Logicians are mathematicians. Well, most of them are.

I have yet to meet a single logician, american or otherwise, who would use the definition without 0.

That said, it seems to depend on the field. I think I’ve had this discussion with a friend working in analysis.

But the vector space of (all) real functions is a completely different beast from the space of computable functions on finite-precision numbers. If you restrict the equality of these functions to their extension,

defined as f = g iff forall x\in R: f(x)=g(x),

then that vector space appears to be not only finite dimensional, but in fact finite. Otherwise you probably get a countably infinite dimensional vector space indexed by lambda terms (or whatever formalism you prefer.) But nothing like the space which contains vectors like

F_{x_0}(x) := (1 if x = x_0; 0 otherwise)

where x_0 is uncomputable.

Functions from the reals to the reals are an example of a vector space with elements which can not be represented as a list of numbers.

Probably ‘proof of concept’

It may have nothing to do with categorization, but has everything to do with categorification which is much more interresting anyway.

Mir hei zum glück üses jährleche alpweekend vo letscht Wuche no chönne uf Disi schiebe. :)

Di letschte drü Täg isch ja eigentlech sehr guet gsi, und ih dene paar Stund wo gwitteret hett, heimer zumindescht äh sehr schöni Ussicht gha.