I keep coming back to “dimensions”. The humanities specialists want to know if “metaphorical” is OK for fourth and fifth perpendiculars and so on. Linear independence is analogous to what we accomplish with XYZ axes in our familiar linear algebra home base.
Then we help ourselves to as many dimensions as we think might help with categorization.
Yes, I lapsed into “machine learning” talk, as many today might recognize, in the sense that X samples of many columns (dimensions), categorized against labels Y, constitute training and testing data.
When studying the machinery itself, it’s fine to generate experimental data sets, and this the practitioners do. I bring this up because higher dimensionality, such as expressed by Euclidean distance, extended beyond three, is considered basic and essential to the algorithms (their teaching and expression).
Have we answered the initial question though? Is “metaphorical” OK? Here the discussion turns philosophical as the ontology of mathematical entities gets called into question, as compared to objects of fantasy and fiction.
Not that metaphors imply anything supernatural is afoot, only that they come across as somewhat surreal if taken too literally. Might linear algebra admit of a penchant for taking a core metaphor too literally, in pushing the analogy of any number of mutual perpendiculars with such fundamentalist rigor?
While we munch on that query, let me toss out another consideration: fractional dimensions, as in fractals.
Pretty patterns now come labeled with dimensionality 2.5 and so forth, but the tip of the iceberg.
However fractals have always suggested recursivity and the reverse (recursivity suggests fractals). The golden ratio, phi (“fie”) is the first fractal in nesting A within A + B in the same ratio as B to A. B : A :: A : (A+B). An actual number falls out, 1.618… or half the sum of one plus the 2nd root of five.
Fractal “zoom ins” suggest perpetually re-encountering an analogous set of circumstances… to some limit, where it stops, turns around, and winds up. I’m thinking of LISP or one of those. Functions are defined in terms of themselves. Lambda calculus.
Then lets not forget imaginary time and the relativistic treatment of 4D (3D + Time). Minkowski space and all that.
Finally, I’ll remind engineers of what we have in the humanities, a somewhat secret garden around a central sculpture of nested shapes. Yet another meaning for “4D” pops up in this namespace and yes I’m speaking of Synergetics, a two-volume enigmatic opus designed to launch new explorations of a philosophical bent, in tandem with geometric metaphors.
With Synergetics we parallel XYZ with IVM, and 3D with 4D. Instead of the “jack” of six spokes, positive and negative, the “caltrop” of four spokes, same signed, maps the surrounding points.
The two systems (frames of reference) swim side by side, convergent for awhile, then veer away from one another, each secure in its identity, and still able to cross-fertilize and breed new hybrids.
As concepts go, “dimension” is on the promiscuous side, eager to make a pact with whatever theorist, advertising rigor and orthodoxy on many channels, flexibility on others.