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

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?

• One of the major goals of structualism is to put math theories to the utmost abstract useful form as pursued by Hilbert, Bourbaki, categorists, etc. Your edge case semantic "theories" such as your identity theory or restricted standard model of PA are models indeed, not even theory, thus not pure theory per structualism. Of course Godel showed even qualified pure math theories under structualism may not be simply categorical (thus his famous penchant for Platonism), so categoricity is moot here regarding structualism... May 12, 2022 at 6:39
• @DoubleKnot, Ok I like this comment, at least its telling me that per maxims of mathematical structuralism such restricted theories are not to be qualified as pure mathematical theories! That's a good and clear response, which is the kind of response I wanted. But my question is that perhaps there is a way to reconcile those restricted semantics theories with structuralism, but honstely I don't know how?.. to be continued May 12, 2022 at 11:31
• @DoubleKnot .....If some of those theories would be judged by mathematicians or math logicians to be purely mathematical then this would indeed be a blow to the ability of mathematical structuralism in providing a full characterization of pure mathematical theories. However, that doesn't mean that it is not fairing well in providing a proper contex for MOST pure mathematical theories, which I think it does! So all in all I think there are counter-examples to mathematical structuralism! Yet those might not be significant. May 12, 2022 at 11:32
• I think one can look at matters from a different angle, Structuralism might succeed in measuring the degree of puarity of a mathematical theory, through the concept of what I call as partial structure, this is just a class of isomorphic systems, so it is a subclass of a structure, I mean if we define structure as equivalence class of systems under isomorphism, then any subclass of it is to be called as partial structure. One can coin the concept of k-partial structure in such a manner that the more k is the more pure is that partial structure. .... May 12, 2022 at 12:51
• @DoubleKnot..., for example we may let k be the cardinality of the partial structure, so the bigger it is the more filtering of individual properties of systems in it is taking place and so the class would stand for a purer structure. That said the purest partial structure is a full structure i.e. the whole iso-class, which is fully pure! So for example my two object theory is speaking about a 1-partial structure, and this is very un-pure, i.e. heavily polluted with individual properties, ... to be continued May 12, 2022 at 12:55

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.

• The point is that if there are some theories considered as pure mathematical and yet its semantics are not closed on isomorphism then this doesn't fit the spirit of Structuralism, that was the purpose of the posting to Mathoverflow, but the purpose here is to know how such a thing suits or doesn't suit Mathematical structuralism, in particular how can Mathematical structuralism account for such systems being called purely mathematical? How can one reconcile that with structuralism. May 12, 2022 at 4:30
• Categoricity agrees strongly with what Mathematical structralism is advancing, if you have a Categorical theory then of course you are speaking about a structure since it has one model up to isomorphism and all those isomorphic models are acceptable in its semantics, so its definitly speaking about structure and actually a single one. The non-Categoricity of a theory does present some challenge to applying Structuralism as a way to characterise what a pure mathematical theory is, but it can be remedied in the way I mentioned, but the problem is when semantics are not closed under isomorphism May 12, 2022 at 4:35
• @Zuhair: Like I jave alrrady hinted above, you are confusing the structures of model theory with the formalised structuralism in category theory. Categoricity has nothing to do with Category theory. That it takes 'structures' upto isomorphism simply shows the progress of mathematical structuralism itself. Think of it thos way, if we take the notion of a group up to isomorphism, this does not make the idea of a group structural. A group is a mathematical object. Likewise with the mathematical notion of categoricity. May 12, 2022 at 10:43
• You said "if there are some theories considered as pure mathematical but whose semantics are not closed on isomorphism this would not fit the spirit of Structuralism". Yes, that would be true. What do you mean by a "pure mathematical"? Presumably a 'structure'. This is the usual term. I would suggest that you stick to the usual terms when they ate there because, as Feynman pointed out, it facilitates communication, rather than using neologisms peculiar to yourself. You said 'if' - can you give me several examples of such structures? Otherwise you are talking about the empty set. May 12, 2022 at 10:51
• Ok lets start from where we agree upon, which is the first the first two sentences in your last above comment. Now, the definition of *pure mathematical theory" is open to judgement, but definitely I don't mean 'structure'. A pure mathematical theory is a theory seen by mathematicians\mathematical logicians to be purely mathematical as such, I take it as a primitive undefined notion. Actually what I want to do is to try provide a definition for it. I've already proposed a restricted PA, this appears to be a pure mathematical theory, yet its semantics are not iso-closed. May 12, 2022 at 11:10

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.

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).

• Thanks for this clear response. About my example of the theory with the restricted semantics to the set {0,{0}}, it doesn't matter if you interpret that set itself as a structure, this is besides the point. I was speaking about the first order theory that extends First order identity theory by the axiom: \exists x \exists y: x \neq y \land \forall z (z = x \lor z=y), In English this says that there exists only two distinct objects. Now, if I restrict the semantics of this theory to just the set {0,{0}}, then this is not closed under isomorphism. Jun 12, 2022 at 20:31
• continuation..., I'm taking ZFC to be the background theory of models here (although this is not necessary), so the Ontology is sets as specified by axioms of ZFC. Now according to that for example the set { {0}, {{0}} } would be shunned from the semantics of this theory (by brute restriction on semantics), so the semantics of that restricted simple extension of identity theory is not closed under isomorphism. So, this should be considered as a theory that is not about structures. In other words, per structuralism, it is not a pure mathematical theory. Jun 12, 2022 at 20:34
• Also my other example with PA with its semantics restricted to the standard models of PA that are hereditarily smaller than some specified cardinal kappa, is also an example of a theory whose semantics is not closed under isomorphism. So, per structuralism, it is not a pure mathematical theory, because it shuns from its semantics models of PA that are isomorphic to those but not hereditarily smaller than kappa. Jun 12, 2022 at 20:45