Two very prominent kinds of phenomenon within mathematics form a sort of reciprocal pair. First, we have that a certain concept or notion is manifest in a wide range of situations. For example, duality appears in: projective geometry between lines and points; platonic solids, e.g., the dodecahedron and icosahedron; Stone duality between certain spaces and algebras; Fourier analysis; Poincaré duality between the homology and cohomology of complementary dimensions; duality between syntactic theories and semantic models; Pontryagin duality for locally compact abelian groups, and so on. Second, the reciprocal kind of phenomenon occurs where a single entity possesses a wide range of properties and structure. It is this latter kind that I wish to discuss in this paper.

An important way of singling out an entity is to define it via some universal property, in particular via freeness. To be the free such-and-such, an entity must possess the structure and properties required in the description and nothing more. Then given any general such-and-such there will be a unique structure preserving map to it from the free one. For instance, the integers form the free abelian group on one generator. This means that given an abelian group and a designated element, g, there is a unique homomorphism from the integers which sends 1 to g.

Another important algebraic entity, Symm, the set of polynomials on a countably infinite number of commuting variables with integer coefficients, is also free – the free ‘lambda-ring’ on one element. More topological examples can also be constructed in this way. For example we can define Tangles as the free X, for some construction. In this case, X = braided monoidal category with duals on one object. What this amounts to is that we take an object of the category to be a finite collection of points sprinkled on a plane. Then an arrow going from a first plane to a second plane of points is a collection of bonds each linking two points either in the same plane or in different planes, and a collection of knots sitting between the two planes. They can be tangled up with each other anyhow, as the name suggests. The freeness of this entity, and the ensuing mapping from it to similarly structured categories, is part of what is called quantum topology.

Now, the crucial observation is that in very many cases when an entity is defined via a universal property, it is found to possess other important properties. Let us illustrate this by reconsidering the integers. Along with their characterisation as the free abelian group on one generator, the integers carry a commutative ring structure, and in fact form the free commutative ring. As such they form the initial object in the category of commutative rings, and so for any such ring there is a unique ring homomorphism from the integers to it. Similarly, Symm displays a huge range of characteristics including algebraic properties relating to the representation of symmetric groups and to the cohomology of an important classifying space. Rather importantly there is a ‘comultiplication present’, so that Symm encodes a decomposition as well as a composition.

Symm bears so many different properties that Michiel Hazewinkel describes it as his ‘star’ example in a very interesting paper entitled “Niceness Theorems.” In this paper, Hazewinkel approaches the issue of explaining why an entity characterised in one regard should possess further properties: “It appears that many important mathematical objects (including counterexamples) are unreasonably nice, beautiful and elegant. They tend to have (many) more (nice) properties and extra bits of structure than one would a priori expect…” [Hazewinkel M., 2008, Niceness Theorems, http://arxiv.org/abs/0810.5691].

Sometimes this is clear. For instance, it is fairly straightforward to show that as a consequence of the universal characterisation of the integers as the free abelian group on one generator, a multiplication can also be defined on the integers in such a way that they possess a ring structure. However, that the integers should be not any ring, but the initial ring, or in other words the free ring on no generators, is not clear from its first characterisation.

We can also define interesting entities by other constructions, for example, the rational numbers as the field of fractions of the integers, and the reals as the Dedekind-MacNeille completion of the rationals. Each possesses a panoply of different properties and structures. Now, the superposition of many interesting features in the same object explains why they crop us so frequently, and suggests an answer to the puzzle as to why when there are many possible structures that mathematicians could study, some of them act almost as ‘attractors’. There is an air of inevitability to some entities as though one cannot fail to encounter them when working in a certain direction.

Now this ‘attractor’ phenomenon might be attributed to many factors. Possibly humans have a limited number of ways of thinking, so work with entities constructed out of choices from a restricted menu. Or perhaps research mathematicians have been socialised to work in a limited set of ways with the same result. The main thrust of this paper is that there is a third viable option – that in the realm of structural possibility, there are privileged members. A strong sense of reality attaches to these members, deriving from, what we might call following William Wimsatt, their robustness. This is a notion that Wimsatt developed in the philosophy of science, which goes by alternative names such as multiple determination. In his chapter “The Ontology of Complex Systems,” Wimsatt explains how he chooses to approach the issue of scientific realism with the concept of robustness: “Things are robust if they are accessible (detectable, measureable, derivable, defineable, produceable, or the like) in a variety of independent ways” [Wimsatt W., 2007, The Ontology of Complex Systems, in Wimsatt, W., 2007, Re-engineering Philosophy for Limited Beings, Harvard UP, on 196]. In this paper I shall explore whether this idea of independent access makes sense in mathematics, and whether it can account for the sense of inevitability presented by certain mathematical entities.