Tanswell and Inglis (2023) searched arXiv for imperatives inside proofs and concluded "in favour of the recipe model of proofs: that proofs are like recipes, giving instructions for mathematical actions to be carried out". Their thirteen common imperatives run at 9,406 per million words inside proofs and 6,854 outside (Table 3). CITED
The script counts the same thirteen verbs at the start of sentences in this series' prose: 832 pages and 1,392,322 words, with formulas and code removed. COMPUTED
| verb | here | per million | arXiv proofs | arXiv non-proof |
|---|---|---|---|---|
| Let | 68 | 48.8 | 4,523 | 4,035 |
| Note | 65 | 46.7 | 929 | 681 |
| Write | 87 | 62.5 | 117 | 39 |
| Take | 48 | 34.5 | 178 | 49 |
| Define | 20 | 14.4 | 272 | 167 |
| Suppose | 8 | 5.7 | 944 | 512 |
| Assume | 5 | 3.6 | 556 | 339 |
| Denote | 0 | 0.0 | 218 | 145 |
The total here is 247 per million, about 38 times fewer than arXiv proofs. The most frequent imperative is not "Let" but "Write", which comes from the courses' instructions to students. This prose reports, argues and corrects. It does not tell the reader what to do. COMPUTED
Arambillete and de Groote (2025) open with a sophism: "Every set of prime numbers is a fortiori a set of coprime numbers. Therefore, every prime number is a coprime number." The trouble is that "coprime is a collective predicate". It is said of a set, not of one number. CITED
Collective.lean makes this exact. The premise is true: any set of primes is pairwise coprime (primes_pairwise_coprime). The conclusion is not false but empty: read about one number, "coprime" is true of every number, because a one-element set is vacuously pairwise anything. So 4 is "a coprime number", and 4 is not prime (four_is_coprime_distributively). KERNEL-CHECKED
This series' title word has the same grammar. "u, v, w are orthogonal" means every pair, and cannot be read along a chain: (1,0) ⊥ (0,1) and (0,1) ⊥ (2,0), but (1,0)·(2,0) = 2 (orthogonal_not_transitive). This is the non-transitivity Chapter 3 found in co-predication, now inside one word. KERNEL-CHECKED
In the pages, the collective form "are orthogonal" appears 11 times. The explicit forms appear less often: "mutually orthogonal" 6 times and "pairwise orthogonal" never. The singular "is orthogonal" appears 35 times. The series mostly writes the ambiguous collective form and leaves the pairwise reading to the reader. COMPUTED
Balkır (2014) models word meaning with density matrices from quantum mechanics: "words are represented by mixed states and each eigenstate represents a sense of the word". An unambiguous word is a pure state, and ambiguity is measured by entropy. CITED
Applied to Chapter 4's operator keys, and reading each gloss as a separate pure state, the two ways of counting disagree: COMPUTED
| key | glosses (pages) | entropy over glosses | entropy over senses |
|---|---|---|---|
| C | 63, 17 | 0.746 bits | 0 |
| K | 51, 26, 12 | 1.369 bits | 0 |
| F | 86 | 0 | 0 |
| U | 68, 3, 10, 3 | 0.956 bits | 0.742 bits |
Over glosses, K is the most ambiguous key: threshold, curvature and gate. But the pages join those three into one sense, so at the level of senses K carries no ambiguity at all. Only U keeps entropy once senses are counted, because union and unification are not joined to unfolding. That is the linguist's distinction between polysemy, one meaning with shades (K), and homonymy, separate meanings under one form (U), measured in bits. The quantum reading is ours. MODEL
Two further sources place this in practice. Orth (2013) argues that "much like learning a foreign language, learning mathematics has been based mostly on oral tradition", and that literacy comes second but cannot be skipped. Corneli, Martin, Pease and colleagues (2017) model mathematicians' dialogues, the arguments people actually have, as "a higher-order nested semantic network". That is the representation Chapter 7 found in the Sanskrit grammarians. CITED