Structuring Mathematical Writing

Setup, claim, example, and justification — and inspecting for logic, not just formatting

A short conceptual reading to accompany the Week 3 work on organizing mathematical writing. The hands-on walkthrough lives in Writing a Claim, Example, and Justification in Quarto.

Week 2 taught you to write mathematical notation from source: inline math, displayed equations, aligned blocks, and the habit of rendering and looking at the PDF. Week 3 takes the next step. The skill this week is not more symbols. It is organizing the mathematics you can already typeset so that a reader can follow it.

By the end of the week, a reader of your writeup should be able to point at the page and say: this is what is being assumed, this is what is being claimed, here is an example of it, and here is the reasoning that supports the claim. That is what it means to write mathematics, as opposed to merely typesetting it.

Mathematical writing has structure, not just equations

A page full of correct equations is not automatically a piece of mathematical writing. Equations are the vocabulary; structure is the grammar. Week 2 gave you the vocabulary. Week 3 is about the grammar: the arrangement of definitions, claims, examples, and arguments that lets a reader reconstruct your thinking.

Think of the difference between a list of true sentences and a paragraph that makes a point. Both can be “correct.” Only one of them communicates. Mathematical writing communicates when its parts are distinguishable and ordered: the reader always knows which kind of sentence they are reading and why it is there.

Four kinds of mathematical sentence

Most short mathematical writeups are built from four kinds of sentence, each doing a different job.

Definition / setup. Fixes meaning. It names the objects you are working with and states the assumptions in force. “Let \(m\) and \(n\) be even integers.” Every symbol you use later should be introduced here first.
Claim (or proposition). Asserts that something is true, stated precisely enough that it could in principle be argued or refuted. “The sum \(m + n\) is even.” A claim is a promise to the reader that you will justify it.
Example. Exhibits a concrete case. “If \(m = 4\) and \(n = 6\), then \(m + n = 10\), which is even.” An example illustrates a claim and builds intuition. It does not prove the claim — one case is not all cases — but it shows the reader what the claim is about.
Justification (or proof). Argues why the claim holds, in general, from the setup. This is the part that does the intellectual work, and the part most students under-write.

A reader who can tell these four apart can follow your writeup. A reader who cannot is lost, no matter how clean your equations are.

What a claim, proposition, or theorem is — lightly

You will hear several words for “a mathematical statement asserted to be true.” They differ mainly in importance, not in kind:

a theorem is a major result;
a proposition or lemma is a smaller supporting result;
a claim is a local assertion you make and then back up.

For this course you do not need to agonize over which word to use. “Claim” or “proposition” is perfectly good for a short writeup. What matters is that the statement is precise: a reader should be able to tell exactly what you are asserting, with no “it depends” left unstated.

What a justification is doing — rhetorically

A justification — or a proof — is an argument aimed at a reader. It is not a pile of symbols and it is not a re-statement of the claim. It starts from what you assumed in the setup, proceeds by steps the reader can follow, and ends by establishing the claim.

The shape is almost always the same:

Since [what we assumed], we have [a consequence]. Therefore [the next consequence]. Hence [the claim].

The connective words — since, we have, therefore, hence — are not filler. They are the logic. They tell the reader that one statement follows from another. A justification that drops them becomes a list of equations with no stated relationship between them, and the reader has to guess at the reasoning. Do not make the reader guess.

A worked shape, for “the sum of two even integers is even”:

Definition. An integer is even if it can be written as \(2k\) for some integer \(k\).

Claim. If \(m\) and \(n\) are even integers, then \(m + n\) is even.

Justification. Since \(m\) and \(n\) are even, we can write \(m = 2a\) and \(n = 2b\) for integers \(a\) and \(b\). Then

\[ m + n = 2a + 2b = 2(a + b). \]

Because \(a + b\) is an integer, \(m + n\) has the form \(2 \times (\text{an integer})\), which is the definition of even. Therefore \(m + n\) is even.

Notice what the displayed equation is doing: it is a step of the argument, not decoration. The prose before it sets it up (“we can write…”), and the prose after it interprets it (“because \(a + b\) is an integer…”). The equation and the prose work together.

Prose matters as much as the equations

In Week 2 the equations were the point. In Week 3 the prose carries the logic, and the equations carry the algebra. Neither alone is a justification.

A useful test: read your writeup with the displayed equations covered up. Can you still follow the argument from the prose alone? If the prose by itself is a sequence of “since… therefore…” steps that reach the claim, you have written a justification. If covering the equations leaves a few disconnected sentences, your prose is doing too little.

Then read it the other way: with the prose covered, is there a displayed equation that does real work — a step the argument actually needs? If the only equations are restatements of the claim, your math is decorative.

How to mark structure: headings and bold labels

You do not need theorem environments, amsthm, or raw .tex to mark structure. The required mechanism for this course is the simplest robust one: ordinary headings plus bold labels. Begin the relevant paragraph with a bold lead-in:

**Definition.** An integer is *even* if it equals $2k$ for some
integer $k$.

**Claim.** If $m$ and $n$ are even, then $m + n$ is even.

**Example.** With $m = 4$ and $n = 6$, $m + n = 10$ is even.

**Justification.** Since $m$ and $n$ are even, write $m = 2a$ and
$n = 2b$ ...

When this renders, each labeled paragraph stands out, and a reader skimming the PDF can find the parts instantly. You can also use section headings (##) to separate a “Setup,” a “Claim and example,” and a “Justification” section — whatever makes the structure most visible. The goal is simply that the parts are distinguishable on the page.

How to avoid “equation dump” writing

The most common Week 3 failure is the equation dump: a column of displayed equations with no prose saying what is being claimed or why each step follows. It is the Week 3 version of Week 2’s “math that renders but is wrong” — it compiles, it looks mathematical, and it communicates nothing.

The cure is structural. Before you write a single equation, write the claim in words. Then, for each displayed equation, make sure there is prose before it that sets it up and prose after it that says what it gives you. If a displayed equation has no sentence explaining why it is there, either add one or delete the equation.

A short, well-explained argument beats a long wall of symbols. One page of reasoning a reader can follow is the target — not length.

Rendering now includes inspecting the logic

Week 2 built the habit of rendering and looking at the PDF to check the formatting. Week 3 extends that habit to a second layer. When you open your rendered PDF, inspect it twice:

Formatting (the Week 2 check). Are the symbols, fractions, subscripts, and aligned = signs the way you intended?
Structure and logic (new in Week 3).
- Can a reader find the definition, the claim, the example, and the justification? Are the parts visually distinct?
- Does the justification argue the claim you actually stated — not a different or weaker one?
- Does every displayed equation support a step of the reasoning?
- Is every symbol you use introduced in the setup before it appears?

A render that succeeds is not an argument that works. The only way to catch a justification that proves the wrong thing is to read the rendered PDF as a skeptical reader would and follow the logic.

AI can help with structure — but cannot verify your reasoning

You may use AI assistants for Week 3 work, as for any technical writing in this course. AI is genuinely useful for:

suggesting an outline (“how should I structure an argument that the product of two odd numbers is odd?”),
explaining how to mark structure clearly in a document,
revising the connective prose so the “since… therefore…” steps read cleanly,
diagnosing a render error,
proposing an example to illustrate a claim.

What AI is not is a verifier of reasoning. AI assistants are fluent at producing proof-shaped text — paragraphs that look like a proof, read smoothly, and are sometimes wrong: they prove a different statement, skip the load-bearing step, or assert “clearly” where the actual work should be.

The course rule from Weeks 1–2 applies with extra force here: AI is a first pass, not a verdict. Any AI-suggested reasoning must be read step by step, checked against the claim you stated, and rewritten in your own words. If you cannot explain a step, you have not verified it — and it does not belong in your writeup.

The three-line AI Use Note (Tool / Purpose / Verification) applies to Week 3 work. For Week 3 the Verification line should describe how you confirmed the reasoning is correct, for example:

Verification: rewrote the justification in my own words and checked that each step followed from the definition; confirmed the final sentence restates the claim I made.

See the AI use guidelines for the full course position.

Optional level-up: Quarto’s theorem blocks

Optional — not required for Week 3

If you want auto-numbered, formally styled statement blocks, Quarto has a native theorem/proof feature using fenced divs, e.g.

::: {#thm-evens}
## Sum of two even integers
The sum of two even integers is even.
:::

::: {.proof}
Let $m = 2a$ and $n = 2b$ ...
:::

These render to numbered “Theorem 1,” “Definition 1,” “Proof” blocks. They are a nice touch, but they are entirely optional for this course and not required. The required, fully supported mechanism for Week 3 is headings plus bold labels, which is simpler and never fights the PDF renderer. Reach for theorem blocks only if you are comfortable and want the extra polish — and verify they render the way you expect, the same as any other markup.

What you’ll do this week

The Week 3 work, in one paragraph:

You will create a Week 3 portfolio subfolder, start a new .qmd source file in VS Code, and write a short structured mathematical writeup on a familiar statement — a setup, a clearly marked claim, an example, and a justification, with at least one displayed equation doing real work in the reasoning. You will render it to PDF and inspect it on both levels: formatting, and the logic and structure. The Writing a Claim, Example, and Justification in Quarto walkthrough takes you through it step by step. The exact assignment prompt and submission details live in the Assignments/LMS space.

A light look ahead: Week 4 includes a required LaTeX checkpoint conference, where you will discuss the scope of your LaTeX Project. The structured-writing habit you build this week is exactly what that project will rely on.