Claude-Instant gave the best answer to “use mathematical analogies to describe an essay and a paragraph”:
Here are mathematical analogies for an essay and paragraph:
An essay is like a function. It takes various inputs (research, experiences, ideas) and through the process of writing, maps them to an output (analysis, argument, insight).
A well-written essay is a continuous function – the paragraphs flow smoothly from one to the next without breaks. There are no jumps in logic or clarity.
An introduction is like defining the domain of the function – it establishes the scope of what will be discussed. The body is evaluating the function – taking each input individually and applying the rules.
A conclusion is finding the range – summarizing the output and showing what was learned. It “solves” the problem or debate set up in the introduction.
A paragraph is like a single step in solving an equation. It takes a set of related ideas, facts or anecdotes (terms) and combines them using transitions/logic (operators) to simplify into a single point or elaboration (solution/result) that advances the overall argument.
Well-connected paragraphs are like steps in integral calculus – each builds seamlessly from the last, with no gaps between levels, to solve the problem statement.