display all the ideas for this combination of texts
4 ideas
| 13008 | Geometry, unlike sensation, lets us glimpse eternal truths and their necessity [Leibniz] |
| Full Idea: What I value most in geometry, considered as a contemplative study, is its letting us glimpse the true source of eternal truths and of the way in which we can come to grasp their necessity, which is something confused sensory images cannot reveal. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.12) | |
| A reaction: This is strikingly straight out of Plato. We should not underestimate this idea, though nowadays it is with us, but with geometry replaced by mathematical logic. |
| 12956 | Only whole numbers are multitudes of units [Leibniz] |
| Full Idea: The definition of 'number' as a multitude of units is appropriate only for whole numbers. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.15) | |
| A reaction: One can also define rational numbers by making use of units, but the strategy breaks down with irrational numbers like root-2 and pi. I still say the concept of a unit is the basis of numbers. Without whole numbers, we wouldn't call the real 'numbers'. |
| 12937 | We shouldn't just accept Euclid's axioms, but try to demonstrate them [Leibniz] |
| Full Idea: Far from approving the acceptance of doubtful principles, I want to see an attempt to demonstrate even Euclid's axioms, as some of the ancients tried to do. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.02) | |
| A reaction: This is the old idea of axioms, as a bunch of basic self-evident truths, rather than the modern idea of an economical set of propositions from which to make deductions. Demonstration has to stop somewhere. |
| 16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
| Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it. | |
| From: Plato (Parmenides [c.364 BCE], 144a) | |
| A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed. |