Evan Chen’s “Remarks on English” is essentially a guide to writing clear, rigorous mathematical proofs, especially for olympiad mathematics. His central message is that a solution should not merely contain correct mathematics; it should make the mathematical ideas easy to see. He contrasts good and bad solutions to show that organization matters enormously: state the answer early, identify the main idea, separate important claims from routine calculations, use displayed equations, define objects precisely, and include diagrams when they are part of the reasoning. A longer solution can actually be faster to read than a short but poorly organized one. Chen particularly emphasizes that mathematical writing must never contain false statements, even temporarily, and that definitions deserve exceptional care because the reader cannot reconstruct an incorrect or incomplete definition for you. 

For proof structure, Chen recommends making the logical architecture explicit. In “find all” problems, first state the proposed solutions, verify that they work, and then prove that there are no others; for minimum/maximum problems, state the claimed value, demonstrate that it is attainable, and separately prove the required bound. Good exposition also means leaving whitespace, separating cases, naming important objects, and writing arguments forward rather than disguising them as backward reasoning. For students trying to become more concise, Chen argues that excessive length is usually caused not by “too much detail” but by disorganized writing; the solution is to rewrite it around a sequence of clear claims rather than simply deleting sentences. The overall philosophy is simple: a mathematical proof should communicate its reasoning, not force the reader to reconstruct it. 

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