The Evolution of Group Theory, Israel Kleiner (Al)
Bibliographic record
Abstract
Editors' Note: Israel Kleiner received his PhD in ring theory under the direction of J. Lambek at McGill University and has been Professor of Mathematics at York University in Ontario since 1965. He became interested in the history of mathematics and has written a long series of articles for MAA and other journals on historical topics. For the one included here he was awarded the Allendoerfer Award in 1987, and then he went on to win another Allendoerfer Award for “Rigor and proof in mathematics: an historical perspective” in 1992; a Lester R. Ford Award (with N. Movshovitz-Hadar) in 1995 for “The role of paradoxes in the evolution of mathematics,” which appeared in the Monthly ; and a Polya Award in 1990 for “Evolution of the function concept: a brief survey” in the College Mathematics Journal . This article gives a brief sketch of the evolution of group theory. It derives from a firm conviction that the history of mathematics can be a useful and important integrating component in the teaching of mathematics. This is not the place to elaborate on the role of history in teaching, other than perhaps to give one relevant quotation: Although the study of the history of mathematics has an intrinsic appeal of its own, its chief raison d'etre is surely the illumination of mathematics itself. For example the gradual unfolding of the integral concept from the volume computations of Archimedes to the intuitive integrals of Newton and Leibniz and finally to the definitions of Cauchy, Riemann and Lebesgue — cannot fail to promote a more mature appreciation of modern theories of integration. — C. H. Edwards [11]
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 0.003 |
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".