5 ideas
13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
Full Idea: Archimedes gave a sort of definition of 'straight line' when he said it is the shortest line between two points. | |
From: report of Archimedes (fragments/reports [c.240 BCE]) by Gottfried Leibniz - New Essays on Human Understanding 4.13 | |
A reaction: Commentators observe that this reduces the purity of the original Euclidean axioms, because it involves distance and measurement, which are absent from the purest geometry. |
9548 | A mathematical object exists if there is no contradiction in its definition [Waterfield] |
Full Idea: A mathematical object exists provided there is no contradiction implied in its definition. | |
From: Robin Waterfield (Introduction to 'Hippias Minor' [1987], p.44), quoted by Charles Chihara - A Structural Account of Mathematics 1.4 | |
A reaction: A rather bizarre criterion for existence. Not one, for example, that you would consider applying to the existence of physical objects! But then Poincaré is the father of 'conventionalism', rather than being a platonist. |
6900 | A prior understanding of beauty is needed to assert that the Form of the Beautiful is beautiful [Westaway] |
Full Idea: If it were asserted that the Form of the Beautiful was itself beautiful, such a statement would require a prior understanding of the concept of beauty, so would immediately lead to an infinite regress, so the Forms can't be self-predicating. | |
From: Luke Westaway (talk [2005]), quoted by PG - Db (ideas) | |
A reaction: This is a nice clear statement of the mess that Plato gets himself into if he wants the Forms to be self-predicating. Clearly the Form of the Beautiful can't be beautiful, but must be that which gives other things their beauty. |
6956 | At what point does an object become 'whole'? [Westaway] |
Full Idea: At what point does an object become 'whole'? | |
From: Luke Westaway (talk [2005]), quoted by PG - Db (ideas) | |
A reaction: This nice question strikes me as the central one in mereology. It is tempting to reply that the matter is entirely conventional, but there seems an obvious fact about something missing if one piece is absent from a jigsaw, or a cube is chipped. |
7335 | The Chinese Room should be able to ask itself questions in Mandarin [Westaway] |
Full Idea: If the Chinese Room is functionally equivalent to a Mandarin speaker, it ought to be able to ask itself questions in Mandarin (and it can't). | |
From: Luke Westaway (talk [2005]), quoted by PG - Db (ideas) | |
A reaction: Searle might triumphantly say that this proves there is no understanding in the room, but the objection won't go away, because the room is presumably functionally equivalent to a speaker, and not just a mere translator (who might use mechanical tricks). |