Bibliographic record
Abstract
Abstract Requiring a blend of logic and spatial imagination, Alcuin’s six problems in geometry—numbers 21, 22, 23, 24, 25, and 26 in the Propositiones—put on display what mathematical thinking involves in its essence. This chapter looks at these problems in the light of the kinds of notions and methods they entail, and the type of thinking processes required to solve them. It also discusses the importance of notions such as the Pythagorean theorem, figurate numbers, and triangular numbers. It looks as well at the nature of proof, illustrated by a well-known early proof for π in the Egyptian Rhind Papyrus, and at the implications of Fermat’s Last Theorem. The geometrical problems in Alcuin’s text would have been truly challenging for his medieval readers, some of which require “outside-the-box” thinking, as it is called today. In Alcuin’s era, geometry was seen to constitute the starting point for a theoretical study of mathematics, since it involved mastering the methods of proof established by Euclid and other ancient mathematicians.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.048 | 0.012 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".