16 ideas
18851 | Pairing (with Extensionality) guarantees an infinity of sets, just from a single element [Rosen] |
Full Idea: In conjunction with Extensionality, Pairing entails that given a single non-set, infinitely many sets exist. | |
From: Gideon Rosen (The Limits of Contingency [2006], 04) |
7557 | To solve Zeno's paradox, reject the axiom that the whole has more terms than the parts [Russell] |
Full Idea: Presumably Zeno appealed to the axiom that the whole has more terms than the parts; so if Achilles were to overtake the tortoise, he would have been in more places than the tortoise, which he can't be; but the conclusion is absurd, so reject the axiom. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.89) | |
A reaction: The point is that the axiom is normally acceptable (a statue contains more particles than the arm of the statue), but it breaks down when discussing infinity (Idea 7556). Modern theories of infinity are needed to solve Zeno's Paradoxes. |
10059 | In mathematic we are ignorant of both subject-matter and truth [Russell] |
Full Idea: Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.76) | |
A reaction: A famous remark, though Musgrave is rather disparaging about Russell's underlying reasoning here. |
7556 | A collection is infinite if you can remove some terms without diminishing its number [Russell] |
Full Idea: A collection of terms is infinite if it contains as parts other collections which have as many terms as it has; that is, you can take away some terms of the collection without diminishing its number; there are as many even numbers as numbers all together. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.86) | |
A reaction: He cites Dedekind and Cantor as source for these ideas. If it won't obey the rule that subtraction makes it smaller, then it clearly isn't a number, and really it should be banned from all mathematics. |
18852 | A Meinongian principle might say that there is an object for any modest class of properties [Rosen] |
Full Idea: Meinongian abstraction principles say that for any (suitably restricted) class of properties, there exists an abstract entity (arbitrary object, subsistent entity) that possesses just those properties. | |
From: Gideon Rosen (The Limits of Contingency [2006], 04) | |
A reaction: This is 'Meinongian' because there will be an object which is circular and square. The nub of the idea presumably resides in what is meant by 'restricted'. An object possessing every conceivable property is, I guess, a step too far. |
18849 | Metaphysical necessity is absolute and universal; metaphysical possibility is very tolerant [Rosen] |
Full Idea: If P is metaphysically necessary, then it is absolutely necessary, and necessary in every real (non-epistemic) sense; and if P is possible in any sense, then it's possible in the metaphysical sense. | |
From: Gideon Rosen (The Limits of Contingency [2006], 02) | |
A reaction: Rosen's shot at defining metaphysical necessity and possibility, and it looks pretty good to me. In my terms (drawing from Kit Fine) it is what is necessitated or permitted 'by everything'. So if it is necessitated by logic or nature, that's included. |
18850 | 'Metaphysical' modality is the one that makes the necessity or contingency of laws of nature interesting [Rosen] |
Full Idea: 'Metaphysical' modality is the sort of modality relative to which it is an interesting question whether the laws of nature are necessary or contingent. | |
From: Gideon Rosen (The Limits of Contingency [2006], 02) | |
A reaction: Being an essentialist here, I take it that the stuff of the universe necessitates the so-called 'laws'. The metaphysically interesting question is whether the stuff might have been different. Search me! A nice test of metaphysical modality though. |
18858 | Sets, universals and aggregates may be metaphysically necessary in one sense, but not another [Rosen] |
Full Idea: It may be metaphysically necessary in one sense that sets or universals or mereological aggregates exist, while in another sense existence is always a contingent matter. | |
From: Gideon Rosen (The Limits of Contingency [2006], 10) | |
A reaction: This idea depends on Idea 18856 and 18857. Personally I only think mereological aggregates and sets exist when people decide that they exist, so I don't see how they could ever be necessary. I'm unconvinced about his two concepts. |
18857 | Standard Metaphysical Necessity: P holds wherever the actual form of the world holds [Rosen] |
Full Idea: According to the Standard Conception of Metaphysical Necessity, P is metaphysically necessary when it holds in every possible world in which the laws of metaphysics (about the form or structure of the actual world) hold | |
From: Gideon Rosen (The Limits of Contingency [2006], 10) | |
A reaction: Rosen has a second meaning, in Idea 18856. He thinks it is crucial to see that there are two senses, because many things come out as metaphysically necessary on one concept, but contingent on the other. Interesting.... |
18856 | Non-Standard Metaphysical Necessity: when ¬P is incompatible with the nature of things [Rosen] |
Full Idea: According to the Non-Standard conception of Metaphysical Necessity, P is metaphysically necessary when its negation is logically incompatible with the nature of things. | |
From: Gideon Rosen (The Limits of Contingency [2006], 10) | |
A reaction: Rosen's new second meaning of the term. My immediate problem is with it resting on being 'logically' incompatible. Are squares 'logically' incompatible with circles? I like the idea that it rests on 'the nature of things'. (Psst! natures = essences) |
18848 | Something may be necessary because of logic, but is that therefore a special sort of necessity? [Rosen] |
Full Idea: It is one thing to say that P is necessary in some generic sense because it is a truth of logic (true in all models of a language, perhaps). It is something else to say that P therefore enjoys a special sort of necessity. | |
From: Gideon Rosen (The Limits of Contingency [2006], 02) | |
A reaction: This encourages my thought that there is only one sort of necessity (what must be), and the variety comes from the different types of necessity makers (everything there could be, nature, duties, promises, logics, concepts...). |
18855 | Combinatorial theories of possibility assume the principles of combination don't change across worlds [Rosen] |
Full Idea: Combinatorial theories of possibility take it for granted ....that possible worlds in general share a syntax, as it were, differing only in the constituents from which they are generated, or in the particular manner of their arrangements. | |
From: Gideon Rosen (The Limits of Contingency [2006], 08) | |
A reaction: For instance, it might assume that every world has 'objects', to which 'properties' and 'relations' can be attached, or to which 'functions' can apply. |
18853 | A proposition is 'correctly' conceivable if an ominiscient being could conceive it [Rosen] |
Full Idea: To a first approximation, P is correctly conceivable iff it would be conceivable for a logically ominiscient being who was fully informed about the nature of things. | |
From: Gideon Rosen (The Limits of Contingency [2006], 05) | |
A reaction: Isn't the last bit covered by 'ominiscient'? Ah, I think the 'logically' only means they have a perfect grasp of what is consistent. This is to meet the standard problem, of ill-informed people 'conceiving' of things which are actually impossible. |
7554 | Self-evidence is often a mere will-o'-the-wisp [Russell] |
Full Idea: Self-evidence is often a mere will-o'-the-wisp, which is sure to lead us astray if we take it as our guide. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.78) | |
A reaction: The sort of nice crisp remark you would expect from a good empiricist philosopher. Compare Idea 4948. However Russell qualifies it with the word 'often', and all philosophers eventually realise that you have to start somewhere. |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
18854 | The MRL view says laws are the theorems of the simplest and strongest account of the world [Rosen] |
Full Idea: According to the Mill-Ramsey-Lewis account of the laws of nature, a generalisation is a law just in case it is a theorem of every true account of the actual world that achieves the best overall balance of simplicity and strength. | |
From: Gideon Rosen (The Limits of Contingency [2006], 08) | |
A reaction: The obvious objection is that many of the theorems will be utterly trivial, and that is one thing that the laws of nature are not. Unless you are including 'metaphysical laws' about very very fundamental things, like objects, properties, relations. |