11 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) |
| 8463 | Maths can be reduced to logic and set theory [Quine] |
| Full Idea: Researches in the foundations of mathematics have made it clear that all of (interpreted) mathematics can be got down to logic and set theory, and the objects needed for mathematics can be got down to the category of classes (and classes of classes..). | |
| From: Willard Quine (The Scope and Language of Science [1954], §VI) | |
| A reaction: This I take to be a retreat from pure logicism, presumably influenced by Gödel. So can set theory be reduced to logic? Crispin Wright is the one the study. |
| 14742 | It can't be indeterminate whether x and y are identical; if x,y is indeterminate, then it isn't x,x [Salmon,N] |
| Full Idea: Insofar as identity seems vague, it is provably mistaken. If it is vague whether x and y are identical (as in the Ship of Theseus), then x,y is definitely not the same as x,x, since the first pair is indeterminate and the second pair isn't. | |
| From: Nathan Salmon (Reference and Essence: seven appendices [2005], App I) | |
| A reaction: [compressed; Gareth Evans 1978 made a similar point] This strikes me as begging the question in the Ship case, since we are shoehorning the new ship into either the slot for x or the slot for y, but that was what we couldn’t decide. No rough identity? |
| 8461 | The category of objects incorporates the old distinction of substances and their modes [Quine] |
| Full Idea: The category of objects embraces indiscriminately what would anciently have been distinguished as substances and as modes or states of substances. | |
| From: Willard Quine (The Scope and Language of Science [1954], §6) | |
| A reaction: This nicely captures Quine's elimination of properties, by presenting them as inseparable from their objects/substances. Armstrong calls this 'Ostrich Nominalism' (for refusing to address the universals problem) but Quineans are unshaken. |
| 8462 | A hallucination can, like an ague, be identified with its host; the ontology is physical, the idiom mental [Quine] |
| Full Idea: A physical ontology has a place for states of mind. An inspiration or a hallucination can, like the fit of ague, be identified with its host for the duration. It leaves our mentalistic idioms fairly intact, but reconciles them with a physical ontology. | |
| From: Willard Quine (The Scope and Language of Science [1954], §VI) | |
| A reaction: Quine is employing the same strategy that he uses for substances and properties (Idea 8461): take the predication as basic, rather than reifying the thing being predicated. The ague analogy suggests that Quine is an incipient functionalist. |
| 18885 | Kripke and Putnam made false claims that direct reference implies essentialism [Salmon,N] |
| Full Idea: Kripke and Putnam made unsubstantiated claims, indeed false claims, to the effect that the theory of direct reference has nontrivial essentialist import. | |
| From: Nathan Salmon (Reference and Essence: seven appendices [2005], Pref to Exp Ed) | |
| A reaction: Kripke made very few claims, and is probably innocent of the charge. Most people agree with Salmon that you can't derive metaphysics from a theory of reference. |