21 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. |
| 10061 | The If-thenist view only seems to work for the axiomatised portions of mathematics [Musgrave] |
| Full Idea: The If-thenist view seems to apply straightforwardly only to the axiomatised portions of mathematics. | |
| From: Alan Musgrave (Logicism Revisited [1977], §5) | |
| A reaction: He cites Lakatos to show that cutting-edge mathematics is never axiomatised. One might reply that if the new mathematics is any good then it ought to be axiomatis-able (barring Gödelian problems). |
| 10065 | Perhaps If-thenism survives in mathematics if we stick to first-order logic [Musgrave] |
| Full Idea: If we identify logic with first-order logic, and mathematics with the collection of first-order theories, then maybe we can continue to maintain the If-thenist position. | |
| From: Alan Musgrave (Logicism Revisited [1977], §5) | |
| A reaction: The problem is that If-thenism must rely on rules of inference. That seems to mean that what is needed is Soundness, rather than Completeness. That is, inference by the rules must work properly. |
| 10049 | Logical truths may contain non-logical notions, as in 'all men are men' [Musgrave] |
| Full Idea: Containing only logical notions is not a necessary condition for being a logical truth, since a logical truth such as 'all men are men' may contain non-logical notions such as 'men'. | |
| From: Alan Musgrave (Logicism Revisited [1977], §3) | |
| A reaction: [He attributes this point to Russell] Maybe it is only a logical truth in its general form, as ∀x(x=x). Of course not all 'banks' are banks. |
| 10050 | A statement is logically true if it comes out true in all interpretations in all (non-empty) domains [Musgrave] |
| Full Idea: The standard modern view of logical truth is that a statement is logically true if it comes out true in all interpretations in all (non-empty) domains. | |
| From: Alan Musgrave (Logicism Revisited [1977], §3) |
| 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. |
| 10058 | No two numbers having the same successor relies on the Axiom of Infinity [Musgrave] |
| Full Idea: The axiom of Peano which states that no two numbers have the same successor requires the Axiom of Infinity for its proof. | |
| From: Alan Musgrave (Logicism Revisited [1977], §4 n) | |
| A reaction: [He refers to Russell 1919:131-2] The Axiom of Infinity is controversial and non-logical. |
| 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. |
| 10062 | Formalism seems to exclude all creative, growing mathematics [Musgrave] |
| Full Idea: Formalism seems to exclude from consideration all creative, growing mathematics. | |
| From: Alan Musgrave (Logicism Revisited [1977], §5) | |
| A reaction: [He cites Lakatos in support] I am not immediately clear why spotting the remote implications of a formal system should be uncreative. The greatest chess players are considered to be highly creative and imaginative. |
| 10063 | Formalism is a bulwark of logical positivism [Musgrave] |
| Full Idea: Formalism is a bulwark of logical positivist philosophy. | |
| From: Alan Musgrave (Logicism Revisited [1977], §5) | |
| A reaction: Presumably if you drain all the empirical content out of arithmetic and geometry, you are only left with the bare formal syntax, of symbols and rules. That seems to be as analytic as you can get. |
| 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! |
| 7658 | Obviously there can't be a functional anaylsis of qualia if they are defined by intrinsic properties [Dennett] |
| Full Idea: If you define qualia as intrinsic properties of experiences considered in isolation from all their causes and effects, logically independent of all dispositional properties, then they are logically guaranteed to elude all broad functional analysis. | |
| From: Daniel C. Dennett (Sweet Dreams [2005], Ch.8) | |
| A reaction: This is a good point - it seems daft to reify qualia and imagine them dangling in mid-air with all their vibrant qualities - but that is a long way from saying there is nothing more to qualia than functional roles. Functions must be exlained too. |
| 7655 | The work done by the 'homunculus in the theatre' must be spread amongst non-conscious agencies [Dennett] |
| Full Idea: All the work done by the imagined homunculus in the Cartesian Theater must be distributed among various lesser agencies in the brain, none of which is conscious. | |
| From: Daniel C. Dennett (Sweet Dreams [2005], Ch.3) | |
| A reaction: Dennett's account crucially depends on consciousness being much more fragmentary than most philosophers claim it to be. It is actually full of joints, which can come apart. He may be right. |
| 7657 | Intelligent agents are composed of nested homunculi, of decreasing intelligence, ending in machines [Dennett] |
| Full Idea: As long as your homunculi are more stupid and ignorant than the intelligent agent they compose, the nesting of homunculi within homunculi can be finite, bottoming out, eventually, with agents so unimpressive they can be replaced by machines. | |
| From: Daniel C. Dennett (Sweet Dreams [2005], Ch.6) | |
| A reaction: [Dennett first proposed this in 'Brainstorms' 1978]. This view was developed well by Lycan. I rate it as one of the most illuminating ideas in the modern philosophy of mind. All complex systems (like aeroplanes) have this structure. |
| 7656 | I don't deny consciousness; it just isn't what people think it is [Dennett] |
| Full Idea: I don't maintain, of course, that human consciousness does not exist; I maintain that it is not what people often think it is. | |
| From: Daniel C. Dennett (Sweet Dreams [2005], Ch.3) | |
| A reaction: I consider Dennett to be as near as you can get to an eliminativist, but he is not stupid. As far as I can see, the modern philosopher's bogey-man, the true total eliminativist, simply doesn't exist. Eliminativists usually deny propositional attitudes. |
| 7654 | What matters about neuro-science is the discovery of the functional role of the chemistry [Dennett] |
| Full Idea: Neuro-science matters because - and only because - we have discovered that the many different neuromodulators and other chemical messengers that diffuse throughout the brain have functional roles that make important differences. | |
| From: Daniel C. Dennett (Sweet Dreams [2005], Ch.1) | |
| A reaction: I agree with Dennett that this is the true ground for pessimism about spectacular breakthroughs in artificial intelligence, rather than abstract concerns about irreducible features of the mind like 'qualia' and 'rationality'. |
| 10060 | Logical positivists adopted an If-thenist version of logicism about numbers [Musgrave] |
| Full Idea: Logical positivists did not adopt old-style logicism, but rather logicism spiced with varying doses of If-thenism. | |
| From: Alan Musgrave (Logicism Revisited [1977], §4) | |
| A reaction: This refers to their account of mathematics as a set of purely logical truths, rather than being either empirical, or a priori synthetic. |