A simple basic axiomatic theory of syntax is developed. It allows us to prove Gödel’s celebrated diagonal lemma in a simple and straightforward way.

A more expressive axiomatic theory of syntax is presented. It is shown that this theory generalises the theory of chapter 5 and allows the derivation of many natural properties of syntax.

Eventful analysis employs the most unfrozen and hence the most exploratory strand of CHA. It employs historical comparisons and explores transformation patterns, that is, patterns of qualitative change. It uses two key tools: historical description and conceptualization. The aim of historical description is to figure out what is going on, to gain a basic understand of a phenomenon before proceeding to explain it. Often this involves de-redescribing a phenomena that has qualitatively changed over time. Historical description, in turn, involves six concrete steps: fact gathering, chronicling, concatenation, periodizing, looking for intercurrence patterns, and rethinking research questions. Conceptualization serves to make historical description more comparativist and to explore broader patterns. The chapter discusses how to replace proper names with broader concepts by defining both the positive and the negative pole of concepts. It lists criteria for assessing the content and temporcal validity of concepts.

We discuss the free monoid and the cofree comonoid on a species (relative to a fixed hyperplane arrangement). In addition, we discuss the free bimonoid on a comonoid, and dually the cofree bimonoid on a monoid. More generally, for any scalar q, we have the free q-bimonoid on a comonoid and the cofree q-bimonoid on a monoid. An important special case is when the starting (co)monoid has trivial (co)product. We employ the terms concatenation and q-(quasi)shuffle for the products, and deconcatenation and q-de(quasi)shuffle for the coproducts. For q = 1, the q-(quasi)shuffle product is commutative, while the q-de(quasi)shuffle coproduct is cocommutative. The concatenation product and deconcatenation coproduct do not depend on q, and do not satisfy any commutativity property. In addition, we also discuss the free commutative monoid and the cofree cocommutative comonoid on a species and related constructions. These have signed analogues. We discuss the q-norm map between free and cofree q-bimonoids. It is an isomorphism when q is not a root of unity. Invertibility of the Varchenko matrix associated to the q-distance function plays a critical role here. We also discuss the (co)free graded (co)monoid on a graded species. Every species can be viewed as a graded species concentrated in degree 1. The free graded monoid on a species has a unique coproduct which turns it into a graded q-bimonoid. This is precisely the q-deshuffle coproduct. Dually, the q-shuffle product is the unique product which turns into a graded q-bimonoid.