6 ideas
| 10245 | One geometry cannot be more true than another [Poincaré] |
| Full Idea: One geometry cannot be more true than another; it can only be more convenient. | |
| From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics | |
| A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate. |
| 9159 | You can't simply convert geometry into algebra, as some spatial content is lost [Burge] |
| Full Idea: Although one can translate geometrical propositions into algebraic ones and produce equivalent models, the meaning of geometrical propositions seems to me to be thereby lost. Pure geometry involves spatial content, even if abstracted from physical space. | |
| From: Tyler Burge (Frege on Apriority [2000], IV) | |
| A reaction: This supports Frege's view (against Quine) that geometry won't easily fit into the programme of logicism. I agree with Burge. You would be focusing on the syntax of geometry, and leaving out the semantics. |
| 5040 | Necessary truths can be analysed into original truths; contingent truths are infinitely analysable [Leibniz] |
| Full Idea: Derivative truths are of two sorts: some are analysed into original truths, others admit of an infinite process of analysis. The former are necessary, the latter are contingent. | |
| From: Gottfried Leibniz (On Freedom [1689], p.108) | |
| A reaction: An intriguing proposal. Hume would presumably see contingent truths as being analysed until you reach 'impressions'. Analysis of necessary truths soon comes to the blinding light of what is obvious, but analysis of contingency never gets there. |
| 13159 | Only God sees contingent truths a priori [Leibniz] |
| Full Idea: Only God sees contingent truths a priori. | |
| From: Gottfried Leibniz (On Freedom [1689], p.95) | |
| A reaction: This because everything is interconnected, and the whole picture must be seen to understand a contingent truth. |
| 5039 | If non-existents are possible, their existence would replace what now exists, which cannot therefore be necessary [Leibniz] |
| Full Idea: If certain possibles never exist, then existing things are not always necessary; otherwise it would be impossible for other things to exist instead of them, and so all things that never exist would be impossible. | |
| From: Gottfried Leibniz (On Freedom [1689], p.106) | |
| A reaction: A neat argument, though it is not self-evident that when possibles came into existence they would have to replace what is already there. Can't something be possible, but only in another world, because this one is already booked? |
| 5041 | God does everything in a perfect way, and never acts contrary to reason [Leibniz] |
| Full Idea: We can regard it as certain that everything is done by God in the most perfect way, that he does nothing which is contrary to reason. | |
| From: Gottfried Leibniz (On Freedom [1689], p.109) | |
| A reaction: The famous optimism which Voltaire laughed at in 'Candide'. I can't help thinking that there is an ideal of God being ABOVE reason. We reason, and give reasons, because we are unsure, and life is a struggle. The highest ideal is mystically self-evident. |