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) |
12066 | Aristotelian and Kripkean essentialism are very different theories [Witt] |
Full Idea: The differences between Aristotelian essentialism and Kripke's essentialism are so fundamental and pervasive that it is a serious distortion of both views to think of essentialism as a single theory. | |
From: Charlotte Witt (Substance and Essence in Aristotle [1989], Intro) | |
A reaction: This seems to me to be very important, because there is a glib assumption that when essentialism is needed for modal logic, that we must immediately have embraced what Aristotle was saying. Aristotle was better than Kripke. |
12067 | An Aristotelian essence is a nonlinguistic correlate of the definition [Witt] |
Full Idea: An Aristotelian essence is a nonlinguistic correlate of the definition of the entity in question. | |
From: Charlotte Witt (Substance and Essence in Aristotle [1989], Intro) | |
A reaction: This is a simple and necessity corrective to the simplistic idea that Aristotle thought that essences just were definitions. Aristotle believes in real essences, not linguistic essences. |
12082 | If unity is a matter of degree, then essence may also be a matter of degree [Witt] |
Full Idea: By holding that the most unified beings have essences in an unqualified sense, while allowing that other beings have them in a qualified sense - we can think of unity as a matter of degree. | |
From: Charlotte Witt (Substance and Essence in Aristotle [1989], 4.3) | |
A reaction: This is Witt's somewhat unorthodox view of how we should read Aristotle. I am sympathetic, if essences are really explanatory. That means they are unstable, and would indeed be likely to come in degrees. |
12089 | Essences mainly explain the existence of unified substance [Witt] |
Full Idea: The central function of essence is to explain the actual existence of a unified substance. | |
From: Charlotte Witt (Substance and Essence in Aristotle [1989], 5 n1) | |
A reaction: She is offering an interpretation of Aristotle. Since existence is an active and not a passive matter, the identity of the entity will include its dispositions etc., I presume. |
12102 | Essential properties of origin are too radically individual for an Aristotelian essence [Witt] |
Full Idea: The radical individuality of essential properties of origin makes them unsuitable for inclusion in an Aristotelian essence. | |
From: Charlotte Witt (Substance and Essence in Aristotle [1989], 6.2) | |
A reaction: Nevertheless, Aristotle believes in individual essences, though these seem to be fixed by definitions, which are composed of combinations of universals. The uniqueness is of the whole definition, not of its parts. |
6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
12085 | Reality is directional [Witt] |
Full Idea: Reality is directional. | |
From: Charlotte Witt (Substance and Essence in Aristotle [1989], 4.5) | |
A reaction: [Plucked from context! She attributes the view to Aristotle] This slogan beautifully summarises the 'scientific essentialist' view of reality, based not on so-called 'laws', but on the active powers of the stuffs of reality. |