- A brief account of the history of logic, from the The Oxford Companion to Philosophy (edited by Ted Honderich), OUP 1997, 497-500.
- A biography of Peter Abelard, published in the Dictionary of Literary Biography Vol. 115, edited by Jeremiah Hackett, Detroit: Gale Publishing, 3-15.
- Philosophy in the Latin Christian West, 750-1050, in A Companion to Philosophy in the Middle Ages, edited by Jorge Gracia and Tim Noone, Blackwell 2003, 32-35.
- Ockham wielding his razor!
- Review of The Beatles Anthology, Chronicle Books 2000 (367pp).
- A brief discussion note about Susan James, Passion and Action: The Emotions in Seventeenth-Century Philosophy.
- Review of St. Thomas Aquinas by Ralph McInerny, University of Notre Dame Press 1982 (172pp). From International Philosophical Quarterly23 (1983), 227-229.
- Review of William Heytesbury on Maxima and Minima by John Longeway, D.Reidel 1984 (x+201pp). From The Philosophical Review 96 (1987), 146-149.
- Review of That Most Subtle Question by D. P. Henry, Manchester University Press 1984 (xviii+337pp). From The Philosophical Review 96 (1987), 149-152.
- Review of Introduction to the Problem of Individuation in the Early Middle Ages by Jorge Gracia, Catholic University of America Press 1984 (303pp). From The Philosophical Review 97 (1988), 564-567.
- Review of Introduction to Medieval Logic by Alexander Broadie, OUP 1987 (vi+150pp). From The Philosophical Review 99 (1990), 299-302.
Friday, December 18, 2009
"Category Theory" by Samuel Eilenberg and Saunders Mac Lane,
Category theory, alongside set theory, serves as a universal language of modern mathematics. Categories, functors, and natural transformations are widely used in all areas of mathematics, allowing us to look uniformly and consistently on various constructions and formulate the general properties of diverse structures. The impact of category theory is irreducible to the narrow frameworks of its great expressive conveniences. This theory has drastically changed our general outlook on the foundations of mathematics and widened the room of free thinking in mathematics.
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Set theory, a great and ingenious creation of Georg Cantor, occupies in the common opinion of the 20th century the place of the sole solid base of modern mathematics. Mathematics becomes sinking into a section of the Cantorian set theory. Most active mathematicians, teachers, and philosophers consider as obvious and undisputable the thesis that mathematics cannot be grounded on anything but set theory. The set-theoretic stance transforms paradoxically into an ironclad dogma, a clear-cut forbiddance of thinking (as L. Feuerbach once put it wittily). Such an indoctrinated view of the foundations of mathematics is false and conspicuously contradicts the leitmotif, nature, and pathos of the essence of all creative contribution of Cantor who wrote as far back as in 1883 that “denn das Wesen der Mathematik liegt gerade in ihrer Freiheit.”
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Samuel Eilenberg (September 30, 1913—January 30, 1998) was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.
He earned his Ph.D. from Warsaw University in 1936. His thesis advisor was Karol Borsuk. His main interest was algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names the Eilenberg-Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory.
Eilenberg took part in the Bourbaki group meetings, and, with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic.
Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules.
Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced by Eilenberg in 1974.
Eilenberg was also a prominent collector of Asian art. His collection mainly consisted of small sculptures and other artifacts from India, Indonesia, Pakistan, Nepal, Thailand, Cambodia, Sri Lanka and Central Asia. In 1991-1992, the Metropolitan Museum of Art in New York staged an exhibition from more than 400 items that Eilenberg had donated to the museum, entitled The Lotus Transcendent: Indian and Southeast Asian Art From the Samuel Eilenberg Collection".
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The X-machine (XM) is a theoretical model of computation introduced by Samuel Eilenberg in 1974.
The X in "X-machine" represents the fundamental data type on which the machine operates; for example, a machine that operates on databases (objects of type database) would be a database-machine.
The X-machine model is structurally the same as the finite state machine, except that the symbols used to label the machine's transitions denote relations of type X→X. Crossing a transition is equivalent to applying the relation that labels it (computing a set of changes to the data type X), and traversing a path in the machine corresponds to applying all the associated relations, one after the other.
Interest in the X-machine was revived in the late 1980s by Mike Holcombe[2], who noticed that the model was ideal for software formal specification purposes, because it cleanly separates control flow from processing. Provided one works at a sufficiently abstract level, the control flows in a computation can usually be represented as a finite state machine, so to complete the X-machine specification all that remains is to specify the processing associated with each of the machine's transitions. The structural simplicity of the model makes it extremely flexible; other early illustrations of the idea included Holcombe's specification of human-computer interfaces,his modelling of processes in cell biochemistry, and Stannett's modelling of decision-making in military command systems.
X-machines have received renewed attention since the mid-1990s, when Gilbert Laycock's deterministic Stream X-Machine was found to serve as the basis for specifying large software systems that are completely testable. Another variant, the Communicating Stream X-Machine offers a useful testable model for biological processes[8] and future swarm-based satellite systems.
.
Set theory, a great and ingenious creation of Georg Cantor, occupies in the common opinion of the 20th century the place of the sole solid base of modern mathematics. Mathematics becomes sinking into a section of the Cantorian set theory. Most active mathematicians, teachers, and philosophers consider as obvious and undisputable the thesis that mathematics cannot be grounded on anything but set theory. The set-theoretic stance transforms paradoxically into an ironclad dogma, a clear-cut forbiddance of thinking (as L. Feuerbach once put it wittily). Such an indoctrinated view of the foundations of mathematics is false and conspicuously contradicts the leitmotif, nature, and pathos of the essence of all creative contribution of Cantor who wrote as far back as in 1883 that “denn das Wesen der Mathematik liegt gerade in ihrer Freiheit.”
====================================
Samuel Eilenberg (September 30, 1913—January 30, 1998) was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.
He earned his Ph.D. from Warsaw University in 1936. His thesis advisor was Karol Borsuk. His main interest was algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names the Eilenberg-Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory.
Eilenberg took part in the Bourbaki group meetings, and, with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic.
Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules.
Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced by Eilenberg in 1974.
Eilenberg was also a prominent collector of Asian art. His collection mainly consisted of small sculptures and other artifacts from India, Indonesia, Pakistan, Nepal, Thailand, Cambodia, Sri Lanka and Central Asia. In 1991-1992, the Metropolitan Museum of Art in New York staged an exhibition from more than 400 items that Eilenberg had donated to the museum, entitled The Lotus Transcendent: Indian and Southeast Asian Art From the Samuel Eilenberg Collection".
=====
The X-machine (XM) is a theoretical model of computation introduced by Samuel Eilenberg in 1974.
The X in "X-machine" represents the fundamental data type on which the machine operates; for example, a machine that operates on databases (objects of type database) would be a database-machine.
The X-machine model is structurally the same as the finite state machine, except that the symbols used to label the machine's transitions denote relations of type X→X. Crossing a transition is equivalent to applying the relation that labels it (computing a set of changes to the data type X), and traversing a path in the machine corresponds to applying all the associated relations, one after the other.
Interest in the X-machine was revived in the late 1980s by Mike Holcombe[2], who noticed that the model was ideal for software formal specification purposes, because it cleanly separates control flow from processing. Provided one works at a sufficiently abstract level, the control flows in a computation can usually be represented as a finite state machine, so to complete the X-machine specification all that remains is to specify the processing associated with each of the machine's transitions. The structural simplicity of the model makes it extremely flexible; other early illustrations of the idea included Holcombe's specification of human-computer interfaces,his modelling of processes in cell biochemistry, and Stannett's modelling of decision-making in military command systems.
X-machines have received renewed attention since the mid-1990s, when Gilbert Laycock's deterministic Stream X-Machine was found to serve as the basis for specifying large software systems that are completely testable. Another variant, the Communicating Stream X-Machine offers a useful testable model for biological processes[8] and future swarm-based satellite systems.
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