17 ideas
| 9967 | 'Impure' sets have a concrete member, while 'pure' (abstract) sets do not [Jubien] |
| Full Idea: Any set with a concrete member is 'impure'. 'Pure' sets are those that are not impure, and are paradigm cases of abstract entities, such as the sort of sets apparently dealt with in Zermelo-Fraenkel (ZF) set theory. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.116) | |
| A reaction: [I am unclear whether Jubien is introducing this distinction] This seems crucial in accounts of mathematics. On the one had arithmetic can be built from Millian pebbles, giving impure sets, while logicists build it from pure sets. |
| 9968 | A model is 'fundamental' if it contains only concrete entities [Jubien] |
| Full Idea: A first-order model can be viewed as a kind of ordered set, and if the domain of the model contains only concrete entities then it is a 'fundamental' model. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.117) | |
| A reaction: An important idea. Fundamental models are where the world of logic connects with the physical world. Any account of relationship between fundamental models and more abstract ones tells us how thought links to world. |
| 9965 | There couldn't just be one number, such as 17 [Jubien] |
| Full Idea: It makes no sense to suppose there might be just one natural number, say seventeen. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.113) | |
| A reaction: Hm. Not convinced. If numbers are essentially patterns, we might only have the number 'twelve', because we had built our religion around anything which exhibited that form (in any of its various arrangements). Nice point, though. |
| 9966 | The subject-matter of (pure) mathematics is abstract structure [Jubien] |
| Full Idea: The subject-matter of (pure) mathematics is abstract structure per se. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.115) | |
| A reaction: This is the Structuralist idea beginning to take shape after Benacerraf's launching of it. Note that Jubien gets there by his rejection of platonism, whereas some structuralist have given a platonist interpretation of structure. |
| 9964 | Since mathematical objects are essentially relational, they can't be picked out on their own [Jubien] |
| Full Idea: The essential properties of mathematical entities seem to be relational, ...so we make no progress unless we can pick out some mathematical entities wihout presupposing other entities already picked out. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.112) | |
| A reaction: [compressed] Jubien is a good critic of platonism. He has identified the problem with Frege's metaphor of a 'borehole', where we discover delightful new properties of numbers simply by reaching them. |
| 9963 | If we all intuited mathematical objects, platonism would be agreed [Jubien] |
| Full Idea: If the intuition of mathematical objects were general, there would be no real debate over platonism. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.111) | |
| A reaction: It is particularly perplexing when Gödel says that his perception of them is just like sight or smell, since I have no such perception. How do you individuate very large numbers, or irrational numbers, apart from writing down numerals? |
| 9962 | How can pure abstract entities give models to serve as interpretations? [Jubien] |
| Full Idea: I am unable to see how the mere existence of pure abstract entities enables us to concoct appropriate models to serve as interpretations. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.111) | |
| A reaction: Nice question. It is always assumed that once we have platonic realm, that everything else follows. Even if we are able to grasp the objects, despite their causal inertness, we still have to discern innumerable relations between them. |
| 9969 | The empty set is the purest abstract object [Jubien] |
| Full Idea: The empty set is the pure abstract object par excellence. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.118 n8) | |
| A reaction: So a really good PhD on the empty set could crack the whole nature of reality. Get to work, whoever you are! |
| 16423 | Conceptual possibilities are metaphysical possibilities we can conceive of [Stalnaker] |
| Full Idea: Conceptual possibilities are just (metaphysical) possibilities that we can conceive of. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 1) |
| 16422 | The necessity of a proposition concerns reality, not our words or concepts [Stalnaker] |
| Full Idea: The necessity or contingency of a proposition has nothing to do with our concepts or the meanings of our words. The possibilities would have been the same even if we had never conceived of them. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 1) | |
| A reaction: This sounds in need of qualification, since some of the propositions will be explicitly about words and concepts. Still, I like this idea. |
| 16421 | Critics say there are just an a priori necessary part, and an a posteriori contingent part [Stalnaker] |
| Full Idea: Critics say there are no irreducible a posteriori truths. They can be factored into a part that is necessary, but knowable a priori through conceptual analysis, and a part knowable only a posteriori, but contingent. 2-D semantics makes this precise. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 1) | |
| A reaction: [Critics are Sidelle, Jackson and Chalmers] Interesting. If gold is necessarily atomic number 79, or it wouldn't be gold, that sounds like an analytic truth about gold. Discovering the 79 wasn't a discovery of a necessity. Stalnaker rejects this idea. |
| 16429 | A 'centred' world is an ordered triple of world, individual and time [Stalnaker] |
| Full Idea: A 'centred' possible world is an ordered triple consisting of a possible world, an individual in the domain of that world, and a time. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 2) |
| 16428 | Meanings aren't in the head, but that is because they are abstract [Stalnaker] |
| Full Idea: Meanings ain't in the head. Putnam's famous slogan actually fits Frege's anti-psychologism better than it fits Purnam's and Burge's anti-individualism. The point is that intensions of any kind are abstract objects. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 2) | |
| A reaction: If intensions are abstract, that leaves (for me) the question of what they are abstracted from. I take it that there are specific brain events that are being abstractly characterised. What do we call those? |
| 16432 | One view says the causal story is built into the description that is the name's content [Stalnaker] |
| Full Idea: In 'causal descriptivism' the causal story is built into the description that is the content of the name (and also incorporates a rigidifying operator to ensure that the descriptions that names abbreviate have wide scope). | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 5) | |
| A reaction: Not very controversial, I would say, since virtually every fact about the world has a 'causal story' built into it. Must we insist on rigidity in order to have wide scope? |
| 16430 | Two-D says that a posteriori is primary and contingent, and the necessity is the secondary intension [Stalnaker] |
| Full Idea: Two-dimensionalism says the necessity of a statement is constituted by the fact that the secondary intensions is a necessary proposition, and their a posteriori character is constituted by the fact that the associated primary intension is contingent. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 2) | |
| A reaction: This view is found in Sidelle 1989, and then formalised by Jackson and Chalmers. I like metaphysical necessity, but I have some sympathy with the approach. The question must always be 'where does this necessity derive from'? |
| 16431 | In one view, the secondary intension is metasemantic, about how the thinker relates to the content [Stalnaker] |
| Full Idea: On the metasemantic interpretation of the two-dimensional framework, the second dimension is used to represent the metasemantic facts about the relation between a thinker or speaker and the contents of her thoughts or utterances. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 4) | |
| A reaction: I'm struggling to think what facts there might be about the relation between myself and the contents of my thoughts. I'm more or less constituted by my thoughts. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |