By Roland Watson

In the last article, I said that the traditional view of evolution is that it occurs gradually. Variations in characteristics are generated, and selected, and tested, slowly if not cautiously. There is, though, an alternative view, and which is gaining greater credence. This is the belief that evolution occurs in fits and starts. It is not a continuous, gradual process.

Evolution and chaos

This perspective in turn reflects the issue of equilibrium. Through evolutionary changes, many species of life do achieve a sustainable equilibrium with their environment. As such, they have no further need to evolve. This is particularly the case with "apex" species, or species that have no natural predators, but it actually extends to every species that inhabits an environmental niche, whereby it, and - ideally - only it, can fulfill a specific ecological requirement.

As an aside, the fact that species are guided by such requirements reinforces the idea that the larger ecology of which they are a part is itself also in some way alive.

In these circumstances, the species are in balance. They have no need to break free, to strive for a new form, and, they can maintain their current form, their physical characteristics and behavioral patterns, essentially unchanged for ages. Their only need to evolve arises when their environment, and hence their balance, is disturbed. Some outside influence changes their condition, the coin flips back to survival mode, and they are forced to adapt, and as quickly as possible. Such an adaptation may result from genetic changes, over which they seemingly have no control, or through will-driven behavioral innovations and reconfigurations. And, if they do not succeed, their evolution, and existence, ends. They die. But, if they do adapt, then a new balance is established, and evolution for the most part stops again, until the introduction of a new outside influence.

This type of evolutionary process can be understood more deeply using chaos theory. As we saw in the series on the subject in Part 1 of the website, this theory has uncovered new layers of order in the universe. It uses a new type of mathematics to understand, among other things, systems that are "non-linear" or "dynamic." Such systems cycle from an ordered state to one of disorder and then back to order. The above description of fits and starts evolution matches this perfectly.

What this suggests is that there are not two distinct evolutionary processes, continuous and gradual and fits and starts, only the latter. Therefore, we must have misinterpreted the former in some way. There must be some element of chaos, some system phase transition, in what we regard as continuous and gradual evolution as well.

Consider the phenomenon in the first mechanism that I described: a germ-line genetic mutation. One can say that there is an element of chance, or chaos, in this.

Indeed, it is not unlikely that there is an underlying, stable system that regulates the genome, which with certain energy additions - such as from direct exposure to a mutagen, or the accumulation in the cells of "free-radicals" - loses its balance, leading to a form of turbulence and a resulting mutation. Or, consider the migration of life to a new niche or a new environment. The question is: what caused the migration? Was it some form of system turbulence? For example, Darwin's famous case of the evolutionary development of finches on the different Galapagos Islands does reflect such a trigger: the geologic chaos that split the islands asunder.

The outside influence that triggers evolution does not have to be an abrupt change. Such an influence can also develop with time, in the case of a geologic change, with a very great amount of time.

Development revisited

One might further comment that the first type of new characteristic development that I described, the one that enhances the competitiveness of an individual in its present environment, is not truly evolutionary. Such a development is insufficient to lead to a complete species transformation. Rather, it only enables a shift in dominance within the species, among its various individuals and groups, which for the latter are for the most part family-based.

However, some of the developments seen in humans now may be evolutionary, even though our environment seemingly has not changed. This is because we have effected changes to the planet that are so great that it can no longer be considered to be the same. As I already said, we have changed our environment so drastically that we now must evolve. We are, in effect, migrating to a new environment, one of our own making.

Introduction to topology

Additional insights into evolutionary processes are available from another branch of mathematics, known as topology. As James Gleick wrote in his seminal book Chaos, Making a New Science: "Topology studies the properties that remain unchanged when shapes are deformed by twisting or stretching or squeezing."

That sounds really abstract, but there is no need to worry. I can assure you that the basic concepts are not as difficult as they sound. For example, a rectangle, a triangle and a circle all share the characteristic of being a continuous two-dimensional shape enclosed by a boundary. There are no "holes" or other discontinuous features within the shape. You can deform a rectangle until it is a circle, and this underlying characteristic remains unchanged.

The same effect can also be seen with cubes, pyramids and spheres, except now it is three dimensions and a boundary. With inanimate matter in physical reality, you must add a fourth dimension, time, but this may or may not have a boundary, or boundaries, into the past and the future.

Finally, animate matter, life, adds a fifth dimension, including with the characteristic of evolution and everything else by which it can be described. But, this specific dimension does have boundaries, both in space and time.

Said another way, topology is the study of continuity. Shapes or forms are considered to be "topologically equivalent" if they can be continuously changed into each other. Indeed, topology requires this two-way process. Form A must be transformable into Form B, and vice-versa.

Furthermore, within the mathematics of topology such transformations and equivalences can occur in space as we normally perceive it, meaning in three dimensions, and also in higher "mathematical dimensions": four spatial dimensions, five spatial dimensions, etc.

However, mathematicians use the word "dimension" in a number of ways other than as we use it to describe everyday experience. For instance, mathematically, a leg has nine dimensions, since there are three dimensions of movement each at the ankle, knee and hip. Such dimensional freedom of movement needs to be isolated for mathematical analysis to be accomplished. Each specific dimension represents a distinct variable, which requires its own coordinate or value. For people, though, our brain is able to consolidate all of our possible options for movement into a "composite" three-dimensional space.

Topology and evolution

Returning to evolution, we can conclude that it is not consistent with topological equivalence. The forms of different species in an evolutionary chain are non-topological, because they cannot be continuously undone. They are one-way.

As I mentioned before, other than via temporary behavioral de-evolution, we cannot revert to the characteristics of our predecessor species. The reason for this is that there are "breaks" in the evolutionary chain. Each period of chaos, when a new species evolves, represents such a break. Therefore, if topology is the study of continuity, and chaos the study of discontinuity, our analysis of the evolution of life, to be complete, must consider both.

Of course, you could say that any change is non-topological, since it involves time, and time itself cannot be undone. Topology as an abstract field of mathematics considers deformations without the presence of time, which in the real world is impossible. So, this is true. My point simply illustrates that there is a further lack of equivalence present through the discontinuous breaks that occur as life evolves.

Topology, by analyzing the characteristics that underlie shapes, by evaluating what changes and what remains constant - scale, distance and measurement are irrelevant, in effect considers which transformations are possible and which are precluded. Therefore, with further insight one can envision using topological concepts to evaluate many different things, including:

- Any underlying restrictions that exist on the forms that life may take. Within evolutionary steps, allowable transformations are dictated by the need for continuity. But, such a need actually survives the chaotic break of evolution. Viewed broadly, the constraint of topological equivalence survives chaos. For example, one species may evolve into another, but many specific changes are precluded. An animal will never evolve into a plant, or a bird into a fish. In this way, topology may encompass multiple forms, even categories of forms, such as through how disparate forms enable one another. A certain habitat enables the evolution of one form of life, but not others. And such life changes the habitat, directly or indirectly leads to its transformation, also in certain ways but not others. Another example is how some species, such as plants and animals, are linked. Topology covers all of this, all of the ways that different forms of life can be connected.

- Topology further governs how thoughts develop. It is no coincidence that it is said that one's thoughts take shape. The underlying process by which neural circuits develop and transform, as through the various circuit overlap mechanisms that I have described, and through which various ideas and states of mind are achieved, can be studied using topology.

- As part of this, topology also governs how emotions change, such as from love to hate, or hate to forgiveness. Furthermore, the concepts apply to how social groupings and institutions develop and evolve.

Indeed, as a conceptual framework for understanding change, both continuous and discontinuous, topology, with chaos, can be applied to any and all forms. Every form must have an underlying topology, such that its allowable transformations can be determined, including information about what would cause them to occur.

Also, every break in form, every evolution from one distinct form to another, should be analyzable using chaos, to determine which aspects of the topological structure break down completely and are lost, and which survive the chaos by being embedded into it, and further how such a non-random pattern could and does serve as the seed, or foundation, of the new form.

In the final article in the series, I will return to the goals of evolution, and also consider the different stages or signposts that life achieves as it undergoes the evolutionary process.

© Roland Watson 2015