As Quoted in "The Discovery of Kepler's Laws," Scientific American: Supplement (Apr 29, 1911) Vol. 71, No. 1843, p. 278 https://books.google.com/books?id=ov4-AQAAMAAJ&pg=PA258.
Mysterium Cosmographicum (1596)
“There can be an infinite number of polygons, but only five regular solids.”
37 min 45 sec
Cosmos: A Personal Voyage (1990 Update), The Backbone of Night [Episode 7]
Context: There can be an infinite number of polygons, but only five regular solids. Four of the solids were associated with earth, fire, air and water. The cube for example represented earth. These four elements, they thought, make up terrestrial matter. So the fifth solid they mystically associated with the Cosmos. Perhaps it was the substance of the heavens. This fifth solid was called the dodecahedron. Its faces are pentagons, twelve of them. Knowledge of the dodecahedron was considered too dangerous for the public. Ordinary people were to be kept ignorant of the dodecahedron. In love with whole numbers, the Pythagoreans believed that all things could be derived from them. Certainly all other numbers.
So a crisis in doctrine occurred when they discovered that the square root of two was irrational. That is: the square root of two could not be represented as the ratio of two whole numbers, no matter how big they were. "Irrational" originally meant only that. That you can't express a number as a ratio. But for the Pythagoreans it came to mean something else, something threatening, a hint that their world view might not make sense, the other meaning of "irrational".
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Carl Sagan 365
American astrophysicist, cosmologist, author and science ed… 1934–1996Related quotes
As quoted by James Gow, A Short History of Greek Mathematics https://books.google.com/books?id=9d8DAAAAMAAJ (1884)
Arithmetica (c. 250 AD)
“A complex system can fail in an infinite number of ways”
John Gall
(1925–2014) American physician
Source: General systemantics, an essay on how systems work, and especially how they fail..., 1975, p. 92, cited in: Erik Hollnagel (2004) Barriers and accident prevention. p. 182
Hugh MacDiarmid
(1892–1978) Scottish poet, pen name of Christopher Murray Grieve
Review of Lady Chatterley's Lover (1928)
“The number of rational hypotheses that can explain any given phenomenon is infinite.”
Source: Zen and the Art of Motorcycle Maintenance (1974), Ch. 9; in Ch. 22 (see below) Pirsig recounts finding that Henri Poincaré had made a similar statement decades earlier.
“Motion at low Reynolds number is very majestic, slow, and regular.”
Edward M. Purcell
(1912–1997) American physicist
"Life at Low Reynolds Number" in the American Journal of Physics (January 1977)
“Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum’s composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysic except he who has passed through this [labyrinth].”
Nam filum labyrintho de compositione continui deque maximo et minimo ac indesignabili at que infinito non nisi geometria praebere potest, ad metaphysicam vero solidam nemo veniet, nisi qui illac transiverit.
Gottfried Leibniz
(1646–1716) German mathematician and philosopher
Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae (Spring 1676)
Source: Leibniz, Leibnizens Mathematische Schriften, Herausgegeben Von C.I. Gerhardt. Bd. 1-7. 1850-1863. Halle. The quotation is found in vol. 7. on page 326 in ”Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae”. Link https://archive.org/stream/leibnizensmathe12leibgoog
Source: Geometry and Monadology: Leibniz's Analysis Situs and Philosophy of Space by Vincenzo de Risi. Page 123. Link https://books.google.no/books?id=2ptGkzsKyOQC&lpg=PA123&ots=qz2aKxAYtp&dq=Dissertatio%20Exoterica%20De%20Statu%20Praesenti%20et%20Incrementis%20Novissimis%20Deque%20Usu%20Geometriae%E2%80%9D&hl=no&pg=PA123#v=onepage&q&f=false
Isaac Asimov
(1920–1992) American writer and professor of biochemistry at Boston University, known for his works of science fiction …
I. Asimov: A Memoir (1994)
Context: If I were not an atheist, I would believe in a God who would choose to save people on the basis of the totality of their lives and not the pattern of their words. I think he would prefer an honest and righteous atheist to a TV preacher whose every word is God, God, God, and whose every deed is foul, foul, foul.
I would also want a God who would not allow a Hell. Infinite torture can only be a punishment for infinite evil, and I don't believe that infinite evil can be said to exist even in the case of Hitler. Besides, if most human governments are civilized enough to try to eliminate torture and outlaw cruel and unusual punishments, can we expect anything less of an all-merciful God?
I feel that if there were an afterlife, punishment for evil would be reasonable and of a fixed term. And I feel that the longest and worst punishment should be reserved for those who slandered God by inventing Hell.
