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What are the components of mathematical discourse?
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Mathematical discourse has three components.

The mathematical register.

When communicating mathematical reasoning and facts, mathematicians speak and write in a special register of the language suitable for communicating mathematical arguments. In this book it is called the mathematical register. Mathematical register uses special technical words, as well as ordinary words, phrases and grammatical constructions with special meanings that may be different from their meaning in ordinary English. It is typically mixed with expressions from the symbolic language.

The symbolic language of mathematics.

This is arguably not a form of English, but an independent special-purpose language. It consists of the symbolic expressions and statements used in calculation and presentation of results. For example, the statement $\frac{d}{d x} \sin x=\cos x$ is a part of the symbolic language, whereas "The derivative of the sine function is the cosine function" is not part of it.

Mathematicians' informal jargon.

This consists of expressions such as "conceptual proof" and "intuitive". These communicate something about the process of doing mathematics, but do not themselves communicate mathematics.

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