# Jorge Luis Borges and mathematics

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Jorge Luis Borges and mathematics concerns several modern mathematical concepts found in certain essays and short stories of Argentinian author Jorge Luis Borges (1899–1986), including concepts such as set theory, recursion, chaos theory, and infinite sequences, [1] although Borges' strongest links to mathematics are through Georg Cantor's theory of infinite sets, outlined in "The Doctrine of Cycles" (La doctrina de los ciclos). Some of Borges' most popular works such as "The Library of Babel" (La Biblioteca de Babel), "The Garden of Forking Paths" (El Jardín de Senderos que se Bifurcan), "The Aleph" (El Aleph), an allusion to Cantor's use of the Hebrew letter aleph (${\displaystyle \aleph }$) to denote cardinality of transfinite sets, [2] and "The Approach to Al-Mu'tasim" (El acercamiento a Almotásim) illustrate his use of mathematics.

## Contents

According to Argentinian mathematician Guillermo Martínez, Borges at least had a knowledge of mathematics at the level of first courses in algebra and analysis at a university – covering logic, paradoxes, infinity, topology and probability theory. He was also aware of the contemporary debates on the foundations of mathematics. [1]

## Infinity and cardinality

His 1939 essay "Avatars of the Tortoise" (Avatares de la Tortuga) is about infinity, and he opens by describing the book he would like to write on infinity: “five or seven years of metaphysical, theological, and mathematical training would prepare me (perhaps) for properly planning that book.” [3]

In Borges' 1941 story, "The Library of Babel", the narrator declares that the collection of books of a fixed number of orthographic symbols and pages is unending. [4] However, since the permutations of twenty-five orthographic symbols is finite, the library has to be periodic and self-repeating. [2]

In his 1975 short story "The Book of Sand" (El Libro de Arena), he deals with another form of infinity; one whose elements are a dense set, that is, for any two elements, we can always find another between them. This concept was also used in the physical book the short-story came from, The Book of Sand book. [1] The narrator describes the book as having pages that are "infinitely thin", which can be interpreted either as referring to a set of measure zero, or of having infinitesimal length, in the sense of second order logic. [5]

In his 1936 essay "The Doctrine of Cycles" (La doctrina de los ciclos), [6] published in his essay anthology of the same year Historia de la eternidad, Borges speculated about a universe with infinite time and finite mass: "The number of all the atoms that compose the world is immense but finite, and as such only capable of a finite (though also immense) number of permutations. In an infinite stretch of time, the number of possible permutations must be run through, and the universe has to repeat itself. Once again you will be born from a belly, once again your skeleton will grow, once again this same page will reach your identical hands, once again you will follow the course of all the hours of your life until that of your incredible death." [7] As usual with many of Borges' ideas and constructions, this line of thought was received as metaphysical speculation, a language and philosophical game. Yet almost one century later theoretical physicists are crossing the same paths, this time as a possible consequence of string theory: "“Well, if the universe is really accelerating its expansion, then we know that it’s going to get infinitely large, and that things will happen over and over and over.” And if you have infinitely many tries at something, then every possible outcome is going to happen infinitely many times, no matter how unlikely it is.". [8]

## Geometry and topology

Borges in "The Library of Babel" states that "The Library is a sphere whose exact center is any hexagon and whose circumference is unattainable". The library can then be visualized as being a 3-manifold, and if the only restriction is that of being locally euclidean, it can equally well be visualized as a topologically non-trivial manifold such as a torus or a Klein bottle. [5]

In his 1951 essay "Pascal's sphere" (La esfera de Pascal), [9] Borges writes about a "sphere with center everywhere and circumference nowhere". A realization of this concept can be given by a sequence of spheres with contained centres and increasingly large radii, which eventually encompasses the entire space. This can be compared to the special point in "The Aleph" by the process of inversion. [1]

## Quantum physics

In "The Garden of Forking Paths", Borges describes a novel by the fictional Chinese scholar Ts'ui Pên, whose plot bifurcates at every point in time. The idea of the flow of time branching can be compared to the many-worlds interpretation of quantum mechanics and the notion of multiverses present in some versions of string theory. [10] Similarly, the infinitude of diverging, infinite universes in mathematical cosmology is reflected Borges' rejection of linear, absolute time. [11] Borges' writings address the nature of entity and the possibility of infinite "realities", as in his essay "New Time Refutations" (1946). [12]

## Chaos theory

Bifurcation theory is a model in chaos theory of order appearing from a disordered system, and is a local theory that describes behavior of systems at local points. Borges anticipated the development of bifurcation theory in mathematics, through "The Garden of Forking Paths" in 1941. In "Garden", Borges captured the idea of a system splitting into multiple, uncorrelated states. For example, if a leaf floating in a river comes across a rock, it must flow across either side of the rock, and the two possibilities are statistically uncorrelated. [13]

## Related Research Articles

Jorge Francisco Isidoro Luis Borges Acevedo was an Argentine short-story writer, essayist, poet and translator regarded as a key figure in Spanish-language and international literature. His best-known works, Ficciones (transl.Fictions) and El Aleph, published in the 1940s, are collections of short stories exploring motifs such as dreams, labyrinths, chance, infinity, archives, mirrors, fictional writers and mythology. Borges's works have contributed to philosophical literature and the fantasy genre, and have had a major influence on the magic realist movement in 20th century Latin American literature.

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, including the complete works of William Shakespeare. In fact, the monkey would almost surely type every possible finite text an infinite number of times. The theorem can be generalized to state that any sequence of events that has a non-zero probability of happening will almost certainly occur an infinite number of times, given an infinite amount of time or a universe that is infinite in size.

"The Library of Babel" is a short story by Argentine author and librarian Jorge Luis Borges (1899–1986), conceiving of a universe in the form of a vast library containing all possible 410-page books of a certain format and character set.

Adolfo Bioy Casares was an Argentine fiction writer, journalist, diarist, and translator. He was a friend and frequent collaborator with his fellow countryman Jorge Luis Borges. He is the author of the Fantastique novel The Invention of Morel.

In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set.

"Pierre Menard, Author of the Quixote" is a short story by Argentine writer Jorge Luis Borges.

The Aleph and Other Stories is a book of short stories by Argentine writer Jorge Luis Borges. The title work, "The Aleph", describes a point in space that contains all other spaces at once. The work also presents the idea of infinite time. Borges writes in the original afterword, dated May 3, 1949, that most of the stories belong to the genre of fantasy, mentioning themes such as identity and immortality. Borges added four new stories to the collection in the 1952 edition, for which he provided a brief postscript to the afterword. The story "La intrusa" was first printed in the third edition of El Aleph (1966) and was later included in the collection El informe de Brodie (1970).

This is a bibliography of works by Argentine short-story writer, essayist, poet, and translator Jorge Luis Borges (1899–1986).

"The Aleph" is a short story by the Argentine writer and poet Jorge Luis Borges. First published in September 1945, it was reprinted in the short story collection, The Aleph and Other Stories, in 1949, and revised by the author in 1974.

Ernesto Sabato was an Argentine novelist, essayist, painter, and physicist. According to the BBC he "won some of the most prestigious prizes in Hispanic literature" and "became very influential in the literary world throughout Latin America". Upon his death El País dubbed him the "last classic writer in Argentine literature".

"The House of Asterion" is a short story by Argentine writer Jorge Luis Borges. The story was first published in 1947 in the literary magazine Los Anales de Buenos Aires and republished in Borges's short story collection The Aleph in 1949. It is based on the Greek myth of Theseus and the Minotaur and is told from the perspective of Asterion, the Minotaur.

Paul-François Groussac was a French-born Argentine writer, literary critic, historian, and librarian.

Guillermo de Torre Ballesteros was a Spanish essayist, poet and literary critic, a Dadaist and member of the Generation of '27. He is also notable as the brother-in-law of the Argentine writer Jorge Luis Borges.

Susana Calandrelli was an Argentine writer and teacher.

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .

Harto the Borges is a documentary film by Eduardo Montes-Bradley. Harto the Borges explores the narcissistic side of Jorge Luis Borges, the author of El Aleph, his frequent and often criticized comments to the press, his distinctive and gentle ironies. Harto The Borges had a theatrical release in Buenos Aires in September 2000, and was well received by the critics. Since then has been frequently exhibited at forums, campuses, and film festivals. On October 4, 2011, Harto the Borges was presented at the University of Salamanca, and made available to the general audience in Argentina through the On Line version of Revista Cultura Ñ, in Buenos Aires. The film was released alongside an article in which the director views the film ten years after its premier at the Cine Cosmos. The film is currently available in Vimeo. Harto the Borges was presented at the Festival Internacional del Nuevo Cine Latinoamericano, Havana, Cuba, 2000 and nominated for Premio Cóndor de Plata.

"The Congress" is a 1971 short story by Argentine writer Jorge Luis Borges. The story is on an utopic universal congress and is seen by critics as a political essay.

Lisa Block de Behar is an Uruguayan professor of Linguistics and researcher in Literary Theory, Comparative Literature and Communication media.

Historia de la eternidad is a book of essays published by Argentine author Jorge Luis Borges, in 1936.

The Unimaginable Mathematics of Borges' Library of Babel is a popular mathematics book on Jorge Luis Borges and mathematics. It describes several mathematical concepts related to the short story "The Library of Babel", by Jorge Luis Borges. Written by mathematics professor William Goldbloom Bloch, and published in 2008 by the Oxford University Press, it received an honorable mention in the 2008 PROSE Awards.

## References

1. Martínez, Guillermo (19 February 2003). "Borges and Mathematics" . Retrieved 4 March 2012.
2. Hayles, N. Katherine (1984). The Cosmic Web: Scientific Field Models and Literary Strategies in the Twentieth Century. Ithaca: Cornell University Press. ISBN   0801492904.
3. "Los avatares de la tortuga", en Sur, nº 63, Buenos Aires, diciembre 1939, pp. 18-23. (Recogido en Discusión, Buenos Aires, Emecé, 1957) Original quote says: "Cinco, sietes años de aprendizaje metafísico, teológico, matemático, me capacitarían (tal vez), para planear decorosamente ese libro."
4. Borges, Jorge Luis (1998). . Viking. ISBN   0-670-84970-7.
5. Bloch, William Goldbloom (2008). The Unimaginable Mathematics of Borges' Library of Babel. Oxford University Press. ISBN   978-0-19-533457-9.
6. La doctrina de los ciclos, en Sur, nº 20, Buenos Aires, mayo 1936, pp. 20-29. (Recogido en Historia de la eternidad, Buenos Aires, Viau y Zona, 1936. Fechado 1934)
7. The original quote at the beginning of the essay says "El número de todos los átomos que componen el mundo es, aunque desmesurado, finito, y sólo capaz como tal de un número finito (aunque desmesurado también) de permutaciones. En un tiempo infinito, el número de las permutaciones posibles debe ser alcanzado, y el universo tiene que repetirse. De nuevo nacerás de un vientre, de nuevo crecerá tu esqueleto, de nuevo arribará esta misma página a tus manos iguales, de nuevo cursaras todas las horas hasta la de tu muerte increíble."
8. Brookman, John (2014). The Universe: Leading Scientists Explore the Origin, Mysteries, and Future of the Cosmos. Harper Perennial. ISBN   978-0062296085.
9. La esfera de Pascal, en La Nación, Buenos Aires, 14 enero 1951, 2.ª sec., p. 1. (Recogido en Otras inquisiciones, Buenos Aires, Sur, 1952)
10. Merrel, Floyd (1991). Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics. West Lafayette: Purdue University Press. ISBN   1-55753-011-4.
11. Thiher, Allen (2005). . University of Missouri Press.
12. Di Marco, Oscar Antonio (2006). "Borges, the Quantum Theory and Parallel Universes" (PDF). The Journal of American Science. Retrieved 10 March 2012.
13. Hayles, N. Katherine (1991). Chaos and order: complex dynamics in literature and science. University of Chicago Press. ISBN   0226321436.