One text is an essay that argues by story. The other is a mathematics paper that argues by numbers anyone can recompute. Read both with the same six boxes, then write the disclosure sentence that each author wrote about the tools used. Written 2026-11-09.
Chapter 6 teaches the Toulmin model: claim, grounds, warrant, backing, qualifier, rebuttal. This page applies it to two real texts published within a week of each other in late 2026. Read each one first, at its own address. This page does not reproduce either text; it gives a reader’s reconstruction, which you should check against the originals and correct where you disagree.
You do not need to follow the mathematics of the second text. You need only its introduction (pages 1 to 4) and section 1.5 on tools.
The table is one reader’s reconstruction. The essay does not label its own parts.
| Box | What the essay offers |
|---|---|
| Claim | AI is changing mathematics, the effect on researchers is bad now, and what happens next depends on mathematicians themselves. The last sentence says so: “It’s up to us!” |
| Grounds | One constructed scenario (a student whose open problem is solved overnight by an unreleased, unverified model), a New Yorker article cited for AI’s advances in mathematics, one mathematician’s reaction, and one blog quotation. No result, count or date of a solved problem is named in the essay. |
| Warrant | As knowledge grows, each new contribution adds proportionally less and newcomers need longer to reach the frontier. If AI removes that delay, it changes who can contribute. This link is implied, not stated. |
| Backing | An adapted diagram of a PhD pushing the edge of knowledge, credited to Matt Might’s Illustrated Guide to a Ph.D., and a quotation from the blog Noahpinion. |
| Qualifier | The author limits her own hope in the first person: she says she is optimistic only for today, uses “maybe” twice in a row, and says she does not know what the future is. |
| Rebuttal | The mathematician’s remark that talking to the AI was no fun, and the author’s own remark that reading AI output is painful and not always correct. Both are the essay’s own counter-evidence to its hopeful reading. |
| Box | What the paper offers (introduction only) |
|---|---|
| Claim | A Sidon set with k elements has diameter at least k² − 1.89 k3/2; for k at least 483 the constant improves to 1.88607. Equivalently, a Sidon set inside [1, N] has at most N1/2 + 0.945 N1/4 + 0.447 elements. |
| Grounds | An explicit “WT-triple” with six random components, (υ, α, β) ≈ (10, 0.944814, 0.941259), whose quantity 2√(αβ) is 1.88607 (appendix A.3). |
| Warrant | The paper’s theorems turn any valid WT-triple into a diameter bound; the sections after the introduction prove that step. |
| Backing | Earlier bounds it improves on: Erdős and Turán (1941), Singer (1938), and an earlier non-explicit constant of 1.98183. |
| Qualifier | “k at least 483,” “explicit,” and a conjecture that the method cannot go much below 1.886. The authors say further improvement “must probably” come from another method. |
| Rebuttal | Stated openly: the exact values are known only for k up to 28, and the real optimum may be much smaller than the bound proved. |
This page does not check the proofs, and its author has not read the sections after the introduction closely enough to vouch for them. What the page does check is that the numbers quoted above match the paper (see the verify script).
| Question | The essay | The paper |
|---|---|---|
| What would show the claim wrong? | Hard to say; the claim is about a community’s future. | A smaller diameter for some Sidon set, or an arithmetic error in the triple. |
| Can a reader recompute it? | No, and there is nothing to recompute. | Yes: the exact rational values and a verification script are public, according to the paper. |
| Which words carry the uncertainty? | maybe, arguably, I can imagine, it’s not clear that | at least, for k ≥ 483, we suspect, there is evidence |
| Does it say what the tools did? | Yes: the images, not the text. | Yes: in detail (Part 5). |
Task (150 words). Choose one row. Explain, using the word warrant, why the essay can be convincing without the kind of evidence the paper gives.
A Sidon set is a set of whole numbers in which every difference between two different members is different from every other such difference. A Golomb ruler is the same thing drawn as a ruler with marks: no two pairs of marks are the same distance apart. The paper’s first page gives the definition (and calls it a 1-Golomb ruler); the picture here is ours.
Worked example. The five marks {0, 1, 4, 9, 11} have ten pairs. Their differences are 1, 4, 9, 11, 3, 8, 10, 5, 7, 2: ten different numbers, all inside 1 to 11, with 6 missing. So it is a Sidon set with k = 5 elements and diameter 11 (largest mark minus smallest). By contrast {0, 1, 2, 3} fails, because 1 − 0 = 2 − 1.
The question the paper studies. For k marks, how short can the ruler be? The paper writes the shortest diameter as k² − bk k3/2 and asks how large bk can be. For our five marks, b5 = (25 − 11)/53/2 ≈ 1.252. The paper reports that b28 = 199/(56√7) ≈ 1.343; since 283/2 = 56√7, that gives diameter 28² − 199 = 585.
| k | Shortest known diameter | Paper’s Corollary 1.1 lower bound | Simplified k² − 1.89 k3/2 |
|---|---|---|---|
| 5 | 11 (the example above) | 6.0 | 3.9 |
| 28 | 585 | 524.2 | 504.0 |
Notice that a lower bound can be true and still loose. The paper’s contribution is a better constant, not the exact answer; for k at least 483 the constant is 1.88607.
Tasks. (1) Check by hand that {0, 1, 4, 9, 11} has no repeated difference. (2) Try to add a sixth mark between 12 and 16 without repeating a difference; say what happens. (3) Explain in two sentences why a ruler with k marks has k(k−1)/2 differences and why that forces the diameter to grow roughly like k².
Both texts say how AI tools were used. They differ in how much.
Many research calls now ask for such a statement. Task (80 words, four sentences). For a paper of your own, write: (1) which tool, (2) for which task, (3) who checked the output and how, (4) what remained entirely yours. If you cannot write sentence 3, say so; that gap is information.
Task. Find three hedges in each text. For each, say what the writer is protecting themselves from.