15 ideas
| 13966 | Analytic philosophy loved the necessary a priori analytic, linguistic modality, and rigour [Soames] |
| Full Idea: The golden age of analytic philosophy (mid 20th c) was when necessary, a priori and analytic were one, all possibility was linguistic possibility, and the linguistic turn gave philosophy a respectable subject matter (language), and precision and rigour. | |
| From: Scott Soames (Significance of the Kripkean Nec A Posteriori [2006], p.166) | |
| A reaction: Gently sarcastic, because Soames is part of the team who have put a bomb under this view, and quite right too. Personally I think the biggest enemy in all of this lot is not 'language' but 'rigour'. A will-o-the-wisp philosophers dream of. |
| 13974 | If philosophy is analysis of meaning, available to all competent speakers, what's left for philosophers? [Soames] |
| Full Idea: If all of philosophy is the analysis of meaning, and meaning is fundamentally transparent to competent speakers, there is little room for philosophically significant explanations and theories, since they will be necessary or a priori, or both. | |
| From: Scott Soames (Significance of the Kripkean Nec A Posteriori [2006], p.186) | |
| A reaction: He cites the later Wittgenstein as having fallen into this trap. I suppose any area of life can have its specialists, but I take Shakespeare to be a greater master of English than any philosopher I have ever read. |
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04) | |
| A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case. |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36) | |
| A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation. |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx). | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62) |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03) |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02) | |
| A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction. |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19) | |
| A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter! |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15) |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48) |
| 13969 | Kripkean essential properties and relations are necessary, in all genuinely possible worlds [Soames] |
| Full Idea: By (Kripkean) 'essential' properties and relations I mean simply properties and relations that hold necessarily of objects (in all genuinely possible world-states in which the objects exist). | |
| From: Scott Soames (Significance of the Kripkean Nec A Posteriori [2006], p.168 n5) | |
| A reaction: This is the standard modern view of essences which I find so unsatisfactory. Kit Fine has helped to take us back to the proper Aristotelian view, where 'necessary' and 'essential' actually have different meanings. Note the inclusion of relations. |
| 13973 | A key achievement of Kripke is showing that important modalities are not linguistic in source [Soames] |
| Full Idea: None of Kripke's many achievements is more important than his breaking the spell of the linguistic as the source of philosophically important modalities. | |
| From: Scott Soames (Significance of the Kripkean Nec A Posteriori [2006], p.186) | |
| A reaction: Put like that, Kripke may have had the single most important thought of modern times. I take good philosophy to be exactly the same as good scientific theorising, in that it all arises out of the nature of reality (and I include logic in that). |
| 13968 | Kripkean possible worlds are abstract maximal states in which the real world could have been [Soames] |
| Full Idea: For the Kripkean possible states of the world are not alternate concrete universes, but abstract objects. Metaphysically possible world-states are maximally complete ways the real concrete universe could have been. | |
| From: Scott Soames (Significance of the Kripkean Nec A Posteriori [2006], p.167) | |
| A reaction: This is probably clearer about the Kripkean view than Kripke ever is, but then that is part of Soames's mission. It sounds like the right way to conceive possible worlds. At least there is some commitment there, rather than instrumentalism about them. |
| 13972 | Two-dimensionalism reinstates descriptivism, and reconnects necessity and apriority to analyticity [Soames] |
| Full Idea: Two-dimensionalism is a fundamentally anti-Kripkean attempt to reinstate descriptivism about names and natural kind terms, to reconnect necessity and apriority to analyticity, and return philosophy to analytic paradigms of its golden age. | |
| From: Scott Soames (Significance of the Kripkean Nec A Posteriori [2006], p.183) | |
| A reaction: I presume this is right, and it is so frustrating that you need Soames to spell it out, when Chalmers is much more low-key. Philosophers hate telling you what their real game is. Why is that? |
| 5996 | Critolaus redefined Aristotle's moral aim as fulfilment instead of happiness [Critolaus, by White,SA] |
| Full Idea: Critolaus reformulated Aristotelian theory by defining happiness as a 'fulfilment' (sumplêrôma) of psychic, physical, and external goods, where virtue vastly outweighs the rest. | |
| From: report of Critolaus (fragments/reports [c.170 BCE]) by Stephen A. White - Critolaus | |
| A reaction: The sounds more like an attempt at clarification than a real change of Peripatetic doctrine. Occasionally 'fulfilment' is offered as a translation for eudaimonia. Maybe we should just take up Critolaus' suggestion when we are discussing Aristotle. |