16 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. |
| 7332 | There is a huge range of sentences of which we do not know the logical form [Davidson] |
| Full Idea: We do not know the logical form of sentences about counterfactuals, probabilities, causal relations, belief, perception, intention, purposeful action, imperatives, optatives, or interrogatives, or the role of adverbs, adjectives or mass terms. | |
| From: Donald Davidson (Truth and Meaning [1967], p.35) | |
| A reaction: [compressed] This is the famous 'Davidson programme', where teams of philosophers work out the logical forms for this lot, thus unravelling the logic of the world. If they are beavering away, some sort of overview should have emerged by now... |
| 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. |
| 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. |
| 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. |
| 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! |
| 7772 | Compositionality explains how long sentences work, and truth conditions are the main compositional feature [Davidson, by Lycan] |
| Full Idea: Davidson's main argument in favour of his truth conditions theory of meaning is that compositionality is needed to account for our understanding of long, novel sentences, and a sentence's truth condition is its most obviously compositional feature. | |
| From: report of Donald Davidson (Truth and Meaning [1967]) by William Lycan - Philosophy of Language Ch.9 | |
| A reaction: This seems to me exactly right. As we hear a new long sentence unfold, we piece together the meaning. At the end we may spot that the meaning is silly, or an unverifiable speculation, or not what the speaker intended - but it is too late! It means. |
| 7327 | Davidson thinks Frege lacks an account of how words create sentence-meaning [Davidson, by Miller,A] |
| Full Idea: Davidson thinks that Frege's model for a theory of semantic value (and thereby for a systematic theory of sense) is unsatisfactory, because it provides no useful or explanatory account of how sentence-meaning can be a function of word-meaning. | |
| From: report of Donald Davidson (Truth and Meaning [1967]) by Alexander Miller - Philosophy of Language 8.1 | |
| A reaction: Put like that, it is not clear to me how you could even start to explain how word-meaning contributes to sentence meaning. Try speaking any sentence slowly, and observe how the sentence meaning builds up. Truth is, of course, relevant. |
| 7769 | You can state truth-conditions for "I am sick now" by relativising it to a speaker at a time [Davidson, by Lycan] |
| Full Idea: Davidson's response to the problem of how you would state truth conditions for "I am sick now" ...is to relativize its truth to a particular speaker and a time. | |
| From: report of Donald Davidson (Truth and Meaning [1967]) by William Lycan - Philosophy of Language Ch.9 | |
| A reaction: Lycan is not happy with this, but it seems a reasonable way to treat the truth of any statement containing indexicals. Never mind the 'truth conditions theory of meaning' - just ask whether "I am sick now" is true. |
| 6179 | Should we assume translation to define truth, or the other way around? [Blackburn on Davidson] |
| Full Idea: The concern of some philosophers has been expressed by saying that whereas Tarski took translation for granted, and sought to understand truth, Davidson takes truth for granted, and seeks to understand translation. | |
| From: comment on Donald Davidson (Truth and Meaning [1967]) by Simon Blackburn - Oxford Dictionary of Philosophy p.82 | |
| A reaction: We can just say that the two concepts are interdependent, but my personal intuitions side with Davidson. If you are going to take something as fundamental and axiomatic, truth looks a better bet than translation. |
| 6000 | The goal is rationality in the selection of things according to nature [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined the goal to be rationality in the selection and rejection of the things according to nature. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: This captures the central Stoic idea quite nicely. 'Live according to nature', but this always meant 'live according to reason', because that is (as Aristotle had taught) the essence of our nature. This only makes sense if reason and nature coincide. |
| 5999 | The good is what is perfect by nature [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined the good as what is perfect by nature. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: This might come close to G.E. Moore's Ideal Utilitarianism, but its dependence on the rather uneasy of concept of 'perfection' makes it questionable. Personally I find it appealing. I wish we had Diogenes' explanation. |
| 6001 | Justice is a disposition to distribute according to desert [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined justice as the disposition which distributes to everyone what he deserves. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: The questions that arise would be 'what does a new-born baby deserve?', and 'what do animals deserve?', and 'does the lowest and worst of criminals deserve anything at all?' |