14 ideas
| 19336 | Wisdom involves the desire to achieve perfection [Leibniz] |
| Full Idea: The wiser one is, the more one is determined to do that which is most perfect. | |
| From: Gottfried Leibniz (On the Ultimate Origination of Things [1697], p.151) | |
| A reaction: Debatable. 'Perfectionism' is a well-known vice in many areas of life. Life is short, and the demands on us are many. Skilled shortcuts and compromises are one hallmark of genius, and presumably also of wisdom. |
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 7696 | Leibniz first asked 'why is there something rather than nothing?' [Leibniz, by Jacquette] |
| Full Idea: The historical honour of having first raised the question "Why is there something rather than nothing?" belongs to Leibniz. | |
| From: report of Gottfried Leibniz (On the Ultimate Origination of Things [1697]) by Dale Jacquette - Ontology Ch.3 | |
| A reaction: I presume that people before Leibniz may well have had the thought, but not bothered to even articulate it, because there seemed nothing to say by way of answer, other than some reference to the inscrutable will of God. |
| 19341 | There must be a straining towards existence in the essence of all possible things [Leibniz] |
| Full Idea: Since something rather than nothing exists, there is a certain urge for existence, or (so to speak) a straining toward existence in possible things or in possibility or essence itself; in a word, essence in and of itself strives for existence. | |
| From: Gottfried Leibniz (On the Ultimate Origination of Things [1697], p.150) | |
| A reaction: Thus 'essence precedes existence'. Not sure I understand this, but at least it places an active power at the root of everything (though Leibniz probably sees that as divine). The Big Bang triggered by a 'quantum fluctuation'? |
| 19428 | Because something does exist, there must be a drive in possible things towards existence [Leibniz] |
| Full Idea: From the very fact that something exists rather than nothing, we recognise that there is in possible things, that is, in the very possibility or essence, a certain exigent need of existence, and, so to speak, some claim to existence. | |
| From: Gottfried Leibniz (On the Ultimate Origination of Things [1697], p.347) | |
| A reaction: I love the fact that Leibniz tried to explain why there is something rather than nothing. Bede Rundle and Dale Jacquette are similar heroes. As Leibniz tells us, contradictions have no claim to existence, but non-contradictions do. |
| 12697 | Indivisibles are not parts, but the extrema of parts [Leibniz] |
| Full Idea: Indivisibles are not parts, but the extrema of parts. | |
| From: Gottfried Leibniz (Pacidius Philalethi dialogue [1676], A6.3.565-6), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 1 | |
| A reaction: This is incipient monadology, that the bottom level of division ceases to be parts of a thing, and arrives at a different order of entity, to explain the parts of things. Leibniz denies that this subdivision comes down to points. |
| 5047 | The world is physically necessary, as its contrary would imply imperfection or moral absurdity [Leibniz] |
| Full Idea: Although the world is not metaphysically necessary, such that its contrary would imply a contradiction or logical absurdity, it is necessary physically, that is, determined in such a way that its contrary would imply imperfection or moral absurdity. | |
| From: Gottfried Leibniz (On the Ultimate Origination of Things [1697], p.139) | |
| A reaction: How does Leibniz know things like this? The distinction between 'metaphysical' necessity and 'natural' (what he calls 'physical') necessity is a key idea. But natural necessity is controversial. See 'Essentialism'. |
| 19343 | We follow the practical rule which always seeks maximum effect for minimum cost [Leibniz] |
| Full Idea: In practical affairs one always follows the decision rule in accordance with which one ought to seek the maximum or the minimum: namely, one prefers the maximum effect at the minimum cost, so to speak. | |
| From: Gottfried Leibniz (On the Ultimate Origination of Things [1697], p.150) | |
| A reaction: Animals probably do that too, and even water sort of obeys the rule when it runs downhill. |
| 19429 | The principle of determination in things obtains the greatest effect with the least effort [Leibniz] |
| Full Idea: There is always in things a principle of determination which is based on consideration of maximum and minimum, such that the greatest effect is obtained with the least, so to speak, expenditure. | |
| From: Gottfried Leibniz (On the Ultimate Origination of Things [1697], p.347) | |
| A reaction: This is obvious in human endeavours. Leibniz applied it to physics, producing a principle that shortest paths are always employed. It has a different formal name in modern physics, I think. He says if you make an unrestricted triangle, it is equilateral. |