The Role of Dice in the Emergence of the Probability Calculus
Bibliographic record
Abstract
Summary The early development of the probability calculus was clearly influenced by the roll of dice. However, while dice have been cast since time immemorial, documented calculations on the frequency of various dice throws date back only to the mid‐13th century. This paper examines the conceptual changes that took place between the ancient and medieval worlds regarding dice rolls. Archaeological data and written source material are used to explore various notions around tesseræ, or six‐sided dice, and tali, which are four‐sided dice made from the anklebones of goat or sheep. Given this evidence, it appears that dice rolls provided little or no incentive in Roman society to carry out numerical probability calculations. The first documented frequency computations on dice appeared circa 1260 in the manuscript De vetula , likely written by the English polymath Roger Bacon. In the same period, King Alfonso X of Castile, León and Galicia commissioned a book on games, Libro de los juegos . It is argued that the conceptual changes present in these publications reflect the emerging empiricism, related to dice, and mathematical interests of Bacon and his Islamic predecessors.
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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 | 0.027 | 0.056 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.005 | 0.004 |
| Science and technology studies | 0.002 | 0.025 |
| Scholarly communication | 0.007 | 0.009 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.008 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".