By Roland Watson

The last article focused on universal symmetries. Many of these symmetries are so self-evident that they are overlooked. However, even they are not the most fundamental forms. Underlying all universal order are a number of characteristics that we take to be fundamental, but which can really only be considered to be assumptions and unproven propositions. And these assumptions and propositions, too, are in plain view, but regularly overlooked; or, they are so obscure that they can only be understood via very abstract logic or through highly sophisticated experiments.

Certainty and universality

The first assumption, which I mentioned before, is that of certainty. Until the discovery of quantum mechanics, physicists believed that it was possible to know enough so that given the circumstances, we could predict what would happen next. This has now been disproven. The future, necessarily, is unpredictable.

This in turn has significant consequences for a broader assumption of science, which is that the universe is consistent and unchanging; that there is such a thing as universality. This is the idea that there is a single structure, which is not undergoing any real change, and, we can find it. Further, this assumption is also a consequence of relativity theory, that the laws are the same everywhere and across all time; and it is even a derivative of conformal invariance, that this order, when discovered, will be shown to apply to all scales, from the smallest to the largest.

At some level one would expect that this does hold, even if the universe is asymmetric. Relativity has been demonstrated, and all experiments with subatomic particles, adjusted for the effects of quantum mechanical uncertainty, consistently yield the same results. Also, space-time homogeneity exists. None of the basic forces ever take a break. I must say, though, that there is the problem of the unknown nature of the universe's laws and structure during its first few moments.

Nonetheless, and even without this difficulty, it is still far from certain whether we will ever decipher the entire underlying structure. Much evidence, from the logic of the limits of knowledge, to a plague of paradoxes - which I will review in a later article, argues that such understanding is, for us, precluded.

Infinity and boundaries

The next assumption is that the universe is infinite, in all space-time directions. This is necessary to ensure self-containment, on which a number of conservation laws and symmetry principles are based. On the other hand, it does raise all sorts of questions, including of the specific nature of infinity, which I will also review later, and of the nature of a boundary, and how can something - anything - not have one?

Conversely, if there were a universal limit, or boundary, wouldn't all of the matter-energy inside the universe be attracted to it, through a force akin to gravity, or repulsed by it, due to something like electrical charge? Indeed, couldn't it be the case that both such forms exist, and that they are actually different sides of the same underlying phenomenon, and that the universe flips from one mode to the other through something analogous to a change in magnetism, creating an eternal cycle of expansion and contraction?

A mathematical universe?

Another assumption about physical reality is that its basic structure can be described using the language of mathematics, which attempts, among many things, to quantify infinity. All physical laws are expressed mathematically, and this would appear to be a reflection, according to Peat, of the "numerical harmony [which] exists throughout the universe. ... Again and again, abstract and beautiful mathematical relationships, explored for their own aesthetic sake, are later discovered to have exact correspondences with the real world."

Mathematics - and perhaps music as well - does appear to be a true language of the universe. It appears to be more than a set of symbols, perhaps even capable, with further development, of accurately describing all of reality. Also, one would expect that any mathematics that pass the test of rigorous proof, must have an analogue, somewhere, in the natural world.

This does not infer, though, that all of our current mathematics have been sufficiently well tested. Euclidian geometry, beginning with the idea of a straight line, which was initially considered to have been proven, meaning to accurately reflect reality, failed, when it was put to the test of confirming astronomical observations. It took Einstein's revolutionary calculations of relativity theory, leading to the concept of curved space in which an abstract object such as a straight line could not exist, to prove that Euclidian geometry did not, in fact, mirror the natural world.

A similar mathematical assumption, that space can be infinitely subdivided into a series of dimensionless points, which is the basis of the mathematics of calculus, and the view that space is continuous, also is coming into question. Within the extremely small spaces that are defined by the mathematics of the uncertainty principle, in other words, within the quantum state, space-time may break down, perhaps not into a specific quantum, but instead into what is referred to as quantum foam. This supposition is still linked to calculus, though, since the mathematics of the uncertainty principle incorporate it.

The language of mathematics, when put to the test, does reveal order, but at the price of a greatly increased complexity, including such things as "imaginary" numbers, "negative" probabilities, and unexpected and unresolvable infinities. Even with the language as our primary tool of scientific exploration, we still have a long way to go. As Brian Green said: "Infinite answers are a signal that a theory is being used to analyze a realm, that is beyond the bounds of its applicability."

At this point, I want to reinforce the idea that the word "symmetry," as it is used by physicists, is much broader than our everyday coinage. It extends to "mathematical" symmetries - gauge symmetry is one such example, meaning the types of harmonies that can exist between equations and forms, including in multi-dimensional "abstract" spaces, and, more generally, to any properties that are unchanging. Through this we can see that there are great sympathies, if not equivalences, between the concepts of symmetry, conservation, and the mathematical field of topology. All three, in effect, are considering the subject of nature, but from slightly varying perspectives.

As an example of this, and again, relying on Richard Feynman, many conservation laws and universal symmetries are linked. Indeed, in quantum mechanics, what is known as symmetry "character" is conserved. Also, translation in space is linked to conservation of momentum; delay in time to conservation of energy, which is a simplified version of the law of conservation of matter-energy; and rotation in space to conservation of angular momentum.

Pressing on with some observations that derive from topology, and with a revisit of chaos math as well, we can recall the idea of similarity across scales. I should note, though, that this is distinct from conformal invariance, in which a physical law, principle, or relationship applies to all time and distance scales. The scaled nature of reality, on the other hand, along with the multiplicity of feedback loops that exist between the different scales, is known as "fractal" order, and we can now see that it is yet another type of symmetry.

The topology of life

Characteristics resurface across varying scales, and this extends to the world of life as well. For instance, the same patterns of human behavior are present in all social scales, from a small collection of people, to a community, to larger and larger cultures, to humanity as a species. Other characteristics, both physical and behavioral, are manifested across all species of life, and even all categories of life, if life is extended to include planets, the universe, indeed, to all matter.

This website, in effect, has attempted to describe these topologies, of how we and life remain the same as we change, in other words, what our constants and underlying conditions are. The study of behavioral form reviewed how we are deformed - twisted and stretched and squeezed - by social influences, and what changes, and does not change, in the process. Also, one can see that ideas have topologies, too. Not only does our stream of consciousness evolve for the most part in a continuous fashion; individual ideas themselves can stretch and curve and twist, yet continue to make the same basic point - to present different ways of saying the same thing. In addition, the more you consider different subjects or areas of life, the more you see similar patterns within them, and connections between them. The more you look, and think laterally, the more parallels and linkages become apparent. You end up with a structure of ideas with the characteristic of a lattice or matrix.

Another topological constant can be found within the transition from stability to change, or equilibrium to chaos. We have seen that systems strive for equilibrium, but there are actually three different alternatives that may be achieved. The first is a system at rest - mathematically, a point. The second is a system with regular cyclical behavior - a cycle with a set periodicity. And the third is a system experiencing chaos - referencing Gleick, a complex, dynamic, non-linear system characterized by turbulence. However, mathematicians have determined that there is order in chaos, that such systems cycle from stability to disorder to stability irregularly or aperiodically, following the topological form, or guide, of the strange attractor. Such attractors exist in all turbulent systems, including the evolution of life. Indeed, if there is a fundamental purpose to such a complex, dynamic system, this is the mathematical expression of it. Also, if the universe has a deeper purpose, it - the universe - including all the systems in it - necessarily is not purely random. In such systems, stability and change, or equilibrium and chaos, are just different stages, or reconfigured forms, of the same, constant, underlying reality.

Every form, extending to every form of life, and every thought, must have an underlying "mathematical" shape, such that it can achieve only a limited set of transformations, including the subatomic particle transformations that are described, and permitted, by symmetry. Further, we must learn this topology; invent a new mathematics of form. However, this knowledge may presently be excluded to us, at least as homo sapiens. Such a development may be equivalent to, it may require, an evolution in our consciousness.

Of course, any system can be described mathematically, using numbers to represent spatial coordinates, but this is not what I mean. Can our current mathematics describe a human, and a cat, and demonstrate how they are different; and why, even given a number of intermediate stages of chaos, a topological transformation between them is impossible? Mathematics can capture scale, density and movement. It technically can describe life. But, it cannot describe what it means to be alive, either as an individual or for a species. Can we mathematically describe the shape or form of love, and how it grows?

All of this leads us to the next assumption of science, which is that the universe is a deterministic "thing," following set laws. It is not a living entity, with will. Indeed, life is only a function of these laws, starting with their effects at the subatomic level. There is no real or qualitative difference between the animate and the inanimate; no spark of life, or life force.

Locality and causality

Another assumption is what is known as "locality." This is also a consequence of relativity theory, of its proof that the speed of light cannot be exceeded. There is no possible way to communicate information from one point in space to another faster than the speed of light. Further, locality underlies the conservation laws. They also apply locally, such that the boundary that is defined by the speed of light constitutes another form of self-containment, within that provided by the infinity of the universe.

However, as we have seen, quantum mechanics is at odds with locality. It seems to demand the existence of non-local, or faster than the speed of light - this is called "superluminous," connections. This in turn implies that we should start our search for order - here I am referencing Peat, not at the local level, but globally, with consideration of the whole. Indeed, in this way quantum mechanics itself leads to the second world view, that of the universe as an interconnected whole.

Related to locality is the assumption of "causality." Feynman said: "Effects cannot precede their causes." This in fact is the basis of order, that one thing follows another, and that actions have consequences. It is a restatement of the idea that there is such a thing as time, and that it proceeds in only one direction. But, just as quantum mechanics conflicts with locality, it also leads to questions about causality. In later articles, I will consider evidence that undermines both of these assumptions, locality and causality. I will also continue to present arguments that the universe is not dead.

Is there anything at all?

The last assumption is that there is such a thing as an external, physical reality. Numerous philosophers, from Descartes to Kant to Berkeley, have disputed that we can state this with certainty. Furthermore, Bertrand Russell commented that "perception," if it exists at all, "must be in some degree an effect of the object perceived." (I should note that this requires causality.) And, it must resemble the object if it is to be a source of knowledge about it. We cannot be certain of the first "if," that there is a distinction between mind and matter, nor of the knowledge that is provided by the one of the other, no matter how consistent the experimental results may be. Even more, if reality is only a function of perception, we must remember that perception can be wrong.

In the next article, I will take a closer look at relativity theory, and space-time.

© Roland Watson 2015