The quest for structure and the alchemy of patterns

Shelah’s conjecture on stable fields

Looking for structure is one of the most natural tasks for human beings. Watching a landscape and identifying patterns; reading poems and guessing where the beat lies; staring at paintings and figuring out the quiet refrain of ideas, inspirations, stories being told. Structure guides us and reassures us in what is perhaps an extraordinary expression of anthropocentrism, as if the universe arranged itself in recognisable ways just for us to enjoy.
While hoping to find patterns in the wild world around us might come off as slightly too optimistic (or even arrogant), for mathematicians this is often the purpose of their work, the end goal of their career: to build structure out of chaos, identify threads of meaning in seemingly abstract objects, single out reason, even when hidden behind a game of smoke and mirrors. Even when they come from far off regions of the mathematical landscape, whether they are arithmetic geometers or numerical analysts, mathematics often presents itself to mathematicians as a complex puzzle to solve, an enigma to decode. Behind all the complex computations, the red herrings and the chaos, stands some painting of order, hard and confusing and abstract as it may be.

Model theorists are no strangers to this phenomenon. While they might see mathematical objects with different lenses than other mathematicians, they ultimately analyse them to seek out patterns, storylines, threads to follow to reach the core engine that governs their behaviour.
The main tool in a model theorist’s toolbox is the idea of a “definable set”. Model theorists look at common mathematical objects like rings or groups in a certain language, a set of symbols that can be used to express the properties of the object in question. The language has to be chosen carefully, an operation akin to writing a poem — the wrong word, and all meaning is lost. When talking about a field, two operations and their neutral elements are necessary: it wouldn’t make sense otherwise, since the core essence of a field is that of a mathematical object where sum and multiplication are well-behaved, much like in the familiar field of real numbers. All basic properties of a field can then be expressed in this “language”, and much more.

Once a language is established, one can consider “formulae”, expressions that can be built from the symbols of the language together with logical connectives, variables and quantifiers. Formulae transform sentences in English into mathematical objects, strings of symbols that can be manipulated and interpreted mathematically. Think of the most famous property of a field: multiplicative inverses exist. Rephrase it as “for all elements x of the field, there is an element y of the field such that multiplying x with y gives 1”. Using the symbols we have chosen, variables for x and y and a quantifier to express “for all x” and “there is y”, we obtain a formula expressing the property we have just talked about.

We can also express properties of generic elements, by leaving the variables free, without quantification. Think of the property “there is an element y of the field such that the square of y is x”. This is a property of the element x, which is not blocked by any quantifier “for all” or “there exists”. It will be true of certain elements x, and false of others. (In the real numbers, for example, it will only be true of x if it is a positive number, or zero).

Once we have chosen a language, we can start thinking about properties that can be expressed in that language, and once we have also chosen a certain mathematical object, we can think about which elements of the object satisfy certain properties. By taking a certain formula ϕ(x) and grouping together all elements of a mathematical object that make ϕ true, we obtain a “definable subset” of our mathematical object. We call it definable because it is identified, or defined, precisely by the property ϕ.
Think, again, of the real numbers, and take ϕ(x) to be the formula “there is an element y of the field such that the square of y is x”. The definable set is exactly the set of non-negative real numbers.
Much like the geometer tries to understand the shape of objects, the information contained in how they can be deformed, a model theorist’s job will often be understanding what definable sets look like in mathematical objects of all kind. Identifying repeating patterns, discerning structure from the seemingly abstract world of formulae. It turns out, however, that this is often an impossible task. One typical problem is that there can be too many quantifiers. The formula ϕ written above is reasonable; it can be read out loud. But once we choose a language, we can write down formulae with as many quantifiers as we like, and there is little hope of understanding formulae with 73 quantifiers, even when the language is very limited (like the one of fields). This can be partially solved through some form of “quantifier elimination,” but otherwise the situation can look dire.
How do we know when the goal we have set for ourselves is unreachable? How can we find out beforehand whether the mathematical object we will try to understand via model theory is, in fact, beyond comprehension?
Enter Shelah.

Saharon Shelah’s works are known to be very technical, and the guiding light is often invisible for many that try to understand them. Nevertheless, his prolific production has unearthed the deep reasons for phenomena all over mathematical logic and mathematics and provided essential answers to many profound questions. One of these phenomena is the emergence of structure in mathematical objects, when looked at through the lenses of model theory.
Shelah’s proposal may seem counterintuitive at first. He poses that chaos is often produced from certain patterns, which like Patient Zeros infect the whole object with irredeemable opaqueness. Finding out some of these criminal patterns among the definable subsets of a certain object is sign that the object itself might be untouchable. Objects, and sets of formulae, where these patterns are omitted should be tame, analysable with our toolbox, safe to approach. He isolated a whole hierarchy of “dividing lines” that discern wild objects from tame objects on the basis of the omission of certain patterns. Structure is achieved by omission, removing what might seem like helpful structure but is instead a mole, responsible for devastating chaos.
One of these dividing lines is stability.

Stability was the first, and to this day perhaps the most well-known. For many model theorists, stable objects are like a paradise: our techniques work perfectly there, our questions have the desired answers. Stable objects — the most famous example being algebraically closed fields, like the complex numbers — exhibit extremely deep structural symmetries, often governed by phenomena of genericity that make proofs elegant and insightful. As a criterion for tameness, stability is quite simple: an object is stable if one cannot define a linear order in it.
Linear orders seem the opposite of chaotic objects; they are, as one expects from a rarely appropriate name, long lines of points, one after the other. If they show up in a mathematical object, however, they bring uninvited guests: too many complete types, an amount of information that is then too much to understand quickly and broadly. Unstable objects can still be analysed, but with more sweat and pain than was necessary before.
Stability is, then, a good rule of thumb for when an object is tractable, when a certain degree of control over its properties can be reached.

The thing is, not many stable fields are known. Indeed, all the ones we know are separably closed — and viceversa, we know all separably closed fields are stable (in the language of fields). This led Shelah to formulate one of his most famous conjectures: the stable fields conjecture. The statement is once again strikingly short — “any infinite stable field is separably closed”.
In the absence of chaos among their definable sets, fields should have very good algebraic properties. After all, their nature is intrinsically algebraic; they are exactly the objects where algebra and arithmetic can be performed ideally. If they are well-behaved through the lenses of model theory, they should be also well-behaved through the lenses of algebra. Separably closed are the best we can hope for: fields where all reasonable polynomial equations admit solutions. In the paradise of stability, all reasonable polynomials should have roots. There is no reason to believe that some polynomials should be better than others, for this arbitrariness is also a form of chaos, and we have banished all chaos from our objects through Shelah’s dividing lines.

Tragically open, the conjecture pushes our desire for structure even further: not only we seek out patterns, alchemically isolating the necessary criteria to make objects intelligible to our minds and the techniques we can invent, but we expect some return from this structure. We expect the guiding light of (healthy) patterns to solicit some positive feedback from the objects we work with, their deep symmetries to force algebraic properties. We dream of unifying phenomena that, much like prisms, present themselves in different shades when moved around, through different angles and lenses.