# Jorge Luis Borges and mathematics

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The literary works of Argentine author Jorge Luis Borges contain references to several ideas in modern mathematics. These include notions such as set theory, recursion, chaos, and infinite sequences,[1] although Borges' strongest links to mathematics are through Georg Cantor's theory of infinite sets. The title of Borges' short story "The Aleph" is an allusion to Cantor's use of the Hebrew letter aleph ($\aleph$) to denote cardinality of transfinite sets.[2] In particular, some of Borges' most popular works such as "The Library of Babel", "The Garden of Forking Paths", "The Aleph", "The Approach to Al-Mu'tasim" illustrate his use of mathematics.

## Background

Jorge Luis Borges was born in Buenos Aires in 1899, and was home-schooled by his parents till the age of eleven,[3] and was bi-lingual in Spanish and English. He spent his childhood in Buenos Aires and Geneva, Switzerland and received very little formal education,[4] before he began his writing career. 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 logical paradoxes, infinity, and basic problems in topology and probability theory. He was also aware of the contemporary debates on the foundations of mathematics.[1] In his essay "Avatars of the Tortoise", Borges says, “five or seven years of metaphysical, theological, and mathematical training would prepare me (perhaps) for properly planning a history of the infinite.”[5]

Visualizing a measure zero set as the limit of a sequence of nested intervals. Pages in "The Book of Sand" have measure zero.

Reviewing a history of mathematics published sometime in the mid-thirties, Borges wrote that it suffered “from a crippling defect: the chronological order of its events doesn’t correspond to its logical and natural order. The definition of its elements very frequently comes last, practice precedes theory, the intuitive labours of its precursors are less comprehensible for the profane reader than those of the modern mathematicians.”[6]

## Infinity and cardinality

Borges' described his feelings towards infinity in "Other inquisitions" (1964) by: "One concept corrupts and influences the others. I am not speaking of the Evil, whose limited sphere is ethics; I am speaking of the infinite".[7] 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.[8] However, since the permutations of twenty-five orthographic symbols is finite, the library has to be periodic and self-repeating.[2] In "The Book of Sand", he deals with another form of infinity; one whose elements are dense, that is, for any two elements, we can always find another "between" them.[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.[9]

## 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.[9] In the essay "Pascal's sphere", 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]

In Borges' one-paragraph short story, "On Exactitude in Science", he describes a 1:1 scale cartographic map: "the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it." This an example of the case in the Brouwer fixed-point theorem where every single point on the Euclidean plane is a fixed-point under this mapping.[citation needed]

Schrödinger's cat: the two outcomes can be interpreted as occurring in alternate universes with equal validity

## 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 being non-linear can be compared to the many-worlds interpretation of quantum mechanics and the notion of multiverses present in some versions of string theory.[7] Similarly, the infinitude of diverging, infinite universes in mathematical cosmology is reflected Borges' rejection of linear, absolute time.[10] Borges' writings address the nature of entity and the possibility of infinite "realities", as in his essay "New Time Refutations" (1946).[11]

## 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.[12]

## References

1. ^ a b c d Martínez, Guillermo (19 February 2003). "Borges and Mathematics". Retrieved 4 March 2012.
2. ^ a b Hayles, N. Katherine (1984). The Cosmic Web: Scientific Field Models and Literary Strategies in the Twentieth Century. Ithaca: Cornell University Press.
3. ^ Bowen, Kate (2 February 2012). "Jorge Luis Borges: The Face of Argentine Literature". The Argentina Independent. Retrieved 6 March 2012.
4. ^ Tóibín, Colm (11 May 2006). "Don’t abandon me". London Review of Books. Retrieved 6 March 2012.
5. ^ Borges, Jorge Luis (1964). Other inquisitions, 1937–1952. University of Texas Press.
6. ^
7. ^ a b Merrel, Floyd (1991). Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics. West Lafayette: Purdue University Press. ISBN 1-55753-011-4.
8. ^ Borges, Jorge Luis (1998). Collected Fictions. Viking. ISBN 0-670-84970-7.
9. ^ a b Bloch, William Goldbloom (2008). The Unimaginable Mathematics of Borges' Library of Babel. Oxford University Press. ISBN 978-0-19-533457-9.
10. ^ Thiher, Allen (2005). Fiction refracts science: modernist writers from Proust to Borges. University of Missouri Press.
11. ^ Di Marco, Oscar Antonio (2006). "Borges, the Quantum Theory and Parallel Universes". The Journal of American Science. Retrieved 10 March 2012.
12. ^ Hayles, N. Katherine (1991). Chaos and order: complex dynamics in literature and science. University of Chicago Press. ISBN 0226321436.