11 ideas
| 11257 | The Pythagoreans were the first to offer definitions [Politis, by Politis] |
| Full Idea: Aristotle praises the Pythagoreans for being the first to offer definitions. | |
| From: report of Vassilis Politis (Aristotle and the Metaphysics [2004]) by Vassilis Politis - Aristotle and the Metaphysics 2.4 | |
| A reaction: This sounds like a hugely important step in the development of Greek philosophy which is hardly ever mentioned. |
| 11235 | 'True of' is applicable to things, while 'true' is applicable to words [Politis] |
| Full Idea: It is crucial not to confuse 'true' with 'true of'. 'True of' is applicable to things, while 'true' is applicable to words. | |
| From: Vassilis Politis (Aristotle and the Metaphysics [2004], 1.4) | |
| A reaction: A beautifully simple distinction which had never occurred to me, and which (being a thoroughgoing realist) I really like. |
| 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) |
| 22293 | Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Hilbert, by Potter] |
| Full Idea: Hilbert proposed to circuvent the paradoxes by means of the doctrine (already proposed by Poincaré) that in mathematics consistency entails existence. | |
| From: report of David Hilbert (On the Concept of Number [1900], p.183) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 19 'Exist' | |
| A reaction: Interesting. Hilbert's idea has struck me as weird, but it makes sense if its main motive is to block the paradoxes. Roughly, the idea is 'it exists if it isn't paradoxical'. A low bar for existence (but then it is only in mathematics!). |
| 11277 | Maybe 'What is being? is confusing because we can't ask what non-being is like [Politis] |
| Full Idea: We may be unfamiliar with the question 'What is being?' because there appear to be no contrastive questions of the form: how do beings differ from things that are not beings? | |
| From: Vassilis Politis (Aristotle and the Metaphysics [2004], 4.1) | |
| A reaction: We can, of course, contrast actual beings with possible beings, or imaginary beings, or even logically impossible beings, but in those cases 'being' strikes me as an entirely inappropriate word. |
| 11248 | Necessary truths can be two-way relational, where essential truths are one-way or intrinsic [Politis] |
| Full Idea: An essence is true in virtue of what the thing is in itself, but a necessary truth may be relational, as the consequence of the relation between two things and their essence. The necessary relation may be two-way, but the essential relation one-way. | |
| From: Vassilis Politis (Aristotle and the Metaphysics [2004], 2.3) | |
| A reaction: He is writing about Aristotle, but has in mind Kit Fine 1994 (qv). Politis cites Plato's answer to the Euthyphro Question as a good example. The necessity comes from the intrinsic nature of goodness/piety, not from the desire of the gods. |