David, I'm curious, how would you intuitively distinguish modules and vector spaces? That might inspire more clues.

In studying the field with one element, I realized that when we count the k-dimensional subspaces of an n-dimensional vector space over a field Fq of characteristic q, we may typically depend on a given basis: e1, e2, e3, ..., en. If we want to count the number of independent choices for constructing a vector space, then the first space would be generated by e1 (multiplied by any scalar). The next dimension would be generated by e2 + f21*e1 where f21 can be any scalar including zero. And then e3 + f32*e2 + f31*e1 and so on. So the analogous counts (weights) are growing: 1 + q + q2 +... The weights can be thought of as labeling the spaces with natural numbers. So intrinsic to a vector space is a notion that its basis is a totally ordered set. When q=1 then that order vanishes. And the reason is that when q=1 we have only one scalar and so there is no real "choice of scalars" made by which to naturally distinguish the order of the basis elements.

I have found a similar combinatorial interpretation of the Gaussian binomial coefficients that they count the k-simplexes in an n-simplex where the vertices of the k-simplex are given weights 1, q, q2, ... qk-1 and the edges all have weight 1/q. The simplexes are then total orders, which is to say, ordered sets. So we are counting the ordered subsets of an ordered set. When q=1 this becomes the uniquely orderable subsets of a uniquely orderable set, which is to say, the unordered subsets of an unordered set.

All of this is to say that it can be argued that fields are defined in such a way to give us "choices" which imply intrinsically that vector spaces are constructed in terms of ordered bases. This is an argument based on the "implicit math", not what gets written on the paper, but what reflects our mental activity.

Whereas modules have me think of expansions of amounts and units. I tutored students that "every answer consists of an amount and a unit", "combine like units to simplify calculation", "list different units to make the answer easier to understand".

When we figure things out in mathematics, there seem to be some times that in our minds we make use of a list, as with the basis of a vector space, for example, when we construct a flag. And at other times we just use a set, as when we equate two expansions and thereby establish equations for each term.

Could this distinction between lists and sets have some bearing here?

John, thank you for the link to multilinearity. I will have to study how multilinearity isn't quite linearity but is related. I will have to think about that.

In studying the field with one element, I realized that when we count the k-dimensional subspaces of an n-dimensional vector space over a field Fq of characteristic q, we may typically depend on a given basis: e1, e2, e3, ..., en. If we want to count the number of independent choices for constructing a vector space, then the first space would be generated by e1 (multiplied by any scalar). The next dimension would be generated by e2 + f21*e1 where f21 can be any scalar including zero. And then e3 + f32*e2 + f31*e1 and so on. So the analogous counts (weights) are growing: 1 + q + q2 +... The weights can be thought of as labeling the spaces with natural numbers. So intrinsic to a vector space is a notion that its basis is a totally ordered set. When q=1 then that order vanishes. And the reason is that when q=1 we have only one scalar and so there is no real "choice of scalars" made by which to naturally distinguish the order of the basis elements.

I have found a similar combinatorial interpretation of the Gaussian binomial coefficients that they count the k-simplexes in an n-simplex where the vertices of the k-simplex are given weights 1, q, q2, ... qk-1 and the edges all have weight 1/q. The simplexes are then total orders, which is to say, ordered sets. So we are counting the ordered subsets of an ordered set. When q=1 this becomes the uniquely orderable subsets of a uniquely orderable set, which is to say, the unordered subsets of an unordered set.

All of this is to say that it can be argued that fields are defined in such a way to give us "choices" which imply intrinsically that vector spaces are constructed in terms of ordered bases. This is an argument based on the "implicit math", not what gets written on the paper, but what reflects our mental activity.

Whereas modules have me think of expansions of amounts and units. I tutored students that "every answer consists of an amount and a unit", "combine like units to simplify calculation", "list different units to make the answer easier to understand".

When we figure things out in mathematics, there seem to be some times that in our minds we make use of a list, as with the basis of a vector space, for example, when we construct a flag. And at other times we just use a set, as when we equate two expansions and thereby establish equations for each term.

Could this distinction between lists and sets have some bearing here?

John, thank you for the link to multilinearity. I will have to study how multilinearity isn't quite linearity but is related. I will have to think about that.