Tres "extractos" en torno al tema de los juegos eruditos

1. De la contraportada de Los orígenes trágicos de la erudición (Fondo de Cultura Económica, 1998, 178 p.) de A. Grafton:

"Usadas profusamente por Kant, rechazadas por Hegel, consideradas ya como una forma excelsa del arte literario, ya como enojosas interrupciones de la lectura, las notas al pie, que a menudo se emplean para polemizar con los colegas, han sido ellas mismas objeto de polémica.

A través de un metódico y ameno rastreo de puntos de vista predominantes en diversas épocas, Anthony Grafton expone las diversas funciones que las notas al pie desempeñaron a lo largo de los siglos: dar legitimidad al texto, evadir la censura, refutar a otros estudiosos, elaborar un cuerpo de datos al que otros investigadores podrán recurrir con provecho.

Aunque aparecen como el sustento empírico de los sucesos relatados y los argumentos expuestos, sin el cual no puede verificarse ni refutarse una tesis histórica, las notas al pie no garantizan nada. Pueden usarse para negar los mismos hechos que otros tratan de confirmar apoyándose en las mismas notas; pueden emplearse para acumular citas y referencias carentes de interés o para atacar nuevas posturas. Sólida apoyatura o recurso ampuloso, la nota al pie constituye una parte crucial e insoslayable de esa mezcla de arte y ciencia que es la historia moderna."

2. Augusto Monterroso en el capítulo "Los juegos eruditos" de su libro La palabra mágica (Biblioteca Era, 1991 [2a. reimpresión], 120 p.):

"Como se sabe, en el Quijote hay errores de bulto claramente debidos al autor, y muchos que son simples erratas o minucias insignificantes que los correctores de pruebas dejaron pasar para la mayor gloria de don Diego Clemencín y otros comentaristas, de Francisco Rodríguez para acá, que han convertido la lectura de sus notas al pie de página en una delicia sólo paralela a la que produce la lectura del texto. Me adelanto a la suposición de que esto es una ironía. En realidad, con un poco que a uno le guste la literatura, uno puede pasarse noches enteras leyendo las objeciones que Clemencín ponía al texto de Cervantes y las defensas de Cervantes a cargo de Rodríguez Marín, no menos enloquecido por un ideal de justicia que el propio Alonso Quijano. Pero hay en el capítulo VI del Quijote un galimatías relacionado con galeras que nadie ha logrado desentrañar. Está en el párrafo que dice: '... con todo, os digo que merecía el que lo compuso, pues no hizo tantas necedades de industria, que lo echaran a galeras por todos los días de su vida'. Se refiere al autor de Tirante el Blanco, y esto, contra lo que parece, estaría dicho en su defensa, si uno tomara galeras por galeras de imprenta. Y sin embargo, lo mejor es leer el párrafo sin preocuparse y seguir adelante: es bien sabido a lo que conducen esas intrincadas razones."

3. Fragmento del diálogo entre Ronit Krushka (el personaje de Rachel Weisz) y Esti Kuperman (el personaje de Rachel McAdams) en la escena de Disobedience en la que Ronit y Rachel llegan a la casa del rabino Shlomo Krushka después de que Ronit se enteró de que su padre no la mencionó en su testamento:

Ronit: Lo que hacía todo el día era estar aquí y leer la Torá... Y los comentarios de la Torá. Y las notas a los comentarios y los debates sobre las notas.

Esti: Lo que significaba que podíamos hacer lo que queríamos.

Un ejercicio de archivística

(13.08.1998) Versión estenográfica del diálogo que el Presidente Ernesto Zedillo tuvo en la residencia oficial de Los Pinos, con participantes en la Olimpiada Internacional de Matemáticas, que se celebró del 13 al 21 de julio último, en Taipei, Taiwán.

- Estudiante Omar Antolín Camarena: Del 13 al 21 de julio fue la Olimpiada Internacional de Matemáticas en Taiwán.

- Presidente E. Z.: Muy bien.

- Omar: A la Olimpiada Internacional van seis alumnos de cada país. Bueno, invitan a seis alumnos de cada país y de algunos van menos. Esta vez a la delegación mexicana no nos fue tan bien como el año anterior pero yo regresé con medalla de plata y un compañero trajo una mención honorífica que la otorgan cuando uno tiene la puntuación más alta en un problema pero no alcanza medalla.

- Presidente E. Z.: Muy bien.

- Omar: El año anterior fue en Mar del Plata, Argentina, la Olimpiada Internacional a la que también fui. He ido a las últimas tres. Allá nos fue bastante mejor. Participaron 82 países y México quedó en el lugar 32. Empatamos con Francia, por ejemplo, que tienen tradición en matemáticas. Entonces, este año hubo una medalla de plata, de un compañero de Guadalajara, y otros tres obtuvimos bronce.

- Presidente E. Z.: Muy bien. ¿En dónde fue este año?

- Omar: En Taipei, Taiwán.

- Presidente E. Z.: Y tú, ¿en qué nivel vas de estudios?

- Omar: Acabo de entrar a la universidad: hoy es mi cuarto día de clases.

- Presidente E. Z.: ¿Ah, sí?

- Omar: Sí, señor Presidente.

- Presidente E. Z.: ¿En qué universidad estás?

- Omar: Estoy en la UNAM, en la Facultad de Ciencias, estudiando matemáticas.

- Presidente E. Z.: Matemáticas, para variar.

- Omar: Sí.

- Presidente E. Z.: Entonces, este torneo era, digamos, a nivel de preparatoria.

- Omar: Sí. No hay requisitos sino, más bien, uno no puede estar inscrito en la universidad y no tener más de 20 años. Usualmente la restricción fuerte es no estar todavía en la universidad.

---oOOOo---

NOTA. Hasta hace algunos años el diálogo se podía encontrar en zedillo.presidencia.gob.mx/pages/disc; al parecer ese sitio ya fue eliminado o algo por el estilo. Opté por pasar el diálogo al blog antes de perder la hoja en la cual lo tengo impreso. Estoy cayendo en la cuenta de que me animé a hacerlo justo en la semana que se cumplieron 25 años de ese memorable encuentro de Omar con el Presidente Zedillo.

Un producto infinito para la constante de Ludolph van Ceulen

Consideremos el producto infinito $$ \left(1- \frac{1}{3}+\frac{1}{3^{2}}-\cdots\right) \left(1+\frac{1}{5}+\frac{1}{5^{2}}+\cdots\right) \left(1-\frac{1}{7}+\frac{1}{7^{2}}-\cdots\right)\cdots$$ Dentro de un par de paréntesis hay alternancia de signos si y sólo el número primo que lidera a los paréntesis respectivos es congruente con $-1$ módulo $4$. Al considerar el producto de los primeros $n$ factores de ese producto infinito, los términos que se obtienen son de la forma $$ \pm \frac{1}{p_{1}^{e_{1}} \cdots p_{n}^{e_{n}}} $$ donde $p_{1}, \ldots, p_{n}$ son los primeros $n$ números primos impares y $e_{1}, \ldots, e_{n}$ son números enteros no negativos. El signo de un término dado es negativo si y sólo si la suma de los exponentes de los primos congruentes con $-1$ módulo $4$ que aparecen en la expresión $p_{1}^{e_{1}} \cdots p_{n}^{e_{n}}$ es un número impar. Puesto que todo número entero positivo $N>1$ se puede expresar como un producto de potencias de números primos de manera única (salvo el orden de las potencias), al desarrollar el producto infinito aparecerán como denominadores todos los números impares exactamente una vez. Tenemos así que \begin{eqnarray*}\left(1- \frac{1}{3}+\cdots\right) \left(1+\frac{1}{5}+\cdots\right)\left(1-\frac{1}{7}+\cdots\right) \cdots &=& 1- \frac{1}{3}+\\ && \frac{1}{5}-\frac{1}{7}+\cdots \end{eqnarray*} Luego, en vista de que $$ 1-\frac{1}{p}+\frac{1}{p^{2}} - \cdots = \frac{p}{p+1},$$ $$ 1+\frac{1}{p}+\frac{1}{p^{2}} + \cdots = \frac{p}{p-1}$$ y $$ 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\cdots = \frac{\pi}{4},$$ se concluye que \begin{eqnarray*}\frac{\pi}{4} = \frac{3}{4} \cdot \frac{5}{4} \cdot \frac{7}{8} \cdot \frac{11}{12} \cdot \frac{13}{12} \cdots \end{eqnarray*} La identidad es fácil de recordar pues, en la derecha, los numeradores son todos los números primos impares listados en su orden natural mientras que el denominador del primo $p$ es igual a $p+1$ cuando $p \equiv -1 \pmod{4}$ e igual a $p-1$ cuando $p \equiv 1 \pmod{4}$.

NOTA. La primera vez que vi este producto infinito para $\pi$ fue en las páginas del libro Alberto Barajas: su oratoria, sus matemáticas y sus enseñanzas (SMM & IMATE UNAM, 2010); no obstante, recuerdo haberlo encontrado después en algún otro texto de teoría de números. Espero volver a dar con esa obra más adelante; por ahora sólo me queda reiterarles mis mejores deseos (atrasados) por el Día de Pi 2023... ¡Hasta la próxima!

Some excerpts from V. I. Arnold's "Yesterday and long ago" on the origins of mathematics

«Nowadays we tend to underestimate the knowledge of the ancients, especially of scientists before the ancient Greeks. For more than a hundred years historians have known about the facts I'll discuss below, but about which mathematicians have never heard.

Thousands of years ago (before Moses), a remarkable mathematician who made a lot of discoveries lived in Egypt. He was a land surveyor (he measured land--from this follows the word "geometry"). He is known as Thot, the name which he got after his death (Thot is the name of the God who carried the souls of the dead in a boat across the Lethe river in ancient Egypt).

His first discovery was the natural series: Thot understood that there is no maximal integer (before him the numbers were bounded by the tax payed to Pharaoh). He learned how to carry out proofs based on the existence of actual infinity. His second discovery is not a mathematical one: it was the first phonetic alphabet. Before there were only hieroglyphics in Egypt, and he decreased the number of symbols to several dozen, having realized that, for example, the sound d could be represented by a simplification of the hieroglyph which mean "dog".

In Plato's "Phaedrus", Ammon (the main Egyptian God) discusses with Thot the creation of this alphabet. Thot says that the ability to write down information makes people cleverer, because there is no need to remember everything. Ammon objects that it is "the other way around, they will be more stupid because by relying on their notes they would lose the habit of remembering". They did not discuss computerization yet.

The Jewish and Phoenician alphabets originate from Thot's alphabet. From the Phoenician alphabet comes the Greek one; and later from that the Latin alphabet and only then our Cyrillic version.

The next discovery of Thot is geometry. To measure plots of land in order to estimate an expected crop, the tax it would yield, and how much water for irrigation was required from the Nile, Thot invented axioms, definitions, theorems, and drawings. The only thing he did not care about was the independence of his axioms and, as a result, he did not reach the modern level. For example, instead of one axiom on parallels, he introduced four or five axioms (each of them actually yield the others). But Thot did not prove this, he just used all of them. The honor to choose one (the fifth postulate) and to convert the others into theorems belongs to Euclid.

The measurements of the Earth's radius is among the remarkable geometrical achievements of that time (which belongs either to Thot himself or to his students). Camel caravans walked from Thebes to Memphis along a meridian, and it was not difficult to count their steps (and hence the total distance). They also measured the difference in altitude of the Sun at noon on the same day at these two Egyptian cities. With this information it is easy to calculate the radius of the Earth. It is remarkable that the relative error in this result had been only 1% (as compared with the modern value).

Greek scientists did not trust the Egyptian data claiming that the Egyptian women were publicly prostituting with crocodiles (as it [has also been] mentioned in the book "De la célébration du dimanche" of Proudhon, 1870, Paris). And thus two hundred years later the Greeks decided to measure the radius again. A ship sailed north from the mouth of the Nile to the island Rhodes. To calculate the distance they multiplied "the speed of the ship driven by a wind of an average force" with the time taken by this voyage. They obtained a radius which was two times larger than the correct one (it is easier to count camel's steps than to estimate whether the force of wind is average).

It is interesting that many centuries later a captain from Genoa came to a Catholic Queen asking for permission to sail to India by a western route (instead of the eastern one followed later by Vasco da Gama). The Queen appointed a committee of experts who said that "for such a long distance it is not possible to build a ship capable of carrying sufficient water for survival". Thus the Queen refused permission for this expedition.

But the captain persisted, and after many discussions with experts he got permission to die of thirst. (It is said that the reason for their incorrect conclusion was that the experts trusted the Greek estimate of the Earth's radius, while the captain believed in the Egyptian one.) That is how, just by chance, America was discovered.

Thot founded celestial mechanics and astrology in Egypt. If not he himself, then his ancient followers, knew the law of inverse squares (for a planets' attraction to the Sun) and Kepler's laws of planetary motion. This knowledge disappeared in the destruction of Alexandria's library by fire, where all scientific records of ancient Egypt were kept. Newton wrote that he only restored this ancient knowledge (cf. "Unknown Newton". St. Petersburg, Aleteia, 1999, pp. 731-757)...

Pythagoras was actually one of the first industrial spies in the world. He lived in Egypt for about twenty years, where Egyptian priests taught him their science. He had to take an oath never to reveal this knowledge, and that is why he did not publish anything.

Returning to Greece he told his students about geometry, and they brought this to Euclid who had not taken an oath. So it was that he published the geometry of Thot. Besides from geometry, Pythagoras brought from Egypt the idea of reincarnation (independent of the Indian version), vegetarianism based on it, and the basics of musical harmony for stringed instruments (including the formula for the tension of strings with various lengths, which have the same frequency--the required tension is proportional to the square of the length; the conditions for an octave, a third, a fifth, ...--actually, the Fourier series)...

There were other "spies" who, like Pythagoras, brought Egyptian secrets to Europe: Plato (logic and philosophy), Eudoxus (number theory [up to] Euclid's algorithm and the theory of irrational numbers including the theory of Dedekind's [cuts] and Grothendieck's rings).

The theory of irrational numbers started from the discovery of incommensurability of the diagonal of a square with its side (that is, the irrationality of √2) which was kept as a secret in the Pythagorean school. The point is that this fact undermined the importance of the arithmetical theory of fractions (and in this way of all mathematics): fractions were not sufficient for day-to-day requirements (such as measurement of lengths). Consequently, mathematicians were afraid that they would be accused of creating nonsense, and that they would be discarded or, at least, not be fed.

Thus it was necessary to create a new science--the theory of real numbers. This task (which is not simple, by the way) was solved by Eudoxus. It is surprising how close his approach is to the modern one (in this matter and also in the ... theory of divisibility). The discovery that facts like the uniqueness of expansion of an integer into prime factors require proof, is actually not less important than the proof itself, which is also not evident at all.»

Apropos of negative reviews

«André Weil, spiritual leader of the IAS for many years, set a high standard. In 1973, Associate Professor Michael S. Mahoney of the History of Science Department at Princeton University had the temerity (or perhaps the bad luck) to write a biography of Pierre de Fermat (1601-1665). At that time, Weil had been studying the history of Fermat for some time. He had given a lecture series on the subject. The depth of his understanding was uncanny: Weil had actually figured out sequences of letters that had been sent, on what dates they were mailed and had arrived, and who was thinking what when. He fancied himself to be the pre-eminent Fermat scholar. And the new biography by his colleague down the road did not strike his fancy. Somehow it was arranged for Weil to review the book for the Bulletin of the AMS. Weil begins the published review by reminding us that "In order to write even a tolerably good book about Fermat, a modicum of abilities is required." He then lists these prerequisites:

• Ordinary accuracy.
• The ability to express simple ideas in plain English.
• Some knowledge of French.
• Some knowledge of Latin.
• Some historical sense.
• Some familiarity with the work of Fermat's contemporaries and of Fermat's own mathematics.
• Knowledge of and sensitivity to mathematics.

André Weil then proceeds to consider each of these attributes one by one, and to demostrate--via annotated quotations from the book under review--that the author apparently possesses none of them.»

References
Steven G. Krantz, Mathematical apocrypha: stories and anecdotes of mathematicians and the mathematical. Published and distributed by the Mathematical Association of America, USA, 2002, pp. 51-52.

The "Verfasser" anecdote in Derbyshire's book on the prime obsession

«I don't think Landau's Handbuch has ever been translated into English. Number theorist Hugh L. Montgomery ... taught himself German by reading his way through the Handbuch, one finger on the dictionary. He tells the following story. The first 50-odd pages of the book are given over to a historical survey, in sections each of which is headed with the name of a great mathematician who made contributions in the field: Euclid, Legendre, Dirichlet, and so on. The last four of these sections are headed "Hadamard," "von Mangoldt," "de la Vallée Poussin," "Verfasser." Hugh was extremely impressed with the contributions of Verfasser, but was puzzled to know why he had not heard the name of this fine mathematician before. It was some time before he learned that "Verfasser" is a German word meaning "author" (ordinary nouns are capitalized in German).»

References
J. Derbyshire, Prime obsession: B. Riemann and the greatest unsolved problem in mathematics. Published by Plume (a member of Penguin Group), USA, 2003, pp. 231-232.

On Dirichlet's attitude toward teaching

"Looking back on my activity at the University of Berlin, I can give myself the attestation that during the twenty-seven years of my teaching there I never allowed myself to fail in the highest effort of which I was capable to facilitate your entrance to the glorious, unmeasurably growing, world of scholarship to which all my powers are dedicated..." --- P. G. L. Dirichlet (c. 1855)

It is with this quote that Robin Wilson's review of Uta C. Merzbach's "Dirichlet: a mathematical biography" ends. This review appeared on pages 84-86 of the December 2020 issue of The Mathematical Intelligencer.

I'll catch up with you later!