# 14: The Anomalies

Now we have a base line for how our cell and its internal energy behave, we need to take some measurements …

… and if we are going to unravel this mystery …

… if we are going to prove that a population free from Darwinian competition and evolution is simply impossible … then there are four things we must do:

1. We have already been alerted to the realization that we must pay close attention to the first of three anomalies: the distinction in internal energy between mechanical and nonmechanical chemical energy.
2. We must, however, note the source of a second anomaly. It deserves our equally close attention. We have already observed it in meeting (A) the population fluxes of mechanical and nonmechanical chemical energy of M and P; and (B) their respective average individual values of and ; along with (C) the realization from the Helmholtz decomposition theorem that the latter two values—i.e. the population's average individual mass and energy fluxes, and —are unique to every population. Therefore, the values over the generation, which are the averages of all the averages over the period, are m̅’ and p̅’, and are also unique per each population. Those generational average values—which are always symbolized with the terminal “’”—are two of the three values that we shall prove define each population.
3. We should ready ourselves to follow up on a third anomaly we shall meet shortly concerning the different ways to measure the generation time.
4. We must borrow an idea from Sir Isaac Newton (who better?!) that helps us to navigate the above three anomalies (i.e. including the one we have yet to meet).

We must now understand clearly that biological populations depend upon both sums and distributions. They depend upon both totals and averages. There are: (1) the things they must all do; and (2) the things they must each do. It is that juxtaposition that creates the anomalies in (1) and (2) above that intersect with each other.

As for (4) above, we can express the idea of Newton's that we are going to use very simply.

We return to planetary motion, the source of and explanation for all energy, including biological internal energy. This must cross boundaries. It must do so as work and as heat. It must enter populations; maintain them; and then leave. Microscopic and macroscopic objects are both subject to the energy and its laws. Those laws are the way to unravel this mystery of evolution.

Suppose a planet is travelling in some set direction with a velocity, v, of 5,000 kilometres per hour and so that v = 5,000 kph. Let it travel for 2 hours so that time t = 2 hours. Then we know it must have travelled the length, l, of 10,000 kilometres, because l = 5,000 x 2 = 10,000. We get that 10,000 from the relationship “distance = velocity times time”, i.e. l = vt.

So far so good.

Newton then made a suggestion. Let us imagine a vanishingly small—an infinitesimally small—period of time. The technique he used seems to have also been known to the German philosopher and mathematician Gottfried Wilhelm Leibniz, who hit on it independently. This stimulated a controversy over priority that bedevilled natural philosophy for centuries.

Partly because of this double discovery, there is no one, unified, notation to depict these processes.This is a little unfortunate for those unfamiliar with it. Newton used his own notation which admittedly has its usefulness, and so is sometimes still used. His notation, however, was not as versatile as Leibniz's, which is therefore more common. There are other notations.

The different notations are useful in different contexts, and for different reasons. The most common and more modern way of writing the infinitesimally small distance Newton imagined is using Leibniz's notation, which is “dt”. Newton wrote exactly the same thing as (t with a dot above it, if your browser does not render it correctly). There is no one unified method for depicting the process because there are situations in which one of the various renditions is clearer and more compact; and there are situations in which another of the many variants and symbols is clearer and much less prone to errors and confusions. Since our use of the idea here is straight forward, and mostly quite simple, we will generally use Leibniz's notation: dt. Since t in this case stands for time, then this means something on the lines of ‘imagine a vanishingly small increment of time that is far smaller than you can imagine; and that is yet smaller than that still, if we need a yet smaller one’. And if we use p to stand for perimeter, or c to stand for circulation, then dp and dc would be ‘an infinitesimal amount of perimeter’, and an ‘infiniteismal amount of circulation’ respectively.

Since we are going to discuss reproduction, we need to refer to the past, the present, and the future. We will therefore adopt the convention of letting ‘-1’ stand for the past, while ‘0’ and ‘1’ stand for the present moment and the future one, respectively. Thus t-1 is the past moment, t0 is the present, and t1 the future; with p-1, p0, and p1 being the perimeters at those moments; and c-1, c0, and c1 being the circulations at those same moments.

Figure 14.1 With those conventions, then the total distance dt in Figure 14.1 is that between t-1 and t0 which tends to the past from our present moment; added to the distance between t0 to t1, which tends to the future, also from the present moment. That infinitesimal span dt is the total time between our selected moment in the past, through the present, to our selected moment in the future. It is a part of the overall length, T.

There is an important difference between Figures 14.1.A and B. Figure 14.1.A takes a more “global” view of our expanse of time, while 14.1.B takes a much more close-up view. They of course both deal with both perspectives, but the former is more associated with Leibniz's view of events, the latter with Newton's.

The two views taken together tell us that all the lines and curves we will be dealing with on this site can always be cut up or divided into a set of extremely short consecutive straight-line segments. At whatever scale we cut—and Leibniz and Newton were both insistent that we can always cut a little smaller and finer—we can always stitch all our straight line segments back together to recreate the original curve.

The segments we cut out of our original curve are each so infinitesimally short that they hug the original curve so very closely yet … they are each identical to some straight line. We can safely treat each segment as a line. We can then express each of those infinitesimally short and straight line segments in terms of whatever values we have on the axes. We can express each such line in terms of whatever basis and measures we want.

The most general way to refer to the straight line hops we can create, using Leibniz and Newton's method, is in terms of the two axes x and y. Those are the most common references. However, those two axes can be anything we want. We could have potatos on one axis, and carrots on other. If we use x and y, then we can call each of the infinitesimal lengths along each of the axes dx and dy.

We can now express the original infinitesimal length we are interested in, which was dt, in terms of the ratio between dy and dx. This gives dt = dydx. That now holds good all around that curve. The original curve is now much easier to handle, because it has become a sequence of straight lines. The straight line we draw at each point on a curve, to replace the curve, is called a “tangent”. So “find the tangent to a curve”, simply means, “find whatever straight line fits the curve and substitutes for it at that point”.

Let's now have another look at our planet moving at 5,000 kilometres per hour. It is going to travel a vanishingly small increment of distance, dl, over that equally infinitesimal span of time, dt. If we want to know that infinitesimally small distance, then we simply do what we did before: we multiply the one by the other because distance still equals velocity times time. This gives us dl = vdt.

… And … that's pretty much all we need to know.

A mathematical aside

We use the following convention on this site. An ‘infinitesimal’ is any quantity that it is so small it could as well be zero, but is not quite zero. Following Leibniz's notation, this is something like dx.

However, when x changes by an infinitesimally small amount to become x + dx, we can generally refer to that same infinitesimally small quantity, dx, as a ‘differential’. The differential, dx, is the very small difference that changes the original. So—on this site—a differential and an infinitesimal amount are the same thing; just that sometimes we mean by differential that some prior quantity is changing by some infinitesimally small amount. That change is the difference between the two, and the differential. Generally, when we use infinitesimal, there is no prior quantity being referenced.

It usually takes an infinitesimally small amount of time, dt, to make any infinitesimally small change, such as dx. The change could be an infinitesimally small change in blah, in which case it would be d-blah; or anything else. But since things change over time, or else change in concert with other things, then whenever an infinitesimally small change such as d-blah takes place, there could also be some accompanying infinitesimally small period of time in which it occurs; or else some change in whatever of d-whatever; or of anything else. In general, that prior d means infinitesimally small amount of whatever follows it.

If we want to know the total change that has occurred over a long stretch of such infinitesimal changes, then we want to sum all those little dx's. We still use the terms and symbols Leibniz introduced. To take the sums of many infinitesimally small amounts is to integrate them back into a whole, which has the symbol, ∫, taken from the Latin summa, and simply means ‘sum’. Therefore, infinitesimals are summed. They are most often used in situations in which we are going to find some total change. That total change is the sum or integral of a long sequence of infinitesimally small changes. An infinitesimally small change in blah would be d-blah, or in say, c, would be dc; and the sum of a set of those infinitesimally small changes over our chosen distance or time span—i.e. the total amount of either blah or c—would be either ∫d-blah, or ∫dc.

One thing, however, is usually changing with respect to something else. We most frequently consider infinitesimal time periods dt, over which we have an infinitesimally small change dx. We very often want to know the rate at which x is changing: i.e. how much one thing changes when another changes. To determine that rate, we want the differential amount by which x changes, which is dx; and the differential time period over which it changes, which is dt. (If we are interested in the rate of change with respect to y or blah, then we would want dy or d-blah).

If we now want to know the rate at which something is changing when something else changes, then we put one differential change over the other. An example would be dx/dt. This is telling us the proportionate change in one infinitesimally small quantity with respect to another.

The placement of one differential quantity over another to find relative rates of change has a formal name: ‘derivative’. So velocity, v, is the derivative formed by taking the differential change in position in a given direction, dx; and placing it over the differential in the time taken to travel that distance, dt; which then gives us the derivative of v = dx/dt. This says that velocity is the derivative of the x-position with respect to time. This is Leibniz's way of writing it. In Newton's convention, this is written as v = . They both say ‘the velocity, v, expressed in terms of x is the rate at which x changes in time’.

We will always, on this site, use ‘derivative’ to refer to a rate of change. We will mostly use ‘infinitesimally small amount’ rather than ‘differential’ to refer to quantities such as dx (it is a little more evocative for those who are not used to it). If we use ‘differential’ it is generally because it is either (a) shorter; or else (b) because we are about to take or produce a derivative (or rate of change) by bringing two differentials (i.e. two very small quantities) together, such as with dy/dx.

So let's think again about our biological population. It has n entities, each with an internal energy composed of an average individual mass or mechanical chemical energy of . There are again things they must all do as a population; and also things they must each do as individuals within the population.

As our biological population goes through its cycle of the generations, it has a mass flux of M kilogrammes of mechanical chemical energy per second of chemical components and other resources. If we want to know how much mechanical chemical energy the whole population uses over an infinitesimally small span of time, we only need to multiply the one, M, by the other, dt. The value would at first sight appear to be as simple as Mdt for the kilogrammes of mechanical chemical energy and resources used.

A mathematical aside

We also know, however, that M = nm̅. So we can surely substitute the one for the other? It would then at first sight look as if that Mdt—the thing they must all do—should at all times be equal to nm̅dt—the things they must each do. So if M = nm̅, then, surely, Mdt = nm̅dt?

A mathematical aside

BUT … as we shall see more clearly in a little while, the presence of that n means that it is not necessarily as simple as that. I.e. We are alerted to an anomaly concerning the difference between what they all do, and what they each do. We now have both Mdt = nm̅dt and Mdtnm̅dt. I.e. we are alerted to the fact that Mdt nm̅dt. (We shall explain those symbols in 16: The Basis ).

A mathematical aside

And, in the same way, if we want to know how much nonmechanical chemical energy our population has used over that same infinitesimally small span of time, then it is going to be Pdt joules of energy.

And, in similar fashion, since P = np̅, then it would also seem that Pdt—what they must all do with that nonmechanical chemical energy—should always be equal to np̅dt—what they must each do with that same energy. Unfortunately, however, the presence of that n again means that it not necessarily so simple. So Pdt np̅dt.

Nevertheless … we now know two important things. At each and every moment, our population is directing Pdt joules of nonmechanical chemical energy at Mdt kilogrammes of mechanical chemical energy. So we have a specific quantity of chemical and transformational energy directed at varied, but equally specific, chemical components.

A mathematical aside

As an associated mathematical aside … we can apply Newton and Leibniz's infinitesimals method to the entire generation, and so produce a sum—more properly an integral, ∫— characteristic of that population. (As above, an integral is really just a fancy and convoluted way of taking a sum over an infinite, or near-infinite, number-set of infinitesimal increments). If we call the generation length T seconds, then since the population's mass flux of mechanical chemical energy at each moment is M kilogrammes per second, then the total quantity of resources and chemical components and mechanical chemical energy used is clearly ∫M dT kilogrammes (which just means ‘sum the mass flux over the set of infinitesimal increments we have across our T seconds’). This is what they must all do. They must all together, as a population, bind that quantity of chemical components by expending that many kilogrammes worth of mechanical chemical energy.

Again note that it is the chemical energy aspect that is important here. The kilogrammes is just an index into that chemical energy's current mechanical effects, and so we can equivalently think of this as a given number of moles of chemical components held within a given volume, and ready for a varied set of chemical interactions.

A mathematical aside

Then for the anomaly in numbers: the juxtaposition between population and individuals. We can of course express the same total we derived a different way. We can calculate the same amount using a different method, and take advantage of the common notion: “things that are equal to the same thing are also equal to each other”.

We now work with the individual entities. If we want to know the total mass, or stock of mechanical chemical energy, we can sum (or, more accurately, integrate) all the masses of each of the distinct entities. Since the number in the population at each moment is n; and since they are each of mass m at each moment; then the total quantity of mechanical chemical energy and resources used can also be expressed as ∫dm dn kilogrammes because we have to separately sum—or integrate—both (a) the mass, and (b) the numbers. We therefore need m × n over them all, which is all that that integral says. This is what they must each do as individuals: expend that equivalent amount of kilogrammes worth of mechanical chemical energy.

Since M = nm̅, then we can in fact substitute the average individual mass—which we have already noted to be the measured divergence for the mass flux of mechanical chemical energy—for each distinct entity. Every time we measure an entity we record its mechanical chemical energy or mass, which keeps changing with each of them. When we have measured all of them we have the value common to all of them, which is a weighted average due to their distribution. This procures for us precisely the same result, and is ∫dm̅ dn. Therefore, the average individual value for mechanical chemical energy, and its characteristic distribution, are again one of any population's signature values. It is the precise measure of what they must each do with that particular energy at each moment in time.

And since, in addition, we have two different ways of producing the same result, then these two would seem to be equal such that ∫M dT = ∫dm̅ dn … but it is not necessarily as simple as that for although the totals of those two integrals are certainly the same, that n means that they do not necessarily infinitesimally increment in the same way and at the same rates, for Mdt nm̅dt. We have an anomaly between what they must all do, and what they must each do.

The same of course goes for the nonmechanical chemical energy. Since the Wallace pressure or nonmechanical chemical energy flux is always P watts for the population over the generation; and since it is p watts per each of the n entities; then the total nonmechanical energy used is ∫P dT = ∫dp dn = ∫dp̅ dn joules, with the population's average individual value again being determinative, and again also being the divergence for the nonmechanical energy flux. But once again, it is not necessarily so simple, in that while the integrals can be equal in total, the n very likely means that they are not equal at each point, with Pdt np̅dt, again highlighting the continuing anomaly between what they must all do, and what they must each do.

These two equalities of (A) ∫M dT = ∫dm̅ dn and (B) ∫P dT = ∫dp̅ dn for each of the mechanical and nonmechanical energy fluxes will be of great significance very shortly … along with the fact that one in each pairing contains infinitesimal increments in time (as dT), while the other contains infinitesimal increments in numbers (as dn). The latter infinitesimal increments are apparently independent of time. In other words, we have two intersecting anomalies. One of these ways of reckoning a population's behaviour depends entirely on—and involves—changes in time and the generation, while the other involves changes in numbers over the same generation. But we at the same time have the ongoing anomaly of the distinction between mechanical and nonmechanical chemical energy, along with a potential anomaly concerning time. We shall come back to these important points very shortly.