Before We Begin

All truths are easy to understand
once they are discovered;
the point is to discover them. Galileo Galilei.

You will find a unique achievement upon this site. You will find a clear, sound, and logical proof of Darwin's theory of evolution. It comes with complete experimental validation. We provide that validation by growing some plants. We then use the measured values to prove that they undergo fitness and competition, and therefore that they can and do evolve in the manner Charles Darwin suggested.

It is true that not every new idea, such as the one we present here, is valid. It is also perfectly possible for an idea to be true, but be neither useful nor scientific. For example, Bishop Nicole d’ Oresme, writing in the fourteenth century, said: “I suppose that local motion can be perceived only when one body alters its motion relative to another … just as it seems to a man in a moving boat that the trees outside the boat are indeed in motion” (Menut and Denomy, 1968). We can be generous and say that this was the first recorded allusion to Einstein's theory of relativity. But although what Oresme said was true, he did not make his ideas either useful or scientific. Aristotle's ideas therefore continued to hold sway for another 500 years.

In his well-known book The Selfish Gene, Richard Dawkins gives an example of something that is as true as Oresme's observation; is perhaps more useful; but that is completely unscientific. As an evolutionary biologist, Dawkins wanted to pinpoint those transmissible ‘units of culture’, such as ideas and forms of behaviour, that can spread through gestures, calls, etc., and so that can affect behaviour, but that are not reproductively inherited. He therefore invented the word ‘meme’, which he took from the Greek mimema, ‘an imitated thing’, or mimeisthai, ‘to imitate’. His premise is that memes can mutate, propagate, respond to selective pressures, and even go extinct. Internet memes are a famous example:

We need a name for the new replicator, a noun that conveys the idea of a unit of cultural transmission, or a unit of imitation. ‘Mimeme’ comes from a suitable Greek root, but I want a monosyllable that sounds a bit like ‘gene’. I hope my classicist friends will forgive me if I abbreviate mimeme to meme. If it is any consolation, it could alternatively be thought of as being related to ‘memory’, or to the French word même. It should be pronounced to rhyme with ‘cream’ (Dawkins, 1989).

Figure 0.1
0-sphere and 1-ball

While Dawkins' idea might seem extremely useful, it is of only limited utility. We can see some of the problems in Figure 0.1.

We can think of the first problem as that of defining the ‘cultural space’ or ‘cultural size’ in which memes operate. For example, the Alur live just north of Lake Albert, in the Nebbi, Zombo, and Arua districts of northwestern Uganda and the northeastern parts of the Democratic Republic of the Congo. They number just under about 500,000 individuals and are a subset of the Luo. We cannot yet quantify this, but it is surely a cultural space of not so great a magnitude.

One of the distinctive musical instruments that Alur musicians play is the five- to ten-stringed harp seen in Figure 0.1.A. It is variously known as the ‘adungu’, ‘ekidongo’, or ‘ennenga’. It originated as a woman's instrument, with early variants being little more than a small musical bow. A string was stretched three times across an arch. An attached calabash served as a resonating chamber. The solo singer then accompanied herself.

There are many highly skilled adungu performers, male and female, some of international renown. Their fame and expertise notwithstanding, the instrument is still little more than a neck, a sound box, and a few strings. A craftsman can produce one in less than half-a-day. They may have originated as solo instruments, but they are now also played in small ensembles.

Figure 0.1.B shows a very much larger performance-oriented orchestral double-action pedal harp. It is not just larger, it is a very much more complex instrument, using more varied technologies. Its complexity bespeaks a far larger nexus of social and cultural interactions, suggesting that it requires the extensive cooperation of a very much large number of people to manufacture. We again cannot yet quantify it, but it is the child of a much larger cultural space. Where the adungu is intended for solo or small ensemble performances, this harp is intended primarily for orchestras.

The difficulty with memes is that both the instruments in Figure 0.1 are harps. The musicians who play them, and the artisans who manufacture them, execute their given sets of memes (a) as material bodies upon material objects, and (b) within their specific cultural contexts. The masses, energies, and social groupings involved are clearly very different, but memes have nothing to offer us in the way of analysis beyond the very broad declaration that they—i.e. memes—exist. That point is granted … but … what then?

Dawkins is a scientist, but does not profess to be a mathematician. He cannot therefore be expected to either recognize or engage with these kinds of problems. The issue, however, is that we as scientists can certainly count and measure the genes that memes explicitly seek to emulate. Memes are supposed to emulate genes even to the extent of deliberately sounding like them. But memes, in contrast, to genes, are not particularly scientific. Their creator has failed to give us anything concrete about them that we can similarly count and measure. They achieve very little for us scientifically for we are given no way to place memes within their cultural contexts. We cannot size or measure them and compare adungus or double-action pedal harps, or those who play and listen to them. Weighing the harps, counting the strings, and measuring the lengths of the performances and the sizes of the audiences the various musicians play in front of will not tell us what memes are; will not help us quantify them; and will do nothing to help us understand how they relate to the societies they exist in.

Since memes are supposed to help us discuss evolution, they would surely be far more helpful if they could help us detect—and preferably measure—evolution by giving us something substantive to count. We know exactly how many genes any human has, but how to count the memes? A quick search of the Internet soon reveals some of their other problems. This is not to deny the usefulness of memes. It is merely to point to their inability to do the job their creator had in mind.

As an example of the kinds of things you can expect to see upon this site, rectifying memes to make them properly scientific takes only a little imagination. We have to appreciate, first, that 3 + 0 = 3; that 3 × 1 = 3; and that there is a difference between a circle and a square. Pointing such things out does not initially appear to be “making a substantial and original contribution to this field of study”, which was a criticism levelled at some of the information on this site. However, looking at such apparently obvious things from a different perspective often takes us by surprise, and forces us to reconsider things we think we know well.

Another thing you will notice on this site is the very careful use of several important terms. For example, nobody alive today has seen a dinosaur, although almost everybody thinks they know what one looks like. This is thanks to people like the well-known scientific illustrator and creature designer Terryl Whitlatch, who has worked for numerous zoos and museums, as well as for the World Wildlife Fund.

Whitlatch refers to herself as a paleontological reconstructionist: “an artist who recreates the appearance of long-extinct and prehistoric animals from the bones up” (Whitlatch, 2010). But she also created and drew almost all the animals in Star Wars, illustrating them in The Wildlife of Star Wars. She additionally provided the animals for the films Men in Black, Jumanji, Dragonheart, Beowulf and numerous others. She might draw both imaginary animals and non-imaginary dinosaurs with great skill, but that is because her scientific skills and knowledge are impeccable.

The best-selling author James Gurney is also famous for his dinosaurs. He even invented his own fictional island, Dinotopia, where human beings and dinosaurs coexist in a symbiotic relationship. There are now over twenty books in his Dinotopia series, as well as a live-action TV mini-series, an animated film, and several video games.

Figure 0.2
0-sphere and 1-ball

Other things that nobody alive today has seen are the dodo and the quagga, the latter being a zebra-like South African animal, the last of which died in the wild in the early 1870s, the last dying in captivity in 1883 in a zoo in Amsterdam. We admittedly know what the dodo looked like in real life, because artists such as the French botanist Carolus Clusius left us drawings like the one he did in 1620 that we see in Figure 0.2.A. But even if we did not have the benefit of these real kinds of illustrations, creature designers and biological reconstructionists such as Gurney and Whitlatch could soon provide them for us.

The circle, line, and ellipse in Figure 0.2.B do not look much like a dodo. But when we arrange them as in Figure 0.2.C, they provide the right framework. Whitlatch wrote Animals Real and Imagined: The Fantasy of What Is and What Might Be, and Gurney wrote Imaginative Realism: How to Paint What Doesn't Exist because there is actually not much difference between drawing dinosaurs and drawing dragons, neither of which anyone currently alive has seen. Most people would agree that drawing dinosaurs extends our knowledge. And most would also surely agree that drawing dragons extends our humanity and imagination, which in the long run extends our knowledge.

It doesn't take much to adapt our dodo framework to create an ostrich. The framework in Figure 0.2.D works well for a quagga or, indeed, for any other horse or zebra-like creature, imaginary or real. We can soon adapt it for a camel or an elephant. And a little imagination will convert the rectangles and the oval in Figure 0.2.E into a knuckle-walking gorilla. It takes only a little imagination to take lines, circles, ellipses, squares and rectangles and apply them to the whole of the plant and animal kingdoms. They may be basic shapes, but reality is never far away from such an exercise in imagination.

Figure 0.3
0-sphere and 1-ball

As Figure 0.3 shows, our basic shapes can soon turn Dawkins' memes into objects of great scientific utility. Although the British anatomist Richard Owen, one of Darwin’s contemporaries, did not coin the word ‘homology’, he had an uncanny knack for interpreting fossils. He gave dinosaurs their name, and was a director of the British Natural History museum. He arranged for it to move out of the British Museum and into its current home, overseeing the purchase of the land, and the construction of the building.

Owen used the word homology to suggest anatomical similarities between species. The human arm, for example, is homologous with a bird's wing. Evolutionary biologists then suggest that they must both have come from a common ancestor. These are exercises in the imagination that the science of cladistics tries to quantify, and that have greatly extended our knowledge.

We see the same issue arise with Owen's homologies as we do with Dawkins' memes. Dawkins insists that his memes are similar to homologies in being handed down through the generations. But neither he nor Owen give any indication how long the relevant time period is. And without a proper unit of measure, it is difficult to make any concept truly quantitative, never mind use it to either prove or disprove evolution.

It is surely fair to say that both memes and homologies must be part of, or otherwise belong to, the members of some society. That society must also contain some set number of souls or biological entities. So let there be a Population 1, as in Figure 0.3, that is a society of 500 individuals.

Every population will increase and decrease, over time, in its numbers and in its energy. We therefore select a cohort of recently born individuals and keep count of the entire population until the ones in our chosen cohort start giving birth to the next generation. That is the oscillating loop with arrows we see in Figure 0.3.

Once we have such a loop, we can determine an average over the entire generation. That is the dashed circle again in Figure 0.3. We then do exactly the same for Population 2 which has 5,000 members.

We can now think of the numbers in a population as a first dimension of biological activity that we can track. It helps to establish—although it does not define—the overall size of the traits or cultural space in which memes operate.

Obviously, a birth rate of 1% per generation means 5 new members per cycle in our first population, but 50 in the second. And one person dying in a population of 500 is not the same, proportionately, as one person dying in a population of 5,000. Nevertheless, if our two populations are culturally equivalent, then they need to maintain a certain proportion of, for example, musicians in every generation.

We are now trying to make sense of the idea that five adungu players in a population of 500 has some form of equivalence to 50 pedal harpists in a population of 5,000. A mathematician calls a multiplicative relationship of this kind an “eigenvalue”. We now have a framework—a homology of sorts—that we can begin applying to biological populations.

Since one of our populations is ten times bigger than the other, our proposed eigenvalue is ten. If we call the numbers in each population P, we can describe the “circulation of the generations” by saying that it is completed when ∫ dP = 0. This simply says that a generation, for either population, is completed when the sum, ∫, of all the increases and decreases in their numbers and/or biological energy centres, dP, is zero so that the original numbers have been restored. The cohort we elected to follow has gone through a set of changes in numbers up and down, and has replaced another cohort whose members have all died.

Now we have figured out that every population must register its ∫ dP = 0, we can describe populations busy going through their cycles of the generations in a much more accessible way as:

“Every biological population is associated with a collection of traits, behaviours, and cultural artefacts and information that change constantly, but that can always be divided into discrete elements all of which have the potential to be mimicked and transferred from one individual entity within the population to another”.

We now have a realistic, useful, scientific, and above all eminently measurable backbone to Dawkins' original idea. And what is very striking is that our ∫ dP = 0 joins the 3 + 0 = 3 and 3 × 1 = 3 we have already met. The advantage of ∫ dP = 0 is that it is far more compact than its verbal rendition, and so helps to focus the mind on the real issue we want. And since it incorporates an integral sign, there is a sneaky trick we can play that—as we shall see very shortly—even the most eminent of scientists all too often forget.

Figure 0.4
0-sphere and 1-ball

Every population has to interact with its environment. If, for example, each population must maintain a certain number of harpists and other musicians, then each must direct the necessary resources to its members to cover all needed biological and cultural-economic purposes: i.e. towards its morphologies and its homologies, as well as towards its culture and its cultural artefacts. The double-action pedal orchestral harp might be larger and more complex than the adungu, but they both need adequate resources. The resources and population numbers, taken together, help encode and spread both that society's genes, and its memes. We must have both more human beings who are capable of becoming adungu and harp performers, and more of the instruments the former will need.

Since there need to be so many harps and harpists in each population, we can express the resources needed for a set of traits and cultures as so many kilogrammes of material resources per each individual. We can therefore think of this as a second dimension in which all populations must operate. But as Figure 0.4 suggests, allocating resources per each member immediately invokes areas. If we keep the two arrows in Figure 0.4 the same length—which we could think of as say four harps in each society—then we will have different angles in each circle, which is different areas, and therefore different resources and cultural spaces within each population. But … if we try to make the areas and musical cultural spaces or proportions the same in each society by using our eigenvalue—which would in this case be four harps in one society but forty in the other—then the societies will have the same angles or proportions, but will have different quantities of resources consumed, and will also have different arrow lengths, and so harp numbers, in each. We could also try to make the areas or resources the same, which would then require both different angles, and so perhaps different harpists and cultural musical spaces, as well as different arrow lengths, which might then be the actual harps.

If a population stretches 1 unit in the number dimension and 1 unit in the environmental resources dimension, then its cultural-economic base is one square unit: 1 × 1 = 1. There are now two ways to quadruple the population: (a) by numbers; (b) by its cultural-economic base. If we want to quadruple the numbers yet keep everything the same, then we must surely also quadruple the environmental resources used. So we go from 1 × 1 = 1 to 4 × 4 = 16. But … we have just increased the cultural-economic base by 16. If we decide that we would rather just quadruple that cultural-economic base, then resources and numbers now only need to double, because 4 = 2 × 2.

We have just learned, from our simple circles and squares, that resource allocation has its separate scaling propensity that we must first recognize, and then investigate, to see which is more applicable to memes and/or to genes. We must therefore give that resource allocation a separate eigenvalue.

When we once pointed the above very simple dichotomy out to a lecturer and researcher based in Cornell University, she did not recognize it as a mere change in perspective: i.e. that it was just a different way of saying the same kinds of things she was already well used to, and as she confronted the many still unresolved issues in evolutionary biology. Since she didn't follow its logic, she retorted with “But you can't compare a scorpion to a whale”.

The evolutionary biologist Ernst Mayr described the Linnaean system of biological classification in current use as “The arrangement of entities in a hierarchical series of nested classes, in which similar or related classes at one hierarchical level are combined comprehensively into more inclusive classes at the next higher level (Mayr & Bock, 2002)”. The Latin names of the Emperor scorpion and the blue whale are Pandinus imperator and Balaenoptera musculus respectively. Those names tell us exactly how similar those two organisms are, but also how they differ, in terms of the classes and families they each belong to. If we can't—or won't—compare a scorpion to a whale so we can tease out their similarities and differences, then we surely cannot do biology. Can scorpions increase their numbers of traits, and/or memes, and/or cultural spaces as rapidly as whales, or vice versa?

Although Owen did not quantify his homologies, unlike our above Cornell University researcher he had clearly recognized the effect of differences in changing numbers on the one hand, and their changing resources on the other. He carefully distinguished between what he called ‘serial homology’, which points to similarities in structures between say the front and the back legs upon the same organism, and what he called ‘special homology’, now more widely known as ‘general homology’, and which considers similarities between structures in completely different organisms. Owen's distinction implicitly recognizes the essential point that populations do not—and cannot—change in the same ways that their individual members do … which is surely a critical aspect of evolution.

If any species or society is to be stable, then the sum of all its resources and materials, M, must also balance out over its generation. Any society or species that wants to perpetuate itself must therefore have ∫ dM = 0: i.e. the sum of all its resources and materials used over a generation must increase and decrease, alongside its entity numbers, and eventually return to the same point. If a population does not have ∫ dM = 0, then its morphology, or else its memes, or perhaps both, are evolving because its resource base is changing. We can express the requirement that ∫ dM = 0 in words as:

“The individual entities in every biological population are genotypes which together encode that population’s genome or collective genetic encoding, which at least some amongst them are able to reproduce and recreate.”

Whether we are dealing with trees, scorpions, whales, or anything else, we have just given ourselves something concrete to measure so that Owen's homologies and Dawkins' memes are a little more explicit. Since we now have two distinct dimensions or properties involved in biological reasoning, we can easily take measurements to see what might be changing with respect to what.

We are using our simple circles, ellipses, and squares to make the difficulty with homologies and memes a little clearer. The general issue is that memes, homologies, and the phrase “maintain a set number of musicians” are all inordinately vague. A synthesizer player in a large industrial society and a sistra or shaker player in a small hunting and gathering one are both musicians, but just like with the two different kinds of harpists, there are obviously great differences between them. Different numbers of people need to interact in different ways to produce the adungu in one case, and the double-action pedal harp in the other. The number of interactions that the members in different societies can have, over a generation, even if their societies are the same size and have the same resource base, introduces a third dimension.

Figure 0.5
0-sphere and 1-ball

The ellipsoids of mass and energy we suggest in Figure 0.5 in their turn suggest the difficulties we face in measuring these different kinds of interactions. Each population's size and/or cultural or memetic space now depends on how far out it stretches in each separate dimension to create that ellipsoid, which is a first eigenvalue interaction. Its interfacing with the surroundings depends upon the surface area each dimension constructs in conjunction with the others. Those are a form of surface forces, and are a second eigenvalue interaction. And, finally, the society's interactions both (a) amongst its members, and (b) with the environment in terms of differences in distributions between its internal and external responses—which is surely the source of all memes and homologies—depends upon its volume, which is its set of interacting internal body forces. Any change will have very different consequences across the various dimensions. Since species and populations are clearly at least three dimensional, they cannot possibly change in the same ways as each other, or at the same rates. We obviously need a third scaling factor or eigenvalue to track these volume style interactions between all three of our proposed dimensions.

We still have not properly quantified anything, but we are surely making progress. We can see that our three axes must be like physical space in being “orthogonal”, or mutually at ninety degrees. We also need some way to measure them so they are equivalent. We must therefore arrange for them each to be “orthonormal”, which means centred around a specific unit of length. We must be able to state what is '1', or a unit, in each direction.

If we now call the number of social interactions in any society S, then a society that wishes to maintain itself must ensure that ∫ dS = 0: i.e. that the sum of all its interactions balances out, so that enough musicians in the next generation meet and learn from enough in the previous one to perpetuate that society's memes and musics. We can express this requirement in words as:

  1. “No individual biological entity can be separated from all possible discrete elements of traits, behaviours, and cultural artefacts and information”.
    (Corollary: prodigious savant are always possible; and even ‘the walls of rude minds are scrawled all over with facts, with thoughts’ (Ralph Waldo Emerson)).
  2. “Not all the discrete elements associated with any one trait, behaviour, or cultural artefact or information can be uniquely attributed to any one biological entity”.
    (Corollary: ‘Bernard of Chartres used to say that we are like dwarfs on the shoulders of giants, so that we can see more than them, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size’ (John of Salisbury, Metalogicon, 1159)).

Dawkins might be correct that biological populations have both homologies and memes, but whether heritable or nonheritable, our little exercise in Gurney's imaginative realism has used some rather simple shapes to draw some interesting conclusions about them. If a society stays the same in certain respects—so that a core is left unchanged in spite of some members departing, and others being introduced—then it surely makes sense to say that it has undertaken an identity operation along the lines of 3 + 0 = 3 or 3 × 1 = 3, and so that ∫ dP = ∫ dM = ∫ dS = 0.

There is surely at least one more dimension—a fourth—that we must contend with. For that, we are unfortunately going to need just a little help from a field of study known as “Riemannian geometry”. The Stanford professor Maryam Mirzakhani, who became the first woman to win the Fields Medal, studied this; and Albert Einstein used it to prove his general theory of relativity. As Mirzakhani describes it: “If you hit a pool ball on a billiard table and it moves forever, what sort of patterns would it trace geometrically on the table? Would it eventually cover the entire surface?” (Mirzakhani, 2014). We can ask, in the same vein, whether it is really possible, as creationism and intelligent design insist, for a population to keep on building a given set of homologies and memes, granted their three dimensional nature; granted the random state of the environment; and granted the surely inevitable flexions, deformations, and changes in shape that any population's cultural space ellipsoid will almost certainly undergo? Fortunately, however, we only need the outlines of the concepts. There is really not much more involved for us than counting from one to four. We must be mathematically precise in what we do, but we do not need nearly Riemann's and Mirzakhani's level of sophistication to solve our little problem.

As we can now envision it, with Mirzakhani's help, our cohort of biological entities at the beginning of some generation can go all around the cycle of the generations; can reproduce; can act as progenitors to their own set of progeny; with that progeny then being able to do the same. We might be asking that our three above sets of changes be the same, i.e. zero, across every generation, but since dP, dM, and dS represent very different sets of dimensional interactions, this is rather a tall order. However … the German mathematical physicist Rudolf Clausius faced similar unresolved issues. He struggled for many years to resolve a problem very like the one we currently face not just with homologies and memes, but in evolutionary biology in general.

Figure 0.6
0-sphere and 1-ball

We can suggest the nature of the fourth dimension we need by noting that Clausius struggled to make sense of the cycle that the French physicist Sadi Carnot had discovered when studying the James Watt steam engine, which had spawned the Industrial Revolution. As in Figure 0.6, Clausius had grappled with the realization that infinitely many different cycles are possible between the same initial and final points.

Another way of looking at the problem Carnot and Clausius tackled is to imagine ourselves throwing a loop of string down on the ground. It will adopt a random shape. What is its area? Is there, also, a perfect circle whose area is identical to the original randomly fallen rope? This is exactly the same as the problem we face with homologies, memes, and the biological cycles of the generations. We are trying to make sense of what goes on inside them as they loop around in time from progenitors to succeeding progeny.

If we borrow Mirzakhani's analogy, we can regard the earth and its environment as the billiard table that provides all the earth's resources, with each population being the billiard ball that bounces around within the limits those resources provide. All the infinitely many successive memes and homologies, over potentially infinitely many generations, must fit within those limits. We can rephrase Mirzakhani's question by asking if that proposed infinite succession of identical or near-identical generations is even remotely possible.

The proposal here, from creationism and intelligent design, is that biological populations can repeat their generations infinitely many times and never change. All the entities creating those populations, within each species, will be somehow different as they each move about each of their distinct circulations of the generations … and yet they will also somehow all be the same as they define their different species. As varied as they might be, they are supposed to share something in common.

As indicated by the double arrows in Figure 0.6, we want our population's genes and memes, their homologies and cultures, to always go in one direction. We want genes and memes to be handed on, and so to go away from progenitors, and towards their progeny. There is not generally anything worthwhile that should travel, or needs to travel, the other way: i.e. backwards from progeny to progenitors. Parents tell their children how to behave. Some individual children might teach their parents how to use some aspect of modern technology, but it tends to be the corporation of adults that creates those objects in the first place. Some children somewhere might take up the new gadgets more quickly, but no child invented that modern technology. It is critical that memes and homologies travel onwards and downwards through the generations, so that progeny receive not just the physiologies and metabolisms they need to continue, but also all traits and knowledge. Again: is this really possible in an infinite and unending manner?

Clausius could not at first reconcile those infinitely many cycles. Fortunately, however, the great Swiss mathematician Leonhard Euler had realized that just as 3 + 0 = 3 and 3 × 1 = 3, then since integration is really just a way of simultaneously adding and multiplying, while differentiation is really just a way of simultaneously dividing and subtracting, then they also have an identity operator. Thanks to Euler's discovery, Clausius saw that we can integrate and differentiate anything we like—including biological cycles—as often as we like using what is now called either an “Euler multiplier” or an “integrating factor”. It will always stay just the same.

Clausius eventually saw the significance of Euler's discovery. He saw that the identity operation incorporated in the Euler multiplier powered what he christened “entropy”, and without which there is no energy … or memes or information.

Clausius realized that just as we can calculate an average population size for any biological cycle, which is a form of identity for that generation all around it; so also could he use an Euler multiplier to unify the different cycles that confronted him. In the same way that the average population number states a set of changes that must occur when summed all around the cycle to satisfy any and all ongoing increases and decreases, and in all the dimensions; so also did Clausius see that no matter how circuitous any system's path might be, we can always replace it with a perfect cycle of exactly the same work and energy capability. It does not matter how randomly a rope is shaped when we toss it on the ground. We can calculate its area. That area will also always match some circle, some square, or both.

Clausius' discovery means that we can replace any energy cycle, no matter how convoluted it might be, with an equilibrium set of interactions, within the same surroundings. We can replace the energy cycle with one that is not circuitous. That perfect cycle might be impossible to realize in practice, but thanks to the zeroing capability of its Euler multiplier, all differences to and from it cancel each other out all about itself. Just like a circle can define an area, so also does the cycle we can create define all essential behaviours for any and all circuits of energy between those specific end points. No matter how different they might all appear to be, any cycles that move between the same two end points are now substantively the same.

Clausius provided a very simple formula that announces his discovery of entropy. If we symbolize entropy by S, temperature by T, and the amount of energy by Q, then he gave it as ΔS = ΔQT where the Greek letter Δ means 'a small amount of'. This formula therefore says “it costs a small amount of entropy, ΔS, to hold a substance at some temperature, T, and change its energy content by a small amount, ΔQ”. We can rewrite it as T = ΔQ⁄ΔS, which says that we can determine anything's temperature simply by measuring how much its entropy changes as we change its energy content.

Clausius is telling us that entropy, S, has the characteristics of a substance. It is measurable. It is quantitative. Any system's entropy is a function of its total energy stock, Q, divided by its temperature, T. A system's entropy is therefore a function of its mass, how it is constructed, its configuration, and of its overall behaviour.

The entropy Clausius discovered plays a critical role in all energy effects. It effectively flows in and out of substances and systems. It is their driving force. It is always seeking to equalize them all. As it does so, it changes the state of any other substance that it admixes with.

Entropy is comparable to energy, but is not the same. It is more quantitative. We now view it as an aspect of molecular behaviour. So if matter contains very small quantities of the entropy Clausius discovered, we perceive it as cold. It either has very few molecules, or they are close to stationary. A thing with little entropy also generally has little in the way of internal energy. Its internal energy is an aspect of its net molecular motions, but is not the same as its heat. Heat is a very specific aspect of overall molecular motions.

Since a sample of matter that is small and cold has only a small amount of internal energy, it can only ever exchange a small amount as heat, which is again not the same as its internal energy. As matter gains in its entropy, it most generally also gains in its internal energy, and can eventually feel warm. It feels warm because of its heat, which is the fact that it is now exchanging its internal energy with other samples of matter, and/or with the surroundings. So heat refers to the exchange interactions internal energy can undertake, and so is distinct from the internal energy.

Since entropy can be substance-like and can flow across boundaries, it is distributed throughout all matter. Entropy refers to the propensity to undertake interactions, and so it has a propensity to cross boundaries … because that is what it measures. When a thing interacts, it also tends to change its capability for further interactions.

Entropy spreads with interactions. Since a thing interacts, it changes that possibility by interacting. Entropy tends to accumulate as it spreads. The total number of interactions a substance has already undertaken tends to increase. The larger and/or the warmer something is, then the more entropy it has … and the more internal energy it also tends to have. Heat is certainly a form of energy in that it is the tendency for molecules to move and to interact with other forms. However, there are lots of different kinds of energies. Thus heat can contribute to the amount of internal energy any substance or system has. But there are also likely to be other forms of energy internal to that system or substance. Everything tends to have heat as a tendency to interact, but each system also has other possibilities.

Deep outer space and the surrounding cosmos are extremely cold. They are at or very near to the absolute zero of temperature. They therefore have very little internal energy, and so very little possibility for the exchanges of internal energy known as heat.

Since deep space is immense, and without boundaries, its lack of internal energy means it has an entropy that is as close to zero as possible. Nothing could have less entropy. Its capacity for acquiring entropy is therefore virtually infinite. Its appetite for entropy is certainly greater than any one of its subparts. The cosmos surrounding us, which is relatively devoid of entropy, is therefore always looking to increase its entropy stock. The only possible source the cosmos has is the entropy and the internal energy inside everything else, which it can procure through heat transactions. Nothing can therefore avoid donating its entropy, as also its internal energy, to the maw of that surrounding universe.

We can certainly make efforts to cordon off the entropy, and so internal energy, that any given substance contains, to try to prevent the cosmos from acquiring it. That is effectively what biological populations do. But success requires a perfectly insulated surface. No such perfect insulation exists.

There is also no possible process that can destroy entropy. The universe is always actively increasing its stock of entropy. Since that surrounding universe is without boundaries, unquenchable, and insatiable in its demand, then the best that we can hope for is to discover some process that can somehow gather back exactly as much entropy and internal energy or heat as the universe removes from it at any given time. This is to be completely and faultlessly reversible in a cyclical set of entropy and internal energy or heat exchanges … which is in practice impossible. Once the cosmos has acquired entropy, it will never return the same amount to any system. That means that every conceivable process—even those at the level of biological populations—is in at least some measure irreversible. Therefore, any process or combination of processes that can produce entropy will surrender it, and the internal energy it implies, sooner or later, to the universe, even if it takes the millions of years that a mountain range does to do so.

Clausius had his realization. As the cycle of interactions that Carnot had first described pushed a piston through a distance, then Clausius saw that he could use the Euler multiplier technique. He could use it to describe the process the cycle went through as it interacted with the surroundings. Clausius saw that as the James Watt steam engine converted heat and so did its work, the universe marched in and helped itself to a stock of the engine's entropy. Clausius had discovered, by deploying Euler's identity operator technique, that the entropy or conversion-cost or tax the universe inevitably extracted could be very simply measured. That was exactly the effect he wanted.

The simplicity with which a substance or system's value for entropy is measured—and it is stated as joules per kelvin, which is the joules of energy held at any given temperature—belies the reason and the purpose for its discovery. It is so simple to measure that even the most experienced of scientists forget the main role Clausius devised for it.

Entropy is the expression of an identity operator. It is a way to make everything the same.

Among those with a lack of appreciation for the reality and simplicity of what Clausius achieved by introducing entropy is an eminent professor at the University of California in Berkeley who could not or would not grasp the implications of Clausius' discovery for memes, for homologies, and for biological organisms and their cycles of T = dQdS, which is a precisely equivalent way of rewriting Clausius' original ΔS = ΔQT. He also could not grasp what entropy says about the equivalent difficulty that populations will inevitably have maintaining their stock of internal energy as memes and homologies. It is as impossible to indefinitely maintain a species or a culture as it is to maintain a mountain range. This particular professor's lack of comprehension was perhaps more understandable when we find that he had unfortunately—and wrongly—written in his book that Carnot had discovered the formula that announces entropy, when it was in fact Clausius who made the discovery that resolved the problem that Carnot had only posed, but could not resolve; for it never seemed to occur to Carnot to use the Euler-cum-Clausius identity operator technique.

Once Clausius had realized that he had a valid but purely mathematical strategy—i.e. setting something to zero—for solving the problem that Carnot had posed, he had to make it useful and practical. He had to figure out what it might represent in the real world. He had to give the Euler multiplier—which allowed arbitrary quantities to be added and subtracted—what he called a “simple mechanical meaning”.

We should notice that Clausius has taken the opposite tack to the one Dawkins took when Dawkins coined the word meme. Clausius first had his idea. He also got his idea by studying, and solving, some mathematical formulae. Once he had his solution, he sought for an expression or representation in the real world. Dawkins, however, firstly saw what he wanted in the real world. Once he had seen what he wanted in the real world, he sought for a word. But he has left us floundering in his wake, searching for an appropriate rendition of memes, magnitudes, and rates of change, which we cannot easily find. Dawkins has suggested a picture, but it is still fuzzy. Clausius, therefore, went completely the other way to Dawkins.

Having found a mathematical solution in the Euler multiplier, it took Clausius quite some time to find a suitable physical and real-world correspondence. Nobody before him had ever even suspected entropy's existence. Nobody before him had ever considered that the universe might consistently remove, if at all possible, each and every substance's ability for further interactions. Entropy tells us how unlike the surrounding universe something in it is in terms of how much of its interaction capability can still be removed from it. Entropy thus indicates how much more something can change until it cannot change any more.

Clausius then saw that since the universe has an insatiable appetite for entropy, every system everywhere will eventually sacrifice all possible internal energy and entropy. This is the meaning of his famous and evocative phrase “the energy of the universe is constant, the entropy of the universe tends to a maximum” with which he introduced his discovery. Once any system has donated too great a proportion of its entropy to the universe, and/or had too much of the universe's featurelessness thrust across its boundaries, then it has also given off so much of its original or its internal energy that whatever remains has no versatility or availability for any other purpose. That remaining stock of internal energy has become configured less like the way it used to be configured, and has become configured far too much like the featureless cosmos pressing remorselessly upon its boundaries. Clausius' entropy therefore measures energy's current availability, or unavailability, for ongoing interactions, granted the effects that the universe's constant demands have had, and are having, upon it. Entropy again tells us how unlike the surrounding universe something is, and so indicates how much further it is likely to change so it becomes ever more like that surrounding universe: featureless and unchanging. As Clausius fully intended, energy and entropy therefore always have a strictly mechanical meaning. This applies even to genes and genomes, and to the memes and cultures that surround them.

The universe might always win in the end, but in the mean time … things also have a tendency to keep their entropy, which is their ability to interact. Everything does its best to remain the same in each moment. They all do this with more or less success. But … nothing is perfectly successful. That is what we also mean by species.

Figure 0.7
0-sphere and 1-ball

The entropy Clausius discovered is inextricably bound up with “the conservation of energy”, illustrated in Figure 0.7, and which is the sine qua non of all science. The conservation of energy is a way of indicating what it is about things that stays the same, throughout all the changes that entropy forces, and that the universe around imposes.

The words “conservation of energy” might be very easy to say but unfortunately, and judging from the somewhat surprising response emanating from the Centre for Mathematical Biology at Oxford University (now the Wolfson Centre for Mathematical Biology)—“But you can't compare a biological organism to a heat pump”!—the implications are considerably harder to grasp. The horses in, for example, Figure 0.7.A are marching round and round and achieving some purpose. They may be converting their walking into the energy of drilling that cannon, but they obviously remain just the same as horses. But we know that they cannot keep this up for ever. They will grow old and tired. But then again … we also know that we can breed more horses and replace the ones that get old and tired with fresher and younger ones. Could we, however, keep that up for ever? Another thing we know is that we will eventually have to replace both the machine drilling the holes in the cannon, and the cannon itself. So … when we see horses marching round and round in this way, exactly what is it that is staying the same; and exactly what is it that is changing?

Figure 0.7.A illustrates Sir Benjamin Thompson's, Count Rumford's famous experiment An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction. He conducted it in 1798 to prove, for the very first time, that there was something intimately connecting the walking horses and the heat the cannon generated. He surrounded the cannon with a bath of water, and then boiled the water. He for the first time converted the mechanical effect of the horses marching around into heat which he used to bring the water to the boil … which, in spite of the above observation from Oxford University is precisely a biological organism acting exactly like a heat pump. A heat pump is the name given to anything that converts energy, for they are ultimately all equivalent. Thompson even gave the first ever value for what is now called the “mechanical equivalent of heat”. Something is staying the same through those conversions.

Thompson's discoveries, in conjunction with those of others, gradually led to James Watt's steam engine which, as illustrated in Figure 0.7.B, could do the reverse and convert heat right back into mechanical work. Something was again staying the same, while other things changed.

The conversion cycle Watt invented was the one Carnot investigated so thoroughly in his Reflexions on the Motive Power of Fire, 1824:

We use here the expression motive power to express the useful effect that an engine is capable of producing. … in order to develop motive power by heat, a cold body is required. … So the production of motive power in a steam engine is due … to its passage from a hot body to a cold one. … Hence the conclusion we must reach is that the maximum amount of motive power gained by the use of steam is also the maximum that can be obtained by any means whatsoever. … The effect can always be expressed in terms of a weight being raised to a certain height. It is measured, of course, by the product of the weight and the height to which the weight is considered to have to be raised (Carnot, 1824).

Carnot observed, for the very first time, that hot things can exert a mechanical effect by becoming colder, but that it does not work the other way. And … as we have observed … it also would not work for progeny to infiltrate their almost complete lack of structure, of coherent information, of homology, or of memes, backwards into the general population, or to import there the general amorphousness they embody. This would adversely affect the progenitors that strive to beget them.

Carnot's compatriot Émile Clapeyron, in his Memoir on the Motive Power of Heat, saw something very new in Carnot's work and redefined his motive power, instead calling it “mechanical action”. It is the source of all forwards directioning, including the biological. Clapeyron was therefore the first to state the capability of all energy. He expressed it as PdV: i.e. as a pressure mechanically moving outwards through a volume, and impressing itself into the surroundings. His important realization eventually came to the attention of one James Prescott Joule, located in Manchester in the United Kingdom, who used it to calculate the mechanical equivalent of heat. Figures 0.7.C, D and E show his original equipment, now on permanent display in the Science Museum in London.

Finally, Figure 0.7.F illustrates the conservation of energy: that it does not matter how many objects any pressure-volume interaction moves; nor what their masses, speeds, or directions might be; nor how they interact; nor whether they are macroscopic or microscopic. If we sum their kinetic energies both before and after, we will get the same value every time. ‘Conservation of energy’ is simply a lot easier to say than “infinitely many different macroscopic or microscopic objects with inertia can move in infinitely many varied directions, but the sum over them all will be the same”. Clausius then added the inevitability in the sense of direction in energy that affects even homologies and memes.

The Yale-based US chemist and physicist Josiah Willard Gibbs proved the mechanical equivalence of all possible chemical reactions, including the biochemical. The ‘Gibbs energy’ is a form of positional energy that accounts for all molecular configurations.

It was then only at the end of the nineteenth century that Max Rubner, the German physicist and physiologist, brought all pertinent information about energy together and at last convinced biologists that the energy that all biological organisms use in metabolic processes exactly equals the food energy they consume. Only then did scientists begin to accept that even biological organisms obey the law of the conservation of energy.

Unfortunately, and as the Centre for Mathematical Biology at Oxford University demonstrates, biologists have proved notoriously slow at accepting this basic point. Any transit or conversion of one form of energy to another is technically called work. That work either increases or decreases the internal energy of the object or system either being worked upon or doing the work … even if that object or system is biological. Work just means energy is being converted. Any such increase in internal energy will also be accompanied by a heat pump effect. That heat pump effect is an increase in the propensity to lose that same internal energy through the process known as heat. The process known as heat is a movement from a greater stock of internal energy to a lesser one. Entropy measures that inevitable tendency of loss. It quantifies precisely how much has occurred. If biological entities are going to convert energy, which they must, if they wish to survive, then they are also going to lose heat and increase entropy. Those are inevitable—and measurable—heat pump effects.

Even though all stocks of internal energy are affected by the two processes known as work and heat, it still required the brilliance and originality of Hans Krebs to unravel the important cyclic processes involved in biochemistry. His 1957 book Energy Transformations in Living Matter, written with Hans Kornberg, was the first important work to examine the thermodynamics of biochemical reactions. The appendix of Krebs’ book contained the first-ever thermodynamic tables containing equilibrium constants for the Gibbs energy that formed different biological substances. Biological organisms do not evade either entropy or the mechanical equivalence of heat. Yet in spite of all of that, when we pointed out to them the inevitable consequences, even the editors of the most august Journal of Theoretical Biology still lost sight of the fact that biological interactions are in no way privileged; questioned who Gibbs was; what the Gibbs energy was; and doubted the relevance of his discoveries to biological theorizing.

All nonmechanical and microscopically based forms of energy—which again includes all biochemical reactions—are precisely equivalent to one or another mechanical form for they are, and are what drive, all energy. They still involve conversions. They are statements of the ability to convert from one form to another. They are therefore all quantifiable, and by the same methods. All biochemical interactions have this ability to move weights in gravitational fields, for that is the useful energy conversion process that is carefully defined as work. All energy is equivalent, and therefore any conversion from one form to another has an equivalence as a mechanical conversion, which was Clapeyron's original discovery.

All usable energy, again including the biological, must contain the work capabilities and properties Carnot first stated. All useful and usable energy is convertible, and so must equally well be capable of converting so it can move a given weight through a given height in a given gravitational field under the equivalence of energy stated in the first law of thermodynamics … and which declares that very energy conversion. If a given tranche of energy does not have that measurably mechanical capability so that it can both substitute itself for, and be substituted by, a heat pump of specified mechanical capabilities, then it is unusable by any biological organism whatever, for any purpose whatever, and is known as heat. It is known as heat for all it does is flow from a hotter to a cooler place without converting into any other form and thus being useful, not even to and for biological organisms. But if it is useful to and usable by biological organisms, then it is only useful and usable because it is convertible. It is therefore more than just heat. It immediately has a mechanical equivalent. That is what conservation of energy means.

We will gradually bring these preliminary observations to an end by returning to that fourth dimension. We note that it is necessary; and we also demonstrate the trick we can play using ∫ dP = ∫ dM = ∫ dS = 0: our requirement for Owen's homologies and Dawkins' memes for any population that wishes to be stable.

We can express the issues the following way. If we were to look at the world from a shrew's point of view, then some first generation of shrews can observe a meerkat, and notice that the meerkat is, for example, 720 times bigger than they are. The shrews record that in their cultural traits and memes.

Since meerkats live for twelve years, it is quite possible for the shrews' great-grand children to observe the same meerkat … and to notice that it is in fact only 700 times bigger than is recorded in those traits and memes. How are those descendant shrews to determine whether they have gotten bigger in the interim; whether the meerkats have gotten smaller in the interim; or whether it is some combination of both?

If the shrews and/or the meerkats want to answer the question of which of them has changed, then they need some form of biological identity operator that they can each flawlessly compare themselves to. And we have carefully described Owen's homologies, Dawkins' memes, and biological circulations of the generations using our zero-izers or identity processes of ∫ dP = ∫ dM = ∫ dS = 0 so that we can investigate the possible existence of biological identity operators, biological Euler multipliers, and/or biological limit cycles. We have also been very careful to ensure that if any such exist, we can run a suitable experiment to demonstrate them in properties that we can very easily measure.

We should recognize, however, exactly what we are doing. Demonstrating these various points of principle—which is all we can do at this time—is simply not the same as setting standards of measurement for the things we describe. This is a point that evaded even a famous professor and head of a research laboratory based at Harvard University. There is again a difference between showing things are equal, and establishing the precise magnitudes involved in that equality.

We go back to Oresme. Those of us alive today have the advantage of Einstein's theory of relativity. We can therefore understand the significance of Oresme's ancient observation about a moving boat. So if we wish to explain relativity to some Harvard or any other professor, it is enough to simply row him or her past a tree, and to explain to him or her that since the speed of light is constant; and since it looks as if one or the other of the tree or the boat is moving; then matter and energy are equivalent. We do not need, in this situation, to row at a specific velocity. Nor does it matter if we discuss the situation using knots per day, feet per second, or, as Galileo did, measure all times using a pulsilogon, a pendulum calibrated to his pulse, which was his only way to measure time. We only have to row two or three strokes so our professor can see the trees move relative to the boat, or else the boat move relative to the tree, and point out the consequences of the fact that it is not always possible to know which has changed. The rest hinges upon explaining that relative motion's significance. That does not require more experiments. It instead requires a change in the perspective of the person being rowed … or, in this case, the person observing the experiment. The problem is: is it the shrews or the meerkats that have changed? And … what are we to use as a stable reference point? As a point of principle, and to establish the fundamentals of evolution, it does not matter which. A change in either one, however we might measure it, means evolution exists as surely as relativity exists.

Even the most famous of scientists can mistake the significance of the apparent constancy of the speed of light, which does not need a value so we can establish its role. Very strictly, the Einstein theory does not claim that it is impossible for anything to move faster than the speed of light because “tachyons”, which are faster-than-light-speed particles, are perfectly possible. What relativity theory declares is that if an observer or actor can bring something to rest—whether it be by speeding it up or slowing it down relative to themselves—then it has an inertia relative to all similar observers. They will all have to expend infinite amounts of energy to accelerate that object up to, or decelerate it down to, the speed of light. But since infinite amounts of energy are not available, this cannot be done.

Any object that can move at the speed of light is effectively massless, because it cannot be accelerated. No amount of swatting a beam of light with a baseball bat is going to change that beam of light's state. It has zero inertia. The theory of relativity now says that if a beam of light behaves this way and so is massless to one observer, then it behaves in the same massless way to them all. Light behaves like light everywhere, and nobody anywhere can swat it with a baseball bat.

Although tachyons are perfectly legitimate as possible particles under relativity, they could not be decelerated down to the speed of light because it would again take infinite amounts of energy to slow them. No credible physical theory, at the present time, incorporates tachyons because every suggestion that tachyons are involved has turned out to be an anomaly explainable by present knowledge. It is therefore in theory untrue, but in practice true, to say that nothing can move faster than the speed of light. What is more strictly true is that infinite amounts of energy are required to accelerate anything to the speed of light, whether from above or from underneath. But since tachyons are not known, it is accurate enough, and for most ordinary purposes, to say that nothing can move faster than the speed of light.

In this context, the above distinction between “nothing can move faster than the speed of light”, and “the speed of light cannot be reached by any object with inertia because it would take infinite amounts of energy” is signally important. The former cannot have any analogue in biological reasoning. The latter, however, can. We only have to think of Mirzakhani's billiard table.

The essence of Darwin's theory is competition: the availability of resources as a function of both the environment and the numbers amongst which those resources must be distributed. This is again the issue of Mirzakhani's billiard table and its infinite number of bounces. Granted that creationism and intelligent design declare that all species and populations are beyond competition, they can now be interpreted as the claim that infinite amounts of energy are available to them all, and therefore that they can bounce around in their species domains an infinite number of times. So creationism and intelligent design are implying an infinite earth and an infinite quantity of energy. Darwinian reasoning says that such premises are intrinsically false.

In Darwinian thinking it is perfectly reasonable to consider what the conditions might be that would allow life to begin. The issue is competition as against the question of what the minimum reasonable conditions are for DNA life forms as we know them to begin. But to those who follow creationism and intelligent design, this is not a reasonable matter to speculate upon. The origin of life is an unapproachable given.

If we take sufficient care in understanding the terms used, then to a relativistically imbued scientist the speed of light is unapproachable because it requires infinite amounts of energy; and creationism and intelligent design are unacceptable as proposals for exactly the same reasons. A biological analogue for the speed of light now makes a little more sense because we are looking for the enigma of infinite energy.

The basis of the theory we present to prove Darwinian evolution; and the experiment we conduct to validate it; is very simple. However, it requires a brief foray into the Riemannian geometry that Mirzakhani studied. This may sound daunting but we can explain the basic motivation by noting that adult reindeer live in the Arctic in very harsh conditions. There is a deficiency in salt. They urinate between ten and twelve times every day, producing about 13 litres overall, which is a considerable loss of valuable salt. Reindeer therefore drink their own urine, and are always on the lookout for more. Siberian sled masters will urinate by their sleds to attract the reindeer, so they can hook them up. And a traditional Inuit hunting method is to cover a suitably large hole with thin slabs of ice, and then to urinate on a trail that leads to the trap. The reindeer, on a constant search for salt and urine, walk right up and fall in.

The traditional Finnish unit of measure known as the poronkusema links reindeer urination times to distances. It is the approximate time-distance that a reindeer can walk before needing to urinate. It is approximately six miles, 9.6 kilometres, for the approximately 1.2 litres between reindeer urination events. We will therefore say, for convenience, that 1 poronkusema is 10 kilometre-litres, where the significance of the rather unusual units of measure should be obvious. Similar associations hold good for any two objects of interest.

We can now propose that a reindeer bull–cow couple produces three reindeer:

  • The first, when an adult, can walk 10 kilometres and provide 1 litre of urine to give the 10 × 1 = 10 kilometre-litres for its 1 poronkusema.
  • The second can walk 20 kilometres, but then produces only a ½-litre to give 20 × ½ = 10 kilometre-litres which is again 1 poronkusema.
  • The third walks only 5 kilometres, but has the largest bladder of the three and produces 2 litres to give 5 × 2 = 10 kilometre-litres for the same 1 poronkusema.

Our three reindeer have given us a set of Darwinian variations. The three at first sight seem very different, but they have the identical walking-urinating capability of 1 poronkusema. We can see that if we integrate each of their urination values with respect to the distance they each walk, we get that same 10 kilometre-litres each time. And since we can integrate them all with respect to distance and see this very clearly, we can conclude that in spite of their variations, they have come from a common source, of 1 poronkusema. They were produced by the same couple that gave them the same potentials and possibilities. Their integrals are clearly a journey into their respective evolutionary histories.

The opposite of integrating is differentiating. If we differentiate the poronkusema with respect to the distance that each reindeer walks, we will recover its distinct distance–walked–and–quantity–of–urine–produced rate. Since variations between the reindeer go away when we integrate, but reappear when we differentiate, then their derivatives are a journey into their respective evolutionary futures, while their integrals are a similar journey into their histories. Owen's homologies, Dawkins' memes, and biological circulations of the generations suddenly take on some distinctly interesting possibilities.

Each dimension we discussed for memes and homologies can have its separate Euler multiplier, its separate identity operation, and its separate set of integrals and derivatives with respect to each of the other two dimensions. These have together become ways of investigating homologies and memes, and evolution, fitness, and competition.

The first dimension we considered, when defining memes, was number. This gave us our integral of ∫ dP = 0, which is promptly a journey into the evolutionary past. A derivative with respect to number is then a journey into the future, and also becomes a statement of some rate of change of biological properties—homologies and memes—with respect to number. But since it is with respect to numbers and changes in numbers, it is promptly a statement of fitness and competition. It could even be that the ideas creationism and intelligent design cling to so fiercely are biology's equivalent of Clausius' limit cycle and identity operation.

It also raises other interesting possibilities. Just as ordinary particles can speed up towards the speed of light but never reach it, while tachyons can slow down towards the speed of light and also never reach it; a biological analogue for the speed of light could now be a population whose differential in distance with respect to number can never be constantly zero for that would mean reaching the invariant speed of zero competition which is then similar to the prohibition on the speed of light. This is something it is very easy to investigate.

Figure 0.8
0-sphere and 1-ball

We can now go back to our circles and our simple shapes, and exercise our imaginations. It does not take much to adapt our two circles and a line from a dodo framework to an ostrich one. Any such adaptation is likely to take a specified amount of time. We may therefore occasionally have to investigate that fourth dimension.

We already know that work is the ability to convert energy from one form to another. As in Figure 0.8, we imagine our biological populations as surrounded by ellipsoids that contain their mass and energy. We can go back to the meerkats and the shrews. How do they know which of them has gotten bigger, and which has gotten smaller?

The work, W, all biological populations do using their ellipsoids of mass and energy is defined, in physics, through its mechanical conversion capability. It is the integral, ∫, of force, F, with respect to displacement, d: W = ∫F dd. But … we have also just noted that integrating or differentiating distance is a way of moving into the evolutionary past or future.

As again in Figure 0.8, we can draw all this together. We can look on a generation as a distance. It is a displacement, d, all around a generation, and between progenitors and their progeny. It also affects the shape, size, and orientation of the circle that a generation traverses. That circle and its perimeter is a statement of the pasts and the futures of the populations that cycle about it relative to each other. Thus the whale and its ellipsoid on the upper circle is moving relatively slowly over the displacement, while the bamboo and the cricket and their ellipsoids are moving much faster. We imagine them each as moving at a speed proportionate to their generation lengths. We can therefore determine an effective generation “velocity” at which each population completes its generation. Thinking of a generation as a distance clearly has some interesting possibilities.

If a generation is a distance, then different populations are clearly going to travel different distances to reproduce. An alternative approach, as at the bottom of Figure 0.8, is to place each population at a set distance from a common centre. The whale is then way out on the periphery. Since its circle is bigger, it takes considerably longer to complete its cycle. Species and populations with shorter generation lengths will be closer to the centre. We can use the Euler and Clausius technique to find their integrals and derivatives over those distances, and so determine their evolutionary pasts and futures.

We also imagined an ellipsoid. Where Aristotle had proposed that fossils were formed and grew within the earth, Robert Hooke was the first to give the more correct modern interpretation. He devised a compound microscope and published the first work, his Micrographia, 1665, to describe various microscopic organisms. His illustrations turned the book into an immediate bestseller. He was also the first scientist to use a microscope with an inbuilt illumination system to examine fossils. He saw the close similarities between ancient petrified wood and living woods and other organisms. He convinced other scientists that fossils were the remains of biological entities that had existed in previous geological ages. His most famous discovery was the cell: i.e. that all living organisms are confined to boxlike constructions that separate them from the environment.

In the manner of Hooke, we surround every population with a unit ellipsoid or unit energy sphere. This ‘Hooke biosphere’ contains all the resources, energy, and interactions each population needs to create its memes and its homologies. We can then measure the numbers, resources, energy, and entropy density each population needs as concentrations at each point. Each entity and population's biosphere size oscillates about a mean value that we can now determine all about the cycle.

These little exercises in imagination indicate that entities and their populations display variations in their masses, energies, times, distances, and rates of travel about their generations as they create, disburse, and recreate their homologies, memes, and progeny. They are equivalent to adapting our two circles and a line from the framework for a dodo to that for an ostrich.

These kinds of changes in spaces, times, distances and quantities are the province of Riemannian geometry, whose premise is exceedingly simple. It was eloquently described by the illustrious fourth century Chinese philosopher Zhuang Zhou:

Once upon a time, Zhuang Zhou dreamed that he was a butterfly, flying about enjoying itself. It did not know that it was Zhuang Zhou. Suddenly he awoke, and veritably was Zhuang Zhou again. He did not know whether it was Zhuang Zhou dreaming that he was a butterfly, or whether it was the butterfly dreaming that it was Zhuang Zhou. Between Zhuang Zhou and the butterfly there must be some distinction. This is a case of what is called the transformation of things.

Einstein turned to Riemannian geometry because in spite of all his efforts, he failed to describe his theory by any other method. He eventually realized that he had to label all events so that they were independent of any observer, and so that it did not matter who viewed or reported them. The information for all observers always had to be accurate. He had to purge all measurements of any tainting by any specified vision of reality. We want to find something that is as true for the shrew as it is for the meerkat.

We are trying to do something very similar to Einstein. He tried to purge all measurements of any association with any specific observer, and we are trying to find a biological description for our Hooke biospheres that is independent of any specific genes, memes or homologies. Einstein eventually turned to Riemannian geometry because it is designed to track exactly these kinds of transformations.

It takes far more work to adapt two circles and a line from a dodo framework to an ostrich one than it does to adapt Riemann's geometry for any situation. Riemann's framework needs no adaptation at all.

Riemann's geometry is designed for such situations. It is as true for the shrew as for the meerkat. It is true for any other. This is why Einstein turned to it. It covers all conditions in which the basis for making the measurement has changed, but there is no easy way to determine exactly what has changed with respect to what. That is exactly the situation we are in when the relative sizes of a shrew and a meerkat change. We need a description that tells us what is going on, but that is independent of each entity's conviction either way, for the shrew and the meerkat could be creationists and/or intelligent design advocates who refuse to believe that it is their own populations that have changed. It is equally pointless to measure changes in reindeer urination rates and distances if we cannot then determine whether an observed change is genuinely some particular reindeer moving itself away from the norm, or if the reindeer is as normal as ever relative to all the others, but they are all together busy changing and evolving in their surroundings, while remaining the same relative to each other.

We recall that neither Owen nor Dawkins gave us any way of determining how long one of their generations actually was, even though they are as convinced as each other that their memes and homologies are handed down through the generations. Without that unit of measure for either the distance or the time concerned, it is impossible to prove anything either way. We therefore have to find that generational unit … and we must also purge it of any taint of association with any one population or species. The underlying idea, after all, is that a metre should be a metre irrespective of what is being measured, or who does the measuring.

Even though this problem that bedevils biology is so very simply stated, the editor of a prominent biological journal dismissed Riemann's geometry by saying that “the general theory of relativity was developed to account for a specific natural phenomenon, gravity”. This is to confuse that geometry with the Einstein theory. It is also to completely misunderstand what Riemannian's geometry achieves. It and the Einstein theory are simply not the same.

Riemannian geometry is, for example, applicable to Zhuang's butterfly problem. It is also applicable to the Hooke mass and energy biospheres we are using to surround our populations. Einstein's relativity is not.

Riemannian geometry is simply the study of a ‘connection’ between two distant points on the same surface or within the same space, but where the points upon that surface or within that space—or whatever might occupy those points—have no direct way of communicating. It is the study of how one dimension changes when the other does. If we broaden a mountain's base, there are consequences on how water flows across it in all its dimensions, not just in that one we have changed. This is their Levi-Civita connection.

The precise nature of these connections between points in a space can only be deduced by studying the space between them. A connection is a way of knowing exactly which direction we happen to be moving in with respect to every other direction, and how quickly, but always independently of all other directions; and because of that space and its changes and transformations. In Zhuang's case, the connection arises through a dream. In a biological case, it arises through the circulation of a generation. Neither one of these connections involves gravity, even though both involve distances and connections.

If we see a mountain change shape, then we know that there is a tilt everywhere. The gradient changes everywhere. All points experience and are affected by that change in gradient, no matter where they are. In the same way, if we can find a way to apply a ‘metric’ to Zhuang's butterfly and dream situation, or to our Hooke biospheres, then we can apply Riemannian geometry. The geometry and the study of connections across surfaces and spaces, and so between things and dimensions, allows us to describe what the universe would look like to creatures living within that space, and who are affected by that space, just as Zhuang and the butterfly are; but since every direction can be measured independently of every other direction, they do not necessarily know about the higher dimensional worlds and possibilities surrounding them … just as Zhuang's presentation fails by not taking account of the phenomenon of “lucid” or “directed” dreaming in which a person is aware that he or she is dreaming, and may also have directed themselves to have a specific dream.

In a lucid or directed dream Zhuang could have “fallen asleep” instructing himself to have his butterfly dream, and would then know both that he was a butterfly and Zhuang at the same time through being neither strictly awake nor strictly asleep. Or, alternatively, Zhuang could have been asleep, and was at first having a perfectly ordinary dream, and then within that dream suddenly realized that he was dreaming. The awareness involved in lucid dreaming varies from the faint suspicion that something of this kind is going on, to an extensive awareness that is unlike anything in ordinary waking life, and much fuller for it has all the possibilities of action of a dream. If we could find a way to establish, for such lucid dreaming, some kind of a “metric tensor” so that we could discuss how asleep, and how awake a person was, and also determine whether the lucid dream arose by being in a dream and realizing it, or else by beginning from being awake and moving to the dream, and so that there is a sense of “how much” of Zhuang is butterfly and “how much” is not butterfly at any moment, and how that arose, all in comparison to the quantity of waking or sleeping; if we can do all that, then we can use Riemannian geometry to discuss it … none of which is anything to do with Einstein and his general relativity.

The general theory of relativity is certainly the best known, and the most successful, exposition of Riemannian geometry's ability to handle connection and measurement situations, but the formal principles it embodies are quite distinct. To say that “there is no reason to assume that this formal framework will also apply to other, non-related phenomena” is to forget that Newton developed the entire integral and differential calculus specifically to deal with gravity, although Leibniz, its co-discoverer, admittedly had a broader interest. The calculus has been successfully applied to every field imaginable, without anyone declaring that we should exercise extreme caution in using it because it was first successfully used to discuss gravity and therefore might not be appropriate elsewhere. Modern science would not exist without the calculus. As Mirzakhani expressed it, “Most problems I work on are related to geometric structures on surfaces and their deformations”. The Riemannian geometry on which relativity theory is based pre-existed the Einstein theory. He used it because it was explicitly designed to handle situations such as the ones we also face in dealing with memes, homologies, and those possible differences between shrews and meerkats that might presage evolution.

When Clausius applied the Euler multiplier technique and successfully found an identity state and identity operator, he used the same formal apparatus Riemannian geometry uses. He carefully emulated Newton's similar quest for an isotropic and homogenous condition which would describe the objects he was studying. He was looking for their Riemannian style connection, which is in this case what it would take for two things to have the same entropy, and so the same configuration condition. This is a connection because (a) the entropy can remain the same while the energy changes, and (b) the energy can stay the same while the entropy changes. What kinds of shapes, lines, angles, structures, and surfaces will biological entities and their populations form if we can think of them as being lines, and as having a connection between their various kinds of possible transformations?

Dawkins, who again does not profess to be a mathematician, unwittingly compounds the difficulties for everything to do with memes when he writes, in his The Blind Watchmaker, that:

What lies at the heart of every living thing is not a fire, not warm breath, not a ‘spark of life.’ It is information, words, instructions. If you want a metaphor, don't think of fires and sparks and breath. Think, instead, of a billion discrete, digital characters carved in tablets of crystal. If you want to understand life don’t think about vibrant, throbbing gels and oozes, think about information technology (Dawkins, 1986).

The author of another well-known book makes the difficulties clearer when he reviews current theories in physics; compares them to the relatively parlous state of theories in biology; builds on the Dawkins paradigm; and then says:

… there ought to be a general, abstract mathematical theory of evolution that captures the essence of Darwin’s theory … and develops it mathematically. … How can we do the same for biology, a very very different kind of science from physics? Well, not by using the differential equations of theoretical physics! To develop a theoretical physics for biology, a fundamental mathematical theory for biology, we must use a different kind of mathematics. Differential equations will not do. Not at all.

Yet … those very differential equations that are the ubiquitous armoury of modern science and that are responsible for its extensive successes … they are the very things we have successfully used to investigate distances into evolutionary pasts and futures, and to at last prove Darwin's theory.

Dawkins' well-intentioned statements of principle, and the support they have received, unfortunately take us into the quagmire of modern information theory, which has its own version of ‘information entropy’. This information entropy, which deals with the converting of messages, is all too often confused with the entropy that Clausius discovered, when all that the two really share is a name.

Information entropy requires that we define a ‘message space’. For example, the Sanskrit letter sequence गच्छति—transliterated into Roman script as gacchati—is meaningless to most people. They find it devoid of information. It does not feature in the message space that the symbols they know create. Those symbols are perfectly clear, however, to someone who speaks Sanskrit. Those symbols then feature in the message space jointly constructed by both the sender and the recipient. Under those conditions, the symbols successfully convey the information “he, she, or it goes”.

It makes no sense to discuss the information entropy of a Sanskrit or other sentence, or the probability with which a message is conveyed, when the recipient speaks no Sanskrit. There is then no possible arrangement of symbols that can convey any information, and all arrangements are then the same. Information may be an abstract property that somehow transcends the medium of the message, but it is not so abstract that it does not follow certain very definite rules. One of those is the coherence of the message space.

In the same way, the earth may be a relatively small planet, but its citizens have created space craft that, like Voyager I and Rosetta, have left the solar system or landed on comets. We understand immediately that the capability and organization of a society is distinct from the size of its home planet. We nevertheless expect every alien civilization to have a home planet. We expect it to be large enough to support the resources and capabilities of space travel, and so would not expect it to be much smaller than ours. But we also do not expect it to be so large that space travel becomes impossible through an overwhelming gravitational attraction. In the same way, information entropy requires that a very definite message space exists. We cannot dissociate the content of messages from the mechanisms that convey those messages; from the noise that the medium might impose as it potentially scrambles those messages … and we also cannot dissociate it from the capabilities of both the sender and the receiver. A whale's genetic information is useless to a mosquito and vice versa; and both those sets of genes are useless without a suitable environment. And … any population-environment interaction whatever involves the conversion of energy, which means entropy.

Figure 0.9
0-sphere and 1-ball

Figure 0.9.A shows a shelf of books. They contain a given amount of memes and information. Figure 0.9.B shows exactly the same books, but all set out in a slightly larger font. Since every book is larger, each book holds more pages and more paper, and we need a bigger and sturdier shelf to hold them. Or, alternatively, and as in Figure 0.9.C, we can leave the books the same size; have the same number of pages in each; and simply have more volumes to accommodate that larger font. Either way, the information content is the same, but the number of books and shelves required has increased.

The British Library currently holds about 150 million items (not all of them books), the United States Library of Congress about 158 million. A smallish library can easily hold somewhere in the region of 350,000 books. The walls and floors must be properly shored up to accommodate that approximately 30,000 kilogramme, 30 tonne, mass of ink and paper that encodes all that information.

If we either scrambled up all the letters in every book, or else placed them all in alphabetical order, we would destroy all the information in the library. That relatively useless mass of paper and ink that we can imagine initially confirms that the information in those books is something more than just a mass of ink and paper.

There is, however, another side to this coin. It is all very well to imply that memes and information either transcend mass, or else have no mass, but we cannot hope to preserve the information in those books, nor hand it on to succeeding generations, without that mass. This raises two linked issues.

For the first issue, it is also all very well to say that the Mona Lisa is something vastly greater than the weight of the pigment and the poplar wood panel it is painted on. However, it would still be nothing without that encoding and that substrate. We would not have the Mona Lisa if Leonardo da Vinci had not made efforts to encode his imagination into that specific material form. The difficulty for information is that it always has some kind of encoding or configuration energy. Information must always be encoded in, and must always be transmitted by, something material. We cannot separate information from both (a) the masses, and (b) the energies that emit and/or carry that information. Information is always nothing without its conveying mass.

The second issue invokes Einstein's theory of relativity. The books in any library are rendered useless not only if we either scramble the letters, or else put them all into alphabetical order; they are also rendered useless if we do not teach anyone in the upcoming generation to read and write … and just as the Mona Lisa is an unprepossessing lump of wood if successive generations are not taught to appreciate art. The first situation has people who have been configured to read, and who have the potential to absorb written information, with the objects they wish to read being unable to convey it; while the second has the information intact, while the recipients are improperly configured. Either way, and by relativity theory, the books are converted into mere masses. In the same way, the genetic information encoded in a mosquito is useless to a blue whale and vice versa. We can use a combination of Riemann's geometry and Einstein's relativity to measure which of these has occurred in any situation.

Lumping Dawkins and his most ardent creationist and intelligent design critics together might be initially surprising. But when it comes to the problem of measurement, these two protagonists are little different from each other. To quote the title of another of Dawkins' best-selling books, they are but variants of “the material delusion”.

As the deficiency with memes should have made clear by now, neither Dawkins nor the advocates of creationism and intelligent design are discussing and debating evolution or information in any useful way. Neither party has indicated how this information supposedly contained either in genes or in memes is to be measured. They do not indicate any units. Neither party gives us any way to measure the information in any biological system—whether homologically or memetically—so that we can settle the points at issue. Neither group indicates how meaningful kinds of information are to be distinguished from non-meaningful kinds. Books with all their letters in alphabetical order are as useless as books with all their letters randomly and evenly scrambled, but they are still measurably different from each other. In the same way, smart phones, which are more advanced cell phones with enhanced computing and connectivity capabilities, may be better at handling information than so-called dumb phones, which do not have such additional features, but smart phones also consume considerably more power. Smart phones need their data plans, which dumb phones do not. Those data plans simply add to the smart phone's energy, expense, and resource requirements. All these can be measured. Homologies and memes, by contrast, suggest no way we can measure them.

The overall strategy behind the theory we shall prove is really very simple. We have already outlined the three dimensional attributes we will work with. They create each population's Hook biosphere. We intend to measure that biosphere, which is all mass, energy, and reources, over the generation length we have drawn in Figure 0.8. Once we have measured those biospheres, we shall calculate each of their averages. We then use those averages as a measuring stick to determine the population interactions with the surroundings. They are a set of Euler multipliers and identity operators. So if, for example, an entity has an initial mass of 2 grams, and a final one of 6 grams, then its average mass, expressed in its biosphere over the generation is 4 grams. We now take that 4 and use it to divide everything. This gives a range for its biosphere from ½ (i.e. 2 ÷ 4) to 1½ (i.e. 6 ÷ 4). The population's Hooke biosphere, which it uses to maintain itself, therefore varies as ½–1–1½. That is its range and its average, all in its suitable, self-contained, generational units. We can do the same for the other two dimensions. If we bring them together, we have the population's activities expressed in terms of its surrounding Hooke biosphere.

We can repeat this for the next two generations. One might give the absolute value range 1 to 8 expressed in some set of units; the other 3 to 5. These are differences in biospheres, which are resources and energies. The former has an average, again in absolute terms, of 4.5. Once we express it in terms of its own mean, it gives us the range 0.222–1–1.778. The other value set repeats the original absolute average of 4, but this time with a range we express as 0.75–1–1.25 once we apply the mean. These three Hooke biospheres are clearly different. One is ½–1–1½, another 0.222–1–1.778, another 0.75–1–1.25. They are all, however, orthonomal, or stated relative to their units. Differences in these Hooke biospheres are differences in the populations that create them. We can therefore compare them. We can contrast them easily to see how they differ. We can compare any population's biosphere to any other, which is the advantage of our orthogonal and orthonormal axes. Since they all range around a unit determined in the same way, we can easily find their intrinsic and extrinsic connections, as well as all their variations.

This may seem a strange way to proceed … but then … if ordinary methods and procedures could have proved Darwin's theory, someone else would have done it already by now. They would also have done it using those ordinary and well-known methods. Since it has not thus far been done using those ordinary and well-known methods, then only something unexpected will do it.

The use of the unexpected also means that different pieces of information, from several different fields, each looked at in a slightly different way, are brought together. It also means we have to be very careful with terms, definitions, and usages. We have used diagrams wherever possible, and used as many analogies—such as Zhuang and the butterfly—as possible.

However … it is a regrettable fact of life that every valid scientific theory must have a solid mathematical foundation. This one is no different. If merely talking about fitness, competition, and evolution could definitively prove Darwin's theory, then Darwin himself would have done it. What is lacking is the mathematical provenance. That is what we provide here.

We have done our best to present some sometimes very complex material clearly and simply, bearing in mind the following criticism made by someone who kindly reviewed an early version of this site: “Imagine that you studied Latin (as I did for four years in high school). Many, many years later you are called upon to read a work in Latin but are at a distinct disadvantage because you only recall one-tenth of what you learned—the occasional word or phrase here and there. Not nearly enough to make sense of what you’re reading but enough to make you feel the frustration of your lack of vocabulary”.

If you come across anything that makes you feel like the above, then you should apply Newton's method. It is here described by his friend Abraham DeMoivre:

he having some difficulties … relating to the solution of quadratick [&] Cubick Equations. Took Descartes's Geometry in hand, tho he had been told it would be very difficult, read some ten pages in it, then stopt, began again, went a little farther than the first time, stopt again, went back again to the beginning, read on till by degrees he made himself master of the whole, to that degree that he understood Descartes's Geometry better than he had done Euclid (Whiteside, 1967-1981).

Books Newton owned are easy to identify. He had a very distinctive method of marking his place. He would dog-ear pages by folding a corner down to point to the precise word he was interested in. So do as Newton did, and mark that troublesome spot. Then back up to where things still made sense to you, and try reading forwards again. If it still doesn't make sense then continue. Go on beyond the spot you are having trouble with. Keep reading until you see the information being put to use. Then continue a little further. Then stop, and do what Newton did. Return to your mark. Go back a little behind it, and try reading forwards again through the bit that did not make sense before. If it still does not, then come back here: to this preliminary chapter. Read this chapter again. It has the overview for what we shall be doing. Then read the next chapter: The Experiment. It says very clearly what we are planning to do in terms of experimental proof. Then if you wish, you can go ahead and read Chapter 21: The Conclusion. You will then know where we are headed, what we are trying to achieve, and see the method we use in action. Then go back and reread Chapter 15 and Chapter 18. You will find there a slightly more detailed outline for the experiment, and the foundation for the conclusions we draw. Then go back and reread Chapter 14: The Anomalies. You will find an explanation for almost everything you will need to understand the analysis there.

The basic idea is very simple. Some of the matters we shall investigate are admittedly complex, but this site has been constructed to be accessible to the average layperson who fairly happily completed the final two years of a secondary or High School education. If you have followed this discussion up until this point, then you should have no trouble following the bulk of what is on this site. You can safely hop over the boxes like the one just underneath (headed A mathematical aside). You will miss very little.

A mathematical aside

Here we place some background mathematical information validating what has just been said, or giving some formulae to show that what is being presented verbally is mathematically rigorous.


The first 19 chapters, including The Experiment, are quite short. They should not cause too much trouble. Chapter 19: The Refutation, is a tad more challenging, but it can still be approached on the above basis. It proves that creationism and intelligent design are simply impossible by highlighting their contradictions of principle and definition. It also provides the four equations we use to analyse the results of the experiment we shall conduct.

Many people will find Chapter 20: The Proof, quite challenging. It is also quite lengthy. It is lengthy because, as an example, most people think of the difference between £10 and £20, or between $10 and $20, as a “value” rather than as a “distance”. Value and income are used for a reason: they are generally sufficient. But the very fact that they are so familiar can obscure what is going on with them, and can prevent us from looking at them in a new way.

Few people think of an exchange rate as a way of “translating” one currency into another. Few people think of “income” as a flow, or as a “time derivative within an economic field”. There is no need. Few people would think of economic transactions between any two currencies as “vectors”. But just because we do not generally think of economic and currency transactions as vectors, does not mean they are not vectors. Whether or not it is useful to think of them that way depends on what you want to do. ‘Building an income’ is an easy concept; and so is ‘sending all the money you earn to your relatives in another country’. These two ideas might help us better understand a “vector flow” and a “flux”. But you then have to take care to state the rules; give counter-examples to make sure there are no mistakes; show that you have followed them all in this new situation; and then by way of final confirmation, demonstrate that the counter-examples are inapplicable to the situation at hand. That can be quite a lengthy process.

We have already given several examples of elementary errors of principle that some extremely eminent, and highly experienced, scholars from such prestigious universities as Caltech, Cambridge, Cornell, Harvard, Oxford, MIT, Princeton, Stanford and Yale—to name but a few—have made over issues of exactly this kind. Time is always pressing for scholars of a certain calibre who always have better things to do than to read things that seem so elementary, and so obvious, that they wonder why anyone would write it … and, in any case, they already know it. Every working scientist believes that he or she knows, for example, what entropy is; and what the unbreakable barrier for the speed of light means. But entropy is an identity operation. And it is not just the speed of light. It is the implication for infinite energy.

To return to our example, most economic transactions go in specific directions. Those that go the other way, such as gifts and reciprocal obligations, can still be quantified, but can also be marked as going “in the opposite direction”. Economic transactions, therefore, have both (a) magnitude and (b) direction. Many people who then think they know what vectors are—because they work with them every day but in a very different context—need to be reminded of what it takes for a thing to meet the definition of a vector. Many such people are heads of major departments and research institutions; and/or have written books; and/or edit some extremely well-respected journals, but have quite simply forgotten, because “familiarity breeds contempt”, the true meaning of the speed of light; or that any difference between any two things can be a “distance”. And when looking on money and currency as vectors allows us to prove something completely unexpected—especially if it is an ancient problem that has not been solved before—it is often necessary to repeat the definitions, and to demonstrate all over again that all the rules have been followed, even though this is an unusual way to proceed.

Things that are born behave differently from the things that gave birth to them. If we think of populations as Hooke biospheres ranging across ½–1–1½, 0.222–1–1.778, or 0.75–1–1.25, then anything with a mass value in its biosphere less than 1 has probably just recently been born. It is moving from a minimum towards a maximum, which means it is heading towards 1, which is that population's average mass. Anything beyond 1 and close to the maximum is probably ready to split itself at some point and reproduce. That is to head back to the minimum. These are magnitudes … linked to directions.

We can now measure every population in its intrinsic units and compare them all, from bacteria and shrews to meerkats and whales. And … they are all also vectors. As we shall see, they follow all the rules.

Newton is again pertinent. So also is the quote from Galileo standing at the head of this chapter: “All truths are easy to understand once they are discovered; the point is to discover them”.

In 1663, when Newton was 20 years old, he knew virtually nothing of mathematics. His teachers had no particular reason to suspect his genius. Then quite by happenstance, he bought a book on astrology. He could not follow it. He did not know the mathematics involved. Since the knowledge he needed was mostly trigonometry, he got himself a book. He found that he could not understand that either. So he decided to take another step back. The foundation of trigonometry is geometry, so he got a copy of Euclid's Elements to learn geometry.

And … that was the difference between Newton and most people. As do most people who read Euclid, Newton was initially deceived by the elementary seeming lists of definitions and propositions. They were so obvious, and so simple to follow, that he wondered what anyone could see in Elements, and why it was so famous and had lasted for so long.

And … that was again the difference between Newton and most people. He read the proof that any two parallelograms that share the same base will always have the same area. This surprised him. It surprised him so much that he went back to the beginning. He then reread all those preceding extremely simple things with a new found and lifelong respect for the method Euclid had used:

In 63 [Newton] being at Sturbridge [international trade] fair bought a book of Astrology, out of a curiosity to see what there was in it. Read in it till he came to a figure of the heavens which he could not understand for want of being acquainted with Trigonometry.
Bought a book of Trigonometry, but was not able to understand the Demonstrations.
Got Euclid to fit himself for understanding the ground of Trigonometry. Read only the titles of the propositions, which he found so easy to understand that he wondered how any body would amuse themselves to write any demonstrations of them. Began to change his mind when he read that Parallelograms upon the same base & between the same Parallels are equal, & that other proposition that in a right angled Triangle the square of the Hypothenuse is equal to the squares of the two other sides.
Began again to read Euclid with more attention than he had done before & went through it (Whiteside, 1967-1981).

The material in the Proof chapter has been written with an equally healthy respect for the canons of mathematical proof. Referring to a generation length as a displacement is far more than just an analogy. We do not know why biology and biologists have thus far failed to properly define a generation length, or to reference each population with its native orthonormal unit so they are all comparable. Almost every book highlights the inconsistencies in the ways biologists handle generation length by offering some variant. Yet … defining a generation length is actually very simple to do. It also has profound consequences. We simply turn to 3 + 0 = 3 and 3 × 1 = 3 … and we then copy that and set ∫ dP = ∫ dM = ∫ dS = 0. This establishes a metric tensor. It gives us the ruler we need. We can then set that measuring system loose on a Hooke biosphere … measure all aspects of fitness and competition … and prove Darwinian evolution. We can even run The Experiment we describe in the next chapter to prove it.