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The geometry of emptiness or waiting for the wine!
To Omar Khayyám (1048 – 1131) and Imre Lakatos (1922-1974)
josé
Imre Lakatos - Proofs and Refutations
Proofs and Refutations is a book by the philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs. This gives mathematics a somewhat experimental flavour. At the end of the Introduction, Lakatos explains that his purpose is to challenge formalism in mathematics, and to show that informal mathematics grows by a logic of "proofs and refutations".
Many important logical ideas are explained in the book. For example the difference between a counterexample to a lemma (a so-called 'local counterexample') and a counterexample to the specific conjecture under attack (a 'global counterexample' to the Euler characteristic, in this case) is discussed.
Though the book is written as a narrative, an actual method of investigation, that of "proofs and refutations", is developed. In Appendix I, Lakatos summarizes this method by the following list of stages:
Primitive conjecture.
Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures).
"Global" counterexamples (counterexamples to the primitive conjecture) emerge.
Proof re-examined: the "guilty lemma" to which the global counter-example is a "local" counterexample is spotted. This guilty lemma may have previously remained "hidden" or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem - the improved conjecture - supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.
He goes on and gives further stages that might sometimes take place:
Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at cross-roads of different proofs, and thus emerge as of basic importance.
The hitherto accepted consequences of the original and now refuted conjecture are checked.
Counterexamples are turned into new examples - new fields of inquiry open up.
wikipedia
Omar Khayyám; born Ghiyāth ad-Dīn Abu'l-Fatḥ ʿUmar ibn Ibrāhīm al-Khayyām Nīshāpūrī (/ˈoʊmɑr kaɪˈjɑːm, -ˈjæm, ˈoʊmər/; Persian: غیاث الدین ابوالفتح عمر ابراهیم خیام نیشابورﻯ, pronounced [xæjˈjɒːm]; 18 May 1048 – 4 December 1131), was a Persian mathematician, astronomer, philosopher, and poet. He also wrote treatises on mechanics, geography, mineralogy, music, and Islamic theology.
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 Islamic Mathematics that was eventually transmitted to Europe. In particular, he derived general methods for solving cubic equations and even some higher orders.
"Cubic equation and intersection of conic sections" the first page of two-chaptered manuscript kept in Tehran University
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". 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 created important works on geometry, specifically on the theory of proportions. His notable contemporary mathematicians included Al-Khazini and Abu Hatim al-Muzaffar ibn Ismail al-Isfizari
Theory of parallels
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." This proposition 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. 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 the first treatment of the parallels axiom not based on petitio principii, but on a 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. In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate,
“There was a water-drop, it joined the sea,
A speck of dust, it was fused with earth;
what of your entering and leaving this world?
A fly appeared, and disappeared.”
― Omar Khayyam
Music:
Omar Khayyam Musical Special
Festival of Fez, Morocco
https://youtu.be/tWnUbjbWykU
The geometry of emptiness or waiting for the wine!
To Omar Khayyám (1048 – 1131) and Imre Lakatos (1922-1974)
josé
Imre Lakatos - Proofs and Refutations
Proofs and Refutations is a book by the philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs. This gives mathematics a somewhat experimental flavour. At the end of the Introduction, Lakatos explains that his purpose is to challenge formalism in mathematics, and to show that informal mathematics grows by a logic of "proofs and refutations".
Many important logical ideas are explained in the book. For example the difference between a counterexample to a lemma (a so-called 'local counterexample') and a counterexample to the specific conjecture under attack (a 'global counterexample' to the Euler characteristic, in this case) is discussed.
Though the book is written as a narrative, an actual method of investigation, that of "proofs and refutations", is developed. In Appendix I, Lakatos summarizes this method by the following list of stages:
Primitive conjecture.
Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures).
"Global" counterexamples (counterexamples to the primitive conjecture) emerge.
Proof re-examined: the "guilty lemma" to which the global counter-example is a "local" counterexample is spotted. This guilty lemma may have previously remained "hidden" or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem - the improved conjecture - supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.
He goes on and gives further stages that might sometimes take place:
Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at cross-roads of different proofs, and thus emerge as of basic importance.
The hitherto accepted consequences of the original and now refuted conjecture are checked.
Counterexamples are turned into new examples - new fields of inquiry open up.
wikipedia
Omar Khayyám; born Ghiyāth ad-Dīn Abu'l-Fatḥ ʿUmar ibn Ibrāhīm al-Khayyām Nīshāpūrī (/ˈoʊmɑr kaɪˈjɑːm, -ˈjæm, ˈoʊmər/; Persian: غیاث الدین ابوالفتح عمر ابراهیم خیام نیشابورﻯ, pronounced [xæjˈjɒːm]; 18 May 1048 – 4 December 1131), was a Persian mathematician, astronomer, philosopher, and poet. He also wrote treatises on mechanics, geography, mineralogy, music, and Islamic theology.
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 Islamic Mathematics that was eventually transmitted to Europe. In particular, he derived general methods for solving cubic equations and even some higher orders.
"Cubic equation and intersection of conic sections" the first page of two-chaptered manuscript kept in Tehran University
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". 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 created important works on geometry, specifically on the theory of proportions. His notable contemporary mathematicians included Al-Khazini and Abu Hatim al-Muzaffar ibn Ismail al-Isfizari
Theory of parallels
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." This proposition 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. 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 the first treatment of the parallels axiom not based on petitio principii, but on a 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. In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate,
“There was a water-drop, it joined the sea,
A speck of dust, it was fused with earth;
what of your entering and leaving this world?
A fly appeared, and disappeared.”
― Omar Khayyam
Music:
Omar Khayyam Musical Special
Festival of Fez, Morocco
https://youtu.be/tWnUbjbWykU
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