“In considering the role of Plato in mathematics in general, and in relation to the Golden Ratio in particular, we have to examine not just his own purely mathematical contributions, which were not very significant, but the effects of his influence and encouragement on the mathematics of others of his and of later generations. (emphasis added) To some extent, Plato may be considered as one of the first true theoreticians. His theoretical inclinations are best exemplified in his attitude toward astronomy, where, rather than observing the stars in the motions, he advocates to “leave the heavens alone” and to concentrate on the more abstract heaven of mathematics. The latter, according to Plato, is merely represented by the actual stars, in the same way that the abstract entities of a point, a line, and a circle are represented by geometrical drawings….
“There is little doubt that Plato’s guidance was far more important than his direct contributions. A text attributed to Philodemus from the first century reads: ‘Great progress in mathematics [was achieved] during that time, with Plato as the director and problem-giver, and the mathematicians investigating them zealously.’
“Nevertheless, Plato himself certainly had an intense interest in the properties of numbers and of geometrical figures…
“Plato, himself, a pupil of the Pythagoreans, was also aware of the fact that the sum of the cubes of the sides of the famous 3-4-5 Pythagorean triangle is equal to 216.
“Plato and the Golden Section are linked mainly through two areas that are particularly close to his heart: incommensurability and the Platonic solids. In Laws, Plato expresses his own feeling of shame for having learned about incommensurable lengths and irrational numbers relatively late in his life, and he laments the fact that many of the Greeks of his generation were still not familiar with the existence of such numbers.”
Saturday, August 10, 2013
GOLDEN RATIO...EXCERPTS...
GOLDEN RATIO: THE STORY OF PHI, THE WORLD’S MOST ASTONISHING NUMBER, by Mario Livio (Broadway Books, NY:2002), PP.64-66
“In considering the role of Plato in mathematics in general, and in relation to the Golden Ratio in particular, we have to examine not just his own purely mathematical contributions, which were not very significant, but the effects of his influence and encouragement on the mathematics of others of his and of later generations. (emphasis added) To some extent, Plato may be considered as one of the first true theoreticians. His theoretical inclinations are best exemplified in his attitude toward astronomy, where, rather than observing the stars in the motions, he advocates to “leave the heavens alone” and to concentrate on the more abstract heaven of mathematics. The latter, according to Plato, is merely represented by the actual stars, in the same way that the abstract entities of a point, a line, and a circle are represented by geometrical drawings….
“There is little doubt that Plato’s guidance was far more important than his direct contributions. A text attributed to Philodemus from the first century reads: ‘Great progress in mathematics [was achieved] during that time, with Plato as the director and problem-giver, and the mathematicians investigating them zealously.’
“Nevertheless, Plato himself certainly had an intense interest in the properties of numbers and of geometrical figures…
“Plato, himself, a pupil of the Pythagoreans, was also aware of the fact that the sum of the cubes of the sides of the famous 3-4-5 Pythagorean triangle is equal to 216.
“Plato and the Golden Section are linked mainly through two areas that are particularly close to his heart: incommensurability and the Platonic solids. In Laws, Plato expresses his own feeling of shame for having learned about incommensurable lengths and irrational numbers relatively late in his life, and he laments the fact that many of the Greeks of his generation were still not familiar with the existence of such numbers.”
“In considering the role of Plato in mathematics in general, and in relation to the Golden Ratio in particular, we have to examine not just his own purely mathematical contributions, which were not very significant, but the effects of his influence and encouragement on the mathematics of others of his and of later generations. (emphasis added) To some extent, Plato may be considered as one of the first true theoreticians. His theoretical inclinations are best exemplified in his attitude toward astronomy, where, rather than observing the stars in the motions, he advocates to “leave the heavens alone” and to concentrate on the more abstract heaven of mathematics. The latter, according to Plato, is merely represented by the actual stars, in the same way that the abstract entities of a point, a line, and a circle are represented by geometrical drawings….
“There is little doubt that Plato’s guidance was far more important than his direct contributions. A text attributed to Philodemus from the first century reads: ‘Great progress in mathematics [was achieved] during that time, with Plato as the director and problem-giver, and the mathematicians investigating them zealously.’
“Nevertheless, Plato himself certainly had an intense interest in the properties of numbers and of geometrical figures…
“Plato, himself, a pupil of the Pythagoreans, was also aware of the fact that the sum of the cubes of the sides of the famous 3-4-5 Pythagorean triangle is equal to 216.
“Plato and the Golden Section are linked mainly through two areas that are particularly close to his heart: incommensurability and the Platonic solids. In Laws, Plato expresses his own feeling of shame for having learned about incommensurable lengths and irrational numbers relatively late in his life, and he laments the fact that many of the Greeks of his generation were still not familiar with the existence of such numbers.”