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Leonardo da Vinci is regarded as a "Renaissance man" and is one of the most recognisable polymaths.

A polymath (Greek: πολυμαθής, polymathēs, "having learned much")^[1] is a person whose expertise spans a significant number of different subject areas. In less formal terms, a polymath (or polymathic person) may simply be someone who is very knowledgeable. Most ancient scientists were polymaths by today's standards.^[2]

The common term Renaissance man is used to describe a person who is well educated or who excels in a wide variety of subjects or fields.^[3] The concept emerged from the numerous great thinkers of that era who excelled in multiple fields of the arts and science, including Leonardo da Vinci, Michelangelo, Galileo Galilei, Copernicus and Francis Bacon; the emergence of these thinkers was likewise attributed to the then rising notion in Renaissance Italy expressed by one of its most accomplished representatives, Leon Battista Alberti (1404–1472): that "a man can do all things if he will."^[4] It embodied the basic tenets of Renaissance humanism, which considered humans empowered, limitless in their capacities for development, and led to the notion that people should embrace all knowledge and develop their capacities as fully as possible. Thus the gifted people of the Renaissance sought to develop skills in all areas of knowledge, in physical development, in social accomplishments, and in the arts. The term has since expanded from original usage and has been applied to other great thinkers before and after the Renaissance such as Aristotle, Johann Goethe, and Isaac Newton.

Related terms[link]

Hildegard of Bingen, a medieval polymath, shown dictating to her scribe in an illumination from Liber Scivias.

A different term for the secondary meaning of polymath is Renaissance man or woman (a term first recorded in written English in the early 20th century).^[5] Other similar terms also in use are Homo Universalis (Latin) and Uomo Universale (Italian), which translate to "universal person" or "universal man". These expressions derived from the ideal in Renaissance Humanism that it was possible to acquire a universal learning^[6] in order to develop one's potential, (covering both the arts and the sciences^[7] and without necessarily restricting this farewell learning to be the academic fields). When someone is called a Renaissance man or woman today, it is meant that they do not have only broad interests or a superficial knowledge of several fields, but rather that their knowledge is profound and often that they also have proficiency or accomplishments in at least some of these fields and in some cases even at a level comparable to the proficiency or the accomplishments of an expert.^[8] The related term Generalist is used to contrast this general approach to knowledge to that of the specialist. The expression Renaissance person today commonly implies only intellectual or scholastic proficiency and knowledge and not necessarily the more universal sense of "learning" implied by Renaissance humanism. Note, however, that some dictionaries use the term "Renaissance man" as roughly synonymous with polymath in the first meaning, to describe someone versatile with many interests or talents,^[9] while others recognize a meaning restricted to the Renaissance era and more closely related to Renaissance ideals.

A more colloquial term for such a person would be a jack of all trades, though this often refers to skill and not necessarily knowledge. The term "jack of all trades" also occasionally has negative connotation (see, for instance, jack of all trades, master of none); such a person may be labeled as a dilettante, while "polymath" typically has a positive connotation.

The term Universal Genius is also used, taking Leonardo da Vinci as a prime example again. The term seems to be used especially when a Renaissance man has made historical or lasting contributions in at least one of the fields in which he was actively involved, and when he had a universality of approach. Despite the existence of this term, a polymath may not necessarily be classed as a genius; and certainly a genius may not display the breadth of knowledge to qualify as a polymath. Isaac Newton, Albert Einstein, and Marie Curie are examples of people widely viewed as geniuses, but who are not generally considered to be polymaths.

Renaissance ideal[link]

Many notable polymaths lived during the Renaissance period, a cultural movement that spanned roughly the 14th through the 17th century, beginning in Italy in the late Middle Ages and later spreading to the rest of Europe. They had a rounded approach to education that was typical of the ideals of the humanists of the time. A gentleman or courtier of that era was expected to speak several languages, play a musical instrument, write poetry, and so on, thus fulfilling the Renaissance ideal. The idea of a universal education was pivotal to achieving polymath ability, hence the word university was used to describe a seat of learning. At this time universities did not specialize in specific areas, but rather trained their students in a broad array of science, philosophy and theology. This universal education, as such, gave them a grounding from which they could continue into apprenticeship to a Master of a specific field. It is important to note that a university education was highly regarded. A person was not considered to need this broad knowledge to apprentice as a carpenter, but to apprentice in the sciences or philosophy it contributed hugely to their being able to comprehend the universe as it was understood at the time. During the Renaissance, Baldassare Castiglione, in his guide The Book of the Courtier, wrote about how an ideal courtier should have polymathic traits.^[10]

Castiglione's guide stressed the kind of attitude that should accompany the many talents of a polymath, an attitude he called "sprezzatura". A courtier should have a detached, cool, nonchalant attitude, and speak well, sing, recite poetry, have proper bearing, be athletic, know the humanities and classics, paint and draw and possess many other skills, always without showy or boastful behavior, in short, with "sprezzatura". The many talents of the polymath should appear to others to be performed without effort, in an unstrained way, almost without thought. In some ways, the gentlemanly requirements of Castiglione recall the Chinese sage, Confucius, who far earlier depicted the courtly behavior, piety and obligations of service required of a gentleman. The easy facility in difficult tasks also resembles the effortlessness inculcated by Zen, such as in archery where no conscious attention, but pure spontaneity, produces better and more noble skill. For Castiglione, the attitude of apparent effortlessness should accompany great skill in many separate fields. In word or deed the courtier should "avoid affectation ... (and) ... practice ... a certain sprezzatura ... conceal all art and make whatever is done or said appear to be without effort and almost without any thought about it".^[10][11]

This Renaissance ideal differed slightly from the "polymath" in that it involved more than just intellectual advancement. Historically (roughly 1450–1600) it represented a person who endeavored to "develop his capacities as fully as possible" (Britannica, "Renaissance Man") both mentally and physically, and, as Castiglione suggests, without "affectation".^[10] For example, being an accomplished athlete was considered integral and not separate from education and learning of the highest order. Leon Battista Alberti, who was a Roman Catholic priest, architect, painter, poet, scientist, mathematician, inventor, and sculptor, was in addition a skilled horseman and archer.^{[citation needed]}

Renaissance men[link]

The list above provides examples of notable polymaths (in the secondary meaning only, that is, Renaissance men). Caution is necessary when interpreting the word polymath (in the second meaning or any of its synonyms) in a source, since there is always ambiguity of what the word denotes. Also, when a list of subjects in relation to the polymath is given, such lists often seem to imply that the notable polymath was reputable in all fields, but the most common case is that the polymath made his reputation in one or two main fields where he had widely recognized achievements, and that he was merely proficient or actively involved in other fields, but, once again, not necessarily with achievements comparable to those of renowned experts of his time in these fields. The list does not attempt to be comprehensive or authoritative in any way. The list also includes the Hakeem of the Islamic Golden Age (also known as the "Islamic Renaissance"), and other polymaths from other parts of the world.

Polymath and polyhistor compared[link]

Many dictionaries of word origins list these words as synonyms or, as words with very similar meanings. Thomas Moore took the words as corresponding to similarly erudite "polys" in one of his poems titled The Devil Among Scholars:^[12]

Off I fly, careering far
In chase of Pollys, prettier far
Than any of their namesakes are
—The Polymaths and Polyhistors,
Polyglots and all their sisters.

According to the Oxford English Dictionary, the words mean practically the same; "the classical Latin word polyhistor was used exclusively, and the Greek word frequently, of Alexander Polyhistor", but polymathist appeared later, and then polymath. Thus today, regardless of any differentiation they may have had when originally coined, they are often taken to mean the same thing.

The root terms histor and math have similar meanings in their etymological antecedents (to learn, learned, knowledge), though with some initial and ancillarily added differing qualities. Innate in historíā (Greek and Latin) is that the learning takes place via inquiry and narrative. Hístōr also implies that the polyhistor displays erudition and wisdom. From Proto-Indo-European it shares a root with the word "wit". Inquiry and narrative are specific sets of pedagogical and research heuristics.

Polyhistoric is the corresponding adjective. The word polyhistory (meaning varied learning), when used, is often derogatory.

Notable polymaths[link]

Shen Kuo was a polymathic Chinese scientist and statesman of the Song Dynasty.

A number of people have been described as "polymaths" by reliable sources, fulfilling the primary definition of the term, although there may not be expert consensus that each is a prime example in the secondary meaning, as "Renaissance men" and "universal geniuses" (see the list of renaissance men above for prime examples of "renaissance men" or "universal geniuses").

Other uses of "polymath"[link]

In Britain, phrases such as "polymath sportsman", "sporting polymath", or simply "polymath" are occasionally used in a restricted sense to refer to athletes that have performed at a high level in several very different sports. (One whose accomplishments are limited to athletics would not be considered to be a "polymath" in the usual sense of the word). Examples would include:

• Howard Baker – "Similar claims to the title of sporting polymath could be made for Howard Baker" (who won high jump titles, and played cricket, football, and water polo):^[13]
• Maxwell Woosnam – "Sporting polymath is a full-time post..."^[14]

The term can also be used loosely in other ways, for example, Rolf Harris (whose fame has come as a popular artist, television presenter and singer) has also been described by the Daily Mail as "the People's Polymath".^[15]

See also[link]

• Competent man
• List of people who have been called a "polymath"
• Opsimath
• Philomath
• Polyglot (person)
• Polymath (novel)

References and notes[link]

Further reading[link]

• Frost, Martin, "Polymath: A Renaissance Man"
• Mirchandani, Vinnie, The New Polymath: Profiles in Compound-Technology Innovations", John Wiley & Sons, 2010

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Douglas R. Hofstadter
Hofstadter in Bologna, Italy, in March 2002
Born	(1945-02-15) February 15, 1945 (age 67) New York, New York
Occupation	Professor of cognitive science
Nationality	United States
Alma mater	Stanford University (B.S.), University of Oregon (Ph.D)
Period	1979–present
Subjects	Cognitive science, philosophy of mind, translation, analogy-making

Douglas Richard Hofstadter (born February 15, 1945) is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics. He is best known for his book Gödel, Escher, Bach: an Eternal Golden Braid, first published in 1979. It won both the Pulitzer Prize for general non-fiction^[1][2] and a National Book Award for Science.^[3][a]

Early life and education[link]

Hofstadter was born in New York City, the son of Nobel Prize-winning physicist Robert Hofstadter. He grew up on the campus of Stanford University, where his father was a professor, and he attended the International School of Geneva in 1958–1959. He graduated with Distinction in Mathematics from Stanford University in 1965. He continued his education and received his Ph.D. in Physics from the University of Oregon in 1975.

Academic career[link]

Hofstadter is College of Arts and Sciences Distinguished Professor of Cognitive Science at Indiana University in Bloomington, where he directs the Center for Research on Concepts and Cognition which consists of himself and his graduate students, forming the "Fluid Analogies Research Group" (FARG).^[4] He was initially appointed to the Indiana University's Computer Science Department faculty in 1977, and at that time he launched his research program in computer modeling of mental processes (which at that time he called "artificial intelligence research", a label that he has since dropped in favor of "cognitive science research"). In 1984, he moved to the University of Michigan in Ann Arbor, where he was hired as a professor of psychology and was also appointed to the Walgreen Chair for the Study of Human Understanding. In 1988 he returned to Bloomington as "College of Arts and Sciences Professor" in both cognitive science and computer science, and also was appointed adjunct professor of history and philosophy of science, philosophy, comparative literature, and psychology, but he states that his involvement with most of these departments is nominal.^[5][6][7] In April 2009 Hofstadter was elected a Fellow of the American Academy of Arts and Sciences^[8] and a member of the American Philosophical Society.^[9]

Hofstadter's many interests include music, visual art, the mind, creativity, consciousness, self-reference, translation and mathematics.

At the University of Michigan and Indiana University, he co-authored, with Melanie Mitchell, a computational model of "high-level perception" – Copycat – and several other models of analogy-making and cognition, including the Tabletop project, co-developed with Robert M. French. The Copycat project was subsequently extended under the name "Metacat" by Hofstadter's doctoral student James Marshall.^[10] The Letter Spirit project, implemented by Gary McGraw and John Rehling, aims to model the act of artistic creativity by designing stylistically uniform "gridfonts" (typefaces limited to a grid). Other more recent models are Phaeaco (implemented by Harry Foundalis) and SeqSee (Abhijit Mahabal), which model high-level perception and analogy-making in the microdomains of Bongard problems and number sequences, respectively, as well as George (Francisco Lara-Dammer), which models the processes of perception and discovery in triangle geometry.^[11][12][13]

Both inside and outside his professional work, Hofstadter is driven by a pursuit of beauty. He seeks beautiful mathematical patterns, beautiful explanations, beautiful typefaces, beautiful sonic patterns in poetry, and so forth. Hofstadter has said of himself, "I'm someone who has one foot in the world of humanities and arts, and the other foot in the world of science." He has had several exhibitions of his artworks in various university art galleries. These shows have featured large collections of his gridfonts, his ambigrams (pieces of calligraphy created with two readings, either of which is usually obtained from the other by rotating or reflecting the ambigram, but sometimes simply by "oscillation", like the Necker Cube or the rabbit/duck figure of Joseph Jastrow), and his "Whirly Art" (music-inspired visual patterns realized using shapes based on various alphabets from India). (The term "ambigram" was invented by Hofstadter in 1984 and has since been taken up by many ambigrammists all over the world.)^[14]

Hofstadter collects and studies cognitive errors (largely, but not solely, speech errors), "bon mots" (spontaneous humorous quips), and analogies of all sorts, and his long-time observation of these diverse products of cognition, and his theories about the mechanisms that underlie them, have exerted a powerful influence on the architectures of the computational models developed by himself and FARG members.^[15]

All FARG computational models share certain key principles, among which are: that human thinking is carried out by thousands of independent small actions in parallel, biased by the concepts that are currently activated; that activation spreads from activated concepts to less activated "neighbor concepts"; that there is a "mental temperature" that regulates the degree of randomness in the parallel activity; that promising avenues tend to be explored more rapidly than unpromising ones. FARG models also have an overarching philosophy that all cognition is built from the making of analogies. The computational architectures that share these precepts are called "active symbols" architectures.

Hofstadter's thesis about consciousness, first expressed in Gödel, Escher, Bach (GEB) but also present in several of his later books, is that it is an emergent consequence of seething lower-level activity in the brain. In GEB he draws an analogy between the social organization of a colony of ants and the mind seen as a coherent "colony" of neurons. In particular, Hofstadter claims that our sense of having (or being) an "I" comes from the abstract pattern he terms a "strange loop", which is an abstract cousin of such concrete phenomena as audio and video feedback, and which Hofstadter has defined as "a level-crossing feedback loop". The prototypical example of this abstract notion is the self-referential structure at the core of Gödel's incompleteness theorems. Hofstadter's 2007 book I Am a Strange Loop carries his vision of consciousness considerably further, including the idea that each human "I" is distributed over numerous brains, rather than being limited to precisely one brain.^[16]

Hofstadter's writing is characterized by an intense interaction between form and content, as is exemplified by the 20 dialogues in GEB, many of which simultaneously talk about and imitate strict musical forms used by Bach, such as canons and fugues. Most of Hofstadter's books are characterized by some kind of structural alternation: in GEB between dialogues and chapters, in The Mind's I between selections and reflections, in Metamagical Themas between Chapters and Postscripts, and so forth. Both in his writing and in his teaching, Hofstadter stresses the concrete, constantly using examples and analogies, and avoids the abstract. Typical of the courses he teaches is his seminar "Group Theory and Galois Theory Visualized", in which abstract mathematical ideas are rendered as concretely as possible. He puts great effort into making ideas clear and visual, and asserts that when he teaches, if his students do not understand something, it is never their fault but always his own.

Hofstadter is passionate about languages. He has studied many of them, and speaks them to varying degrees. In addition to English, his mother tongue, he speaks French and Italian fluently (the language spoken at home with his children is Italian). At various times in his life, he has studied (in descending order of level of fluency reached) German, Russian, Spanish, Swedish, Mandarin, Dutch, Polish, and Hindi.^{[citation needed]} His love of sounds pushes him to strive to minimize, and ideally get rid of, any foreign accent.

Le Ton beau de Marot: In Praise of the Music of Language is a long book devoted to language and translation, especially poetry translation, and one of its leitmotifs is a set of some 88 translations of "Ma Mignonne", a highly constrained poem by 16th-century French poet Clément Marot. In this book, Hofstadter jokingly describes himself as "pilingual" (meaning that the sum total of the varying degrees of mastery of all the languages that he's studied comes to 3.14159...), as well as an "oligoglot" (someone who speaks "a few" languages).^[17][18]

In 1999, the bicentennial year of Russian poet and writer Alexander Pushkin, Hofstadter published a verse translation of Pushkin's classic novel-in-verse Eugene Onegin. It is highly constrained and filled with many types of sonic pattern. Aside from Eugene Onegin, Hofstadter has translated many other poems (always respecting their formal constraints), and two other novels (in prose): La Chamade (That Mad Ache) by French writer Françoise Sagan, and La Scoperta dell'Alba (The Discovery of Dawn) by Walter Veltroni, the then head of the Partito Democratico in Italy. The Discovery of Dawn was published in 2007, and That Mad Ache was published in 2009, bound together with Hofstadter's essay Translator, Trader: An Essay on the Pleasantly Pervasive Paradoxes of Translation.

Public image[link]

Hofstadter has said that he feels "uncomfortable with the nerd culture that centers on computers". He admits that "a large fraction [of his audience] seems to be those who are fascinated by technology", but when it was suggested that his work "has inspired many students to begin careers in computing and artificial intelligence" he replied that he was pleased about that, but that he himself has "no interest in computers."^[19][20] In that interview he also mentioned a course he has twice given at Indiana University, in which he took a "skeptical look at a number of highly-touted AI projects and overall approaches."^[7] For example, upon the defeat of Kasparov by Deep Blue, he commented that "It was a watershed event, but it doesn't have to do with computers becoming intelligent".^[21]

Provoked by predictions of a technological singularity (the hypothetical moment at which artificial intelligence will surpass human intelligence), Hofstadter has both organized and participated in several public discussions of the topic. At Indiana University in 1999 he organized such a symposium, and in April 2000, he organized a larger symposium entitled "Spiritual Robots" at Stanford University, in which he moderated a panel consisting of Ray Kurzweil, Hans Moravec, Kevin Kelly, Ralph Merkle, Bill Joy, Frank Drake, John Holland, John Koza. Hofstadter was also an invited panelist at the first Singularity Summit, held at Stanford in May 2006. Hofstadter expressed doubt about the likelihood of the singularity coming to pass in the foreseeable future.^{[22][23][24][25][26][27]}

In April 2009, when asked about his Wikipedia article, he commented, "[It] is filled with inaccuracies, and it kind of depresses me." When asked why he didn't correct it, he replied, "The next day someone will fix it back."^[28] In Aug. 2011 Hofstadter said he is happy with the article and the only inaccurate part of his Wikipedia entry was the previous sentence because the inaccuracies were subsequently fixed.^{[citation needed]}

In 1988 Dutch director Piet Hoenderdos created a docudrama about Hofstadter and his ideas entitled Victim of the Brain based on The Mind's I. It includes interviews with Hofstadter about his work.^[29]

In 2010: Odyssey Two, Arthur C. Clarke's first sequel to 2001: A Space Odyssey, HAL 9000 is described by Dr. Chandra as being caught in a "Hofstadter-Möbius loop".

Hofstadter's book Fluid Concepts & Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought was the first book ever sold by Amazon.com.^[30]

Columnist[link]

When Martin Gardner retired from writing his "Mathematical Games" column for Scientific American magazine, Hofstadter succeeded him in 1981–1983 with a column entitled Metamagical Themas (an anagram of "Mathematical Games"). An idea he introduced in one of these columns was the concept of "Reviews of This Book", a book containing nothing but cross-referenced reviews of itself which has an online implementation.^[31] One of Hofstadter's columns in Scientific American concerned the damaging effects of sexist language, and two chapters of his book Metamagical Themas are devoted to that topic, one of which is a biting analogy-based satire entitled "A Person Paper on Purity in Language", in which the reader's presumed revulsion at racism and racist language is used as a lever to motivate an analogous revulsion at sexism and sexist language.^[32] Another column reported on the discoveries made by University of Michigan professor Robert Axelrod in his computer tournament pitting many iterated prisoner's dilemma strategies against each other, and a follow-up column discussed a similar tournament that Hofstadter and his graduate student Marek Lugowski organized. The "Metamagical Themas" columns ranged over many themes, and included, to name just three, one on patterns in Frederic Chopin's piano music (particularly the études), another on the concept of superrationality (choosing to cooperate when the other party/adversary is assumed to be equally intelligent as oneself), and one on the self-modifying game of Nomic, based on the way in which the legal system modifies itself, and developed by philosopher Peter Suber.^[33]

Personal life[link]

Hofstadter was married to Carol Ann Brush. They met in Bloomington, and married in Ann Arbor in 1985. They had two children, Danny and Monica. Carol died in 1993 from the sudden onset of a brain tumor – glioblastoma multiforme – when their children were five and two. The Carol Ann Brush Hofstadter Memorial Scholarship for Bologna-bound Indiana University students was established in 1996 in her name.^[34] Hofstadter's book Le Ton beau de Marot is dedicated to their two children and its dedication reads "To M. & D., living sparks of their Mommy's soul".

Hofstadter has composed numerous pieces for piano, and a few for piano and voice. He created an audio CD with the title DRH/JJ, which includes all these compositions performed primarily by pianist Jane Jackson, but with a few performed by Brian Jones, Dafna Barenboim, Gitanjali Mathur and himself.^[35]

Hofstadter is related by marriage to the evolutionary biologist Stephen Jay Gould: Hofstadter's paternal aunt was married to Gould's maternal uncle.^{[citation needed]}

The dedication for I Am A Strange Loop is: "To my sister Laura, who can understand, and to our sister Molly, who cannot."^[36] Hofstadter explains in the preface that his younger sister Molly never developed the ability to speak or understand language.^[37]

Hofstadter is a vegan.^[38]

Published works[link]

Books[link]

The books published by Hofstadter are (the ISBNs refer to paperback editions, where available):

• Gödel, Escher, Bach: an Eternal Golden Braid (ISBN 0-465-02656-7) (1979)
• Metamagical Themas (ISBN 0-465-04566-9) (collection of Scientific American columns and other essays, all with postscripts)
• Ambigrammi: un microcosmo ideale per lo studio della creatività (ISBN 88-7757-006-7) (in Italian only)
• Fluid Concepts and Creative Analogies (ISBN 0-465-02475-0)
• Rhapsody on a Theme by Clement Marot (ISBN 0-910-15311-6) (1995, published 1996; volume 16 of series The Grace A. Tanner Lecture in Human Values)
• Le Ton beau de Marot: In Praise of the Music of Language (ISBN 0-465-08645-4)
• Eugene Onegin: A Novel Versification (ISBN 0-465-02094-1)
• I Am a Strange Loop (ISBN 0-465-03078-5) (2007)
• The Discovery of Dawn (ISBN 9780847831098) (2007) (a translation of a novel by Walter Veltroni)
• That Mad Ache, co-bound with Translator, Trader: An Essay on the Pleasantly Pervasive Paradoxes of Translation (ISBN 9780465010981) (2009)
• Surfaces and Essences, co-authored with Emmanuel Sander (ISBN 0465018475) (expected on Sept. 11, 2012)

Papers[link]

Hofstadter wrote, among many others, the following papers:

• "Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields", Phys. Rev. B 14 (1976) 2239.
• "A non-deterministic approach to analogy, involving the Ising model of ferromagnetism", in E. Caianiello (ed.), The Physics of Cognitive Processes. Teaneck, NJ: World Scientific, 1987.
• "To Err is Human; To Study Error-making is Cognitive Science" (co-authored by David J. Moser), Michigan Quarterly Review, Vol. XXVIII, No. 2, 1989, pp. 185-­215.
• "Speechstuff and thoughtstuff: Musings on the resonances created by words and phrases via the subliminal perception of their buried parts", in Sture Allen (ed.), Of Thoughts and Words: The Relation between Language and Mind. Proceedings of the Nobel Symposium 92, London/New Jersey: World Scientific Publ., 1995, 217-267.
• "On seeing A's and seeing As", Stanford Humanities Review Vol. 4, No. 2 (1995) pp. 109–121.
• "Analogy as the Core of Cognition", in Dedre Gentner, Keith Holyoak, and Boicho Kokinov (eds.) The Analogical Mind: Perspectives from Cognitive Science, Cambridge, MA: The MIT Press/Bradford Book, 2001, pp. 499–538.

Hofstadter also wrote over 50 papers that were published through the Center for Research on Concepts and Cognition.^[39]

Involvement in other books[link]

Hofstadter wrote forewords for or edited the following books:

• The Mind's I: Fantasies and Reflections on Self and Soul (co-edited with Daniel Dennett) (ISBN 0-465-03091-2 and ISBN 0-553-01412-9) (ISBN 0-553-34584-2) 1981
• Sparse Distributed Memory by Pentti Kanerva (Bradford Books/MIT Press, 1988). (ISBN 0262111322)
• Are Quanta Real? A Galilean Dialogue by J.M. Jauch (ISBN 0-253-20545-X) 1989 Indiana University Press; Hofstadter wrote the foreword.
• Alan Turing: The Enigma by Andrew Hodges. (Preface)
• Gödel's Proof (2002 revised edition) by Ernest Nagel and James R. Newman, edited by Hofstadter (ISBN 0-8147-5816-9). In the foreword, Hofstadter explains that the book (originally published in 1958) exerted a profound influence on him when he was young.
• Who invented the computer? The legal battle that changed computing history. (2003) by Alice Rowe Burks.
• Alan Turing: Life and Legacy of a Great Thinker by Christof Teuscher (Editor)
• Jason Salavon: Brainstem Still Life (ISBN 981-05-1662-2) 2004 (Introduction)
• Masters of Deception: Escher, Dali & the Artists of Optical Illusion 2004 by Al Seckel. Hofstadter wrote the foreword.
• King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker and Company, 2006. Hofstadter wrote the foreword.

Students[link]

Some of Hofstadter's former students include:

• David Chalmers – philosopher of mind
• Harry Foundalis – creator of Phaeaco, an architecture to solve Bongard problems^{[citation needed]}
• Robert M. French – cognitive scientist
• Scott A. Jones – inventor of voicemail
• Melanie Mitchell – computer scientist and complexity theorist

See also[link]

• American philosophy
• BlooP and FlooP
• Egbert B. Gebstadter
• Hofstadter's butterfly
• Hofstadter's law
• List of American philosophers
• Platonia dilemma
• Hofstadter points

Notes[link]

References[link]

External links[link]

Wikiquote has a collection of quotations related to: Douglas Hofstadter

• Stanford University Presidential Lecture - site dedicated to Hofstadter and his work
• List of publications from the DBLP Bibliography Server.
• Profile at c2.com
• Profile at Resonance Publications
• NF Reviews - bibliographic page with reviews of several of Hofstadter's books
• Radio interview on Philosophy Talk

Omar Khayyám
عمر خیام

A depiction of Omar Khayyám, in the works of Edward FitzGerald
Born	18 May[1] 1048[2]
Died	1131 (aged 82/83)[2]
School	Persian mathematics, Persian poetry, Persian philosophy
Main interests	Mathematics, Philosophy, Astronomy, Poetry
Influenced by Abū Rayhān al-Bīrūnī, Avicenna
Influenced Attar of Nishapur

Omar Khayyám (1048–1131; Persian: ‏عمر خیام‎) was a Persian polymath: philosopher, mathematician, astronomer and poet. He also wrote treatises on mechanics, geography, mineralogy, music, climatology and Islamic theology.^[3]

Born in Nishapur, at a young age he moved to Samarkand and obtained his education there, afterwards he moved to Bukhara and became established as one of the major mathematicians and astronomers of the medieval period. He is the author of one of the most important treatises on algebra written before modern times, the Treatise on Demonstration of Problems of Algebra, which includes a geometric method for solving cubic equations by intersecting a hyperbola with a circle.^[4] He contributed to a calendar reform.

His significance as a philosopher and teacher, and his few remaining philosophical works, have not received the same attention as his scientific and poetic writings. Al-Zamakhshari referred to him as “the philosopher of the world”. Many sources have testified that he taught for decades the philosophy of Avicenna in Nishapur where Khayyám was born and buried and where his mausoleum today remains a masterpiece of Iranian architecture visited by many people every year.^[5]

Outside Iran and Persian speaking countries, Khayyám has had an impact on literature and societies through the translation of his works and popularization by other scholars. The greatest such impact was in English-speaking countries; the English scholar Thomas Hyde (1636–1703) was the first non-Persian to study him. The most influential of all was Edward FitzGerald (1809–83),^[6] who made Khayyám the most famous poet of the East in the West through his celebrated translation and adaptations of Khayyám's rather small number of quatrains (Persian: رباعیات‎ rubāʿiyāt) in the Rubaiyat of Omar Khayyam.

Omar Khayyám died in 1131 and is buried in the Khayyam Garden at the mausoleum of Imamzadeh Mahruq in Nishapur. In 1963 the mausoleum of Omar Khayyam was constructed on the site by Hooshang Seyhoun.

Early life[link]

Khayyám's full name was Ghiyāth ad-Din Abu'l-Fath 'Umar ibn Ibrāhīm an-Nīshāpūrī al-Khayyāmī (Persian: غیاث الدین ابو الفتح عمر بن ابراهیم خیام نیشاپوری‎). He was born in Nishapur, modern-day Iran, but then a Seljuq capital in Khorasan,^[7][8][9] which rivaled Cairo or Baghdad in cultural prominence in that era. He is thought to have been born into a family of tent-makers (khayyami "tent-maker"), which he would make this into a play on words later in life:

Khayyám, who stitched the tents of science,
Has fallen in grief's furnace and been suddenly burned,
The shears of Fate have cut the tent ropes of his life,
And the broker of Hope has sold him for nothing!

— Omar Khayyám^[4]

He spent part of his childhood in the town of Balkh (present northern Afghanistan), studying under the well-known scholar Sheikh Muhammad Mansuri. He later studied under Imam Mowaffaq Nishapuri, who was considered one of the greatest teachers of the Khorasan region. Throughout his life Omar Khayyám was dedicated to his efforts and abilities, in the day he would teach Algebra and geometry in the evening he would attend the Seljuq court as an adviser of Malik-Shah I^[10] and at night he would study Astronomy and complete the important aspects of the Jalali calendar.

Omar Khayyám's years in Isfahan were very productive ones but after the death of the Seljuq Sultan Malik-Shah I (presumably by the Assassins sect), the Sultan's widow turned against him as an adviser and soon therefore Omar Khayyám set out on his Hajj or pilgrimage to Mecca and Medina. He was then allowed to work as a court astrologer and was permitted to return to Nishapur where he was famous for works and continued to teach mathematics, astronomy and even medicine.^[1]

Mathematician[link]

Khayyám Sikander was famous during his times as a mathematician. He wrote the influential Treatise on Demonstration of Problems of Algebra (1070), which laid down the principles of algebra, part of the body of Persian Mathematics that was eventually transmitted to Europe. In particular, he derived general methods for solving cubic equations and even some higher orders.

In the Treatise he wrote on the triangular array of binomial coefficients known as Pascal's triangle. In 1077, Khayyám wrote Sharh ma ashkala min musadarat kitab Uqlidis (Explanations of the Difficulties in the Postulates of Euclid) published in English as "On the Difficulties of Euclid's Definitions".^[11] An important part of the book is concerned with Euclid's famous parallel postulate, which attracted the interest of Thabit ibn Qurra. Al-Haytham had previously attempted a demonstration of the postulate; Khayyám's attempt was a distinct advance, and his criticisms made their way to Europe, and may have contributed to the eventual development of non-Euclidean geometry.

Omar Khayyám had notable works in geometry, specifically on the theory of proportions, his notable contemporary mathematicians included Al-Khazini and Abu Hatim al-Muzaffar ibn Ismail al-Isfizari^[12]

Theory of parallels[link]

"At the Tomb of Omar Khayyam", by Jay Hambidge

Khayyám wrote a book entitled Explanations of the difficulties in the postulates in Euclid's Elements. The book consists of several sections on the parallel postulate (Book I), on the Euclidean definition of ratios and the Anthyphairetic ratio (modern continued fractions) (Book II), and on the multiplication of ratios (Book III).

The first section is a treatise containing some propositions and lemmas concerning the parallel postulate. It has reached the Western world from a reproduction in a manuscript written in 1387-88 AD by the Persian mathematician Tusi. Tusi mentions explicitly that he re-writes the treatise "in Khayyám's own words" and quotes Khayyám, saying that "they are worth adding to Euclid's Elements (first book) after Proposition 28."^[13] This proposition^[14] states a condition enough for having two lines in plane parallel to one another. After this proposition follows another, numbered 29, which is converse to the previous one.^[15] The proof of Euclid uses the so-called parallel postulate (numbered 5). Objection to the use of parallel postulate and alternative view of proposition 29 have been a major problem in foundation of what is now called non-Euclidean geometry.

The treatise of Khayyám can be considered as the first treatment of parallels axiom which is not based on petitio principii but on more intuitive postulate. Khayyám refutes the previous attempts by other Greek and Persian mathematicians to prove the proposition. And he, as Aristotle, refuses the use of motion in geometry and therefore dismisses the different attempt by Ibn Haytham too.^[16] In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate,^[17]

Geometric algebra[link]

Omar Khayyám's geometric solution to cubic equations.

This philosophical view of mathematics (see below) has had a significant impact on Khayyám's celebrated approach and method in geometric algebra and in particular in solving cubic equations. In that his solution is not a direct path to a numerical solution and in fact his solutions are not numbers but rather line segments. In this regard Khayyám's work can be considered the first systematic study and the first exact method of solving cubic equations.^[19]

In an untitled writing on cubic equations by Khayyám discovered in 20th century,^[18] where the above quote appears, Khayyám works on problems of geometric algebra. First is the problem of "finding a point on a quadrant of a circle such that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal." Again in solving this problem, he reduces it to another geometric problem: "find a right triangle having the property that the hypotenuse equals the sum of one leg (i.e. side) plus the altitude on the hypotenuse.^[20] To solve this geometric problem, he specializes a parameter and reaches the cubic equation Failed to parse (Missing texvc executable; please see math/README to configure.): x^3+200x=20x^2+2000 .^[18] Indeed, he finds a positive root for this equation by intersecting a hyperbola with a circle.

This particular geometric solution of cubic equations has been further investigated and extended to degree four equations.^[21]

Regarding more general equations he states that the solution of cubic equations requires the use of conic sections and that it cannot be solved by ruler and compass methods.^[18] A proof of this impossibility was plausible only 750 years after Khayyám died. In this paper Khayyám mentions his will to prepare a paper giving full solution to cubic equations: "If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared."^[18]

This refers to the book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, part of the body of Persian Mathematics that was eventually transmitted to Europe.^[19] In particular, he derived general methods for solving cubic equations and even some higher orders.

Binomial theorem and extraction of roots[link]

This particular remark of Khayyám and certain propositions found in his Algebra book has made some historians of mathematics believe that Khayyám had indeed a binomial theorem up to any power. The case of power 2 is explicitly stated in Euclid's elements and the case of at most power 3 had been established by Indian mathematicians. Khayyám was the mathematician who noticed the importance of a general binomial theorem. The argument supporting the claim that Khayyám had a general binomial theorem is based on his ability to extract roots.^[23]

Khayyám-Saccheri quadrilateral[link]

The Saccheri quadrilateral was first considered by Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.^[24] Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyám was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle):

Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.^[25]

Khayyám then considered the three cases (right, obtuse, and acute) that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.

It wasn't until 600 years later that Giordano Vitale made an advance on Khayyám in his book Euclide restituo (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.

Astronomer[link]

The Jalali calendar was introduced by Omar Khayyám alongside other Mathematicians and Astronomers in Nishapur, today it is one of the oldest calendars in the world as well as the most accurate solar calendar in use today. Since the calendar uses astronomical calculation for determining the vernal equinox, it has no intrinsic error, but this makes it an observation based calendar.^{[26][27][28][29]}

Like most Persian mathematicians of the period, Khayyám was famous as an astronomer. In 1073, the Seljuq Sultan Jalal al-Din Malik-Shah Saljuqi (Malik-Shah I, 1072–92), invited Khayyám to build an observatory, along with various other distinguished scientists. According to some accounts, the version of the medieval Iranian calendar in which 2,820 solar years together contain 1,029,983 days (or 683 leap years, for an average year length of 365.24219858156 days) was based on the measurements of Khayyám and his colleagues.^[30] Another proposal is that Khayyám's calendar simply contained eight leap years every thirty-three years (for a year length of 365.2424 days).^[31] In either case, his calendar was more accurate to the mean tropical year than the Gregorian calendar of 500 years later. The modern Iranian calendar is based on his calculations.

Heliocentric Theory[link]

It is sometimes claimed that Khayyam demonstrated that the earth rotates on its axis^[32] by presenting a model of the stars to his contemporary al-Ghazali in a planetarium.^[33] Whether or not the story is apocryphal, it would only demonstrate the mathematical equivalence of a rotating earth to rotating spheres, as was well known to Khayyam's immediate predecessors, e.g. al-Biruni, and says nothing about heliocentrism, as a spinning earth can be made entirely consistent with geocentric models.

The other source for the claim that Khayyam believed in heliocentrism are Edward Fitzgerald's popular but anachronistic renderings ^[34] of Khayyam's poetry, in which the first lines are mistranslated with a heliocentric image of the Sun flinging "the Stone that puts the Stars to Flight".^[35]

Calendar Reform[link]

Khayyám is claimed to be a member of a panel that introduced several reforms to the Iranian calendar.^{[citation needed]} On March 15, 1079, Sultan Malik Shah accepted this corrected calendar as the official Persian calendar.^[36]

This calendar was known as the Jalali calendar after the Sultan, and was in force across Greater Iran from the 11th to the 20th centuries. It is the basis of the Iranian calendar which is followed today in Iran and Afghanistan. While the Jalali calendar is more accurate than the Gregorian, it is based on actual solar transit, similar to Hindu calendars, and requires an ephemeris for calculating dates. The lengths of the months can vary between 29 and 31 days depending on the moment when the sun crosses into a new zodiacal area (an attribute common to most Hindu calendars). This meant that seasonal errors were lower than in the Gregorian calendar.

The modern-day Iranian calendar standardizes the month lengths based on a reform from 1925, thus minimizing the effect of solar transits. Seasonal errors are somewhat higher than in the Jalali version, but leap years are calculated as before.

Poet[link]

Omar Khayyám was a notable poet during the reign of the Seljuk ruler Malik-Shah I and his contributions to the developments of Mathematics, Astronomy and Philosophy inspired later generations.

He is believed to have written about a thousand four-line verses or rubaiyat (quatrains). In the English-speaking world, he was introduced through the Rubáiyát of Omar Khayyám which are rather free-wheeling English translations by Edward FitzGerald (1809–1883). Other English translations of parts of the rubáiyát (rubáiyát meaning "quatrains") exist, but FitzGerald's are the most well known.

Ironically, FitzGerald's translations reintroduced Khayyám to Iranians "who had long ignored the Neishapouri poet." A 1934 book by one of Iran's most prominent writers, Sadeq Hedayat, Songs of Khayyam, (Taranehha-ye Khayyam) is said to have "shaped the way a generation of Iranians viewed" the poet.^[37]

Khayyam's poetry is translated to many languages.

Khayyám's personal beliefs are not known with certainty, but much is discernible from his poetic oeuvre.

Poetry[link]

A well decorated plaque containing poems from the Rubáiyát of Omar Khayyám.

Omar Khayyám's poems have been translated many times in many languages and many translators have claimed that their translations of the Rubáiyát of Omar Khayyám literal, poetic and less controversial than that of Edward Fitzgerald.^{[citation needed]}

The Moving Finger writes; and, having writ,
  Moves on: nor all thy Piety nor Wit,
Shall lure it back to cancel half a Line,
  Nor all thy Tears wash out a Word of it.

But helpless pieces in the game He plays,
  Upon this chequer-board of Nights and Days,
He hither and thither moves, and checks ... and slays,
  Then one by one, back in the Closet lays.

And, as the Cock crew, those who stood before
  The Tavern shouted - "Open then the Door!
You know how little time we have to stay,
  And once departed, may return no more."

A Book of Verses underneath the Bough,
  A Jug of Wine, a Loaf of Bread--and Thou,
Beside me singing in the Wilderness,
  And oh, Wilderness is Paradise enow.

If chance supplied a loaf of white bread,
  Two casks of wine and a leg of mutton,
In the corner of a garden with a tulip-cheeked girl,
  There'd be enjoyment no Sultan could outdo.

Myself when young did eagerly frequent
  Doctor and Saint, and heard great Argument
About it and about: but evermore
  Came out of the same Door as in I went.

With them the Seed of Wisdom did I sow,
  And with my own hand labour'd it to grow:
And this was all the Harvest that I reap'd -
  "I came like Water, and like Wind I go."

Into this Universe, and why not knowing,
  Nor whence, like Water willy-nilly flowing:
And out of it, as Wind along the Waste,
  I know not whither, willy-nilly blowing.

And that inverted Bowl we call The Sky,
  Whereunder crawling coop't we live and die,
Lift not thy hands to It for help - for It
  Rolls impotently on as Thou or I.

.........................................

Views on religion[link]

There have been widely divergent views on Khayyám. According to Seyyed Hossein Nasr no other Iranian writer/scholar is viewed in such extremely differing ways. At one end of the spectrum there are nightclubs named after Khayyám, and he is seen as an agnostic hedonist. On the other end of the spectrum, he is seen as a mystical Sufi poet influenced by platonic traditions.

The verse: "Enjoy wine and women and don't be afraid, Allah has compassion," suggests that he was not an atheist. He^[who?] further believes that it is almost certain that Khayyám objected to the notion that every particular event and phenomenon was the result of divine intervention. Nor did he zealously believe in an afterlife with a Judgment Day or rewards and punishments.^[38] Instead, he supported the view that laws of nature explained all phenomena of observed life. One contemporary writes: "I did not observe that he had any great belief in astrological predictions or speculations; nor have I seen or heard of any of the great (scientists) who had such belief."^{[citation needed]}

The following two quatrains are representative of numerous others that serve to reject many tenets of religious dogma:

An excerpt from the "al-Risålah fil-wujud" (Treatise on Being), one of Omar Khayyám's major works on Islamic theology.

خيام اگر ز باده مستى خوش باش

با ماه رخى اگر نشستى خوش باش

چون عاقبت كار جهان نيستى است

انگار كه نيستى، چو هستى خوش باش‎

which translates in FitzGerald's work as:

And if the Wine you drink, the Lip you press,

End in the Nothing all Things end in — Yes —

Then fancy while Thou art, Thou art but what

Thou shalt be — Nothing — Thou shalt not be less.

A more literal translation could read:

If with wine you are drunk be happy,

If seated with a moon-faced (beautiful), be happy,

Since the end purpose of the universe is nothing-ness;

Hence picture your nothing-ness, then while you are, be happy!

آنانكه ز پيش رفته‌اند اى ساقى

درخاك غرور خفته‌اند اى ساقى

رو باده خور و حقيقت از من بشنو

باد است هرآنچه گفته‌اند اى ساقى‎

which FitzGerald has erroneously and mistakenly interpreted as:

Why, all the Saints and Sages who discuss’d

Of the Two Worlds so learnedly — are thrust

Like foolish Prophets forth; their Words to Scorn

Are scatter’d, and their Mouths are stopt with Dust.

A literal translation, in an ironic echo of "all is vanity", could read:

Those who have gone forth, thou cup-bearer,

Have fallen upon the dust of pride, thou cup-bearer,

Drink wine and hear from me the truth:

(Hot) air is all that they have said, thou cup-bearer.

But some specialists, like Seyyed Hossein Nasr who looks at the available philosophical works of Khayyám, maintain that it is really reductive to just look at the poems (which are sometimes doubtful) to establish his personal views about God or religion; in fact, he even wrote a treatise entitled "al-Khutbat al-gharrå˘" (The Splendid Sermon) on the praise of God, where he holds orthodox views, agreeing with Avicenna on Divine Unity.^[5] In fact, this treatise is not an exception, and S.H. Nasr gives an example where he identified himself as a Sufi, after criticizing different methods of knowing God, preferring the intuition over the rational (opting for the so-called "kashf", or unveiling, method):^[5]

"... Fourth, the Sufis, who do not seek knowledge by ratiocination or discursive thinking, but by purgation of their inner being and the purifying of their dispositions. They cleanse the rational soul of the impurities of nature and bodily form, until it becomes pure substance. When it then comes face to face with the spiritual world, the forms of that world become truly reflected in it, without any doubt or ambiguity.

This is the best of all ways, because it is known to the servant of God that there is no reflection better than the Divine Presence and in that state there are no obstacles or veils in between. Whatever man lacks is due to the impurity of his nature. If the veil be lifted and the screen and

obstacle removed, the truth of things as they are will become manifest and known. And the Master of creatures [the Prophet Muhammad]—upon whom be peace—indicated this when he said: “Truly, during the days of your existence, inspirations come from God. Do you not want to follow them?” Tell unto reasoners that, for the lovers of God, intuition is guide, not discursive thought."

—Omar Khayyám^[39]

The same author goes on by giving other philosophical writings which are totally compatible with the religion of Islam, as the "al-Risålah fil-wujud" (Treatise on Being), written in Arabic, which begin with Quranic verses and asserting that all things come from God, and there is an order in these things. In another work, "Risålah jawåban li-thalåth maså˘il" (Treatise of Response to Three Questions), he gives a response to question on, for instance, the becoming of the soul post-mortem. S.H. Nasr even gives some poetry where he is perfectly in favor of Islamic orthodoxy, but expressing mystical views (God's goodness, the ephemerical state of this life, ...):^[5]

Thou hast said that Thou wilt torment me,

But I shall fear not such a warning.

For where Thou art, there can be no torment,

And where Thou art not, how can such a place exist?

The rotating wheel of heaven within which we wonder,

Is an imaginal lamp of which we have knowledge by similitude.

The sun is the candle and the world the lamp,

We are like forms revolving within it.

A drop of water falls in an ocean wide,

A grain of dust becomes with earth allied;

What doth thy coming, going here denote?

A fly appeared a while, then invisible he became.

Considering misunderstandings about Khayyám in the West and elsewhere, S.H. Nasr concludes by saying that if a correct study of the authentic rubaiyat is done, but along with the philosophical works, or even the spiritual biography entitled Sayr wa sulak (Spiritual Wayfaring), we can no longer view the man as a simple hedonistic wine-lover, or even an early skeptic, but a profound mystical thinker and scientist whose works are more important than some verses.^[5] C.H.A. Bjerregaard earlier summarised the situation:

"The writings of Omar Khayyam are good specimens of Sufism but are not valued in the West as they ought to be, and the mass of the people know him only through the poems of Edward Fitzgerald which is unfortunate. It is unfortunate because Fitzgerald is not faithful to his master and model, and at times he lays words upon the tongue of the Sufi which are blasphemous. Such outrageous language is that of the eighty-first quatrain for instance. Fitzgerald is doubly guilty because he was more of a Sufi than he was willing to admit."^[40]

Abdullah Dougan, a modern Naqshbandi Sufi, provides commentary^[41] on the role and contribution of Omar Khayyam to Sufi thought. Dougan says that while Omar is a minor Sufi teacher compared to the giants – Rumi, Attar and Sana’i – one aspect that makes Omar’s work so relevant and accessible is its very human scale as we can feel for him and understand his approach. The argument over the quality of Fitzgerald’s translation of the Rubaiyat has, according to Dougan, diverted attention from a fuller understanding of the deeply esoteric message contained in Omar’s actual material – "Every line of the Rubaiyat has more meaning than almost anything you could read in Sufi literature".

Philosopher[link]

Tomb of Omar Khayyám Neishapuri in Nishapur, Iran

An Ottoman Era inscription of a poem written by Omar Khayyám at Morića Han in Sarajevo, Bosnia and Herzegovina.

Khayyám himself rejects to be associated with the title falsafi "philosopher" in the sense of Aristotelianism and stressed he wishes "to know who I am". In the context of philosophers he was labeled by some of his contemporaries as "detached from divine blessings".^[42]

It is now established that Khayyám taught for decades the philosophy of Avicena, especially "the Book of Healing", in his home town Nishapur, till his death.^[5] In an incident he had been requested to comment on a disagreement between Avicena and a philosopher called Abu'l-Barakāt al-Baghdādī who had criticized Avicena strongly. Khayyám is said to have answered "[he] does not even understand the sense of the words of Avicenna, how can he oppose what he does not know?"^[42]

Khayyám the philosopher could be understood from two rather distinct sources. One is through his Rubaiyat and the other through his own works in light of the intellectual and social conditions of his time.^[43] The latter could be informed by the evaluations of Khayyám's works by scholars and philosophers such as Abul-Fazl Bayhaqi, Nizami Aruzi, and al-Zamakhshari and Sufi poets and writers Attar of Nishapur and Najm-al-Din Razi.

As a mathematician, Khayyám has made fundamental contributions to the Philosophy of mathematics especially in the context of Persian Mathematics and Persian philosophy with which most of the other Persian scientists and philosophers such as Avicenna, Abū Rayḥān al-Bīrūnī and Tusi are associated. There are at least three basic mathematical ideas of strong philosophical dimensions that can be associated with Khayyám.

1. Mathematical order: From where does this order issue, and why does it correspond to the world of nature? His answer is in one of his philosophical "treatises on being". Khayyám's answer is that "the Divine Origin of all existence not only emanates wujud "being", by virtue of which all things gain reality, but It is the source of order that is inseparable from the very act of existence."^[43]
2. The significance of axioms in geometry and the necessity for the mathematician to rely upon philosophy and hence the importance of the relation of any particular science to prime philosophy. This is the philosophical background to Khayyám's total rejection of any attempt to "prove" the parallel postulate, and in turn his refusal to bring motion into the attempt to prove this postulate, as had Ibn al-Haytham, because Khayyám associated motion with the world of matter, and wanted to keep it away from the purely intelligible and immaterial world of geometry.^[43]
3. Clear distinction made by Khayyám, on the basis of the work of earlier Persian philosophers such as Avicenna, between natural bodies and mathematical bodies. The first is defined as a body that is in the category of substance and that stands by itself, and hence a subject of natural sciences, while the second, called "volume", is of the category of accidents (attributes) that do not subsist by themselves in the external world and hence is the concern of mathematics. Khayyám was very careful to respect the boundaries of each discipline, and criticized ibn al-Haytham in his proof of the parallel postulate precisely because he had broken this rule and had brought a subject belonging to natural philosophy, that is, motion, which belongs to natural bodies, into the domain of geometry, which deals with mathematical bodies.^[43]

Legacy[link]

• A lunar crater Omar Khayyam was named after him in 1970.
• A minor planet called 3095 Omarkhayyam, discovered by Soviet astronomer Lyudmila Zhuravlyova in 1980, is named after him.^[44]

See also[link]

	Iran portal
	Biography portal
	Poetry portal

• Astronomy in medieval Islam
• Mathematics in medieval Islam
• Nozhat al-Majales
• Omar Khayyam (film)
• The Keeper: The Legend of Omar Khayyam

Notes[link]

References[link]

External links[link]