Theorems in pure mathematics can be proved right but the models used in applied mathematics, natural and social science, as well as in engineering, can at most be “not yet proved wrong”
Bibliographic record
Abstract
Canadian Journal of Earth Sciences might seem a strange place to publish this commentary, of which the title expresses a familiar widely accepted idea in epistemology—the theory of knowledge. The reason I write here is that I have lately heard talks and read papers by Earth scientists, who while working in science, gave me the impression that they think of their work as having a property characteristic only of pure mathematics. They claim that they have proved their results right. Reviewers, program managers, and editors of published work containing those claims appear to adopt a similar position. I offer this commentary for publication here because I think that it can be useful for both the Earth scientists who think that they have proved their results right and for the rest of us if we can all recognize and make use of the distinction of the title. Pure mathematics has been made entirely by humans. In contrast applied mathematics, engineering, and the sciences, all of which represent a second category of human activity, involve the properties of the universe which humans have encountered since we first evolved but which, because we did not make the universe, we can never fully understand. Paradoxically we cannot get very far in interpreting the universe without applying results and formulations from pure mathematics. The most significant difference between pure mathematics and models constructed in the second category is that while theorems in pure mathematics can be proved right, models in the second category can only either be proved wrong or not yet proved wrong. Our models are falsifiable. Those of us working in the second category have the task of building models and of testing those models. We know that all our models will eventually, because of new understanding, be shown to be wrong. The …
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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".