10 ideas
7085 | The main problem of philosophy is what can and cannot be thought and expressed [Wittgenstein, by Grayling] |
Full Idea: The 'Tractatus' concerns the theory of what can be expressed by propositions (and, which comes to the same thing, can be thought), and what cannot be expressed by propositions, but can only be shown; which, I believe, is the main problem of philosophy. | |
From: report of Ludwig Wittgenstein (Letters to Russell [1919]) by A.C. Grayling - Wittgenstein Ch.2 | |
A reaction: This contains what a I consider the heresy of making thought depend on language, but his main question remains, of the limits of thought. It is dramatised nicely in the 'mysterian' view of the mind-body problem (e.g. Idea 2540). |
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) |
23463 | Atomic facts correspond to true elementary propositions [Wittgenstein] |
Full Idea: Atomic fact [Sachverhalt] is what corresponds to an elementary proposition [Elementarsatz] if it is true. | |
From: Ludwig Wittgenstein (Letters to Russell [1919], CL 125) | |
A reaction: This is perhaps the key to the Tractatus, because it is the binding point between world and language. A true realist would allow for atomic facts that may go beyond even possible propositions. |
23490 | A thought is mental constituents that relate to reality as words do [Wittgenstein] |
Full Idea: Does a Gedanke [thought] consist of words? No! But of psychical constituents that have the same sort of relation to reality as words. | |
From: Ludwig Wittgenstein (Letters to Russell [1919], p.125), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus 4B | |
A reaction: This is roughly my view of propositions, as non-lingustic mental events. The 'psychical constituents' seem to be concepts, in a psychological rather than a Fregean sense. This idea allowed transfer of his representation theory from thought to language. |
2594 | A true cause must involve a necessary connection between cause and effect [Malebranche] |
Full Idea: A true cause as I understand it is one such that the mind perceives a necessary connection between it and its effects. | |
From: Nicolas Malebranche (The Union of Body and Soul [1675], p.116) |