display all the ideas for this combination of texts
8 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. |
| 10573 | Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K] |
| Full Idea: Because of Dedekind's definition of reals by cuts, there is a bizarre modern doctrine that there are many 1's - the natural number 1, the rational number 1, the real number 1, and even the complex number 1. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) | |
| A reaction: See Idea 10572. |
| 10575 | Why should a Dedekind cut correspond to a number? [Fine,K] |
| Full Idea: By what right can Dedekind suppose that there is a number corresponding to any pair of irrationals that constitute an irrational cut? | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) |
| 10574 | Unless we know whether 0 is identical with the null set, we create confusions [Fine,K] |
| Full Idea: What is the union of the singleton {0}, of zero, and the singleton {φ}, of the null set? Is it the one-element set {0}, or the two-element set {0, φ}? Unless the question of identity between 0 and φ is resolved, we cannot say. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) |
| 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. |
| 10560 | Set-theoretic imperialists think sets can represent every mathematical object [Fine,K] |
| Full Idea: Set-theoretic imperialists think that it must be possible to represent every mathematical object as a set. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1) |
| 10568 | Logicists say mathematics can be derived from definitions, and can be known that way [Fine,K] |
| Full Idea: Logicists traditionally claim that the theorems of mathematics can be derived by logical means from the relevant definitions of the terms, and that these theorems are epistemically innocent (knowable without Kantian intuition or empirical confirmation). | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) |