[hintymad]

I was also puzzled by det(0x0) being 1, because I had built an intuition that determinant of a matrix was the volume of the parallelepiped represented by the matrix. I made my peace by accepting that my intuition on volume implies that volume is defined in a space that has positive dimensions, and by treating zero space as an algebraic construct.

[pointedset]

Now you're reminding me of a wacky math conversation I had at Mathcamp [1] with a much smarter guy, who was talking about more esoteric definitions of volume in euclidean space. Something like:

- n-dimensional volume is a function from (some) subsets of space to real numbers

- it should be additive under union

- it should scale by t^n when you scale the space by a factor of t

I think the upshot of the conversation was that 0-dimensional volume of a shape should be its Euler characteristic. In the simple case of a finite set of points, the "volume" would be the number of points.

And by your earlier comment, span({}) consists of a single point, so its volume should be 1. It all works!

[1] https://www.mathcamp.org/