Per Mathematical Structuralism, can a pure mathematical theory have semantics that is not closed on isomorphism?

Question

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?

Answer 1

I really don't understand where this question is driving at. It's a very peculiar mix of various terms from set/model theory - like categoricity which has no place in what is understood as structuralism.

Mathematical structuralism began with the discovery of homomorphism in abstract algebra, particularly in Noether's school. Eventually, after the discovery of category theory by Eilenberg & MacLane whilst theorising about structures discovered in algebraic topology, it was understood that this offered a formalisation of structuralism per se. In a sense, it turned set theory on its head. Set theory begins with sets and derives morphisms whilst category theory begins with morphisms and derives objects, of which some are sets.

Your term 'pure mathematical theory' appears to be your own neologism for 'structure' in model theory. As I have explained in the comments, it facilitates communication to use standard terms. If you are looking for a structuralist inspired model theory, then one example of such a theory, is the theory of Institutions.

Answer 2

Short Answer

Per Mathematical Structuralism, can a pure mathematical theory have semantics that is not closed on isomorphism?

No. Structuralism is defined in terms of isomorphisms, and there are not exceptions by definition.

Long Answer

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.

You're in good company. According to Linnebo in his Philosophy of Mathematics, p.167:

Structuralism is the belief that all matters in mathematics is preserved under isomorphism.

Work on theorems regarding isomorphisms does indeed go back 100 years to Noether and her teachings about algebraic structure. Unlike Frege's intuitions that led him to logicism which also brought the nature of the abstraction to the fore in the philosophy of mathematics, structuralism doesn't attempt to reduce mathematical propositions to logical ones. What it does do methdologically is deal in fundamental equivalences, and in fact, the equivalence between equivalence classes of Cauchy sequences and Dedekind cuts is precisely a good example of structuralism. So that you define structuralism as equivalence classes of systems under isomorphism is not in the least controversial.

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.

I find the claim that this set-theoretic isn't a structure confusing. You certainly can claim that {0, {0}} defines a set of two distinct objects, but to claim that somehow you have outwitted structure is not possible. First and foremost, if you don't define the symbol "0", then anything can be 0. Thus, the structure {0, {0}} can represent systems, for instance, 0 can represent empty space, and {0} can represent a well-delimited volume of empty space. Membership necessitates intension or extension, and either is predication. Predication is the essence of structure. And, 0 can be the origin on a line, and {0} represent the neighborhood about the origin. So, the string "{0,{0}}" is the same structure whether it represents a metric appraising n-space or a location in n-space. In fact, your structure can be represented conveniently as such:

1. 0
2. S(x) != 0
3. ∃!S(x) : S(x):={0}

This is because while you have expressed your structure using set theory which eschews highlighting the order of elements, the elements themselves exhibit ontological dependence. In set theory, {0} cannot exist until 0 exists, in our formulation meaning that 3 cannot be declared until 1 is declared. What the mathematical structure captures IS that ontological dependence, and hence if you have a predication, you necessarily have a structure. That's a property that inheres to abstraction by definition.

Furthermore, the categoricity theorem as I understand it merely says that any two infinite systems are isomorphic. That does not logically imply that non-infinite systems are not isomorphic. That would be conflating the contrapositive with the inverse. So, to me your claims about "pure" mathematical systems sounds an appeal to purity of sorts since your are mucking about with an inappropriate counterexample, since your string itself isn't really a counterexample at all. You seem to be throwing both "finite" and "semantic restriction" against structuralism hoping to somehow efface the predication inherent in your own example, and yet you are stuck with three: that of substitution, that of membership, and that of ontological dependence.

Ultimately, what I think you are driving at is "is it possible to cook up an example of a system in which I can use semantics to eliminate the possibility of having objects which I can abstract a structure from, and still have structuralism". Despite your attempts at legerdemain, the quintessence of your ask is no, and by definition. Mathematical structures exist when rules are abstracted from mathematical claims (which are often but not necessarily understood to be abstractions of other things, such as concrete objects).