15 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) |
| 17304 | As causation links across time, grounding links the world across levels [Schaffer,J] |
| Full Idea: Grounding is something like metaphysical causation. Just as causation links the world across time, grounding links the world across levels. Grounding connects the more fundamental to the less fundamental, and thereby backs a certain form of explanation. | |
| From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], Intro) | |
| A reaction: Obviously you need 'levels' for this, which we should take to be structural levels. |
| 17306 | If ground is transitive and irreflexive, it has a strict partial ordering, giving structure [Schaffer,J] |
| Full Idea: By treating grounding as transitive (and irreflexive), one generates a strict partial ordering that induces metaphysical structure. | |
| From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], Intro) | |
| A reaction: Schaffer's paper goes on to attach the claim that grounding is transitive, but I didn't find his examples very convincing. |
| 22049 | Transcendental idealism aims to explain objectivity through subjectivity [Bowie] |
| Full Idea: The aim of transcendental idealism is to give a basis for objectivity in terms of subjectivity. | |
| From: Andrew Bowie (German Philosophy: a very short introduction [2010], 1) | |
| A reaction: Hume used subjectivity to undermine the findings of objectivity. There was then no return to naive objectivity. Kant's aim then was to thwart global scepticism. Post-Kantians feared that he had failed. |
| 22055 | The Idealists saw the same unexplained spontaneity in Kant's judgements and choices [Bowie] |
| Full Idea: The Idealist saw in Kant that knowledge, which depends on the spontaneity of judgement, and self-determined spontaneous action, can be seen as sharing the same source, which is not accessible to scientific investigation. | |
| From: Andrew Bowie (German Philosophy: a very short introduction [2010]) | |
| A reaction: This is the 'spontaneity' of judgements and choices which was seen as the main idea in Kant. It inspired romantic individualism. The judgements are the rule-based application of concepts. |
| 22054 | German Idealism tried to stop oppositions of appearances/things and receptivity/spontaneity [Bowie] |
| Full Idea: A central aim of German Idealism is to overcome Kant's oppositions between appearances and thing in themselves, and between receptivity and spontaneity. | |
| From: Andrew Bowie (German Philosophy: a very short introduction [2010], 2) | |
| A reaction: I have the impression that there were two strategies: break down the opposition within the self (Fichte), or break down the opposition in the world (Spinozism). |
| 22056 | Crucial to Idealism is the idea of continuity between receptivity and spontaneous judgement [Bowie] |
| Full Idea: A crucial idea for German Idealism (from Hamann) is that apparently passive receptivity and active spontaneity are in fact different degrees of the same 'activity, and the gap between subject and world can be closed. | |
| From: Andrew Bowie (German Philosophy: a very short introduction [2010], 3) | |
| A reaction: The 'passive' bit seems to be Hume's 'impressions', which are Kant's 'intuitions', which need 'spontaneous' interpretation to become experiences. Critics of Kant said this implied a dualism. |
| 17308 | Explaining 'Adam ate the apple' depends on emphasis, and thus implies a contrast [Schaffer,J] |
| Full Idea: Explaining why ADAM ate the apple is a different matter from explaining why he ATE the apple, and from why he ate THE APPLE. ...In my view the best explanations incorporate ....contrastive information. | |
| From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.3.1) | |
| A reaction: But why are the contrasts Eve, or throwing it, or a pear? It occurs to me that this is wrong! The contrast is with anything else which could have gone in subject, verb or object position. It is a matter of categories, not of contrasts. |
| 17305 | I take what is fundamental to be the whole spatiotemporal manifold and its fields [Schaffer,J] |
| Full Idea: I myself would prefer to speak of what is fundamental in terms of the whole spatiotemporal manifold and the fields that permeate it, with parts counting as derivative of the whole. | |
| From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.1.1) | |
| A reaction: Not quite the Parmenidean One, since it has parts, but a nice try at updating the great man. Note the reference to 'fields', suggesting that this view is grounded in the physics rather than metaphysics. How many fields has it got? |
| 17307 | Nowadays causation is usually understood in terms of equations and variable ranges [Schaffer,J] |
| Full Idea: The leading treatments of causation work within 'structural equation models', with events represented via variables each of which is allotted a range of permitted values, which constitute a 'contrast space'. | |
| From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.3.1) | |
| A reaction: Like Woodward's idea that causation is a graph, this seems to be a matter of plotting or formalising correlations between activities, which is a very Humean approach to causation. |