47. Finding the way to test the validity of natural selection

Although we seem to have made some progress, it is as well to recall that there are those who are intransigently opposed to Darwin and his theory. They are opposed because although Darwin saw a problem and addressed it, they instead saw no “problem” in need of a “solution” in the first place. But since, like Aristotle, Darwin lacked a sound model that could be rendered in coherent mathematical terms, he could not silence his critics, who are still abroad.

There is an unfortunate difficulty with those who oppose Darwin. The basis of their claims is not easy to pin down, and they are rarely amenable to logic:

An alert reader will have noticed that … we have not been presented with a summary of ID theory: though books entitled Intelligent Design have appeared. ID creationists are remarkably evasive about what their theory is besides vague claims such as intelligence guides organic evolution or that ID has something to do with the transfer of information (Sarkar, 2007, p. 20).

Nevertheless, without clarifying the basis of that opposition in suitable terms, we cannot take measurements and come to determinations … if that is possible. We therefore seek a simple, clear, and robust way to characterize that opposition, and that is also susceptible to measurements using the model we are busy constructing.

The model we are constructing is of course based on mathematics … a word that is in itself subject to the etymological fallacy. Its meaning has changed substantially over historical time. Back in the days of Pythagoras, for example, the famous Pythagorean Brotherhood was divided into two groups. There were the mathematikoi who seem to have made the principal mathematical discoveries in what we would now call pure mathematics. Then there were the akousmatiki or ‘hearers’, who sought to find meaning in those discoveries (Kahn, 2001 p. 15):

The Pythagorean brotherhood’s conception of mathematics was broad and human. All of their philosophy, of which mathematics was only a subordinate if important part, was directed to but the one end of sane, civilized living. Arithmetic, geometry, astronomy, and music were the four divisions of their mathematics. This tetrad was to survive for centuries, passing through the Middle Ages in the attenuated quadrivium which formed four-sevenths of a liberal education, the rest being the trivium of grammar, rhetoric, and logic (Bell, 1992, p. 60).

By the time of St. Augustine of Hippo, the mathematicus was still little different from what we would now think of as an astrologer. Once having computed the movements of the planets, the mathematician’s duty was to find meaning in those numbers—something we now consign to astrology. Augustine was therefore being mathematical, and was carefully distinguishing between these two different uses when he said, somewhat scornfully:

Quid ergo vanius, quam ut illas constellationes intuens mathematicus, ad eumdem horoscopum, ad eamdem lunam, diceret unum eorum a matre dilectum, alterum non dilectum? So ponder what could therefore be more vain than the mathematician who intuits from those constellations—from the very same horoscope, from the self-same moon, indeed—that one of those born from the one mother is to be loved, while the other is not? [Translation by author] (Augustine, 408, II.xvii.36).

A principal difficulty here is not, in fact, those who oppose Darwin. A principal difficulty is Darwin himself, along with those who regard his theory as a true explanation for biology. The difficulty is that the mainstream of biological thought has thus far failed to frame Darwin’s theory with the clarity now accorded to Newton, Joule and Faraday. Yet in spite of that manifest deficiency, Darwin’s has become the backbone of the science in which he worked. Theodosius Dobzhansky even went so far as to write an entire paper entitled: Nothing in biology makes sense except in the light of evolution to indicate its status (Dobzhansky, 1973).

It is worth noting the difference between the clear, precise, and mathematical manner in which Newton’s, Joule’s and Faraday’s answers to the various problems they addressed have been expressed. This has proved a tremendous advantage when compared to the manner in which Darwin’s have been expressed. As knowledge of the various phenomena those three each described has increased, so also has the explanatory power of the ideas they were the first to provide. In modern terms, for example, the essence of the Joule experiment is that as a gas expands adiabatically against a vacuum—i.e. with no external or opposing pressure—then its internal energy, U, is at all times a function of its temperature, T. Any loss in internal energy immediately implies a fall in temperature. Joule in other words sought to measure (T/V)U—i.e. the rate of change of temperature with volume, as internal energy is held constant. And once the proposal is expressed that way, the cyclic rule of partial derivatives immediately converts it as follows:

(T/V)U = -(T/U)V (U/V)T = -(1/CV) (U/V)T

and where, in the case of an ideal gas, (U/V)T = 0—i.e. there is no change in internal energy with respect to volume when temperature is held constant, which is what Joule was trying to determine. Darwin’s theory does not seem to lend itself to such clear and rigorous treatments.

The problems we currently face in dealing with Darwin and his theory, and in trying to succinctly characterize the nature of the opposition, could hardly be made clearer than by contrasting his relatively respectful opinion of Aristotle, and of Aristotle’s biology, with the following starkly opposite assessment:

“Aristotle ‘was a man of science’ in the modern sense. He was a careful collector and observer of an enormous range of acts … much of his work is still regarded with respect by scientists who care to study it”. These two sentences by the humanist Goldsworthy Lowes Dickinson betray an almost majestic incomprehension of the character of science and of Aristotle’s influence on science in the modern sense … (Medawar and Medawar, 1985).

This assessment of Aristotle shows a grave misunderstanding not only of Aristotle, but also of the true source of the admittedly malign influence being attributed to him. That complete misunderstanding of Aristotle is in fact made clearer a little later in the Medawar’s continuing analysis:

Moreover, although Aristotle’s philosophical origins—on teleology, for instance—command respectful attention, the pioneers of or spokesmen for the new science of the seventeenth century (men such as Robert Boyle of The Sceptical chymist, the Reverend Dr. Joseph Glanvill, Francis Bacon the Lord Chancellor of England, and the poet Abraham Cowley) repudiated the doctrinal authority of Aristotle with weary exasperation (Medawar and Medawar, 1985).

And that makes the essential point. It also establishes the problem Darwin faces in our own era. Aristotle was not responsible for the “doctrinal authority” being referred to. It is the same doctrinal authority—and one more associated with St. Augustine—that Darwin confronts. Aristotle was surely not responsible for the unquestioned, and unquestionable, command over all intellectual theorizing that came to be the responsibility of his every utterance. He after all died several centuries before the birth of Christ: in 322 bce. He spoke as clearly as was incumbent on any philosopher. The historical events the most clearly associated with his name, and that the Medawars describe, did not occur until the age of the Scholastics nearly a millennium and a half after his death.

To confuse Aristotle with what was done in his name is to do both Aristotle and the clarity of thought we seek a singular disservice, for it makes it impossible to find and to properly characterize what that opposition might be. Aristotle was again not responsible for the stifling of debate upon the various issues on which he spoke. That was a historical accident that cemented itself around his various writings. It surely had nothing whatever to do with him personally—except in so far as, like Plato, he left a coherent and coordinated, not to say voluminous, body of works behind him around which that kind of dictatorial authoritarianism could congeal. There is, again, a big difference between what Aristotle wrote, and the meaning others insisted on imputing to it, simply because there was nothing and no-one else to fufill the tasks they wanted fulfilled.

It is also true that the Medawar’s assessment of Aristotle’s more specifically biological writings is in many ways valid, but a reason for the events alluded to is needed, particularly when they impact on Darwin’s theory which is in many ways the replacement for Aristotle’s. The Medawars make scant effort to provide that reason, instead saying:

The biological works of Aristotle are a strange and generally speaking rather tiresome farrago of hearsay, imperfect observation, wishful thinking, and credulity amounting to downright gullibility. What empirical evidence can have convinced Aristotle that the semen of youths between puberty and the age of twenty-one is “devoid of fecundity” and that young men and women produce undersized and imperfect progeny? Prolapse of the uterus and menstruation as often as three times within a month are said to be symptoms of “excessive desire” (Medawar and Medawar, 1985).

Aristotle’s thoughts and conclusions would certainly seem fanciful and without any conceivable justification … if the Medawars did not immediately, and also, provide us with the model Aristotle was using—the background thinking that motivated his thoughts and speculations, and that therefore makes them understandable. Aristotle’s was far from a mathematically precise model, when judged by present day standards, but it was still logical, and it was a model of sorts. It certainly shows that he did not work in a vacuum or without some form of intellectual guidance:

Aristotle was a firm believer, for example, in the Hebdomadal rule, that everything goes in sevens. Man has seven ages, each seven years long, so anyone with a true understanding of nature would realize that human semen must be infertile between the ages of fourteen and twenty-one, for would not fertility in this period contravene the Hebdomadal rule? And so what ought to be became what was. Sometimes of course Aristotle is right; his writings were so voluminous he could hardly fail to be correct sometimes (irreverent thoughts of monkeys and typewriters steal into the mind) (Medawar and Medawar, 1985).

It is easy to mock, but the use of sevens in that way, and as a guide to thought, was common in his day. It is also very easy to mock Aristotle for not using the scientific method, but it took the combined efforts of many astute minds, including Bacon and Galileo, to mathematicize and systematize that method into the coherent system we enjoy today. Aristotle, like Plato, used the best mathematical and geometric philosophies and methods:

Aristotle uses mathematics and mathematical sciences in three important ways in his treatises. Contemporary mathematics serves as a model for his philosophy of science and provides some important techniques, e.g., as used in his logic. Throughout the corpus, he constructs mathematical arguments for various theses, especially in the physical writings, but also in the biology and ethics. Finally, Aristotle’s philosophy of mathematics provides an important alternative to platonism (Mendell, 2008).

The preference for textual study, rather than for observation, was certainly not Aristotle’s invention. He after all wrote the very texts that others sanctified after his death, so he can hardly be accused of only working textually. He observed … and far more so than Plato. Outside his biology, he might not have observed quite enough for modern tastes, but he cannot be accused of not observing at all, nor of not reflecting on those observations.
The real issue with Aristotle—and so therefore also with Darwin—is more that such traits as the use of sevens opened him up to some very specific Christianized interpretations of his work. And … this is the major point. Aristotle was used to justify positions for which informed scientific debate or questioning was simply impossible. He was used to justify and to find meaning in, and for, already entrenched positions. This doctrinal imperative then grievously affected issues—such as with Darwin—which a steadily expanding knowledge base threw into question.

It was not only Aristotle’s biology that was eventually questioned on these grounds. So also were his dynamics and his cosmology. As far as his cosmology was concerned, it is again important to distinguish Aristotle’s own position from the one he was conscripted to defend.

There were, traditionally, two main ideas on the origins of the universe. Neither could be rationally debated. There was no real evidence for either. Aristotle’s view—which simultaneously justified and was based upon his vision of indefinite and perpetual circular motions for the planets—was that the universe had no specific moment of creation. The cosmos at large had existed forever in a stable and unchanging condition. It would persist in that condition eternally (Halvorson and Kragh, 2011).

The second option—which was obviously the Roman Catholic Church’s preferred position in its quest to find meaning—was that the universe had been created at some very definite date and point in the finite past. It had then remained in that condition and would do so indefinitely. It would only either cease to be, or else undergo some fundamental change, when specific instructions were given by its Author. Thus Bishop Ussher calculated that Sunday 23 October 4004 bce, was the first day of Creation; that Adam and Eve were driven from Paradise on Monday 10 November 4004 bce; and that Noah’s Ark touched down upon Mount Ararat on Wednesday 5 May 2348 bce (Ussher, 1658). Not to be outdone his contemporary John Lightfoot, Vice-Chancellor of Cambridge University, in fact preceded him in calculating that Creation began at nightfall, with the creation of humanity taking place at 9 am (but both possibly in 3929 bce?) (Lightfoot, 1644).

Aristotle says that the universe was never created … whereas Christian doctrine declares that it was. Indeed, the fact that precise dates could even be proposed for events previously considered timeless or mythological should highlight the contrast between these two more traditional cosmologies. They clearly reflect very different viewpoints. As with many other such pronouncements, Biblical doctrine was clear and dogmatic.

Since it took a considerable intellectual effort by some of the greatest Scholastic philosophers—such as St. Albertus Magnus, William of Ockham and St. Thomas Aquinas—to square Aristotle’s quite contrary eternally-existing-and-so-no-origin-is-necessary cosmology with fundamental points of Christian doctrine, based as those were on a specific moment of Creation, then it was surely understandable that once Aristotle's teachings had become the foundations of Christian practice, they were not going to be easily relinquished. Aristotle was therefore repeatedly invoked to defend the Christian orthodoxy of a static and changeless universe that already had all its essentials in place … and that neither developed nor evolved in any way.

The question of the universe’s origin was not the only outstanding one. There were some very particular difficulties with the different ways in which Aristotle, on the one hand, and the Church, on the other, understood the words psyche, or anima, or soul. Since their views on time and creation were different, then theologians of course understood “soul” quite differently. That different perspective in its turn influenced the biological perspective they took upon the human animal. Their view was—again—quite different from Aristotle’s. Theirs was based upon a more dualistic conception. Soul, to them, was something distinct. It was something quite separate from matter. Soul could undergo entirely distinct processes such as salvation, redemption, and life after death. To Aristotle, however, soul and body formed an inseparable amalgam. Soul was more, to him, a vital force that exhibited itself in particular ways and as a distinct principle or force contained within each organism, and that then guided that organism. Plants exhibited the most basic form of this power:

The soul of an animate organism, in this framework, is nothing other than its system of active abilities to perform the vital functions that organisms of its kind naturally perform, so that when an organism engages in the relevant activities (e.g., nutrition, movement or thought) it does so in virtue of the system of abilities that is its soul. …

Aristotle seems to think that all the abilities that are constitutive of the souls of plants, beasts and humans are such that their exercise involves and requires bodily parts and organs. This is obviously so with, for instance, the abilities for movement in respect of place (e.g., by walking or flying), and for sense-perception, which requires sense-organs. Aristotle does not, however, think that there is an organ of thought, and so he also does not think that the exercise of the ability to think involves the use of a bodily part or organ that exists specifically for this use. Nevertheless, he does seem to take the view that the activity of the human intellect always involves some activity of the perceptual apparatus, and hence requires the presence, and proper arrangement, of suitable bodily parts and organs; for he seems to think that sensory impressions [phantasmata] are somehow involved in every occurrent act of thought, at least as far as human beings are concerned (De Anima 3.7, 431a14-7; 3.8, 432a7-10; cf. De Memoria 1, 449b31ff.). If so, Aristotle in fact seems to be committed to the view that, contrary to the Platonic position, even human souls are not capable of existence and (perhaps as importantly) activity apart from the body (cf. De Anima 1.1, 403a3-25, esp. 5-16) (Lorenz, 2009).

In spite of these marked differences in conception and perception between Aristotle and Christian orthodoxy, it did not take too long before Aristotle’s entire biology, along with his vision of an uncreated yet eternal universe, had been transformed into the Western theological conception of the soul as a unique attribute of the human. That conviction drives biological discourse.

The theologians also of course annexed Aristotle’s taxonomy, his systematics, into their service:

Aristotle was a keen observer and a seminal logician, and he exercised both of these talents in his taxonomy. Thus he spelled out the key idea—that in order to classify in a formal, reproducible manner, you have to rely on something more than the gestalt that enables songbirds to differentiate between pigeons and hawks. You have in fact to define the criteria of your classification. In practice, you have to carefully describe the things in question, and then create groupings according to the features they have in common.

But Aristotle also showed that in the case of living creatures, this process is far from straightforward. You can produce different classifications depending on which features—or characters, as modern biologists prefer to say—you choose to focus on. But he also showed (another crucial insight) that some sets of characters seem to give more satisfactory results than others. Thus he quickly saw that a classification based on numbers of legs produces obvious anomalies; for example, human beings become lumped with birds. His alternative proposal—to divide animals into oviparous (egg-layers) and viviparous (by which he meant exclusively mammals)—has a more satisfactory feel even though, taken alone, it does lead to further problems (and it helps that he knew nothing about the duck-billed platypus, one of the few surviving mammals that lays eggs).

All in all, then, Aristotle did not arrive at anything resembling a modern classification, but he did confront the crucial issues: the need to state clearly the criteria for classification; the fact that different criteria produce different results; and the inescapable need, therefore, to choose between different criteria [emphasis in original] (Tudge, 2000).

The distinctively theological vision of soul had now fused with Aristotle’s biology to became an expression of a divine drama with a unique point, intent, and time of origination, complete with sacerdotal and purposive drama.

Perhaps the best way to answer the charges laid against Aristotle, and also to put Darwin in a clearer context, is to show how much they deflect from the real issue at stake. We need only compare Aristotle to Newton, generally reckoned as the greatest scientist of any age:

Newton cannot be read as some kind of proto-deist because of his voluntaristic conception of the attributes of God and, especially, because of his philosophical methodology concerning how man arrived at a rational understanding of God’s power and dominion. Certainly, Newton promoted the design argument, but he utilized it in conjunction with the argument from prophecy to emphasize the nature and the extent of God’s power and argued that, consequently, mankind ought to fear his wrath. In addition to interpreting the Book of Nature, Newton also interpreted the Book of Scripture. The Bible, which Newton acknowledged to have many human accretions and errors, contained a central core of truth in key prophecies in Daniel and Revelation and the Newtonian scientific exegete must continue to record fresh instances of fulfilled historical prophecies as a foundation for inductively justified human expectations regarding the future (which must, of course, be subject to revision owing to God’s power to alter nature’s course) (Force and Popkin, 1990, p. 135).

Newton thus used the Bible to both justify his physical theories, and to for example work out precisely when the Roman Catholic Church—which he abhorred for its complete prostitution, in his opinion, of the words of Jesus Christ, Saviour—would at last be driven from the face of the Earth. At that auspicious time, or so Newton believed, God’s true intent for humanity would be realized. A divinely inspired era of peace—personally headed by the Lord Jesus Christ, who is God’s Supreme Servant—would at last begin. Newton was generally reluctant to set an exact date for this event, but he is nevertheless on record as saying:

… the period … if dated from the complete conquest of the three kings A.C. 800, will end A.C. 2060. It may end later, but I see no reason for its ending sooner. This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fancifull men who are frequently predicting the time of the end, & by doing so bring the sacred prophesies into discredit as often as their predictions fail. Christ comes as a thief in the night, & it is not for us to know the times & seasons wch God hath put into his own breast (Newton, 1700).

Although Newton’s most favoured date would seem to be 2060 as he gives it above, his calculations show that anyone living between 2035 and 2954 might in fact have the opportunity to witness this transformation of the Earth. This will be the ending of time as we know it and as he described it … and also, perhaps, its real beginning?

And lest Newton’s true intentions for his scientific corpus be in any way doubted, then when he released the second edition of Principia in 1713, during Queen Anne’s reign, he took great care to add to his General Scholium—his ‘comment’ or ‘lecture’—his declaration that the Principia is not, and was never intended to be, ‘just’ a physical theory. The Scholium was Newton’s intended bridge so his readers could properly understand his own true intentions, and so be suitably undistractedly imbued with the presence and the power of the Supreme Author of the universe in which he so firmly believed:

This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. … This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God pantokratwr or Universal Ruler ; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures for ever, and is every where present; and by existing always and every where, he constitutes duration and space (Newton, 1689).

Newton had a precise mathematical model to discuss the one set of events. He may have used mathematical techniques to calculate values for the others, but he had no similarly sound mathematical model to back them. Therefore, his pronouncements on those other topics have largely been ignored by scientists. They may be of historical curiosity to nonscientists and to scientific biographers and historians, but there are many other such millenarian predictions to go along with Newton’s, and there is no way to verify any of them … until it is too late.

The larger and more relevant point is that, thanks to Newton’s model, what to keep and what to ignore in his corpus is relatively easy for a scientist to decide. For scientific work, we keep the model and discard the rest with whatever attitude—be it amused indulgence or irritated scorn—each given scientist chooses to adopt.

Aristotle and Darwin present a quite different problem. Neither of them left a mathematical model remotely comparable to Newton’s for gravity. This therefore raises the issue of how we pick what to heed and what to ignore in either one of them. Lacking a sound model, what criteria should we use to declare specific aspects of either of these two’s proposals valid, particularly when it comes to things that are not easy to measure … such as claims about the origins of species? Lacking a sound model, this decision is surely arbitrary, as the current field of discourse around Darwin attests.

Darwin is similar to both Joule and to Faraday in that he, too, was a pioneer. All three indicated the overall direction from which suitable answers should come to the questions they posed: namely, the phenomena they each studied so thoroughly. None could frame their answers appropriately, but all three were scientists of the first rank who all agreed that—as far as possible—the answers should come directly from within the self-same phenomena that had first provided the questions.

Faraday was initially as unclear as Darwin, but Maxwell later provided all the clarity Faraday needed with his vector calculus and his four justly famous Maxwell equations of the electromagnetic field: E = ρ/ε0; B = 0; x E = -(B/t); and x B = µ0 (j + ε0(E/t)). And the PV T that Boyle first discovered, and that Mayer and Joule both later exploited, went through a similar process as later researchers such as Carnot, Helmholtz, Thomson (Lord Kelvin) and Clausius—as well as Maxwell—provided the thoroughly comprehensive equations we enjoy today. For this reason Joule and Faraday now have specific and easily quantifiable terms and references, along with behaviours associated with those terms and references that are equally clear. Everything is measurable and predictable. Darwin at present lacks that luxury.

The answer to this enigma lies with Newton. Although his answers to his particular gravitational questions were always mathematical, they could not be complete. There were tensions. Those tensions are the same ones we see in Darwin. In Newton’s case they are linked to the assumptions Newton made about gravity, and that he also described in his Scholium:

Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.

  1. Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.
  2. Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is vulgarly taken for immovable space; such is the dimension of a subterraneaneous, an æreal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be perpetually mutable (Newton, 1689).

Newton’s definitions of absolute space and time are almost as mythological as those he sought to replace. There is no way to validate them. But he needed those now scientifically outdated concepts of absolute space, absolute time, as well as of the instantaneous transmission of gravitation, to make his theory work. Leibniz was but one who thought Newton’s proposal ridiculous:

In his letter to Newton, Leibniz praises the tremendous accomplishment of the Principia; he then contends, unsurprisingly, that Newton’s theory of gravity fails to indicate not only the cause of gravity, as was later acknowledged by Newton himself in the General Scholium — which appeared in the second edition of the Principia in 1713 — but also the cause of the phenomena treated by Newton’s theory, especially the planetary orbits. Leibniz of course argued that the phenomena in question must ultimately be understood as following from some cause that meets what he took to be the strictures of the mechanical philosophy. … Hence Leibniz writes: “You have made the astonishing discovery that Kepler’s Ellipses result simply from the conception of attraction or gravitation and passage in a planet. And yet I would incline to believe that all these are caused or regulated by the motion of a fluid medium, on the analogy of gravity and magnetism as we know it here. Yet this solution would not at all detract from the value and truth of your discovery” (Janiak, 2009).

Another issue was that Newton could not allow either an ‘edge’ or a ‘centre’ to his universe. Gravity as he conceived it is always attractive. But this straight away causes a problem. If the universe has an edge then stars upon that edge will feel a gravitational attraction on one side, and not upon the other. All stars would therefore feel a mutual gravitational attraction away from all edges and would rush towards each other. This, however, was not observed. The stars appeared stationary. Therefore, the only way a mutual indrawing of stars could be prevented was to propose a Euclidean style vastness and infinitude—i.e. a universe without an edge. All stars would then have balancing forces on each side, and so would remain stationary. This might be mathematically and scientifically ‘necessary’, but that necessity did not, in and of itself, constitute a proof.

Newton’s universe also could not have a centre. If there were a centre then objects at that centre would have nowhere to fall, and would gradually draw all other stars further away from the centre towards themselves. Only stars at the centre would be balanced. And just as planets could only avoid gravitational attraction by orbiting about a sun, so also could stars further away from the centre only avoid a similar fate by processing about that centre. But this, also, was not observed. Therefore, an infinite universe with neither an edge nor a centre neatly avoided both problems, and that is what he proposed with his Scholium definitions. But unfortunately, neither mathematical nor scientific “necessity”, no matter how logical and consistent the world view derived, constitutes a proof.

The similarity with Darwin is that Newton’s model, as he proposed it, conflicted with some central tenets of existing Christian doctrine. There is a certain sense in which none of the three are verifiable. Newton might have done Christian doctrine a service by confirming the notion of ‘eternity’, how ever that might be defined, but he went far beyond anything even Copernicus—to whom the Church had been opposed—had ever proposed. Indeed, Newton’s cosmology agreed, in essential respects, with that proposed by Giordano Bruno who had been burned at the stake, in 1600. Bruno had also proposed that the Sun was essentially a star, with the universe then containing an infinite number of inhabited worlds, all potentially populated by other intelligent beings. This was problematic because Christ could surely only have but the one Resurrection.

Newton was saved from his own confrontation with ecclesiastical dogma by his model’s immense success. And when challenged, on more scientific grounds, by Leibniz, amongst others, on questions he could not answer—such as on the vexing question of how his gravity was magically and instantaneously transmitted across free space—he simply said: hypothesis non fingo or “I frame no hypotheses”:

Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding from the sun decreases accurately in the duplicate proportion of the distances as far as the orb of Saturn, as evidently appears from the quiescence of the aphelions of the plants; nay, and even to the remotest aphelions of the comets; if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phænomena, and I frame no hypotheses; for whatever is not deduced from the phænomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phænomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.

And now we might add something concerning a certain most subtle Spirit which pervades and lies hid in all gross bodies; by the force and action of which Spirit the particles of bodies mutually attract one another at near distances, and cohere, if contiguous; and electric bodies operate to greater distances, as well repelling as attracting the neighbouring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies; and all sensation is excited, and the members of animal bodies move at the command of the will, namely, by the vibrations of this Spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic Spirit operates (Newton, 1689).

The success of Newton’s model allowed the implicit tension with the alternative one proposed by Roman Catholic and Anglican dogma to live together without overt confrontation. But those tensions were rudely exposed—although albeit again without confrontation—almost as soon as Einstein produced his general theory of relativity in 1916. In 1917 he submitted his paper Cosmological Considerations on the General Theory of Relativity to the Berlin Academy of Sciences. In it, he presented a vision that followed his philosophical orientation … but that was surprisingly conservative. Einstein saw a traditional universe. It was static, eternal and finite … and remarkably similar to Aristotle’s:

Aristotle’s cosmology belonged to the class of steady-state theories in so far that his universe was changeless and eternal. When Einstein in 1917 proposed the first relativistic model of the universe, he unwittingly pictured a universe which had qualitative features in common with Aristotle’s: it was finite in space, but infinite in time (Halvorson and Kragh, 2011).

Einstein had produced a cosmology very similar to Aristotle’s by introducing a “cosmological constant”, Λ, into his equations of relativity to “save the appearances” and to allow the universe to remain static, as it was widely thought to be. George Gamow later remarked how Einstein came to feel that “the introduction of the cosmological term was the biggest blunder of his life” (Gamow, 1970).

Einstein may have birthed the general theory, but he was not the first to notice its deeper implications: that it could (a) show that the universe was dynamic rather than static; and (b) put questions of the origin of the universe firmly in the hands of scientists. And … these are the identical effects of Darwin’s earlier theory. These are what lie at the root of the entrenched opposition to him. But thanks to the Inquisition’s over-enthusiastic response to Galileo, cosmology is free from such confrontation, whereas biology has yet to earn that luxury. But then … biology lacks a coherent model whereas when Einstein sought to succeed to Newton—or perhaps, and more accurately, to extend Newton—he did so with a proper model.

The Russian physicist and mathematician Alexandr Alexandrovich Friedmann saw no need to follow Einstein. He did not introduce any potentially restrictive cosmological constant into his own work. Unlike Einstein, he was happy to go wherever the equations took him, for he wanted to understand the long-term development and behaviour of the universe. Friedmann was thus the first to realize that Einstein’s general relativity constituted a thorough, complete, self-contained, and totally consistent system for discussing that long-term behaviour of the universe. He also saw that Einstein’s new theory could be used to show that, purely on mathematical grounds, the universe simply must be expanding out into the future.

Friedmann presented his formulation of a completely new mathematical model for the universe in 1922. He used his model to speculate upon the universe’s present rate of expansion … and also to show that if it was expanding, then it must have had a beginning at a given point in time. This vision of a Big Bang, as it is now popularly called, evolved gradually and eventually became the new symbol for the origin of the universe. The essential message was that by studying gravitation—i.e. physical phenomena—scientists could speculate about the possible origin of the universe, and so invade traditionally theological territory.

And … it is this similar invasion into avowedly theological territory that truly marks the opposition to Darwin. That invasion into a previously exclusive religious domain is the logical consequence of his natural selection, and of his claim that all the differing anatomical and physical characteristics of all biological entities everywhere and at all times can be inherited—the one from another and in reproductive succession—with only slight variations between progenitors and their progeny. In the long run, he asserts, such variations will originate an entire species. Darwin therefore made it very clear—and long before Einstein—that to examine the manifest historical world is to examine the history of all organisms, and so again to invade theological territory. Darwin, however, set off on that invasion without a valid model as we have come to understand it:

Authors of the highest eminence seem to be fully satisfied with the view that each species has been independently created. To my mind it accords better with what we know of the laws impressed on matter by the Creator, that the production and extinction of the past and present inhabitants of the world should have been due to secondary causes, like those determining the birth and death of the individual. When I view all beings not as special creations, but as the lineal descendants of some few beings which lived long before the first bed of the Silurian system was deposited, they seem to me to become ennobled. Judging from the past, we may safely infer that not one living species will transmit its unaltered likeness to a distant futurity. And of the species now living very few will transmit progeny of any kind to a far distant futurity; for the manner in which all organic beings are grouped, shows that the greater number of species of each genus, and all the species of many genera, have left no descendants, but have become utterly extinct (Darwin, 1859, pp. 488-489).

There is now little point in simply restating these claims and counter-claims if it does not take us any further forward. And … we cannot really proceed without something rigorous to test these contrasting ideas. So … what is there to test?

We recollect our two population equations, one Euler and one Gibbs-Duhem. We also recollect our function with its gradient, and that gives us entry into the vector calculus. We then turn first to Darwin. His claims are surprisingly easily phrased so we can test them with our equations, and especially through our newly derived vector model based upon fluxes and divergences. Darwin’s view is centred upon his “slight variations”, and on his idea that changes in numbers will affect future populations through the masses and the energies inherited by progeny. Since our fundamental function for a population is φ(nqw), then Darwin asks only that we provide differentials and partial differentials such as dn/dt, or n/t, dφ/dn and so forth. We also have ready and available the two terms Σi(S/ui)U,V,{Nj=/i} dui and Σi(S/vi)U,V,{Nj=/i} dvi from our Euler-style population equation. These two represent the abundance, C, and the accreativity, Y, respectively, of comparative development, L. These are the changes in mass and in energy, again respectively, applicable to individual entities, and as they are inserted and remived, rather than to entire populations. We only need to find a way to measure these factors, and we can then put some numbers and some teeth onto Darwin.

We now turn to providing a suitable characterization of the alternative view. For want of a better expression, we call it “the proposal of the Aristotelian template”. We base our understanding of it on John Ray’s original definition of a species, the first ever offered in biology:

In order that an inventory of plants may be begun and a classification (divisio) of them correctly established, we must try to discover criteria of some sort for distinguishing what are called “species”. After long and considerable investigation, no surer criterion for determining species has occurred to me than the distinguishing features that perpetuate themselves in propagation from seed. Thus, no matter what variations occur in the individuals or the species, if they spring from the seed of one and the same plant, they are accidental variations and not such as to distinguish a species … Animals likewise that differ specifically preserve their distinct species permanently; one species never springs from the seed of another nor vice versa (Ray, 1686).

We now seem to have successfully identified the contrary position, because we have an immediate conflict of principle. Darwin said:

Many years ago, when comparing, and seeing others compare, the birds from the separate islands of the Galapagos Archipelago, both one with another, and with those from the American mainland, I was much struck how entirely vague and arbitrary is the distinction between species and varieties. On the islets of the little Madeira group there are many insects which are characterized as varieties in Mr. Wollaston’s admirable work, but which it cannot be doubted would be ranked as distinct species by many entomologists. Even Ireland has a few animals, now generally regarded as varieties, but which have been ranked as species by some zoologists. Several most experienced ornithologists consider our British red grouse as only a strongly-marked race of a Norwegian species, whereas the greater number rank it as an undoubted species peculiar to Great Britain. A wide distance between the homes of two doubtful forms leads many naturalists to rank both as distinct species; but what distance, it has been well asked, will suffice? if that between America and Europe is ample, will that between the Continent and the Azores, or Madeira, or the Canaries, or Ireland, be sufficient? It must be admitted that many forms, considered by highly-competent judges as varieties, have so perfectly the character of species that they are ranked by other highly competent judges as good and true species. But to discuss whether they are rightly called species or varieties, before any definition of these terms has been generally accepted, is vainly to beat the air (Darwin, 1859, p. 48-49).

If we really want to test Ray's ideas then we need to put some mathematical teeth to them, which means pinpointing these claims being made. Ray was a prominent exponent of ‘natural theology’, a prominent intellectual movement of the seventeenth, eighteenth and early nineteenth centuries. The Collins English Dictionary and the The Encyclopedia of Evolution describe the movement’s principal aims as follows:

natural theology: n. the attempt to derive theological truth, and esp the existence of God, from empirical facts by reasoned argument (Collins, 2009).

NATURAL THEOLOGY. Religious Science. As developed by such outstanding practitioners as the Reverend John Ray and the Reverend William Paley, it was a synthesis of science and religion that admitted no conflict between the two. Any regularities, perceived patterns or designs in nature were taken as evidence of a supreme being or designer.

Modern “creationism” and “creation science” spring directly from the old natural theology tradition: Their aim is to discern the hand of the Creator in nature. An underlying premise is that there can be no conflict between observed nature and revealed religion, since both are traceable to the same source of truth.

Science, according to this view, is in the service of providing evidence for religious beliefs (Milner, 1990).

Perhaps the most famous proponent of natural theology was the above-mentioned Paley who gave the following well-known and oft-quoted argument in its support:

In crossing a heath, suppose I pitched my foot against a stone, and were asked how the stone came to be there, I might possibly answer, that for anything I know to the contrary it had lain there for ever; nor would it, perhaps, be very easy to show the absurdity of this answer. But suppose I had found a watch upon the ground, and it should be inquired how the watch happened to be in that place, I should hardly think of the answer which I had before given, that for anything I know the watch might have always been there. Yet why should not this answer serve for the watch as well as for the stone; why is it not as admissible in the second case as in the first? For this reason, and for no other, namely, that when we come to inspect the watch, we perceive—what we could not discover in the stone—that its several parts are famed put together for a purpose, e.g. that they are so formed and adjusted as to produce motion, and that motion so regulated as to point out the hour of the day; that if the different parts had been differently shaped from what they are, or placed after any other manner or in any other order than that in which they are placed, either no motion at all would have been carried on in the machine, or none which would have answered the use that is now served of it. … This mechanism being observed—it requires indeed an examination of the instrument, and perhaps some previous knowledge of the subject, to perceive and understand it; but being once, as we have said, observed and understood, the inference, we think, is inevitable, that the watch must have had a maker—that there must have existed, at some time and at some place or other, an artificer or artificers who formed it for the purpose which we find it actually to answer, who comprehended its construction, and designed its use (Paley, 1802).

We can be certain we are at last in the right general vicinity because we have an immediate counter-argument from Darwin:

It can hardly be supposed that a false theory would explain, in so satisfactory a manner as does the theory of natural selection, the several large classes of facts above specified. It has recently been objected that this is an unsafe method of arguing; but it is a method used in judging of the common events of life, and has often been used by the greatest natural philosophers. The undulatory theory of light has thus been arrived at; and the belief in the revolution of the earth on its own axis was until lately supported by hardly any direct evidence. It is no valid objection that science as yet throws no light on the far higher problem of the essence or origin of life. Who can explain what is the essence of the attraction of gravity? No one now objects to following out the results consequent on this unknown element of attraction; notwithstanding that Leibnitz formerly accused Newton of introducing “occult qualities and miracles into philosophy”.

I see no good reason why the views given in this volume should shock the religious feelings of any one. It is satisfactory, as showing how transient such impressions are, to remember that the greatest discovery ever made by man, namely, the law of the attraction of gravity, was also attacked by Leibnitz, “as subversive of natural, and inferentially of revealed, religion” (Darwin, 1872, pp. 421–422).

We may have discerned the germ of an idea, but it is still far from mathematical. We may also now be close to our target, but we still have to bear the etymological fallacy in mind: the tendency for words to change their meanings over historical time. It is difficult to bring the precision of mathematics to bear in such conditions. We must exercise the greatest of care because, for example, St. Augustine is reputed to have said, in a supposedly a well-known and seemingly anti-mathematical dictum:

… the good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell (Kline, 1953).

However, a close examination of Augustine’s original statement leads to a rather different understanding (Taylor, 1982). The relevant verse occurs immediately after the one we have already quoted above, and in which he is railing against divination in general, and thus against the mathematician when seen as astrologer, as distinct from the mathematician when seen as astronomer. This is an important distinction. And once we appreciate that Augustine is railing against divination in general, then ‘numerologist’ would arguably be what he meant because mathematicians only calculate, whereas numerologists go on to impute meanings to those numbers. That leads to a very different translation and interpretation of the above verse quoted by Kline:

Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium, maxime dicentes vera, cavendi sunt, ne consortio daemoniorum animam deceptam, pacto quodam societatis irretiant. And so for this reason, the good Christian should be wary not only of numerologists, but also all those making mischievous predictions, particularly when what they say is true. If not so, then they may deceive the mind, entrapping it in an unrevealed partnership with the unworldly [Translation by author] (Augustine, 408, II.xvii.37).

We cannot produce a theory when there are such shifts in meaning. A sound mathematical model is the only thing that allowed Newton’s celestial mechanics to be separated from his religious views on the movements of those same planets. Those religious views are generally categorized along with the numerological and the astrological. That model allowed his predictions to demonstrate their accuracy and validity, and quite separately from any “meanings” that could be attributed either to those planets themselves, or to their existence and behaviour. A similar model was what also allowed the dynamical system Galileo and Newton founded to be separated from Aristotle’s search for a deeper meaning and from all his metaphysical speculations about sensations and perceptions; and about whether it was perceptions that really mved, or the objects that really moved. The same happened in the partnership between Faraday and Maxwell.

As the events surrounding Darwin surely demonstrated, without a suitable model it is surely impossible to separate Darwin’s ideas about fitness, evolution, and natural selection from the whole question of human existence and the more “spiritual” urge to find “meaning” in biology.

It is all very well for Darwin to say:

There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved (Darwin, 1859, p. 490)

… but without a suitable model this is simply an opinion. It is hard to see how Darwin’s proposals extricate biology from mere suppositions in the same way that numerology is distinct from mathematics, or astronomy from astrology. Even St. Augustine is on record as praising the ability of numbers to convey simple and absolute truths:

St. Augustine was thoroughly educated in both pagan and Christian learning. He could see much that was good, true, and irrefutable in the work of pagan scholars and accepted truth where he found it. “We may well say that St. Augustine defends … the principle of logic as the inviolable foundations of knowledge. … Side by side with logic we find the truths of mathematics … all those truths are necessarily and unconditionally true; they cannot be contested”. Augustine himself said (a.d. 386), in illustration of the reality of absolute truth: “However, if there are one world and six worlds, it is clear that there are seven worlds, no matter how I may be effected … for even if the whole human race were fast asleep, it would still be necessarily true that three times three are nine and that this is the square of intelligible numbers.” (Swetz, 1994, p. 230)

It is surely, therefore, imperative to find a rigorous mathematical model in support of Darwin.

Now Ray and Paley have given us proposal of the Aristotelian template, we must characterize it in the same style we have characterized Darwin's. It must become testable. We can therefore say:

the proposal of the Aristotelian template is that there exists an ideal, for each species, that keeps it immune to, and so free from, all the ‘slight variations’ and ‘preservations’ Darwin mentions.

On this view each biological organism is an expression of its ideal type or template which constantly influences its morphology, its physiology, its behaviour, and its reproducibility. This proposed Aristotelian-style biological entity and/or population is therefore not fully described by its DNA; by its morphology; by its bipedality or by its opposable thumb or lack thereof of one or both; nor by any other such feature. It is instead described by its abstract permanence … its Platonic-Aristotelian style fixity in type. This permanence expresses itself in an imperviousness to any proposed changes in the numbers of any similar entities. It is also independent of whether any given organism does, or does not, leave progeny. Irrespective of progeny or lack thereof, the integrity of each organism is maintained, as also is that of its entire species. The permanence shows itself in the distinctive characteristics or “design” each organism maintains and which again stands apart from any changes in number densities at any point, and whether extinction does or does not arise.

If this characterization of the essentials of these two positions is correct, then we can certainly now measure this proposed independence of the population values exhibited by a group of entities from the individual ones exhibited by each. We can do it with our Euler and Gibbs-Duhem equations. We have our essential development, λ. This is the probundance of γ = (S/U)V,{Ni} dU, and the procreativity of ψ = (S/V)U,{Ni} dV . If the proposal of the Aristotelian template holds, then these two partial differentials simply must be the only values ever measured in any population. The partial differentials we gave previously of Σi(S/ui)U,V,{Nj=/i} dui and Σi(S/vi)U,V,{Nj=/i} dvi must at all times register zero.

Testing these two proposals is now solidly based upon numbers. All four partial differentials return values in kilogrammes of mass and joules of energy. This is given numbers of chemical components, along with their binding energies at each time point in that generation. We can now—and at last—see whether or not the partial differentials for individuals are or are not zero; and whether or not Darwin’s ideas can at last be supported by a sound and rigorous mathematical model.It should now be quite straight forward to devise tests and to run experiments … such as we did with Brassica rapa.