| Title | Why Does Plato's Element Theory Conflict With Mathematics (Arist. Cael. 299a2-6)? |
| Type | Article |
| Language | English |
| Date | 2003 |
| Journal | Rheinisches Museum für Philologie |
| Volume | 146 |
| Issue | 3/4 |
| Pages | 328-345 |
| Categories | no categories |
| Author(s) | Kouremenos, Theokritos |
| Editor(s) | |
| Translator(s) |
| In Cael. 3.1 Aristotle argues against those who posit that all bodies are generated because they are made from, and dissolve into, planes, namely Plato and perhaps other members of the Academy who subscribed to the Timaeus physics (cf. Simplicius, In Cael. 561,8-11 [Heiberg]). In his Timaeus Plato assigns to each of the traditional Empedoclean elements a regular polyhedron: the tetrahedron or pyramid to fire, the cube to earth, the octahedron to air, and the icosahedron to water. Each regular polyhedron can be anachronistically called a molecule of the element in question, and, as is suggested by the analogy between the regular solids and molecules, Plato also posits that the regular polyhedra are made from 'atoms': the faces of the tetrahedron, octahedron, and icosahedron are made from scalene right-angled triangles, whose hypotenuses are double the length of the smaller sides, whereas the faces of the cube consist of isosceles right-angled triangles. Since fire, air, and water consist of polyhedral molecules whose elementary constituents are of the same type, they can freely change into one another. Any of these three elements turns into another when its molecules break down into their elementary constituents, and these building blocks recombine into molecules of another element. Aristotle has in mind the reshuffling of elementary triangles when he refers to all bodies being made from, and dissolving into, planes. His first objection to this fundamental assumption in Plato's element theory is set out in Cael. 299a2-6: as is easily seen, constructing bodies from planes runs counter to mathematics whose 'hypotheses' should be accepted, unless one comes up with something more convincing. Contrary to Aristotle's claim, it is not easy to see why Plato's element theory runs counter to mathematics because it constructs the polyhedral molecules from the triangular planes in the faces of these molecules. Aristotle presumably implies that this violates some mathematical 'hypotheses' which should be better left as they stand but does not explain what the 'hypotheses' in question are. Nor is it any clearer whether Plato commits himself to the rejection of these 'hypotheses' or some aspect of Plato's element theory entails their rejection by Aristotle's own lights. I will attempt to answer these questions after a critique of Simplicius who identifies the hypotheses in Cael. 299a2-6 with the Euclidean definitions of point, line, and plane but also thinks that Aristotle sets out further mathematical objections to Plato's element theory in Cael. 299a6-11: contrary to the commentator, there is only one such objection in Cael. 299a6-11, namely that Plato's element theory introduces indivisible lines, and, as is suggested by an allusion to Cael. 299a2-6 in the treatise On Indivisible Lines, the same objection is also implicit in Cael. 299a2-6. That in this passage Plato's element theory is said to conflict with mathematics because it entails the existence of indivisible lines is borne out not only by Cael. 299a6-11 but also by 299a13-17. After interpreting the 'hypotheses' in Cael. 299a2-6 consistently with this fact, I will show that, when Aristotle charges Plato with introducing various sorts of indivisibles in his element theory, he actually brings out the untenability of this theory by arguing that Plato ought to introduce such entities which are, though, ruled out by mathematics. Aristotle's implicit objection in Cael. 299a2-6 follows from a similar argument which I will attempt to reconstruct in the final sections of this paper. [introduction p. 328-329] |
| Online Resources | https://uni-koeln.sciebo.de/s/9EHiPSWuW9oh0c4 |
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| Title | Why Does Plato's Element Theory Conflict With Mathematics (Arist. Cael. 299a2-6)? |
| Type | Article |
| Language | English |
| Date | 2003 |
| Journal | Rheinisches Museum für Philologie |
| Volume | 146 |
| Issue | 3/4 |
| Pages | 328-345 |
| Categories | no categories |
| Author(s) | Kouremenos, Theokritos |
| Editor(s) | |
| Translator(s) |
In Cael. 3.1 Aristotle argues against those who posit that all bodies are generated because they are made from, and dissolve into, planes, namely Plato and perhaps other members of the Academy who subscribed to the Timaeus physics (cf. Simplicius, In Cael. 561,8-11 [Heiberg]). In his Timaeus Plato assigns to each of the traditional Empedoclean elements a regular polyhedron: the tetrahedron or pyramid to fire, the cube to earth, the octahedron to air, and the icosahedron to water. Each regular polyhedron can be anachronistically called a molecule of the element in question, and, as is suggested by the analogy between the regular solids and molecules, Plato also posits that the regular polyhedra are made from 'atoms': the faces of the tetrahedron, octahedron, and icosahedron are made from scalene right-angled triangles, whose hypotenuses are double the length of the smaller sides, whereas the faces of the cube consist of isosceles right-angled triangles. Since fire, air, and water consist of polyhedral molecules whose elementary constituents are of the same type, they can freely change into one another. Any of these three elements turns into another when its molecules break down into their elementary constituents, and these building blocks recombine into molecules of another element. Aristotle has in mind the reshuffling of elementary triangles when he refers to all bodies being made from, and dissolving into, planes. His first objection to this fundamental assumption in Plato's element theory is set out in Cael. 299a2-6: as is easily seen, constructing bodies from planes runs counter to mathematics whose 'hypotheses' should be accepted, unless one comes up with something more convincing. Contrary to Aristotle's claim, it is not easy to see why Plato's element theory runs counter to mathematics because it constructs the polyhedral molecules from the triangular planes in the faces of these molecules. Aristotle presumably implies that this violates some mathematical 'hypotheses' which should be better left as they stand but does not explain what the 'hypotheses' in question are. Nor is it any clearer whether Plato commits himself to the rejection of these 'hypotheses' or some aspect of Plato's element theory entails their rejection by Aristotle's own lights. I will attempt to answer these questions after a critique of Simplicius who identifies the hypotheses in Cael. 299a2-6 with the Euclidean definitions of point, line, and plane but also thinks that Aristotle sets out further mathematical objections to Plato's element theory in Cael. 299a6-11: contrary to the commentator, there is only one such objection in Cael. 299a6-11, namely that Plato's element theory introduces indivisible lines, and, as is suggested by an allusion to Cael. 299a2-6 in the treatise On Indivisible Lines, the same objection is also implicit in Cael. 299a2-6. That in this passage Plato's element theory is said to conflict with mathematics because it entails the existence of indivisible lines is borne out not only by Cael. 299a6-11 but also by 299a13-17. After interpreting the 'hypotheses' in Cael. 299a2-6 consistently with this fact, I will show that, when Aristotle charges Plato with introducing various sorts of indivisibles in his element theory, he actually brings out the untenability of this theory by arguing that Plato ought to introduce such entities which are, though, ruled out by mathematics. Aristotle's implicit objection in Cael. 299a2-6 follows from a similar argument which I will attempt to reconstruct in the final sections of this paper. [introduction p. 328-329] |
| Online Resources | https://uni-koeln.sciebo.de/s/9EHiPSWuW9oh0c4 |
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