9 ideas
22820 | Early Romantics sought a plurality of systems, in a quest for freedom [Hösle] |
Full Idea: It was an early Romantic idea that there is necessarily a plurality of systems in which individuality is expressed; for a complete system would destroy freedom. | |
From: Vittorio Hösle (A Short History of German Philosophy [2013], 7) | |
A reaction: I'm not clear why you are free because you are locked into system that differs from that of other people. True freedom seems to be either no system, or continually remaking one's own system. Why is such freedom valuable? Freedom v truth? |
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) |
16668 | Modes of things exist in some way, without being full-blown substances [Gassendi] |
Full Idea: Modes are not nothing but something more than mere nothing; they are therefore 'res' of some kind, not substantial of course, but at least modal. | |
From: Pierre Gassendi (Disquisitions [1644], II.3.4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 260 | |
A reaction: This is the great modern atomist talking pure scholastic metaphysics. He's been reading Suárez. Gassendi seems to accept more than one type of existence. |
22819 | In the 18th century history came to be seen as progressive, rather than cyclical [Hösle] |
Full Idea: The turning point in the history of the philosophy of history occurs in the eighteenth century, when the ancient cyclical model of Vico is superseded by the idea of progress. | |
From: Vittorio Hösle (A Short History of German Philosophy [2013], 6) | |
A reaction: He says that Hegel merely inherited this progressive view, rather than creating it. I'm not sure how widely held the cyclical view was. I don't recognise it in Shakespeare. Science and technology must have suggested progress. |