A note on nature of spacetime dimensions (and more!)

If there are more than three dimensions, can I get there by meditating?

I’ve often heard people fascinated by the idea that reality is actually 4-dimensional, or that physics beyond the standard model proposes that there really are 10-dimensions (or more!). And this is fine: it’s pretty interesting stuff. The problem is that all too often comments about 4D spacetime or the dimensions suggested by string theory (or some successor to it like M-theory) are accompanied by comments about how these extra-dimensions suggest that there could be an astral plane dimension, or elicit questions like “what if one of these dimensions is a spiritual dimension?” and so forth.

My problem isn’t that people are curious about mysticism, spirituality, etc. (I am too, albeit from a much more skeptical foundational perspective than many). The issue is the conflation of very, very different senses of the word “dimension”.

Dimensions in physics are mathematical, even when they aren’t

In common parlance (which doesn’t include the use of the word “parlance”), dimensions are individual, or separate from one another. Hence one might refer to a spiritual dimension vs. a material or physical dimension, or speak of the different dimensions of some business problem, or refer to the racial dimension of a sociocultural problem, etc.

Unfortunately, a lot of mathematics education unintentionally reinforces this misconception of dimensions being “separable” like this. Students of pre-college algebra learn about the x,y-plane (or Cartesian plane). They get used to working with problems in coordinate geometry, trigonometry, even calculus in this “2-dimensional” plane. Only it isn’t actually 2-dimensional.

Some (elementary) mathematics that is pointless if you know it and probably incomprehensible if you don’t- you can skip it

The truth is that even physics students have a hard time here, because they are initially taught about spatial dimensions (defined in terms of x,y, & z axes), and learn to understand this “space” in terms of special vectors i, j, & k (vectors are mathematical “objects” that are, simplistically, composed of individual values; you can think of a vector in the Cartesian plane as quite similar to a point in the plane, only instead of a pair of values (x,y) we have a single vector x which consists of an x-value and a y-value and instead of a point the vector can be thought of as a line from the point (0,0) to the values of the x,y-components of the vector). The vector i, for example, has an x-value of 1, a y-value of 0, and a z-value of 0, j has an x-value of 0, a y-value of 1, and a z-value of 0, and k has 0,0,1. This is very useful for doing mathematics required for classical mechanics, because in classical mechanics things like velocity are vectors in our familiar 3-dimensional space. Then the physics student takes a course in linear algebra, or moves into more advanced differential geometry, or studies dynamical systems, or studies electromagnetism, or is otherwise introduced into the “real” world of mathematical spaces in which there are no special vectors for the x-axis, the y-axis, and the z-axis because frequently one has to work in 1-dimensional space or 10,000-dimensional space (which would require 10,000 vectors like i, j, & k, each one having 0’s except on the “axis” for that “dimension”). At this point, all the familiarity with these special 3-dimensional vectors and how to work with them become more of a conceptual stumbling block than a step on a the path to more sophisticated topics.

…And we’re back: How to think about dimensions

All this is overly complicated, though, and I mention it briefly only to demonstrate that even so brief a treatment as this one can still be overly complicated. A much better way to think about dimensions is to consider why coordinate geometry problems worked out on a dry-erase board or chalkboard by the teacher or on paper by the student aren’t actually 2-dimensional. As chalk is scraped across a chalkboard, it leaves behind a 3-dimensional residue on a 3-dimensional surface. No matter how “thin” you make a sheet of paper, it is a 3-dimensional object. NOTHING truly 2-dimensional exists in a 3-dimensional world. Mathematically, a 2-D plane in a 3-D space has 3-dimensions, and in reality there are no 2-D planes in a 3-D reality.

You can also think of how a line is defined in the x,y-plane. Every point on the line has 2-dimensions (each has an x-value and a y-value). In fact, the “real” 1D space is just the number line (technically this is a 1D Euclidean space, and not all spaces with the same dimensions are the same, but for simplicity they are here).

The point is that the 3-dimensional world we experience isn’t composed of the separate dimensions in any way that most can readily conceptualize and certainly not that any can experience. Adding an extra dimension for time means that everything described in this spacetime is always and everywhere described by single points or sets of points that are defined by 4-dimensions. If there are really 10-dimensions, or 26, etc., then we are always and everywhere living as 10-dimensional or 26-dimensional beings. There is no special status accorded to these dimensions such that one could be the astral plane, heaven, hell, Nirvana, higher “universal” consciousness, or whatever, any more than there is some special status accorded to the 3rd spatial dimension z compared to x and y (technically, this isn’t true, but the special status of extra dimensions has to do with manifolds, topology, and other mathematical notions that the surgeon general cautions can lead to headaches, psychotic breakdowns, permanent social disorders, and writing blogs nobody in their right mind would read).

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