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17. Let natural selection be augmented
““The areas which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same immovable planes, and are proportional to the times in which they are described. … Now let the number of those triangles be augmented, and their breadth be diminished in infinitum … their ultimate perimeter ADF will be a curved line (Newton, 1686)”.
It was also in 1679 that Newton learned of Kepler’s second law: the radius from the sun to a planet sweeps out equal areas in equal times. It seems strange that Newton would have learned of the third law as a student in 1664, but not about the second until years later. The explanation is that the third law was generally accepted in the scientific community because it could be empirically verified, whereas the second was much more of a conjecture.
In August 1684, Edmund Halley (1656–1743) visited the 41year old Lucasian Professor at Cambridge. He asked the question that had been consuming him and his friends Hooke and Wren at the Royal Society in London. What path would the planets describe if they were attracted to the sun with a force varying inversely as the square of the distance between them? Newton replied at once that the orbits would be ellipses. Since this was the expected answer, Halley asked Newton how he knew. Newton astonished him by answering that he had calculated it. Halley asked to see Newton’s computation, but as Newton seemingly saved ever scrap of paper he ever wrote on, he (not surprisingly) could not find it. … In November of 1684, Newton did send the computation to Halley in London, who was so excited that he prompted Newton to expand his work (Swetz, 1994, p. 503)”.
The Joule experiment, and most others of yore, are much easier to understand using modern tools and equipment. Is there then—using modern tools such as Newton provided above—a way to rephrase the issues Darwin addressed?
Although the claim has been lodged that differential equations are incapable of producing a simple and abstract biological theory of precision and generality, the fundamental theorem of calculus is one of the most powerful of all tools in the scientist’s toolkit. It is how Newton defeated the planets. It is the singular reason modern science can be so clear in its pronouncements. Without it there can be no quantum theory. If we are going to improve on Darwin’s statements so they can be better answered—which is our ambition—then it needs to be expressed in terms that make it approachable with these best possible tools. This also means making sure we do not breach the parameters of those tools.
If we for example return to the suggestion made by Harte et al that an area should be substituted for a volume, we surely do not need the etymological fallacy to tell us that this procedure is unlikely to succeed (Harte et al, 2008; Harte, 2011). This is nothing to do with any avowed differences in scope or reference between the thermodynamic and the information theories. Drawing such an analogy is potentially unsound simply because these are mathematically very different entities, serving very different purposes. An area is twodimensional meaning that one mechanical mode for the containing molecules has been absented, while a volume is threedimensional and successfully encompasses them all. Thermodynamic systems are very specifically brought together with two intensive and two extensive variables to overcome such issues. The interiors and boundaries of areas and volumes are completely different; they grow and increase in very different ways; and they embrace different molecular mechanical modes. It is thus hard to see how areas and volumes could be compatible, or could be substitutible for each other if thermodynamics is to be the means of approaching biological and ecological issues. The fact that we indeed end up with something that Harte et al admit is neither intensive nor extensive is surely an indicator of those incompatibilities (Harte et al, 2008).
One thing we must certainly do is respect the dimensionalities of our objects. If we are going to make natural selection susceptible to calculus in particular, then points need to relate impeccably to distances … which then need to relate impeccably to areas … which then need to relate equally impeccably to volumes. If we are going to make natural selection susceptible to calculus in particular, then points need to relate impeccably to distances … which then need to relate impeccably to areas … which then need to relate equally impeccably to volumes. That last step must certainly be explicitly taken, no matter how unlikely it might initially seem that there is no ‘volume’ of any relevance in biology. But without appropriate dimensional interdependencies, there is no possibility of referencing molecules.
It should certainly be possible to attain our ambitions because at its very simplest, the fundamental theorem of calculus states that if we have a given function G(t) whose derivative is G’(t), then the integral of that derivative is directly related to the values of G(t) at the boundary points of any suitably described interval: i.e. ∫abG’(t) = G(b)  G(a). We have everything we need in that theorem, and it is only necessary to respect it. Our planimeter is a very good start.
If we first consider points, then in a strict sense, a fundamental theorem exists for a ‘zerodimensional integral’. If one exists, then we simply evaluate a function or line at each end point or whatever lies across our location. Beyond that, fundamental theorems exist for onedimensional curves (i.e. lines), twodimensional regions, and threedimensional volumes. In every case, there is a boundary. A onedimensional object has zerodimensional points for its boundaries. By the same token, a volume can be evaluated simply by examining the surfaces at either side. In general, any ndimensional object has a boundary of (n  1) dimensions.
Boundaries are important for they are how we access whatever lies at either end of them, or within them, or that they enclose, or else that they describe and stretch between. Since vector calculus is what we want to use, then we need to respect its fundamental theorems as we proceed. The fundamental theorems of vector calculus state that there is always some kind of an integral, of some kind of derivative, over some object, that is equal to the values of whatever function describes that object’s boundary. For this reason, there are four such fundamental theorems within the vector calculus (Nykamp, 2011):
The gradient theorem of line integrals relates a given line to its two endpoints through a gradient, ‘∇’ (pronounced “del”):
∫C ∇f • ds = f(q)  f(p)
where the line integral of the gradient is simply the difference in the function’s values at those two points. We simply take the difference of a properly constituted line, and we have a gradient which tells us something about the behaviour of whatever exists or acts across that line, and between the two points. If that line is in fact the expression of a force, then we can evaluate the gradient and rate of change of that conservative vector force field, F, between those end points. Since F = ∇f, then instead of the above we have:
∫C F • ds = f(q)  f(p),
with the integral of any conservative F over a closed curve of course being zero (because we are right back where we started from, and the sum or net of all the changes is zero).
Green’s theorem, which we have again already met, concerns lines and areas and justifies our planimeter. We assume an x and a yaxis such that any given force F can be analysed into two components—F1 and F2—along each of those orthogonal axes, x and y (for example, a blowing wind can be divided into two components). This granted, then Green’s theorem immediately relates the double integral that evaluates the region’s area to the line integral of the curve that loops around and forms that region’s boundary … which we did with our planimeter. Thus:
∫C F • ds = ∬D(∂F2/∂x  ∂F1/∂y) dA
where C is the properly oriented curve that forms the boundary of the region D it encloses; ds is an infinitesimal increment of arc about the boundary; and F is a force that is continuously differentiable everywhere inside D into F1 and F2. If we take care how we describe everything we meet, then we can use this theorem.

Stokes’ theorem now introduces a third aspect. It relates a line integral about a closed curve, C, to a surface integral, S, over the actual behaviour of the entire surface whose boundary is that curve, C. This is a little more than simply reckoning the area inside the boundary. Thus the surface concerned may have objects or points implanted in it, microscopic or otherwise, that respond to any forces imposed upon them by rotating or spinning or curling independently about themselves because of that force—i.e. about an axis perpendicular to them and that passes through the surface. For example, a wind farm could easily have weather vanes across it which measured the strength and direction of the wind by placing a counter or meter on the axes of those vanes, to evaluate the speed as they turned. We could then also try to harness their power with a current that then pushed a large belt around the wind farm’s perimeter. Then if C is a positively oriented path or a boundary of that surface S; and if we use the symbol ‘∇ x’ to represent a curl or vector cross product that relates the net rotational behaviour of those implanted objects as they move about their axes to any net contribution they make to that line or surface boundary by their spinnings or rotations; then we have:
∫C F • ds = ∬S ∇ x F • ds
as the description of the net force around the boundary due to those curls or spinnings and rotations.
Gauss’ theorem or the divergence theorem concerns volumes. We can use two or more surfaces and construct an arbitrary shape with a volume. This theorem then relates the double integral of an area or surface to the triple integral taken for a volume. If we now use ‘∇ •’ to represent the flow or ‘divergence’ through that surface and into or out of that volume; then if some surface, S, is the boundary of that solid formed, we have:
∬S F • ds = ∭W ∇ • F dV
where dV represents each infinitesimally small volume element into which we slice that solid, W.
These are some very powerful statements. If we start carefully by looking on biological entities the right way, and we always respect these theorems, then we can derive some equally powerful results. We will also certainly build the foundations for a quantized biology, for these are the selfsame components of the electromagnetic theory.
The most important thing is never to misunderstand any terms—and never to fail to see the possibilities of Darwin’s position in the light of these modern developments. We will then be able to understand the issues that he raised in these terms, and then approach them.