Is there an analogue in category theory of the distinction between structures/objects and material/aggregate, a distinction reflected in the grammar associated with count and non-count nouns in English?
This question is motivated by the related comments in the post "Spivak on Category Theory" at the n-Category Cafe. A discussion of the intricacies of the linguistic classification is given in this post.
The paper by Reyes et al. cited in the blog post and below doesn't, to me, capture the essence of everyday intuition guiding distinctions between countable nouns (CNs) referring to countable structures/objects and uncountable mass nouns (MNs) referring to substances/materials (of which structures are made). Substances are quantified by 'volume' or weight (or number of actual or potential containers--ingots, tanks, cups, etc.) rather than count.
The paper states:
The usual way of distinguishing MNs from CNs is by specifying semantical properties that MNs have but CNs lack.
Two or more dogs do not together constitute a larger dog; whereas two or more quantities of matter together constitute a larger quantity of matter. It is customary, following Quine [18, p91], to refer to this property of the extension of MNs as cumulative reference.
. . .
The transformations mentioned before between CNs and MNs are described by functors. One example is the plural formation that takes the CN ‘dog’ into ‘dogs’. Since the extension of this term obviously has the property of cumulative reference, we categorize ‘dogs’ as a MN.
I would not say that 'the dogs', referring to a specific group of dogs, has a cumulative reference and I can certainly count the members of any actual group or groups of dogs. Only 'dogs' as a class of animals as in 'Dogs are sociable' has a MN quality as in 'Liquids are fluid' and an associated syntax in English.
Examples that don't seem to fit the scheme: A 'drop of water' suspended in the air in an environment without gravity has a cumulative (and divisive) reference--I can easily add another drop to obtain a single larger drop (and divide it into smaller drops) yet it's regarded by most people, I believe, as a structure and, therefore, a countable entity/CN and not an uncountable MN, plural or otherwise. Same for a 'pile of sand" (or coins) or a 'line of sand' (or coins). If I add (accumulate) other piles or lines in a suitable way, I maintain a single line or pile yet a pile/piles or a line/lines are regarded as countable structures rather than an uncountable substance. This is in contrast to the discussion of 'dog' as a CN with no cumulative reference--adding dogs doesn't give you a larger dog (nor does dividing give you two smaller dogs, except in reproduction).
Consequently, my cursory review of the paper leaves me skeptical, but I found it thought-provoking, motivating me to ask if the notion of structure versus substance has been categorified (though this wouldn't be suficient to understand all countable versus noncountable nouns, e.g, shirts versus pants, shorts, eyeglasses, or scissors, which are counted as 'pairs of'.)
