15: Measuring the Anomalies

We spotted two (out of the three) anomalies in the last chapter. Spotting them, however, is not the same as being able to measure them. One of those anomalies is the contrast, within a biological entity's internal energy, between the mechanical and the nonmechanical chemical energies; the other is the similar contrast between individuals and their populations, and so in the summing and partitioning of their internal energies. (The third will factor in when we have to decide how we should measure the time it takes to navigate the circulations of the generations). We are nevertheless making good progress.

Our prototype cell gives us a basis we can use to measure things. Our mass and energy fluxes of mechanical and nonmechanical chemical energy, M and P, give us population values: things they must all do jointly that we can measure. But there are also things they must do individually that we can also measure.

A mathematical aside

We know, in addition to all of that, that if we have a mass flux of M kilogrammes per second and an energy flux of P watts, then our population will use Mdt kilogrammes of various chemical components and mechanical chemical energy over any Newton-style infinitesimal time span of length dt, as well as Pdt joules of nonmechanical chemical energy. Since the Mdt is the mass of kilogrammes over the infinitesimal time interval, dt, it measures the population's net “impact” or “forcefulness”. And since Pdt is the energy used, it measures the “biological power” used. Those two together are the complete effect on the surroundings.

That is all very well, but we are actually rather more interested in the population's transformations. Those are the key to separating out what they must all do, from what they must each do. And … measuring those kinds of transformations can be rather tricky. They involve M = nm̅ and P = np̅. The first in each coupling is for the population, the second for the collected individuals. We therefore have two different ways of measuring what happens to either mechanical or nonmechanical energy over any infinitesimal time interval, dt. We can have either Mdt or nm̅dt for the mechanical chemical energy, and either Pdt or np̅dt for the nonmechanical variety.

Since M = nm̅ and P = np̅, then it would at first seem that they would give the same value. But … we must first establish that equivalence. Each of nm̅dt and np̅dt involves n. They therefore involve numbers over time, which are distributions. Since dt is an infinitesimal period of time the two quantities can only be the same if numbers stay the same across the entire interval we measure. This is the heart of Darwinian fitness, competition, and evolution. It is what we have to settle.

There is the added complication of the difference between the mechanical and the nonmechanical chemical energies. A molecule of H2O, for example, needs chemical bonds to hold it together. It is not so easy to decide how much of that chemical bond energy is responsible for the mere presence and inertia of those water molecules, as simple masses and inertias, and how much of it is responsible for maintaining water's distinctive chemistry.

The problem we are facing is that the ideal biological cycle we have described is very far from realistic. Our traditional African hunter-gatherers may try to leave the environment around them the same, but the sun is not perfect; free space does not hold itself constantly at the absolute zero of temperature; and the surroundings do not faultlessly provide all organisms with the resources and chemical components that they need. Just to begin with, the environment is all too likely to degrade. These many imperfections are the basis for Darwinian competition. Since this is science, we of course want to quantify all of them.

We will follow Albert Einstein (who better, once again, to emulate?!!). We will borrow the same idea for studying such problems that he also eventually borrowed

Even though we never measure the ideal metre or kilogramme, we measure lengths and weights with no trouble, In the same way, while our ideal biological cycle may be unrealistic, it proves to be extremely useful. The ideal biological cycle tells us what is important.

Figure 15.1

The solution to our problem of measuring transformations—which is the method Einstein also used—lies in the object in Figure 15.1. It is a 4 × 4 grid that mathematicians call a “tensor”, which is taken from the Latin tensere, to stretch. Tensors are useful because they allow us to stretch one thing out to match another, and thus compare them.

Ours looks like a standard 4 × 4 tensor. We will, however, spend more time investigating the 3 × 3 one located inside it: i.e. the bottom and rightmost set of rows and columns.

To understand what we are using a tensor for, imagine I come to your house. I measure you, your significant other(s), your surroundings, and your 2.4 children (or however many of you and your partner(s) there may be, and however many wards you exercise your parental guardianship over). I can now give all those initial measurements a value of ‘1’, because I am using all those measurements I make of you as my base line or standard. That gives me a “basis”. I now have a reference point or standard I can compare everything else to. Or … to put it another way … I can now stretch any other thing I measure out to match you, by comparing it to you.

Now let's randomly select a country on another continent. Let's also randomly use the phone directory to select some other household. I go there and I measure them, their household values, their 2.4 children or whatever. I can now express all those new measurements in terms of the values I measured, first, for you, and using you as a base line and reference against them.

However … I could just as well turn this around. There is nothing to stop me changing my mind and using THEIR measurements as my basis instead, and then using that as my standard or reference against which I assess the values I originally got when I measured you. It really doesn't matter which way round I do it. Tensors are designed to accommodate both possibilities.

And now that I have done that, let me come back to your house. Let me hang around and observe your youngsters for the generation. Or … I could stay over there with the alternate household and do this first with them. Whichever way I do it, I keep taking all my measurements for both sets. I do all this, whichever way round I want to do it, until both sets of children have produced their own sets of 2.4 children, turning you original householders into grandparents. And once those grandchildren appear, that finishes my measurement process, and the analysis and comparisons begin.

I can now express all of my measurements whichever way I want to, because it is easy enough to convert one set into the other, and so express one set in terms of the other.

Since I have measured everyone in their surroundings, I can gradually work out what changes, and what transformations, have been caused internally and so by DNA, and relatively so, in each case; and what has been caused externally and by the surroundings. I can also distinguish between differences in my measurements caused by a change in the phenomenon itself, and differences caused simply by a change in the basis I have used to measure. That's the clear advantage of tensors This is why Einstein used them in his general theory.

There is a difference between an average and a mean. Indeed, there is a difference between the median, the mean, and the mode. The median tends to be the midpoint in the range of values in a data set, no matter how many entities might possess any particular value. The mode is the value that most of the entities in that data set hold. The mean tends to reflect a weighting. If we measure any two things; then express them in terms of each other in this tensor; and see a 1:1 upon the diagonal; then we know that no matter how the values and averages are expressed, they are identical in their distribution. Their means, modes, medians, and averages all match, so that their distributions are the same. All other grid-positions upon any matching row, and in any corresponding column, will be zero or empty. They will be empty because they have contributed no change to that value or property. That diagonal position's power to transform is then equal to its power to sustain itself. We have the same basis and measures between any two things we are comparing. One measure could be what they must all do, and another what they must each do; or else the mechanical as against the nonmechanical.

If there is any change in the property we measure that is not caused solely by the change in the basis on which we are measuring, then its value on the diagonal will not be 1:1. There will be some value somewhere in one or more of the off-diagonal positions in the grids in the accompanying row or column in that tensor … and we will gradually be able to figure out what is causing that change, and why. This is what tensors can do. That is why they are used. We are simply taking advantage of the systematic method they give us for separating changes in basis from changes in the things measured, and then applying them to both inter-generational and cross-generational transformations in biology.

Another aspect of tensors is that the values we read in the vertical columns are similar to, but not quite the same as, the values along the rows. The columns represent that fraction of an object's behaviour that—like the mechanical and gravitational and planetary energy of a rock falling upon a piston—is available for conversion into other forms.

Figure 15.2
Falling objects

As in Figure 15.2, we can take up a soccer ball and a soft cushion, each of the same mass. We then raise them up to the same height. They will have the same potential energy. If we now drop them, they will convert their potential energy into kinetic energy at the same rates. They will experience the same acceleration, a, and will share the same velocity, v, at all times. As long as they are flying through the air, they keep the same values for both as each other. They will do this until they both hit the ground, which they will do at the same instant.

The difference between the soccer ball and the soft cushion comes when they hit the ground. They will initially have the same momentum, p, and the same kinetic energy, E. For the first infinitesimal moment after impact, they will go through the same transformations and maintain those same values. But after that first infinitesimal moment, they will each start going through the kinds of transformations that tensors are extremely good at recording.

The difference between the soccer ball and the soft cushion is that the soccer ball's stock of kinetic energy is very much more easily converted into the reverse motion of bouncing. The soft cushion's on the other hand, is largely converted into some wasted heat and a change in shape. Most of that kinetic energy is simply absorbed.

The difference between the soccer ball and the soft cushion arises because of what they each do with their momentums after impact. The soccer ball retains more of its kinetic energy and momentum as a continuing motion. As it experiences the force of impact, it starts undergoing some elastic transformations. Those will work to restore it to its previous shape. They work so that it can use its elasticity of transformations to start moving in the reverse direction. The cushion, on the other hand, has no intention of going anywhere. It absorbs all that impact as a non-reversing change in its shape, and as a change in its internal energy stocks. This is a slightly more permanent change in its configuration.

The values we put in our tensor's rows are not the same as we put in the columns. The values we will see, instantaneously, along the rows represent that fraction of an object's behaviour that are more externally directed. They can have an external impact. They are more mechanical. These will not be the same from one moment to the next for the soccer ball as the cushion. We see more of the ball's momentum manifest on the rows. The cushion's energy, on the other hand, is not so easily convertible, via the work done via the momentum, upon its impact with the floor. More of the cushion's energy is therefore found in the columns.

The soccer ball and the cushion will gradually differ, after impact, in the values they each put in the rows and columns. One undergoes more rapid, reversible, internal changes in state than does the other.

The values in the columns show the overall kinetic energy. They are what is convertible vis-a-vis how they would display along the rows, if they were in fact converted into momentum. What is not convertible or converted shows along the columns, rather than the rows.

As with the soccer ball and the cushion, biological organisms do not have the same impacts upon their surroundings. They do not thrust themselves forwards through time and along their generations, or into their ecologies, in the same way because—like a soccer ball and a cushion—they have different energetic behaviours and responses from one moment to the next. As per our first and second laws of biology, they will not handle their mechanical and nonmechanical chemical energies and responses in the same way, and from moment to moment … and we must eventually frame some further laws to handle the possibilities for this diversity.

Biological populations are not at all “like” either soccer balls or cushions, but they do show moment to moment differences in behaviour. They differ in the rates at which they convert energy from the usable to the unusable. If, for example, shrews observe that they are now only 700 times smaller than the neighbouring meerkats, whereas their great-grandparents used to be 720 times smaller, then they can use tensors to gradually figure out whether or not they have gotten bigger; whether the meerkats have gotten smaller; of if it has been a little bit of both. One or another population has changed its expression in the surroundings and tensors can record those differences.

Tensors are ideal when it comes to settling questions of the relationships and differences between things, particularly when the standards of measurement used can vary. If one or another population changes a basis so we move from a range of ½–1–1½, to 0.222–1–1.778, to 0.75–1–1.25 or any other, then tensors can distil such variations. Transformations within both populations and their environments, across generations and circulations, are of deep concern in biology. Tensors can easily measure all such differences. Soccer balls and cushions are just a way of giving a reasonably familiar example to demonstrate their capabilities: how we can use them to measure the transfomations we are interested in as populations navigate their way through time and across their various generations.

Tensors are really just a way of measuring values in changing situations and/or when our basis or reference keeps changing, and so that we can have confidence in our measurements. Einstein used them extensively, but he did not invent them. He was introduced to them by his good friend Marcel Grossman; saw their value; and then put them to very good use. But they are just a way of measuring transformations. Tensors are ways of accurately measuring things that are liable to undertaking extensive changes and transformations, even as we measure them … which is a pretty accurate description of biological organisms.

Tensors are also not specific to physics. They are certainly not specific to the general theory of relativity. They transcend both those subjects. The germ of the idea was first thought of by people like the German mathematicians Karl Friedrich Gauss and Bernhard Riemann, the latter of whom virtually created what is called “absolute differential calculus”. This was further developed by Elwin Bruno Christoffel, and then most thoroughly conceived of by Tullio Levi-Civita and Gregorio Ricci-Curbastro.

A tensor's power is its exquisite ability to measure any transformations, even throughout all changes in the bases—i.e. the references and the standards—used to take those measures. That is again why Einstein decided to use them in his very famous general theory of relativity.

Tensors and their power over transformations are how we can prove that a population free from Darwinian competition and evolution is impossible. We shall use them and their ability to record instantaneous transformations to frame the two further laws of biology, and the two further maxims of ecology, that we need, as well as the three constraints upon biological populations (which we met briefly in Before We Begin). We shall use tensors to separate out the mechanical from the nonmechanical; and to distinguish what biological entities must all do from what they must each do, again jointly and severally. We shall also use them to navigate between the two different ways we shall soon find of how to measure time. All populations must navigate their ways through these different kinds of transformations, and tensors will help us keep track of them all … and do so from each population's point of view.

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