13 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) |
22142 | In future, only logical limits can be placed on divine omnipotence [Anon (Par), by Boulter] |
Full Idea: The Condemnation stipulated that all portions of the ancient intellectual heritage that placed non-logical limits on divine omnipotence were no longer to be tolerated. ...Philosophers now had to entertain the wildest ideas with all seriousness. | |
From: report of Anon (Par) (The Condemnation of 1277 [1277]) by Stephen Boulter - Why Medieval Philosophy Matters 3 | |
A reaction: Boulter identifies this as 'the ultimate source of Hume's philosophical delirium'. Presumably the angels-on-a-pinhead stuff originated with this. It is crazy to think that the only limit on possible existence is logic. Can God make a planet of uranium? |
14288 | 'If A,B' affirms that A⊃B, and also that this wouldn't change if A were certain [Jackson, by Edgington] |
Full Idea: According to Jackson, in asserting 'If A,B' the speaker expresses his belief that A⊃B, and also indicates that this belief is 'robust' with respect to the antecedent A - the speaker would not abandon A⊃B if he were to learn that A. | |
From: report of Frank Jackson (On Assertion and Indicative Conditionals [1979]) by Dorothy Edgington - Conditionals (Stanf) 4.2 | |
A reaction: The point is that you must not believe A⊃B solely on the dubious grounds of ¬A. This is 'to ensure an assertable conditional is fit for modus ponens' - that is, that you really will affirm B when you learn that A is true. Nice idea. |
13769 | Conditionals are truth-functional, but should only be asserted when they are confident [Jackson, by Edgington] |
Full Idea: Jackson holds that conditionals are truth-functional, but are governed by rules of assertability, rather like 'but' compared to 'and'. The belief must be 'robust' - the speaker would not abandon his belief that A⊃B if he were to learn that A. | |
From: report of Frank Jackson (On Assertion and Indicative Conditionals [1979]) by Dorothy Edgington - Conditionals 17.3.2 | |
A reaction: This seems to spell out more precisely the pragmatic approach to conditionals pioneered by Grice, in Idea 13767. The idea is make conditionals 'fit for modus ponens'. They mustn't just be based on a belief that ¬A. |
16716 | It is heresy to require self-evident foundational principles in order to be certain [Anon (Par)] |
Full Idea: Heresy 151: 'To have certainty regarding any conclusion, it must be founded on self-evident principles'. | |
From: Anon (Par) (The Condemnation of 1277 [1277], 151), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.3 | |
A reaction: The correct view is obviously to found certainty on faith and authority. It is one thing to be told that foundationalism is a poor theory, but another to be told it is a heresy, and thus a potential capital crime! |
1866 | It is heresy to teach that history repeats every 36,000 years [Anon (Par)] |
Full Idea: It is heresy to teach that with all the heavenly bodies coming back to the same point after a period of thirty-six thousand years, the same effects as now exist will reappear. | |
From: Anon (Par) (The Condemnation of 1277 [1277], §92) |
1865 | It is heresy to teach that natural impossibilities cannot even be achieved by God [Anon (Par)] |
Full Idea: It is heresy to teach that what is absolutely impossible according to nature cannot be brought about by God or another agent. | |
From: Anon (Par) (The Condemnation of 1277 [1277], §17) |
1864 | It is heresy to teach that we can know God by his essence in this mortal life [Anon (Par)] |
Full Idea: It is heresy to teach that we can know God by his essence in this mortal life. | |
From: Anon (Par) (The Condemnation of 1277 [1277], §9) |