12 ideas
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 3622 | The Cogito is not a syllogism but a self-evident intuition [Descartes] |
| Full Idea: When someone says 'I am thinking, therefore I am, or I exist', he does not deduce existence from thought by means of a syllogism, but recognises it as something self-evident by a simple intuition of the mind. | |
| From: René Descartes (Reply to Second Objections [1641], 140) |
| 6387 | A minimum requirement for a theory of meaning is that it include an account of truth [Davidson] |
| Full Idea: Whatever else it embraces, a theory of meaning must include an account of truth - a statement of the conditions under which an arbitrary sentence of the language is true. | |
| From: Donald Davidson (Reality without Reference [1977], p.132) | |
| A reaction: It is a moot point whether we can define meaning if we assume truth, or if we can define truth by assuming meaning. Tarski seems to presuppose meaning when he defines truth (Idea 2345). I like Davidson's taking of truth as basic. |
| 6391 | A theory of truth tells us how communication by language is possible [Davidson] |
| Full Idea: A theory of truth lets us answer the underlying question how communication by language is possible. | |
| From: Donald Davidson (Reality without Reference [1977], p.137) | |
| A reaction: If, instead, you explain communication by understood intentions (á la Grice), you have to say more about what sort of intentions are meant. If you use reference, you still have more to say about the meaning of sentences. Davidson looks good. |
| 6388 | Is reference the key place where language and the world meet? [Davidson] |
| Full Idea: The essential question is whether reference is the, or at least one, place where there is direct contact between linguistic theory and events, actions, or objects described in nonlinguistic terms. | |
| From: Donald Davidson (Reality without Reference [1977], p.134) | |
| A reaction: How do you 'describe objects in nonlinguistic terms'? The causal theory of reference (e.g. Idea 4957) is designed to plug language straight into the world via reference. It simplifies things nicely, but I don't quite believe it. |
| 6390 | With a holistic approach, we can give up reference in empirical theories of language [Davidson] |
| Full Idea: I defend a version of the holistic approach, and urge that we must give up the concept of reference as basic to an empirical theory of language. | |
| From: Donald Davidson (Reality without Reference [1977], p.136) | |
| A reaction: He proposes to connect language to the world via the concept of truth, rather than of reference. It is a brilliant idea, and is the key issue in philosophy of language. I go back to animals, which seem to care about situations rather than things. |
| 6389 | To explain the reference of a name, you must explain its sentence-role, so reference can't be defined nonlinguistically [Davidson] |
| Full Idea: It is inconceivable that one should be able to explain the relationship between 'Kilimanjiro' and Kilimanjiro without first explaining the role of the word in sentences; hence there is no chance of explaining reference directly in nonlinguistic terms. | |
| From: Donald Davidson (Reality without Reference [1977], p.135) | |
| A reaction: I point at the mountain, and a local says 'Kilimanjiro'? There is a 'gavagai'-type problem with that. The prior question might be 'what is it about this word that enables it to have a role in sentences?' Unlike whimpering or belching. |