This question is the philosophical side of a question that I've recently posted to MathOverflow. Here, I'm specifically asking about the output of Mathematical Structuralism on that question that I'll re-post here. However, I'll include the brief account on the essential claims of mathematical structuralism, in order to have a clear view about what I'll ask about here, and also to relief any confusion about the terminology used.
Mathematical structuralism advances the view that pure mathematical theories are about what they call as 'structures', to be warned here is that what they call as structure is not the meaning commonly used in mathematical logic. Mathematical structuralists' define a "system" (also called "relational system") as set[s] with relations on it[them], this can be captured as a tuple of the general form (M_1, M_2,..; R_1, R_2,..) where each R_i is a relation whose domains and co-domain are among the M_i sets. The M_is' are to be called the domains of the system, while the R_is' are the relation sets of the system. Clearly what we usually call a "model" of a theory qualifies for being a 'system' as defined here.
Now structure is defined as a kind of universal that range over all isomorphic systems. My personal attempt of capturing that in terms of classes is that a structure is an equivalence class of systems under system isomorphism. Where the last is a bijection between the domains of the systems that preserves the relation sets of them. I think there is also another way of capturing the notion of structure, which is as the image of a system under a function from systems that is sensitive to isomorphisms, i.e. sends isomorphic systems in its domain to a common image. However, I'll mostly keep to the first definition of structures, that of being equivalence classes of systems under isomorphism.
Now matters are straightforward if we are working with a categorical theory, like for example second order arithmetic under full semantics. Here, this theory is about a SINGLE structure, and so, per structurlism, is a perfect pure mathematical theory. But the problem is that not all theories are categorical. All first order theories with infinite domains are not categorical. So a theory like first order PA would be a recipe for lots of structures, say a SPECTRUM of structures. However, this still can be reconciled with Mathematical Structuralism, since its about structures after all, in other words its semantics is closed under system isomorphism, that is if PA speaks about some system A, then every system that is system isomorphic to A is still enrolled in its semantics, so although PA is about multiple structures, yet its about structures and not something else.
Now what if we have a formal theory that employs a restriction on its semantics as not to be closed under system isomorphism, what that theory still qualify of being purely mathematical from the stand point of Mathematical structuralism?
Examples of what I mean is a theory whose semantics is closed under isomorphism to system A, but has a system B among its semantics such that not any system isomorphic to B is acceptable among its semantics, or even worse having its semantics being a proper subclass of a structure. A trivial example is a theory extending identity theory that stipulates existence of two distinct objects, and forbid others from existing. Now, if we restrict the models of that theory to say just the set {0, {0}}, then this would be a stringent restriction on the semantics of this theory that is heavily against structuralism. We can say that this theory is not a pure mathematical theory, since its about a particularity that bears no structural genre at all. It can be considered as an applied formal theory to that particularity, or even applied mathematical theory to that restriction, but not purely mathematical, since its semantics doesn't display respect to structures.
It appears to me that such formal systems cannot be justified of being purely mathematical from the view point of Mathematical Structuralism, since it cannot be said to be speaking about structure[s]?
Now my question here: are there examples of known theories that Mathematical Structuralism qualify as being purely mathematical and yet employ restrictions on their semantics as to have the class of all acceptable models of them not being closed under model (system) isomorphism?
I have some example in my mind, take a theory say with the axioms of PA, but restrict its semantics to just standard models of PA that are say "hereditarily < kappa", for some fixed infinite cardinal kappa. Is such a theory considered as purely mathematical theory? I mean from the view point of Mathematical Structuralism!
The question is about the ability of Mathematical Structuralism to characterise what a pure mathematical theory is, the expected answer is that those are theories speaking about structure(s) in the way defined above. But if we have a theory that is seen as purely mathematical and yet its semantics is not closed under isomorphism, then how this fits the program of Structuralism? How structuralism would explain this theory being about structures in the sense used in Structuralism. That's the point from this question. Would they for example reason them as partially exemplified structures??? And how such a concept would be defined?