them to do work both upon themselves and upon the environment and so to move through the biological cycle and produce progeny.

However, the biological entities cannot do work without energy. By the second law of thermodynamics, this is always degrading. Since we need to properly define and measure that energy, we need a third axis that is again orthogonal to the above two. As in Figure 59 we therefore measure that energy with this third axis. Its usable or Helmholtz component will curl about the entities which use it to manipulate the mass of chemical components they capture from the environment, and that they also curl about themselves in an equally measurable mass flux, courtesy of the two divergences of m̅ and p̅.

As is required for this three-dimensional vector space, every point within it can now be described by the triple F(F1, F2, P) where F1 + F2 = nm̅ = M, and P = np̅. Our three axes are therefore mutually perpendicular and have an orthonomal basis. Each has its unit normal. Our three-dimensional vector basis is therefore standard or canonical. As is further required, the mass flux, M, occupies one entire orthogonal plane, and is divided between F1 and F2, which form two axes and as are again the mechanical and the nonmechanical work being done at any time; while the energy derived from their interaction, and so the work done along with the heat absorbed and evolved in all interactions with the environment—being δW and δQ—each occupy the appropriate axis and as according to Maxim 4 of ecology, the maxim of apportionment; and as we again demonstrate in Figure 59.

And with Figure 59, we at last have available the complete description—and so specification—of any and all biological populations, Darwinian and Aristotelian alike. The vector space we have created follows Fleming’s right-hand rule for such spaces, and as required. If we raise our thumb upwards along the energy axis, then our fingers will naturally curl in the direction of the circulation the mass takes around the plane, and along which all biological activity is induced by that energy flux, and which has its positive direction upwards through the plane. Thus energy now passes through the population along that energy axis, and according to the magnitudes of the capturing and escaping tendencies expressed through its chemical configurations, and also as precisely given by the partial differential equation (∂S/∂E)q,β in which, as ever, if the number of constraints, β, and the moles of the components, q, hold constant, while energy, E, changes, then entropy, S, must also change (Encyclopaedia Britannica, 2002; Atkins, 1990). We can now see that if entropy is changing while the visible presence, V, is being held constant, then the population simply must be exercising its mechanical chemical energy and moving positively along the F1 axis. The average individual mass over the population, m̅, is increasing and a positive divergence is in effect; or else n is increasing while that divergence remains constant; or else some combination of both. But if energy is changing while the mechanical chemical energy now holds constant, then the visible presence must instead be changing; or else there is some counterbalancing change in n and V—i.e. in the numbers of entities, how they are configured, and in their divergence, and that serves to leave that mass flux constant. All other possible changes in energy or in mass or in numbers are similarly and easily recognized. They can be rapidly and rigorously apportioned across our three axes, and so within this biological canonical vector space we have now discovered. All movements around or along the M1-M2 axis, which are all in response to F acting about the boundary, are now exact differentials. They reflect either changes in quantities in mass, or else in how that mass is configured. All changes along the energy axis instead reflect the inexact differentials of appropriate work done and/or heat evolved in the environment. This is again now straight forwards and utterly rigorous.

Figure 60 is a clearer way of seeing the circulation of the generations and the behaviour of biological populations under energy. The energy flux again moves upward; and the right hand can again curl around within the vector space and in the direction of the circulation in mass within that space, and entirely according to a measurable vector curl as rigorously defined. The length of the circuit is again T, the generation length. The mechanical chemical work done by the population now absorbs the mass, M, from the environment. It is measured by the mass flux axis stretching orthogonally from right to left. The third axis then measures the energy density—and the nonmechanical work—and again projects directly into and out of the page. Each individual biological entity then forms a trail across the generation length, and as shown in the figure. Each entity forms a part of the circulation of mass, curling its mass and its energy about itself as it does work and configures the chemical components made available to it. Reproduction then consists, as seen, of progeny inheriting mass, energy, and components from their progenitors over a given time span.

Given the Liouville and Helmholtz theorems, then every population can be uniquely specified by stating its two totals of the mass and the energy fluxes, and its two averages of its curl and its divergence again in both mass and energy. And since the energy flux depends upon the energy density, then simply stating the mass flux, its divergence, and its curl, along with the energy density at every point will also suffice.

The population totals in mass and energy form—and as we have already determined—the boundary conditions for both the Liouville and the Helmholtz theorems namely: that their curls and divergences must approach zero more rapidly than 1/r2 as they each approach infinity, and such that when r = ∞ then [∇ • P]∞ = 0 for the divergence and [∇ x P]∞ = 0 for the curl (Weisstein, 2011). Those boundary conditions must be met over a given time span, T, which is the time period for the generation which is the time in which some components degrade and dissipate to be replaced by others, and as according to Proviso 1 of Maxim 1, the maxim of dissipation, which is ∫dm < 0.

A ‘unit engenetic population’ is now one that averages one biomole over the cycle of this substitution. The entities concerned distribute themselves over the cycle, in a specified number density, to act first as progeny and then as progenitors to produce more like themselves. This gives us the “absolute generation length”, Z, expressed in seconds per biomole, or seconds biomole-1. This absolute generation length immediately implies that an average of one biomole of entities over the given time span can successfully balance its escaping and its capturing tendencies such that some entities are lost, but which are duly replaced, and so that the given average is maintained over the stipulated time span, with the net gradient being zero. The average number stays at one biomole over the generation, and there is no net increase or decrease in numbers. What therefore matters is, rather, the maintenance of a given distribution and number densities, and not merely the absolute time span which is simply a function of boundary length. By the Liouville theorem, this is free to vary as the phase volume is transformed to remain constant. Therefore, when several generations are taken together, then the absolute generation length, Z, will appear as the average boundary length for that energy volume and over all those time spans, for it is entirely permissible for the population to on some occasions work at a faster rate to complete a more rapid cycle or part of cycle, only to slow down, and so to take a slightly longer time period to do the similar work on the same quantity in biomoles of entities and and moles components, N and q. Since we are using our cylindrical and/or spherical coordinates to measure pertinent transformations, then it is the ratio between n and q that establishes the circuit, and not a given period of time. The absolute generation length is therefore a weighted time interval and average over many generations and is again a distribution of entities for a one biomole average.

The other two properties needed to define a population come directly from the Helmholtz theorem. They are the two divergences in mass and energy, m̅ and p̅ in so far as these state both the curl and the divergence under the same boundary conditions. Therefore, each and every population is uniquely specified by stating:

- its absolute generation length, Z, in time units per biomole;
- its average individual mass over that time period, m̅; and
- its average individual energy or Wallace pressure, p̅, also over that same time period.

By all the laws of science; and also by the canons of the vector calculus and its vector bases; then no two populations can share those same three values without also being members of the same population or species. These three are together the only specification needed. They are wholly sufficient to define any population and no other is required.