display all the ideas for this combination of texts
7 ideas
| 23460 | To count, we must distinguish things, and have a series with successors in it [Morris,M] |
| Full Idea: Distinguishing between things is not enough for counting. …We need the crucial extra notion of a successor in a series of a certain kind. | |
| From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro) | |
| A reaction: This is the thinking that led to the Dedekind-Peano axioms for arithmetic. E.g. each series member can only have one successor. There is an unformalisable assumption that the series can then be applied to the things. |
| 23451 | Counting needs to distinguish things, and also needs the concept of a successor in a series [Morris,M] |
| Full Idea: Just distinguishing things is not enough for counting (and hence arithmetic). We need the crucial extra notion of the successor in a series of some kind. | |
| From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.5) | |
| A reaction: This is a step towards the Peano Axioms of arithmetic. The successors could be fingers and toes, taken in a conventional order, and matched one-to-one to the objects. 'My right big toe of cows' means 16 cows (but non-verbally). |
| 23452 | Discriminating things for counting implies concepts of identity and distinctness [Morris,M] |
| Full Idea: The discrimination of things for counting needs to bring with it the notion of identity (and, correlatively, distinctness). | |
| From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.5) | |
| A reaction: Morris is exploring how practices like counting might reveal necessary truths about the world. |
| 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] |
| 21224 | Pure mathematics is the relations between all possible objects, and is thus formal ontology [Husserl, by Velarde-Mayol] |
| Full Idea: Pure mathematics is the science of the relations between any object whatever (relation of whole to part, relation of equality, property, unity etc.). In this sense, pure mathematics is seen by Husserl as formal ontology. | |
| From: report of Edmund Husserl (Formal and Transcendental Logic [1929]) by Victor Velarde-Mayol - On Husserl 4.5.2 | |
| A reaction: I would expect most modern analytic philosophers to agree with this. Modern mathematics (e.g. category theory) seems to have moved beyond this stage, but I still like this idea. |
| 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) |