By the proposal here, there was an event in human evolution by which a particular algebra became accessible to cognition. This algebra was and is what is now known in computer science, information theory, critical path analysis, and current generative linguistics as the algebra of rooted, planar, binary-branched trees. On my current understanding of the evidence, the simplest possible statement of this tree has to have been primordial. This must have been how some human ancestor or ancestors started to differentiate themselves from others sharing the same ancestry with modern chimpanzees. This must have been such that it could be stated in a form which could be read by the biology so that it could become part of the human genome. By the proposal here, this algebra developed in seven steps, each building on all the previous steps, each useful for thought and communication, but only transmissible by virtue of the mathematical basis. Crucially, the algebra is recursive, allowing the grammar to generate the infinite set of structures we know as language.

Rooted, panar, binary-brancjed trees are mostly represented with the root at the top and the leaves or ‘terminal nodes’ at the bottom. The terminology and the graphical representation are thus not in line with one another. The representation is upside down. Here I am just following linguistic convention. Crucially this represents a process known as ‘derivation’. By the framework here, these trees represent structures, not sequences of elements.

Intuitively, when we look for a word or want to say something the structures come to mind more or less instantaneously and reliably, often with subtle shades of meaning. Or so it seems. But ChatGPT notwithstanding, there is a substantial task here of finding appropriate words and parts of words and assembling them in a particular order at the tempo of everday speech, language and conscious thought.

By the framework here, this process of retrieval and assembly is universal across human languages, despite the obvious differences between them. Thus in English we say ‘a clever person’. And in German and Dutch, the equivalent words are sequenced the same way. But in French, Italian, and Welsh, part of the ordering is reversed. And that seems like a significant difference between two types of language. But by the framework here, these differences, and others far more extensive and much more difficult to teach to adult second language learners, are in fact quite superficial. Underlyingly there is a universal notion corresponding to what is said in English as a clever person. It just surfaces differently according to comparatively minor language particulars.

Sometimes this process of retrieval and assembly breaks down, and we can’t thing of a particular word or name. It is on the ‘tip of the tongue’. Or we can’t find the right way of saying what we want to say. Or the small child wants to make some deep observation, seemingly beyond his or her years, and it comes out with one or more grammatical defects. Or the older child with a developmental language defect struggles to articulate some insight in a way which can be clearly and reliably understood. Or the normally competent speaker chooses a word or name which makes a nonsense of what her or she means.

But by and large the human faculty of language is remarkable for its everyday robustness and reliability, rather than its failures or the period of around ten years during which speech and language are normally being acquired. This robustness and reliability depend on the simplicity of the underlying code, in the sense of machine code in software. At the moment our best way of stating this is by an algebra. It represents a deep and powerful general resource for human cognition.

As well as allowing the development of human language, the recursive property given by this algebra can be, and is, usefully applied to all branches  of mathematics, including arithmetic and the way we count.