49. “A fine hypothesis”

The Darwinian and the Aristotelian proposals now need closer examination. We need to contrast their properties. We shall do this by specifying a hybrid: an attempt to satisfy them both. We shall construct a population that respects Darwinian fitness, Darwinian competition, and Darwinian natural selection … but that still does not evolve.

A Darwinian population that has no evolutionary potential must simply return the same values for Z, and every generation. It is completely free to distribute its entities, its components, and its energies howsoever it wishes within each generation. It must simply respect the constancy of those average values. Thus where there is a greater number of entities, n, at any t, and/or where they have a greater mass, and/or where they are configured to possess greater energy; then there must be a contrary distribution or subpopulation of sufficiently lesser magnitude to maintain the average, and so the distribution. For every variation tending to carry the population in one direction, then there is another carrying it in another, and the sum of those variations is insufficient to establish a long-term evolutionary trend. The values for the three constraints are unchanged. A reciprocating population of this kind shows Darwinian natural selection, competition and fitness … but since the given average values and totals are maintained between one T and the next, then no evolution occurs.

There is, however, a difficulty. The population we have just described may not evolve, but it is still completely dependent on numbers and on its responses to variations in numbers. It remains intricately linked to its responses to numerical variations. The question now is whether a population of this kind is Aristotelian or Darwinian.

The difficulty here is that we have given the exact description of a population undergoing Darwinian fitness and competition. Any numbers lost at one point must be made up at another through the exercise of a suitable potential, which we can also measure as µ. And since this population still depends on numbers, then it does not satisfy the premises for the proposal of the Aristotelian template which requires zero dependency on numbers. We still therefore lack a suitably rigorous definition for an Aristotelian population.

If we want to build a population that respects the proposal of the Aristotelian template, we must first understand its behaviour. We have constructed a canonical vector space. From it we can now draw the following conclusions about any and all changes in n:

  1. The Wallace pressure, P, of a biological system is simply its energy flux. It is, at each moment, the sum of all the energies being used and emitted by an entire population. Since a given population is of size n, then the average individual Wallace pressure—which is the divergence in energy—is immediately dh̅ /dt = = P/n. Therefore, if ever n changes, then either one of the Wallace pressure, P, which is a total; or else the divergence, which is the average, must also change. But unfortunately … the Liouville and the Helmholtz theorems together tell us that both the total and the average are defining properties of any population. The former defines the phase volume, and therefore the boundary around which force is applied; while the latter uniquely specifies any vector such as P(nq) through the divergence and the curl for each and every population. But those values are unique for every t over T for each population. Thus if n changes then either P or must change, with both being definitive. The population has therefore immediately changed and so breached the template by showing a dependency on n. There is no other possibility. And … we can easily measure both.
  2. If n changes, then either the total mass, M, over the population, or the divergence, , in mass, must also change. But if M changes then we immediately have a quantitative difference in the number of chemical components being maintained over that generation, and over those entities. But … being one of the two totals it is a defining property of any population. And if changes then we have a qualitative change as the form and/or type of entity changes. We also and again observe, in this latter case, and through the Helmholtz theorem, that the divergence in mass, which is the average individual mass, is definitive through the Avogadro constant, which works through the same P(nq) as above. Something definitive therefore changes whether it comes through or M, as soon as n changes, for it is a mathematical given that one or the other simply must change along with it. Either one, however, involves a fundamental change in the definition of that species or population. There is again no other possibility; and we can again measure both.
  3. If n changes then either P/M or / must change. It is not possible for all four to remain the same. But unless both P and M or and change by exactly the same measure, then we immediately have a change in energy density, which is a change in the way chemical components are configured, and so a change in the quantity of work done and heat evolved per unit time per entity and per unit mass. But this is the very definition of ‘adaptation’. So not only do we have an immediate conflict, in this change, between the constraints of constant propagation and constant size; but the constraint of constant equivalence also changes. If it indeed happened that either P and M or and changed by identical amounts then we simply return to the joint and already stated conundrums in (1) and (2) for both are in any case active. There are no other possibilities.

We can straight away note that the Darwin principle respects the fact that the values of and X are in all cases linked through n and that they have consequences across all three constraints … whereas the Aristotelian template proposal does not. This instead conjectures that P and on the one hand, and M and on the other, can each be independently—and without consequence—altered, which premise is demonstrably false. These numbers and their requirements now attest that the proposal of the Aristotelian template is simply impossible.

Any change in a population can also be expressed using our function f(nqw) with the gradient then being f = (f/n, f/q, f/w). It is indeed notable that this function is itself another vector for it can be written as f(P(nq), w) with each value for w being distinct and following all the before-stated rules for vector spaces and bases. Our proposed non-evolving population containing variations also contains fitness, competition and natural selection for it responds to any and all changes imposed by the environment … including those based on number and so on f/n. Since this partial differential is separable, we can always determine its precise value.

A Darwinian population responds with a directional directive that always points along the line of maximum gradient. An Aristotelian population is free to respond to f/q and f/w … but it may by definition not respond to any stimulus from f/n. It is therefore forsworn from following the directional derivative and its line of maximum gradient. And since it cannot actualize that potential, then it cannot restore any changes caused by n. It is thus impossible for the other two components to make up the deficiency, and the proposal of the Aristotelian template again fails for this population is doomed to become extinct.

The problem we now face is that supporters of this failed proposal always have a counter-argument ready. They argue that not even a proof of the logical impossibility of the template proposal is enough to invalidate it. They argue that it is the template itself—i.e. its character—that guarantees that it holds. No other factor is required. They therefore argue that a theoretical proof is completely beside the point.

The supporters of the Aristotelian template insist that there is a larger point at stake: the question of the ‘meaning’ of biological existence and events. That meaning overtly or covertly ‘participates’ in the manifest world through whatever force first initiated, and therefore oversees, biological events and the template. It is impervious to rationalist arguments including these about impossiblity based on numbers and gradients:

Kepler, for example, saw himself as a priest ‘of the Most High God with respect to the Book of Nature’ who, by discovering the pattern which God had imposed on the cosmos was ‘thinking God’s thoughts after Him’. Francis Bacon described his plans for the reform of natural philosophy as a work of preparation for the Sabbath. The Sabbath he had in mind was the ultimate, everlasting Sabbath after the Day of Judgement, which he believed would be ushered in, according to biblical prophecy, after the augmentation of the sciences. The natural philsophies of Pierre Gassendi, René Descartes, Robert Boyle, Isaac Newton and Gottfried Liebniz were each carefully developed in order to provide support for the individual theological views of their respective authors. …

One of the major concerns of the mechanical philosophers, for example, was to show how God interacted with the mechanical world (Henry, 2002, p.86).

Newton’s gravitational system serves as a notable example of these arguments and counter-arguments. It worked extremely well, but still had its anomalies … certain ‘irregularities’, as Newton called them, in planetary motions (Thorp, 1802, p. 264). The Moon, for example, was never exactly where his calculus predicted it should be. He could not explain the irregularities so he attributed the planets' observed orderly behaviour to a higher and divine intervention:

Upon this head, I think it not improper to mention a reflection made by our excellent author upon these small inequalities in the planets motions, which contains in it a very strong philosophical argument against the eternity of the world. It is this, that these inequalities must continually increase by slow degrees, till they render at length the present frame of nature unfit for the purposes it now serves. And a more convincing proof cannot be desired against the present constitution’s having existed from eternity than this, that a certain period of years will bring it to an end. I am aware, that this thought of our author has been rendered even as impious, and as no less than casting a reflection upon the wisdom of the Author of nature for framing a perishable work. But I think so bold an assertion ought to have been made with singular caution : for if this remark upon the increasing irregularities in the heavenly motions be true in fact, as it really is, the imputation must return upon the assertor, that this does not detract from the divine wisdom. Certainly we cannot pretend to know all the omniscient Creator’s purposes in making this world, and therefore cannot pretend to determine how long he designed it should last; and it is sufficient if it endure the time designed by the Author. The body of every animal shows the unlimited wisdom of the Author, no less, nay, in many respects more, than the larger frame of nature : and yet we see they are all designed to last but a small space of time (Encyclopaedia Britannica, 1823, Vol. III, p. 124-125).

What happened next is notable. It has not yet happened to Darwin’s theory. The idea of divine intervention suffered a mortal blow some forty-five years after Newton’s death when Laplace decided to take those irregularities in hand. Those irregularities did not arise, in his view, simply so divine intervention could make itself evident. Nor did they arise because cosmic phenomena were intractable and irreducably complex. Laplace's view was that the irregularities arose because the calculus itself needed improving.

Inspired by his vision Laplace transformed the use of differentials and developed the idea of a potential. His Laplace operator—which we used to prove the Helmholtz theorem—is the sum of many specified second-order derivatives. It even allowed him to link the macroscopic to the microscopic for it allowed him to attribute a suitable potential to every large body, every one of which he saw as being composed of minute corpuscles, all moving under gravity's potential. He extended his operator to embrace those minuscule corpuscles of which each small terrestrial body was composed. In conjunction with Lavoisier, he extended his ideas into heat and developed the now outmoded caloric theory.

In his work on an improved gravitational theory, Laplace made Newton's divine intervention theory moribund. He produced a slew of theorems that affirmed the innate constancy and invariability of planetary motion. He not only proved that the mean distances of the planets were constant, he accounted for their variations. He proved that the solar system was inherently stable and that it did not need divine intervention to regularize it. He then solved the outstanding problem of the irregularity of Jupiter’s and Saturn’s orbits, proving that their inclinations and eccentricities were always small, constant, and completely self-correcting. He even solved the last major problem facing the astronomers and mathematicians of the day: the Moon’s irregularities. He found a periodic motion whose cycle was in the millions of years.

Laplace's achievements highlighted the tussle between the scientific and the religious approaches to, and explanations of, phenomena:

The story of Laplace’s encounter with Napoleon over the Mécanique céleste shows the mathematician as he really was. Laplace had presented Napoleon with a copy of the work. Thinking to get a rise out of Laplace, Napoleon took him to task for an apparent oversight. “You have written this huge book on the system of the world without once mentioning the author of the universe.” “Sire,” Laplace retored, “I had no need of that hypothesis.” When Napoleon repeated this to Lagrange, the latter remarked “Ah, but that is a fine hypothesis. It explains so many things.” [emphasis in original] (Bell, 1986, p. 181)

Those who support the proposal of the Aristotelian template put exactly the same argument Lagrange puts here in favour of divine intervention: “it explains so many things”. They argue that it is through the template’s interactions that ‘oversight’ in the biological world world occurs … and so therefore also in human destiny and behaviour. But no matter what Lagrange’s retort, the power of the mechanical philosophy—which took centre stage in the scientific revolution—proved able to explain “so very many things” that resorting to templates, divine interventions, and other non-measurable “things” with respect to gravity became simply redundant and unnecessary.

We have followed Laplace and we have used his operator. We must now turn to a practical demonstration … such with our Brassica rapa experiment. We must use our vector calculus method to explain “so many things” in the behaviour of those B. rapa entities that resorting to alternative explanations such as “templates” and “divine intervention” becomes equally redundant and unnecessary—not so say inaccurate—because the vector calculus also succeeds in explaining “so many things” in both biology and ecology.