D. Rosenthal et al.: A Readable Introduction to Real Mathematics
Why this work is in the frame
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Bibliographic record
Abstract
This book comes with a public health warning: very hard to put down.Contrary to the impression given by its possibly over-loud title -probably deriving from the excitement of the bridging course in Toronto -this book is basically a new and refreshing introduction to number theory.In outline, in twelve chapters, it progresses from the natural numbers and induction through modular arithmetic, the "fundamental theorem of arithmetic" and the Euclidean algorithm to the RSA method of public key encryption.After a look at complex numbers and the cardinality of infinite sets, it goes on to discuss "plane geometry", ruler-and-compass constructibility, and surds.But the heart of it is the number theory: prime factorization, the φ function, the Euclidean algorithm, Fermat's and Wilson's theorem, here put to work in the service of RSA and public key encryption.This reviewer has always been a little afraid of number theory: all that stuff about phi functions and prime number density has seemed arcane and irrelevant, very far away from "real mathematics".Now. after immersing himself in this little book, number theory begins to make sense."Uncle Petros", fronting for Apostolos Doxiadis [1], tells his nephew that "addition is natural, but multiplication is artificial".To see what he is getting at just, on day two of your introductory course, write out the first few natural numbers in factorized form: 1, 2, 3, 2 2 , 5, 2 • 3, 7, 2 3 , 3 2 , 2 • 5, 11, 2 2 3, 13, 2 • 7, 3 • 5, 2 4 , 17, 2 • 3 2 , 19 : is there any discernible pattern?Number theory is born out of the attempt to answer that question.
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.
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.004 |
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 it