On The Practice of Mathematics
The question is not whether mathematics should be applied. Most of us agree that it should. The concern is rather that our subject is sometimes being used as a mystifying smoke screen to protect pseudo-applications against the scrutiny of the general public and of the scientific colleagues in adjacent disciplines. We need to ensure that applications themselves be maximally effective, not clouded by misunderstanding.
In mathematics we never use “properties” that are defined on the universe of “everything”. There is the “universe of discourse” principle which is very important: for example, any given group, (or any given topological space, etc.) acts as a universe of discourse. As these examples suggest, a universe of discourse typically carries a structure which permits interesting properties and constructions on it. As the examples also show, there are typically many objects of a given mathematical category and also many categories, so transformation is an essential part of the content.
As quantity includes zero, so structure includes the case of no structure, which Cantor considered one of his most profound and exciting discoveries. Dedekind, Hausdorff, and most of 20th century mathematics followed the paradigm whereby structures have two aspects, a theory and an interpretation of it in such a featureless background. The background thus contributes minimal distortion to the assumptions of the theory. Such is “set theory” in the practice of mathematics; it is part of the essence from which organization emerges.