20 ideas
12780 | We can grasp the wisdom of God a priori [Leibniz] |
Full Idea: We can grasp the wisdom of God a priori, and not from the order of the phenomena alone. ... For the senses put nothing forward concerning metaphysical matters. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.05.29) | |
A reaction: Nice instance of the aspirations of big metaphysics, before Kant cut it down to size. The claim is not far off Plato's, that by dialectic we can work out the necessities of the Forms, to which even the gods must bow. Are necessities really kept from us? |
13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3 | |
A reaction: [see Hart 73-74 for an explication of this] |
12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11 |
18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}). | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals. | |
From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3 |
12774 | Without a substantial chain to link monads, they would just be coordinated dreams [Leibniz] |
Full Idea: If that substantial chain [vinculum substantiale] for monads did not exist, all bodies, together with all of their qualities, would be nothing but well-founded phenomena, like a rainbow or an image in a mirror, continual dreams perfectly in agreement. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1712.02.05) | |
A reaction: [The first appearance, apparently, of the 'susbtantial chain' in his writings] I take this to be a hugely significant move, either a defeat for monads, or the arrival of common sense. Spiritual monads must unify things, so they can't just be 'parallel'. |
12777 | Monads do not make a unity unless a substantial chain is added to them [Leibniz] |
Full Idea: Monads do not constitute a complete composite substance, since they make up, not something one per se, but only a mere aggregate, unless some substantial chain is added. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1712.05.26) | |
A reaction: This is the clearest statement in the Des Bosses letters of the need for something extra to unite monads. Since the main role of monads was to replace substances, which are only postulated to provide unity, this is rather a climb-down. |
12782 | Monads control nothing outside of themselves [Leibniz] |
Full Idea: Monads aren't a principle of operation for things outside of themselves. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.05.29) | |
A reaction: This is why Leibniz has got into a tangle, and is proposing his 'substantial chain' to join the monads together. I suspect that he would have dumped monads if he had lived a bit longer. |
12778 | There is active and passive power in the substantial chain and in the essence of a composite [Leibniz] |
Full Idea: I do not say there is a chain midway between matter and form, but that the substantial form and primary matter of the composite, in the Scholastic sense (the primitive power, active and passive) are in the chain, and in the essence of the composite. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.05.29) | |
A reaction: Note that this implies an essence of primitive power, and not just a collection of all properties. This is the clearest account in these letters of the nature of the 'substantial chain' he has added to his monads. |
12783 | Primitive force is what gives a composite its reality [Leibniz] |
Full Idea: The first entelechy of a composite is a constitutive part of the composite substance, namely its primitive force. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.05.29) | |
A reaction: For me, Leibniz's most interesting proposal is to characterise Aristotelian 'form' as an active thing, which offers an intrinsic account of movement, and a bottom level for explanations. There always remains the inexplicable. Why anything? Why this? |
12775 | Things seem to be unified if we see duration, position, interaction and connection [Leibniz] |
Full Idea: Important relations are duration (order of successive things) and position (order of coexisting things) and interaction. Position without a thing mediating is presence. Beyond these is connection when things move one another. Thus things seem to be one. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1712.02.05) | |
A reaction: [compressed] This is the best account I can find of his epistemological angle on the unity of things. They are symptoms of the inner power of unification, and he says that God sees these relations most clearly. |
12776 | Every substance is alive [Leibniz] |
Full Idea: Every substance is alive. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1712.02.05) | |
A reaction: The most charitable interpretation of this is that substances are what have unity, and the best model of unity that we can grasp is the unity of an organism. The less charitable view is that he literally thinks a pebble is 'alive'. Hm. |
12753 | A substantial bond of powers is needed to unite composites, in addition to monads [Leibniz] |
Full Idea: Some realising thing must bring it about that composite substance contains something substantial besides monads, otherwise composites will be mere phenomena. The scholastics' active and passive powers are the substantial bond I am urging. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.01.13), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 9 | |
A reaction: [compressed] This appears to be a major retreat, in the last year of Leibniz's life, from the full monadology he had espoused. How do monads connect to matter, and thus unify it? He is returning to Aristotelian hylomorphism. |
12781 | A composite substance is a mere aggregate if its essence is just its parts [Leibniz] |
Full Idea: An aggregate, but not a composite substance, is resolved into parts. A composite substance only needs the coming together of parts, but is not essentially constituted by them, otherwise it would be an aggregate. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.05.29) | |
A reaction: The point is that there is more to some things than there mere parts. Only some unifying principle, in addition to the mere parts, bestows a unity. Mereology is a limited activity if it has nothing to say about this issue. |
12779 | There is a reason why not every possible thing exists [Leibniz] |
Full Idea: There is a reason why not every possible thing exists. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.05.29) | |
A reaction: This is the kind of wonderful speculative metaphysical remark that we are not allowed to make any more. Needless to say, he doesn't tell us what the reason is. Overcrowding, perhaps. |
12785 | Truth is mutually agreed perception [Leibniz] |
Full Idea: In the mutual agreement of perceivers consists the truth of the phenomena. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.05.29) | |
A reaction: This remark is startling close to the 'perspectivism' that crops up in the late notebooks of Nietzsche. Leibniz was keen on relativism in many areas, starting with the nature of space. I personally think Leibniz meant 'knowledge' rather than 'truth'. |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
Full Idea: Archelaus was the first person to say that the universe is boundless. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |
12784 | Allow no more miracles than are necessary [Leibniz] |
Full Idea: Miracles should not be increased beyond necessity. | |
From: Gottfried Leibniz (Letters to Des Bosses [1715], 1716.05.29) | |
A reaction: Leibniz defends miracles (where Spinoza dismisses them). This remark is, of course, an echo of Ockham's Razor, that 'entities' should not be multiplied beyond necessity. It is hard to disagree with his proposal. Zero might be result, though. |