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Structures are sets together with some constants, relations, and functions on that set. They are studied in many areas of mathematics: For example, universal algebra studies algebraic structures (i.e. structures with no relations, only functions and constants) in general. Also, model theory studies the connection between a formal logic and structures. As a last example, let me mention abstract algebra. This field studies more concrete structures, such as groups, fields, rings, monoids, and so on.

Okay. Structures are very important in mathematics. I am interested in the history and the motivation of the definition of structure.

  • What makes structures so interesting?
  • Why gives us the study of structures so powerful tools?
  • What is the motivation behind the definition of the term structures? (For example, why can't structure have more than one carrier set?)
  • How did the term structure historically develop? What were the first motivating examples?
  • Why did one choose the name "structure"? I personally interpret this word ("structure") as being some kind of pattern. What has this to do with the mathematical notion of structures?
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    $\begingroup$ For an introduction to philosophical aspects of mathematical structuralism see the IEP page, which has many references. $\endgroup$ – Bill Dubuque Aug 20 '16 at 18:24
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    $\begingroup$ @BillDubuque I'm not really sure that's relevant to the OP. I think they're less interested in structuralism (vs., say, set-theoretic foundationalism) than they are in why the classical model-theoretic definition of "structure" is what it is. $\endgroup$ – Noah Schweber Aug 20 '16 at 18:40
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    $\begingroup$ A key book on this subject is Leo Corry's Modern Algebra and the Rise of Mathematical Structures. $\endgroup$ – symplectomorphic Aug 20 '16 at 18:58
  • $\begingroup$ @NoahSchweber I think anyone asking such questions would learn much from being familiar with all viewpoints. $\endgroup$ – Bill Dubuque Aug 20 '16 at 19:36
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    $\begingroup$ @BillDubuque While I do think that the OP could benefit from knowing about as many viewpoints on the foundations of math as possible, I don't really think that any of those viewpoints are relevant here (note that I didn't push set-theoretic foundations either, in my comment to you or in my answer below); I think they're just asking a concrete question about the history and value of a particular definition. I think right now, foundational perspectives of any kind are mostly irrelevant. $\endgroup$ – Noah Schweber Aug 20 '16 at 19:51
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This is really just a comment - see my postscript below - but this is way, way too long:

So the main idea is this: we're looking for an idea which generalizes all or many of the already-very-general kinds of mathematical object studied so far. For instance, groups, rings, fields, vector spaces, etc. Whatever definition we settle on should have two properties:

  • It should be sufficiently general to cover all, or at least a wide variety, of the mathematical objects we've already become interested in.

  • It should be specific enough that we can prove things about it. Too much generality is not inherently good!

Now, let me point out that this question has been answered in different ways! E.g.

  • In the context of universal algebra, we're interested in sets equipped with functions - no relation symbols allowed!

  • In the context of model theory, the definition of structure is as you've given it. Note that this generalizes beyond the universal algebraic setting by allowing relations.

  • What about times when we care about topology? Topological groups, rings, fields etc. make sense, but aren't captured by the classical notion of "structure" from model theory. If you want to talk about topological structures, you need an even more general definition: a topological structure is a structure (in the usual sense) together with a topology on its underlying set. See e.g. the book Continuous Model Theory by Chang and Keisler. Now, maybe we also want to add some compatibility conditions on how the structure and topology interact; this is the approach taken in continuous logic, which has so far been more effective than the unrestricted version (at least, that's my impression).

  • And, of course, sometimes we're interested in things that look like structures "from the outside" - e.g., a group object in some category. We can develop categorical versions of model theory or universal algebra (see e.g. Lawvere theories), which is even more general than what I've described so far.

  • Going in a different direction, we could allow infinitary relations and functions. I believe this was looked at by Addison a long time ago, but I don't have a citation. Note that there are very natural examples of infinitary operations and relations - for example, the relation "converges as an infinite series!"

So that's my pushback against the idea that there's a "right" notion of structure. That said, the classical model-theoretic notion of structure has clearly been extremely useful. So, what made it so?

It's difficult to make a convincing argument here, but let me list a few points in favor of this definition.

  • It is sufficiently general to capture basically every structure studied in (the non-topological parts of) abstract model theory.

  • It provides a semantics for set theory, which in turn lets us talk about the more general approaches. Remember that ZFC, despite talking about the set-theoretic universe, is really a first-order theory! Note that we run into trouble here if we want to talk about e.g. class-sized objects, but I don't really think that's fatal to this idea (see e.g. NBG or universes).

  • This definition is sufficiently narrow that we can prove theorems about it: e.g. compactness, Lowenheim-Skolem, Herbrand's theorem, etc. These theorems in turn have been applied to specific mathematical problems - see e.g. the Ax-Kochen theorem or proof mining. Note that these theorems are really theorems of the underlying logic (first-order logic) rather than the notion of structure per se; but these aren't really that different. The completeness theorem gives a sense in which the notion of structure is in correspondence with the underlying logic. (Speaking of logic, note that we could also ask why first-order logic is the "right" logic to use, and how that decision was made; see e.g. this paper by Ferreiros. Also see Lindstrom's theorem.)

Note that this answer completely avoids any real historical discussion. This reflects my lack of knowledge. While I know a little bit about the philosophical arguments which surrounded the adoption of first-order logic, I know nothing about the history of the definition of "structure." That's why everything in this answer is retrospective: knowing what we know now mathematically, what can we say about "structure"? I hope someone with actual knowledge will give a better answer.

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    $\begingroup$ Something that I find ugly is that even fields and vector spaces can't be naturally viewed as structures (in the sense of model theory): the function $(-)^{-1}\colon\mathbb K\setminus\{0\}\to\mathbb K$ is not defined at $0$! And a vector space should be a structure for a many-sorted signature: we have scalars and vectors. But the standard definitions only allow on sort. $\endgroup$ – ajsdkf Aug 20 '16 at 21:17
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    $\begingroup$ @ajsdkf Actually, many-sorted structures can be reduced to single-sorted ones via unary predicates. This only causes a problem when we consider many-sorted theories with infinitely many sorts, because then the compactness theorem guarantees that there will be models with "unsorted" elements. However, many-sorted first-order logic isn't very different from classical first-order logic, and is well-understood and used in model theory (see e.g. Shelah's $M^{eq}$ construction). Similarly, a function can be thought of as the relation giving its graph; this allows for partial functions as well. $\endgroup$ – Noah Schweber Aug 20 '16 at 21:26
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    $\begingroup$ Of course, you are right. But that are just artificial solutions to this problem. That's why I call it ugly. $\endgroup$ – ajsdkf Aug 20 '16 at 21:30
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    $\begingroup$ @ajsdkf yes, that would be a meaningful statement in this context. Now, presumably(!) you included axioms asserting that the sorts are distinct, and that they exhaust the universe (this latter is expressible if there are only finitely many sorts, but not otherwise, due to compactness), so those statements are killed off. You could also just develop the theory of many-sorted first-order logic. This doesn't really require much work; basically everything goes through with a couple obvious caveats. $\endgroup$ – Noah Schweber Aug 20 '16 at 21:43
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    $\begingroup$ @ajsdkf "even fields and vector spaces can't be naturally viewed as structures". That is true about fields, but not about vector spaces. And vector spaces over fields (or modules over rings) can be viewed as two-sorted algebras as you ask, but the scalar multiplication can also be viewed as a family of unary operations (as I suppose you know), and it is good that there are these two options since we can use whatever definitions suits better in each context. $\endgroup$ – amrsa Aug 21 '16 at 11:30
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I'll try to answer to some of your questions.

What makes structures so interesting? Why give us the study of structures so powerful tools?

One very common and useful thing in mathematics is abstraction, a.k.a. generalization. Thinking in the language of structures allows mathematicians to generalize many properties to lot of different objects: for instance Cauchy's theorem tells us that for every prime divisor of the cardinality of a finite group there is an element that has order that prime, no matter if it is a group of permutations or isometries or matrixes. Basically it is the maths version of the "two birds one stone"...although "lots of birds one stone" would be more appropriate, at least in my opinion.

Another important thing given by the notion of structure is the notion of (homo)morphism. Via morphisms we are able to relate and confront different objects having similar structure and this is useful for discoverying and proving properties of different objects: for instance in group theory we can study properties of different groups through their homomorphism in groups of permutation or groups of matrices (there is an entire field in maths that deals with these objects called representation theory, it has important applications outside mathematics for instance in physics).

Note that without the notion of structure the notion of morphism cannot be stated.

We could go on for very long answering just these two questions but I'd rather stop here to avoid being too boring (feel free to ask in the comments for addional stuff).

What is the motivation behind the definition of the term structures? (For example, why can't structure have more than one carrier set?)

Probably you should be more specific on which notion of structure you are referring to because there are multi-sorted structures which have families of sets as carrier. Examples of these multi-sorted structures are modules over generic rings (or if you like a more specific example vector spaces over different fields). Another example that I cite (because I'm a little biased) is that of categoy, which is a multi-sorted structure.

How did the term structure historically develop? What were the first motivating examples?

I'm not sure about that but I think that it came first during the 1800 when english algebraists observed the similar....structure undelying different kind of algebras and that through that notion they could prove a lot of intersting properties (like existace of solutions to equations) for large class of structures at once by reasoning in term of these abstract structures, instead of redoing the same work for all the concrete examples. These observations led to what is called modern or abstract algebra.

Why did one choose the name "structure"? I personally interpret this word ("structure") as being some kind of pattern. What has this to do with the mathematical notion of structures?

This question is itself your answer. The various notion of structures capture different common pattern in large class of objects. For instace the notion of group capture a pattern common to symmetries, isometries and various kind of number-systems: namely the fact of having a binary operation that is associative, with unit and inverses. Similar arguments apply to other kind of structures (rings, vector spaces, etc).

Hope this (maybe too long) answer may help.

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I grasp 'structure' as relations on sets, sets which should be perceived as structureless per se, and that the modern use of it is tied to the set theory.

All bijections on a set is a concrete example of a group and at first all subsets of those bijections which were closed under composition where thought of as 'groups'. Then it was found that the groups were uniquely defined by the four classic axioms of groups. This might have been one of the first modern structures.

Having axioms of structures on sets is a big advantage as a foundation for further investigations.

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