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) |
| 594 | Speusippus suggested underlying principles for every substance, and ended with a huge list [Speussipus, by Aristotle] |
| Full Idea: Speusippus suggested principles for each substance, including principles for numbers, magnitude and the soul. He thus arrived at no mean list of substances. | |
| From: report of Speussipus (thirty titles (lost) [c.367 BCE]) by Aristotle - Metaphysics 1028b |
| 19722 | We could know the evidence for our belief without knowing why it is such evidence [Mittag] |
| Full Idea: While one might understand the proposition entailed by one's evidence, one might have no idea how or why one's evidence entails it. This seems to imply one is not justified in believing the proposition on the basis of one's evidence. | |
| From: Daniel M. Mittag (Evidentialism [2011], 'Evidential') | |
| A reaction: An example might be seen if a layman tours a physics lab. This looks like a serious problem for evidentialism. Once you see why the evidence entails the proposition, you are getting closer to understanding than to knowledge. Explanation. |
| 19723 | Evidentialism can't explain that we accept knowledge claims if the evidence is forgotten [Mittag] |
| Full Idea: If one came to believe p with good evidence, but has since forgotten that evidence, we might think one can continue to believe justifiably, but evidentialism appears unable to account for this. | |
| From: Daniel M. Mittag (Evidentialism [2011], 'Forgotten') | |
| A reaction: We would still think that the evidence was important, and we would need to trust the knower's claim that the forgotten evidence was good. So it doesn't seem to destroy the evidentialist thesis. |
| 19720 | Evidentialism concerns the evidence for the proposition, not for someone to believe it [Mittag] |
| Full Idea: Evidentialism is not a theory about when one's believing is justified; it is a theory about what makes one justified in believing a proposition. It is a thesis regarding 'propositional justification', not 'doxastic justification'. | |
| From: Daniel M. Mittag (Evidentialism [2011], 'Preliminary') | |
| A reaction: Thus it is entirely about whether the evidence supports the proposition, and has no interest in who believes it or why. Knowledge is when you believe a true proposition which has good support. This could be internalist or externalist? |
| 19721 | Coherence theories struggle with the role of experience [Mittag] |
| Full Idea: Traditional coherence theories seem unable to account for the role experience plays in justification. | |
| From: Daniel M. Mittag (Evidentialism [2011], 'Evidence') | |
| A reaction: I'm inclined to say that experience only becomes a justification when it has taken propositional (though not necessarily lingistic) form. That is, when you see it 'as' something. Uninterpreted shape and colour can justify virtually nothing. |
| 2632 | Speusippus said things were governed by some animal force rather than the gods [Speussipus, by Cicero] |
| Full Idea: Speusippus, following his uncle Plato, held that all things were governed by some kind of animal force, and tried to eradicate from our minds any notion of the gods. | |
| From: report of Speussipus (thirty titles (lost) [c.367 BCE]) by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') I.33 |