14 ideas
| 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) |
| 14212 | A consistent theory just needs one model; isomorphic versions will do too, and large domains provide those [Lewis] |
| Full Idea: A consistent theory is, by definition, one satisfied by some model; an isomorphic image of a model satisfies the same theories as the original model; to provide the making of an isomorphic image of any given model, a domain need only be large enough. | |
| From: David Lewis (Putnam's Paradox [1984], 'Why Model') | |
| A reaction: This is laying out the ground for Putnam's model theory argument in favour of anti-realism. If you are chasing the one true model of reality, then formal model theory doesn't seem to offer much encouragement. |
| 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. |
| 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. |
| 14213 | Anti-realists see the world as imaginary, or lacking joints, or beyond reference, or beyond truth [Lewis] |
| Full Idea: Anti-realists say the only world is imaginary, or only has the parts or classes or relations we divide it into, or doubt that reference to the world is possible, or doubt that our interpretations can achieve truth. | |
| From: David Lewis (Putnam's Paradox [1984], 'Why Anti-R') | |
| A reaction: [compression of a paragraph on anti-realism] Lewis is a thoroughgoing realist. A nice example of the rhetorical device of ridiculing an opponent by suggesting that they don't even know what they themselves believe. |
| 14210 | A gerrymandered mereological sum can be a mess, but still have natural joints [Lewis] |
| Full Idea: The mereological sum of the coffee in my cup, the ink in this sentence, a nearby sparrow, and my left shoe is a miscellaneous mess of an object, yet its boundaries are by no means unrelated to the joints of nature. | |
| From: David Lewis (Putnam's Paradox [1984], 'What Might') | |
| A reaction: In that case they do, but if there are no atoms at the root of physics then presumably their could also be thoroughly jointless assemblages, involving probability distributions etc. Even random scattered atoms seem rather short of joints. |
| 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. |
| 14215 | Causal theories of reference make errors in reference easy [Lewis] |
| Full Idea: Whatever happens in special cases, causal theories usually make it easy to be wrong about the thing we refer to. | |
| From: David Lewis (Putnam's Paradox [1984], 'What Is') | |
| A reaction: I suppose the point of this is that there are no checks and balances to keep reference in focus, but just a requirement to keep connected to an increasingly attenuated causal chain. |
| 14209 | Descriptive theories remain part of the theory of reference (with seven mild modifications) [Lewis] |
| Full Idea: Description theories of reference are supposed to have been well and truly refuted. I think not: ..it is still tenable with my seven points, and part of the truth of reference [7: rigidity, egocentric, tokens, causal, imperfect, indeterminate, families]. | |
| From: David Lewis (Putnam's Paradox [1984], 'Glob Desc') | |
| A reaction: (The bit at the end refers to his seven points, on p.59). He calls his basic proposal 'causal descriptivism', incorporating his seven slight modifications of traditional descriptivism about reference. |
| 18006 | Chomsky's 'interpretative semantics' says syntax comes first, and is then interpreted [Chomsky, by Magidor] |
| Full Idea: Chomsky and his followers (whose position was labelled 'interpretative semantics') claimed that a sentence is first assigned a syntactic structure by an autonomous syntactic module, and this structure is then provided as input for semantic interpretation. | |
| From: report of Noam Chomsky (Aspects of the Theory of Syntax [1965]) by Ofra Magidor - Category Mistakes 1.3 | |
| A reaction: This certainly doesn't fit the experience of introspecting speech, but then I suppose good pianists focus entirely on the music, and overlook the finger movements which have obvious priority. But I don't know the syntax of the sentence when I begin it. |