display all the ideas for this combination of philosophers
13 ideas
| 13163 | Circles must be bounded, so cannot be infinite [Leibniz] |
| Full Idea: An infinite circle is impossible, since any circle is bounded by its circumference. | |
| From: Gottfried Leibniz (Dialogue on human freedom and origin of evil [1695], p.114) | |
| A reaction: This is interesting if one is asking what the essence of a circle must be. If is tempting to say merely that the radii must be equal, but can they have the length of some vast transfinite number? The circumference must be 2π bigger. |
| 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. |
| 12920 | There is no multiplicity without true units [Leibniz] |
| Full Idea: There is no multiplicity without true units. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30) | |
| A reaction: Hence real numbers do not embody 'multiplicity'. So either they don't 'embody' anything, or they embody 'magnitudes'. Does this give two entirely different notions, of measure of multiplicity and measures of magnitude? |
| 9147 | Number cannot be defined as addition of ones, since that needs the number; it is a single act of abstraction [Fine,K on Leibniz] |
| Full Idea: Leibniz's talk of the addition of ones cannot define number, since it cannot be specified how often they are added without using the number itself. Number must be an organic unity of ones, achieved by a single act of abstraction. | |
| From: comment on Gottfried Leibniz (works [1690]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §1 | |
| A reaction: I doubt whether 'abstraction' is the right word for this part of the process. It seems more like a 'gestalt'. The first point is clearly right, that it is the wrong way round if you try to define number by means of addition. |
| 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'. |
| 19390 | Everything is subsumed under number, which is a metaphysical statics of the universe, revealing powers [Leibniz] |
| Full Idea: There is nothing which is not subsumable under number; number is therefore a fundamental metaphysical form, and arithmetic a sort of statics of the universe, in which the powers of things are revealed. | |
| From: Gottfried Leibniz (Towards a Universal Characteristic [1677], p.17) | |
| A reaction: I take numbers to be a highly generalised and idealised description of an aspect of reality (seen as mainly constituted by countable substances). Seeing reality as processes doesn't lead us to number. So I like this idea. |
| 19406 | I strongly believe in the actual infinite, which indicates the perfections of its author [Leibniz] |
| Full Idea: I am so much for the actual infinite that instead of admitting that nature abhors it, as is commonly said, I hold that it affects nature everywhere in order to indicate the perfections of its author. | |
| From: Gottfried Leibniz (Reply to Foucher [1693], p.99) | |
| A reaction: I would have thought that, for Leibniz, while infinities indicate the perfections of their author, that is not the reason why they exist. God wasn't, presumably, showing off. Leibniz does not think we can actually know these infinities. |
| 13190 | I don't admit infinite numbers, and consider infinitesimals to be useful fictions [Leibniz] |
| Full Idea: Notwithstanding my infinitesimal calculus, I do not admit any real infinite numbers, even though I confess that the multitude of things surpasses any finite number, or rather any number. ..I consider infinitesimal quantities to be useful fictions. | |
| From: Gottfried Leibniz (Letters to Samuel Masson [1716], 1716) | |
| A reaction: With the phrase 'useful fictions' we seem to have jumped straight into Harty Field. I'm with Leibniz on this one. The history of mathematics is a series of ingenious inventions, whenever they seem to make further exciting proofs possible. |
| 19375 | The continuum is not divided like sand, but folded like paper [Leibniz, by Arthur,R] |
| Full Idea: Leibniz said the division of the continuum should not be conceived 'to be like the division of sand into grains, but like that of a tunic or a sheet of paper into folds'. | |
| From: report of Gottfried Leibniz (works [1690], A VI iii 555) by Richard T.W. Arthur - Leibniz | |
| A reaction: This from the man who invented calculus. This thought might apply well to the modern physicist's concept of a 'field'. |
| 18081 | Nature uses the infinite everywhere [Leibniz] |
| Full Idea: Nature uses the infinite in everything it does. | |
| From: Gottfried Leibniz (works [1690]), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.1 | |
| A reaction: [The quote can be tracked through Kitcher's footnote] He seems to have had in mind the infinitely small. |
| 18080 | A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz] |
| Full Idea: We have only to keep in mind that to find a tangent means to draw a line that connects two points of a curve at an infinitely small distance. | |
| From: Gottfried Leibniz (works [1690]), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.1 | |
| A reaction: [The quote can be tracked through Kitcher's footnote] |
| 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. |
| 23026 | We know mathematical axioms, such as subtracting equals from equals leaves equals, by a natural light [Leibniz] |
| Full Idea: It is by the natural light that the axioms of mathematics are recognised. If we take away the same quantity from two equal things, …a thing we can easily predict without having experienced it. | |
| From: Gottfried Leibniz (Letters to Queen Charlotte [1702], p.189) | |
| A reaction: He also says two equal weights will keep a balance level. Plato thinks his slave boy understands halving an area by the natural light, but that is just as likely to be experience. It is too easy to attribut thoughts to a 'natural light'. |