Yes. Obviously higher dimensions in physics require a different kind of intuition, but data folks deal with multidimensional tabular data all the time without ever seeing the underlying structure. Seeking a spatial explanation often hinders rather than helps.
Instead, there’s this notion of a “theory of coordinatized data” [1] where one understands that dimensions (doesn’t matter if they are continuous, discrete, categorical) are essentially coordinates for values. This is a powerful way of thinking about tidy multidimensional tabular data.
Once you realize dimensions are coordinates, a certain mathematical intuition emerges. For instance, most people have a hard time understanding pivot/unpivot operations. But they really are analogous to matrix transposes, but instead on a row/col axis, they rotate on the “coordinate” dimensions which are invariants.
Once somehow understands this, their understanding of SQL and Tableau and of data frames becomes a lot deeper. Aggregations and filtering and window operations take on a new meaning.
Sure, encoding an extra dimension in a vector is just an additional element, but for the exception of categorical data this view is very restrictive. If you want to do things like describe embedded-space and projective spaces you can't just add a term to your formulas and expect everything to work. Like an ant walking on a ball in your room on earth in spacetime projected on your computer screen.
In geometric algebra there is a way to encode every element and transformation in such space and those correspond to shuffling around terms in an equation.
Instead, there’s this notion of a “theory of coordinatized data” [1] where one understands that dimensions (doesn’t matter if they are continuous, discrete, categorical) are essentially coordinates for values. This is a powerful way of thinking about tidy multidimensional tabular data.
Once you realize dimensions are coordinates, a certain mathematical intuition emerges. For instance, most people have a hard time understanding pivot/unpivot operations. But they really are analogous to matrix transposes, but instead on a row/col axis, they rotate on the “coordinate” dimensions which are invariants.
Once somehow understands this, their understanding of SQL and Tableau and of data frames becomes a lot deeper. Aggregations and filtering and window operations take on a new meaning.
[1] https://winvector.github.io/FluidData/RowsAndColumns.html