> For example, ZFC is fine for calculus (most of engineering), but completely inadequate for computer science. For CS you have to allow self-reference and that is expressly verboten in ZFC.
I assume you are referring to the axiom of foundation which implies, for example, that you cannot find three sets A, B and C such that A ∈ B, B ∈ C, C ∈ A. This is not an obstacle to talking about self-reference, recursion or cyclic graphs, etc. The axiom of foundation keeps the membership relation from having cycles, but it doesn't keep you from talking about other relations that might have cycles! Indeed, defining functions recursively is completely standard practice in mathematics, and is formalizable in ZFC with no problem. Also, the most common definition of graph is explicitly in terms of set (a graph is an ordered pair (V,E) where E is a set whose elements are pairs of elements of V)and easily formalizable in ZFC. That you can't have ∈-cycles never comes up in, say, the study of cycles in graphs or of recursively defined functions.
The axiom of foundation forbids cycles in the membership relation, you are of course allowed to talk about other relations that do have cycles. For example, R={(1,2),(2,3),(3,1)} is a relation defined in ZFC that relates 1 to 2, 2 to 3 and 3 to 1. If you want to study cycles in directed graphs it never matters that you can't have membership cycles and I'm confused about why anyone would think it does.
I believe they are pointing out that the constraints on the construction of ZFC are not meaningfully/practically constraining the objects build atop the system that you would want to study and manipulate in CS.
I can't see how your implication works either, can you make it explicit?
Those tricky relationships arise from the most fundamental circuits that define CS, yes.
What I am trying to say is that you do not need mathematical formulas to explain mathematical concepts, but certain vocal academics have managed to push a view that math is nothing but formulas AND that there is no controversy here. There is, in fact, a lot of controversy and this does lead back to Grothendieck since his work does not support the formulas-only view.
> but certain vocal academics have managed to push a view that math is nothing but formulas AND that there is no controversy here.
This isn't really a thing, which may explain why I'm having trouble understanding what you are trying to say. To be more concrete: from what I can understand of your somewhat incoherent characterization it does not match any mathematical community I know of - it may exist, but if so it is not mainstream.
I assume you are referring to the axiom of foundation which implies, for example, that you cannot find three sets A, B and C such that A ∈ B, B ∈ C, C ∈ A. This is not an obstacle to talking about self-reference, recursion or cyclic graphs, etc. The axiom of foundation keeps the membership relation from having cycles, but it doesn't keep you from talking about other relations that might have cycles! Indeed, defining functions recursively is completely standard practice in mathematics, and is formalizable in ZFC with no problem. Also, the most common definition of graph is explicitly in terms of set (a graph is an ordered pair (V,E) where E is a set whose elements are pairs of elements of V)and easily formalizable in ZFC. That you can't have ∈-cycles never comes up in, say, the study of cycles in graphs or of recursively defined functions.