Mathematics Paper Writing Conventions: Theorem-Proof Structure and Style

Mathematics papers do not read like papers in the biomedical or social sciences. There is usually no Methods section, no Results section, and no discussion of “findings” in the empirical sense. Instead, a research-level mathematics paper is organized around a sequence of formally stated claims — theorems, lemmas, propositions, corollaries — each followed by a proof that establishes it beyond doubt from stated assumptions and prior results. This page covers the structural conventions research administrators, early-career mathematicians, and co-authors from adjacent fields need to recognize: how the theorem-proof-lemma-corollary hierarchy works, how notation is conventionally introduced and reused, how LaTeX’s amsthm package encodes these conventions in typesetting, and why the whole apparatus looks so different from IMRaD.

Why mathematics papers don’t follow IMRaD

IMRaD (Introduction, Methods, Results, and Discussion) organizes a paper around an empirical study: a method is applied to data, results are observed, and the discussion interprets what was found. Pure and much applied mathematics doesn’t have that shape. A mathematics paper isn’t reporting what was observed when a procedure was run — it is establishing, by proof, that a precisely stated claim is true given precisely stated assumptions. There’s no “results may vary” or a Methods section describing an experimental protocol, because there’s no experiment to replicate: the entire evidentiary burden is carried by the proof itself, which any competent reader is expected to be able to verify by following the logical steps. CASRAI’s broader guide on how IMRaD varies across disciplines covers clinical, engineering, social-science, and humanities departures from the biomedical default; mathematics is a more fundamental case — not a variant of IMRaD with different section labels, but a genuinely different organizing logic built around formal claims and their justifications.

That said, most math papers still open with an ordinary-prose introduction (motivation, informal statement of the main result, how it relates to prior work, and often a short outline of the paper’s structure) before the formal theorem-proof material begins. The introduction is where a non-specialist reader gets the most value; the body is written for a reader willing to follow the argument step by step.

The theorem-proof-lemma-corollary hierarchy

These terms are not interchangeable stylistic choices — each signals something specific about the claim’s role in the paper’s logical structure, and readers rely on that signal to decide how much attention a given statement deserves:

• Definition — introduces a term or notation precisely, fixing its meaning for the rest of the paper. Definitions are not proved; they are stipulated.
• Theorem — a major result, usually the paper’s central contribution or one of a small number of headline claims. Reserved for statements the author considers significant, not every provable fact along the way.
• Proposition — a result of real interest but of lesser weight than a theorem; some authors use it for results that are true and useful but not the paper’s main point.
• Lemma — a supporting result proved specifically because it is needed as a step toward a theorem. A lemma’s value is instrumental: it exists to make a later proof work, not usually to stand on its own.
• Corollary — a claim that follows quickly from a theorem (or proposition) already proved, typically with little or no additional argument beyond applying the preceding result.
• Conjecture — a claim the author believes to be true, and states explicitly as unproved, often with supporting evidence or partial results.
• Remark — informal commentary, context, or an observation that doesn’t rise to the level of a stated result — often not proved at all, or proved only informally.

The American Mathematical Society’s LaTeX documentation for the amsthm package describes this same family explicitly, noting that theorems and proofs are the most common elements in mathematical writing but that authors also routinely use “lemmas, propositions, axioms, corollaries, conjectures, definitions, remarks, cases, steps, and so forth” as part of the same general class of numbered, formally-styled statements (American Mathematical Society, Using the amsthm Package).

How a proof is conventionally structured

A proof follows the statement it justifies, typically introduced with the word “Proof” and closed with an end-of-proof marker — historically “Q.E.D.” (Latin quod erat demonstrandum, “which was to be demonstrated”), and in modern typeset papers usually a small hollow or filled square (□ or ■), sometimes called a “tombstone” or “halmos” symbol after mathematician Paul Halmos, who popularized its use. AMS’s own amsthm documentation specifies that authors should use the package’s built-in proof environment rather than defining their own, since it handles the italicized “Proof” label, the spacing conventions, and the end-of-proof symbol consistently.

Within the proof itself, several recurring conventions make arguments easier to follow across the field:

• Explicit proof technique signposting — authors commonly state up front whether a proof proceeds directly, by contradiction, by induction, or by contrapositive, so the reader knows what kind of argument to expect.
• “Without loss of generality” (WLOG) — used to justify considering only one case among several that are equivalent by symmetry, avoiding repetitive argument.
• Case-based structure — complex proofs are often broken into explicitly labeled cases (“Case 1,” “Case 2”) or steps, each closed before moving to the next.
• Cross-referencing prior results by number — a proof will typically invoke “by Lemma 3.2” or “by the theorem above” rather than restating an earlier argument, which is why consistent, stable numbering across the paper matters more in mathematics than in most other empirical writing.

Notation conventions

Because mathematical arguments depend on symbols carrying exact, unambiguous meaning, notation conventions get more explicit attention in mathematics than in most other fields:

• Define before use. Every non-standard symbol or piece of notation is expected to be defined at (or before) its first use, usually within a Definition, Notation section, or inline in the introduction — readers should never have to guess what a symbol means from context alone.
• Avoid overloading. Using the same symbol for two different things within a paper, even in different sections, is a common and disruptive error; conventionally, if a symbol must be reused with a different meaning, the paper says so explicitly.
• Follow field-standard notation where it exists. Many symbols and conventions are effectively fixed by long usage within a subfield (e.g. ℝ for the real numbers, ℝⁿ for n-dimensional real space, ∈ for set membership); departing from field-standard notation without a stated reason creates friction for expert readers and is generally avoided.
• A dedicated “Notation” or “Preliminaries” section is common in longer papers, collecting definitions and conventions that will be used throughout, ahead of the theorem-proof material proper.
• New macros/definitions should aid the reader, not obscure ordinary prose. AMS’s own author guidance specifically cautions against defining shorthand macros for terms that appear in ordinary text (rather than mathematical expressions), partly because it can prevent search/indexing tools from recognizing the underlying terms in an online article (American Mathematical Society, Using the amsthm Package).

LaTeX and AMS-style formatting norms

Mathematics is one of the few fields where the typesetting toolchain and the writing conventions are inseparable — the theorem-proof structure described above is directly encoded in LaTeX through the AMS document classes (amsart, amsbook) and the amsthm package, which essentially every math journal and preprint expects authors to use, either directly or through a journal-specific class file built on top of it. Key mechanics worth knowing:

• Theorem environments are declared, not hand-formatted. A newtheorem declaration ties a name (“Theorem,” “Lemma,” “Definition”) to a numbering scheme; authors then write begin{theorem}...end{theorem} rather than manually bolding a label and adding a number, which keeps numbering and cross-references consistent even as the paper is edited and reordered.
• Three predefined theorem styles. AMS’s documentation defines plain (extra vertical space, italicized body text — the default for theorems, lemmas, propositions, corollaries), definition (extra space but upright, non-italic text — used for definitions and examples), and remark (no extra spacing, upright text — used for remarks and notes), giving readers a consistent visual cue for which kind of statement they’re looking at before they even read it.
• Consistent numbering across statement types. Journals commonly number theorems, lemmas, propositions, and corollaries from a single shared counter within each section (e.g. “Theorem 3.1,” “Lemma 3.2,” “Corollary 3.3” in sequence) rather than numbering each type separately, so that “the result stated just before this one” is unambiguous regardless of its label.
• Equations are numbered and referenced, not just displayed. Displayed equations that will be referred to later carry an equation number (via begin{equation} or the align environment), and later text refers back to “equation (4)” or similar rather than restating it.
• Preprint-first norms. Papers are conventionally posted to arXiv (specifically its math and math-ph sections, which for a newcomer require an arXiv endorsement from an established author before a first submission) before or during journal submission — a discipline norm CASRAI covers in more detail in its guide to preprint servers, and one with direct implications for how mathematics attributes and orders authorship, covered separately in CASRAI’s guide to mathematics’ alphabetical author-order tradition.

Research administrators supporting collaborative math projects — particularly multi-institution or multi-national collaborations working in a shared Overleaf project — may also find CASRAI’s guide to LaTeX and Overleaf workflows for collaborative research writing useful for the practical tooling side of producing a paper with this structure.

Prose style conventions in mathematical writing

Beyond formal structure, mathematical prose has its own register, distinct from both empirical-science and humanities writing:

• Economy over elaboration. Mathematical writing favors precision and brevity; a proof states exactly what’s needed to justify each step and no more. Widely used style guidance in the field (e.g. MIT mathematician Bjorn Poonen’s frequently cited “Practical Suggestions for Mathematical Writing”) emphasizes cutting anything that doesn’t advance the logical argument.
• The mathematical “we.” Even single-authored papers conventionally use “we” (“we show,” “we conclude”) rather than “I,” a long-standing convention across the field rather than a claim about multiple authors.
• Precise logical connectives. Words like “if,” “only if,” “iff” (if and only if), “for all,” and “there exists” are used in their exact logical sense, not loosely as in ordinary prose — a math reader treats “if” and “if and only if” as meaningfully different claims, so authors are expected to use each deliberately.
• Present tense for established results. Once proved, a result is referred to in the present tense (“Theorem 2 shows that…”) regardless of when the proof was written, since a proved mathematical fact doesn’t change with time the way an empirical finding might be revised.
• Abstracts are typically shorter and less structured than in empirical fields — usually a single unstructured paragraph stating the main result and its significance, without the structured Background/Methods/Results/Conclusions subheadings common in biomedical abstracts.

Frequently asked questions

What’s the difference between a lemma and a theorem?

A theorem is a result the author considers a main contribution of the paper; a lemma is a supporting result proved primarily because it’s needed as a step toward proving a theorem. The distinction is about the claim’s role in the paper’s argument, not its mathematical difficulty — some lemmas require far more work to prove than the theorem they support.

Does a mathematics paper need a Methods section?

No. There is no experimental protocol to describe in most pure and applied mathematics papers; the equivalent of “methods” is the sequence of definitions, lemmas, and proof techniques used to establish the results, which appear inline as the paper’s main body rather than in a separate section.

What does the symbol at the end of a proof mean?

It marks the end of the proof, functioning the same way “Q.E.D.” historically did. Modern typeset papers commonly use a small square (□ or ■) instead of writing out “Q.E.D.,” a convention often credited to mathematician Paul Halmos.

Why do math papers use “we” even with one author?

It’s a long-standing stylistic convention in the field, not an indication of multiple authors or an editorial “royal we” borrowed from other disciplines — single-authored math papers routinely use “we show,” “we conclude,” and similar constructions.

Is there a standard structure for a mathematics paper’s introduction?

There’s no single mandated structure, but introductions commonly cover the same ground: the problem’s motivation, an informal statement of the main result, its relationship to prior work, and often a short outline of how the rest of the paper is organized. This is the section written most accessibly for a non-specialist reader, since the formal theorem-proof material that follows assumes closer engagement.

Should notation be defined even if it’s “standard” in the field?

Widely standard notation (like ℝ for the real numbers) is often used without redefinition, but anything non-standard, reused with a different meaning than elsewhere in the field, or specific to the paper’s own constructions should be defined explicitly at first use — the convention is to err toward over-defining rather than assume the reader shares every unstated convention.

This page covers general, field-wide conventions common across mathematics research writing. Individual journals and subfields (e.g. applied vs. pure mathematics, or fields like theoretical computer science that sit adjacent to mathematics) may specify their own house style on top of these norms — always check a target journal’s specific author guidelines before submission.