Generalized Clifford algebra: Difference between revisions

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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 theory|chaos]], and [[infinite sequences]],<ref name=martinez>{{cite web|last=Martínez|first=Guillermo|title=Borges and Mathematics|url=http://logicophilosophicus.org/guillermo-martinez/borges-and-mathematics.html|date=19 February 2003|accessdate=4 March 2012}}</ref>  although Borges' strongest links to mathematics are through [[Georg Cantor]]'s theory of infinite sets. The title of Borges' short story "[[The Aleph (short story)|The Aleph]]" is an allusion to Cantor's use of the Hebrew letter [[aleph]] (<math>\aleph</math>) to denote cardinality of [[set theory|transfinite set]]s.<ref name=cosmicweb>{{cite book|last=Hayles|first=N. Katherine|title=The Cosmic Web: Scientific Field Models and Literary Strategies in the Twentieth Century|year=1984|publisher=Cornell University Press|location=Ithaca|url=http://books.google.com/books?id=2Ey5nmCC_BcC&pg=PA218&lpg=PA218&dq=hayles+the+cosmic+web&source=bl&ots=lk0JfUXD5N&sig=uDYCW2oFV9ZkPsslwhf0x6T91XA&hl=en&sa=X&ei=1u5TT_7pOcPiiAKk7M20Bg&ved=0CEIQ6AEwAg#v=onepage&q=hayles%20the%20cosmic%20web&f=false}}</ref> 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.
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==Background==
Jorge Luis Borges was born in [[Buenos Aires]] in 1899, and was home-schooled by his parents till the age of eleven,<ref name=katebowen>{{cite news|last=Bowen|first=Kate|title=Jorge Luis Borges: The Face of Argentine Literature|url=http://www.argentinaindependent.com/top-story/jorge-luis-borges-poster-boy-of-argentine-literature/|accessdate=6 March 2012|newspaper=The Argentina Independent|date=2 February 2012}}</ref> and was bi-lingual in Spanish and English. He spent his childhood in Buenos Aires and [[Geneva]], [[Switzerland]] and received very little formal education,<ref name=toibincolm>{{cite news|last=Tóibín|first=Colm|title=Don’t abandon me|url=http://www.lrb.co.uk/v28/n09/colm-toibin/dont-abandon-me|accessdate=6 March 2012|newspaper=London Review of Books|date=11 May 2006}}</ref> before he began his writing career. According to Argentinian mathematician [[Guillermo Martínez (writer)|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.<ref name=martinez /> 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.”<ref name=otherinquisitions>{{cite book|last=Borges|first=Jorge Luis|title=Other inquisitions, 1937–1952|year=1964|publisher=University of Texas Press|url=http://books.google.com/books/about/Other_inquisitions_1937_1952.html?id=xvycg3RMAW4C}}</ref>
[[Image:Illustration nested intervals.svg|300px|thumb|right|Visualizing a [[measure (mathematics)|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.”<ref> url=http://pt.scribd.com/doc/165889923/A-History-Of-Reading </ref>
 
==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".<ref name=fmerrel>{{cite book|last=Merrel|first=Floyd|title=Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics|year=1991|publisher=Purdue University Press|location=West Lafayette|url=http://books.google.com/books/about/Unthinking_thinking.html?id=3IY67Ck-TFQC|isbn=1-55753-011-4}}</ref> In Borges' 1941 story, "The Library of Babel", the narrator declares that the collection of books of a fixed number of [[orthography|orthographic]] symbols and pages is unending.<ref name=collfiction>{{cite book|last=Borges|first=Jorge Luis|title=Collected Fictions|year=1998|publisher=Viking|isbn=0-670-84970-7}}</ref> However, since the [[permutation]]s of twenty-five orthographic symbols is finite, the library has to be [[periodic function|periodic]] and self-repeating.<ref name=cosmicweb /> In "[[The Book of Sand]]", he deals with another form of infinity; one whose elements are [[Dense set|dense]], that is, for any two elements, we can always find another "between" them.<ref name=martinez /> The narrator describes the book as having pages that are "infinitely thin", which can be interpreted either as referring to a set of [[measure (mathematics)|measure zero]], or of having [[infinitesimal]] length, in the sense of [[second order logic]].<ref name=wgbloch>{{cite book|last=Bloch|first=William Goldbloom|title=The Unimaginable Mathematics of Borges' Library of Babel|year=2008|publisher=Oxford University Press|isbn=978-0-19-533457-9|url=http://books.google.com/books/about/The_unimaginable_mathematics_of_Borges_L.html?id=ebILwSGlUlgC}}</ref>
 
==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 geometry|euclidean]], it can equally well be visualized as a topologically non-trivial manifold such as a [[torus]] or a [[Klein bottle]].<ref name=wgbloch /> 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 transformation|inversion]].<ref name=martinez />
 
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.{{cn|date=April 2013}}<!---Blog source removed--->
 
[[Image:Schroedingers cat film.svg|thumb|right|250px|[[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 [[people of China|Chinese]] scholar Ts'ui Pên, whose plot bifurcates at every point in time. The idea of the flow of time being [[Nonlinear system|non-linear]] can be compared to the many-worlds interpretation of [[quantum mechanics]] and the notion of [[multiverse]]s present in some versions of [[string theory]].<ref name=fmerrel /> Similarly, the [[infinity|infinitude]] of diverging, infinite universes in mathematical [[cosmology]] is reflected Borges' rejection of linear, absolute time.<ref name=thiher>{{cite book|last=Thiher|first=Allen|title=Fiction refracts science: modernist writers from Proust to Borges|year=2005|publisher=University of Missouri Press|url=http://books.google.com/books/about/Fiction_refracts_science.html?id=uNZsdTLq1UUC}}</ref> Borges' writings address the nature of [[entity]] and the possibility of infinite "realities", as in his essay "New Time Refutations" (1946).<ref name=jams>{{cite journal|last=Di Marco|first=Oscar Antonio|title=Borges, the Quantum Theory and Parallel Universes|journal=The Journal of American Science|year=2006|url=http://www.jofamericanscience.org/journals/am-sci/0201/01-marco-0117.pdf|accessdate=10 March 2012}}</ref>
 
==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.<ref name=chaosnorder>{{cite book|last=Hayles|first=N. Katherine|title=Chaos and order: complex dynamics in literature and science|year=1991|publisher=University of Chicago Press|isbn=0226321436}}</ref>
 
==References==
<references/>
{{Jorge Luis Borges}}
[[Category:Jorge Luis Borges]]
[[Category:Mathematics and culture|Borges, Jorge Luis]]

Latest revision as of 20:30, 24 November 2014

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