Talk:Mathematical structure

Ambiguities in the intro
An article about 'Mathematical structure' should be crystal clear about the relationship between said structure, the set to which it is "attached", and perhaps a resulting object that is comprised of the set and the structure. But the intro seems to muddle some of this together.

(1)In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance. What is the "some manner" of attachment?

(2) A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.

So... groups and fields are structures? Is this stating that you add structure to a set to create a structure? That seems circular, or at least uses two different meanings of the word "structure".

(3) Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. Not my main point, but this sentence suggests that adding a structure to a set causes a change in mathematicians' studying abilities. Seems to me that the thing that gets richer (has more complex characteristics) is the new object formed from combining a set with some structure. The mathematicians retain their original studying abilities, and can apply those to the study of the new, more complex object.

(4) For example, an ordering imposes a rigid form, shape, or topology on the set. As another example, if a set has both a topology and is a group, and these two structures are related in a certain way, the set becomes a topological group. Again, this seems to use the word "structure" in two different ways: As something that a group can have, and something that a group can be. Gwideman (talk) 02:06, 29 November 2019 (UTC)