10 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) |
| 9354 | Why should necessities only be knowable a priori? That Hesperus is Phosporus is known empirically [Devitt] |
| Full Idea: Why should we accept that necessities can only be known a priori? Prima facie, some necessities are known empirically; for example, that water is necessarily H2O, and that Hesperus is necessarily Phosphorus. | |
| From: Michael Devitt (There is no a Priori [2005], §2) | |
| A reaction: An important question, whatever your view. If the only thing we can know a priori is necessities, it doesn't follow that necessities can only be known a priori. It gets interesting if we say that some necessities can never be known a priori. |
| 9353 | We explain away a priori knowledge, not as directly empirical, but as indirectly holistically empirical [Devitt] |
| Full Idea: We have no need to turn to an a priori explanation of our knowledge of mathematics and logic. Our intuitions that this knowledge is not justified in some direct empirical way is preserved. It is justified in an indirect holistic way. | |
| From: Michael Devitt (There is no a Priori [2005], §2) | |
| A reaction: I think this is roughly the right story, but the only way it will work is if we have some sort of theory of abstraction, which gets us up the ladder of generalisations to the ones which, it appears, are necessarily true. |
| 9356 | The idea of the a priori is so obscure that it won't explain anything [Devitt] |
| Full Idea: The whole idea of the a priori is too obscure for it to feature in a good explanation of our knowledge of anything. | |
| From: Michael Devitt (There is no a Priori [2005], §3) | |
| A reaction: I never like this style of argument. It would be nice if all the components of all our our explanations were crystal clear. Total clarity about anything is probably a hopeless dream, and we may have to settle for murky corners in all explanations. |
| 18278 | Kant showed that our perceptions are partly constructed from our concepts [Reichenbach] |
| Full Idea: It was Kant's great discovery that the object of knowledge is not simply given but constructed, and that it contains conceptual elements not contained in pure perception. | |
| From: Hans Reichenbach (The Theory of Relativity and A Priori Knowledge [1965], p.49), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap |