In the vernacular the term “dimension” usually refers to a hidden realm. In the scientific literature the meaning of “dimension” depends on the branch of science.
In physics “dimension” usually refers to spatial dimensions: the 3 spatial dimensions of external space that we are familiar with as well as the hypothetical compact (rolled-up) spatial dimensions of certain theories. Some physicists interpret the “time” in Einstein’s theories of Special Relativity and General Relativity as a “dimension” also.
In 2011, I wrote an article titled “What is Dimension?” where I listed some of the uses of the term “dimension” in science: 1) object dimensions 2) external space dimensions 3) compact space dimensions 4) time dimension 5) fractal dimensions 6) statistical factors as dimensions 7) degrees of freedom as dimensions.
In science, there is another use of the term “dimension”. Physicists always check whether an equation or inequality has the same physical units on its left and right sides. This is known as dimensional analysis. The “dimension” in “dimensional analysis” refers to physical units. In this article I will not discuss physical units.
Most general definition of dimension
The most general definition of “dimension” is “degree of freedom”. This definition needs refinement. I propose that we define dimension as a degree of freedom that comes with a metric.
Dimensionless
I have an inspirational piece titled Dimensionless where I describe the concept of “dimensionless” by these words: Soul is dimensionless. Love is dimensionless. Consciousness is dimensionless. Dimensionless is endless, formless, unbounded, uncountable, innumerable, immeasurable and eternal. Dimensionless is beyond the bounds of time, space and person. Dimensionless does not come within the scope of measurement. Dimensionless is free of attributes and dimensions.
Dimensionless does not mean zero degrees of freedom. On the contrary, dimensionless refers to infinite freedom. The concept of infinite freedom is meaningless in the scientific context but very meaningful in the context of spiritual philosophy.
Existential mystery
By “Soul/Love/Consciousness” I mean Divinity. One handicap of using descriptive words for Divinity is that we eventually run into paradoxes like this one: God and Godhead is One. This is very difficult to understand intellectually. Confusion arises because God has infinite number of names, attributes and dimensions. Godhead is nameless, dimensionless and free of attributes.
We cannot resolve the confusion intellectually but we try. Spiritual philosophy teaches that a drop of the infinite ocean of Consciousness transformed into a phase where two aspects appeared within the transformed portion: the cognitive and the creative aspects of Consciousness.
What followed next, how divine transformation proceeded, the details, nobody knows. There are speculations, metaphysical theories. There is also science. Humanity is making slow progress understanding some aspects of the transformations of Consciousness.
When I contemplate the big picture from the “dimension” perspective, I see this:
infinite freedom (dimensionless) –> infinite degrees of freedom (dimensions) –> cognitive cores –> interactions –> attributes
There are many other perspectives on the big picture, of course. My point here is that dimensions precede the appearance of attributes.
A “degree of freedom” is subject to constraints. “Infinite freedom”, on the other hand, is free of all constraints. That’s the definition of “infinite freedom”.
Metric
We said dimensions are degrees of freedom; then we said dimensions are subject to constraints. Is there a contradiction here?
Well…without these constraints cognitive cores and interactions cannot be initiated. Besides…the word “degree” in “degree of freedom” implies an association with the concept of metric.
Consider distance in space. Distance is a metric. The metric of space reflects the curvature of space as well as the distance measure. The speed of light and the path of light in empty space defines the metric of space.
The concept of “dimension” is more meaningful in the context of geometry. In “Geometrical versus Topological” I explained that topology is about connectivity which can be represented mathematically as a graph. When we impose a metric on a graph we obtain geometry. Dimension is a geometrical concept because of its association with the concept of metric.
Attributes are not dimensions
In this speculative post I said: Dimensions are degrees of freedom before the interaction (measurement) takes place. Attributes are the specific outcomes from the interaction. Since then I have written about “cognitive cores” and “interactions” and think that the progression is as follows:
degrees of freedom in the cognitive aspect of Consciousness –> confinement mechanisms –> cognitive cores –> interactions –> attributes
Attributes are the resultants of interactions. Attributes manifest in the arena of space and time. Attributes are phenomena.
Are explanatory factors dimensions? No, not in general.
Models/theories have to have at least 2 explanatory factors. We can come up with more explanatory factors, of course. The logical requirement is that the explanatory factors be orthogonal (independent). Don’t forget, however, that orthogonality degrades as we increase the number of explanatory factors.
“Explanatory factor” and “degree of freedom” are similar concepts. Can we call explanatory factors dimensions? No, not in general. Why not? Because explanatory factors can be agents, attributes or dimensions. We cannot restrict explanatory factors to the “dimension” category only.
Also, explanatory factors are not necessarily associated with a metric.
Are statistical factors dimensions? Yes.
In finance, economics, politics, sociology and many other fields (climatology, artificial intelligence, machine learning and many more) statistical modeling is very important. In statistical models we have multiple explanatory factors which can be described as statistical dimensions. If the explanatory factors are statistical in nature then we can categorize them as dimensions. This is consistent with the definition of dimension as a degree of freedom. Statistical factors are statistical degrees of freedom.
Statistical factors are not attributes because they do not result from an interaction. Statistical factors are not agents either because they do not have a mind of their own.
Dimensions are degrees of freedom subject to certain constraints (metric). Statistical factors fulfill the metric requirement as well.
External and internal dimensions
Previously, in this blog, I talked about two distinct ontological categories: 1) cognitive core category 2) interaction category.
cognitive core category: {charge, spin, …, conservation laws}. The elements of the cognitive core category other than the conservation laws can be described as internal degrees of freedom, in other words, internal dimensions. Elements of the cognitive core category are space-invariant and time-invariant.
interaction category: {dynamical laws, attributes}. The elements of this category are necessarily time-dependent and space-dependent. Attributes are not dimensions but “space” and “time” are. In other words, the parameters of the interaction category (space and time) can be described as external dimensions.
External dimensions cannot be converted into internal dimensions
Internal and external dimensions belong to distinct categories: the cognitive core category and the interaction category, respectively. This is why we cannot explain the electron charge (an internal dimension) by compact extra spatial (external) dimensions. That would be a category mistake.
Suresh Emre
Renaissance Universal 