14 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) |
| 10735 | Abstraction from objects won't reveal an operation's being performed 'so many times' [Geach] |
| Full Idea: For an understanding of arithmetic the grasp of an operation's being performed 'so many times' is quite indispensable; and abstraction of a feature from groups of nuts cannot give us this grasp. | |
| From: Peter Geach (Abstraction Reconsidered [1983], p.170) | |
| A reaction: I end up defending the empirical approach to arithmetic because remarks like this are so patently false. Geach seems to think we arrive ready-made in the world, just raring to get on with some counting. He lacks the evolutionary perspective. |
| 10732 | If concepts are just recognitional, then general judgements would be impossible [Geach] |
| Full Idea: If concepts were nothing but recognitional capacities, then it is unintelligible that I can judge that cats eat mice when neither of them are present. | |
| From: Peter Geach (Abstraction Reconsidered [1983], p.164) | |
| A reaction: Having observed the importance of recognition for the abstractionist (Idea 10731), he then seems to assume that there is nothing more to their concepts. Geach fails to grasp levels of abstraction, and cross-reference, and generalisation. |
| 10731 | For abstractionists, concepts are capacities to recognise recurrent features of the world [Geach] |
| Full Idea: For abstractionists, concepts are essentially capacities for recognizing recurrent features of the world. | |
| From: Peter Geach (Abstraction Reconsidered [1983], p.163) | |
| A reaction: Recognition certainly strikes me as central to thought (and revelatory of memory, since we continually recognise what we cannot actually recall). Geach dislikes this view, but I see it as crucial to an evolutionary view of thought. |
| 10733 | The abstractionist cannot explain 'some' and 'not' [Geach] |
| Full Idea: The abstractionist cannot give a logically coherent account of the features that are supposed to be reached by discriminative attention, corresponding to the words 'some' and 'not'. | |
| From: Peter Geach (Abstraction Reconsidered [1983], p.167) | |
| A reaction: I understand 'some' in terms of mereology, because that connects to experience, and 'not' I take to derive more from psychological experience than from the physical world, building on thwarted expectation, which even animals experience. |
| 10734 | Only a judgement can distinguish 'striking' from 'being struck' [Geach] |
| Full Idea: To understand the verb 'to strike' we must see that 'striking' and 'being struck' are different, but necessarily go together in event and thought; only in the context of a judgment can they be distinguished, when we think of both together. | |
| From: Peter Geach (Abstraction Reconsidered [1983], p.168) | |
| A reaction: Geach seems to have a strange notion that judgements are pure events which can precede all experience, and are the only ways we can come to understand experience. He needs to start from animals (or 'brutes', as he still calls them!). |
| 8412 | A causal interaction is when two processes intersect, and correlated modifications persist afterwards [Salmon] |
| Full Idea: When two processes intersect, and they undergo correlated modifications which persist after the intersection, I shall say that the intersection is a causal interaction. I take this as a fundamental causal concept. | |
| From: Wesley Salmon (Causality: Production and Propagation [1980], §4) | |
| A reaction: There may be a problem individuating processes, just as there is for events. I like this approach to causation, which is ontologically sparse, and fits in with the scientific worldview. Change of properties sounds precise, but isn't. Stick to processes. |
| 8413 | Cause must come first in propagations of causal interactions, but interactions are simultaneous [Salmon] |
| Full Idea: In a typical cause-effect situation (a 'propagation') cause must precede effect, for propagation over a finite time interval is an essential feature. In an 'interaction', an intersection of processes resulting in change, we have simultaneity. | |
| From: Wesley Salmon (Causality: Production and Propagation [1980], §8) | |
| A reaction: This takes the direction of time as axiomatic, and quite right too. Salmon isn't addressing the real difficulty, though, which is that the resultant laws are usually held to be time-reversible, which is a bit of a puzzle. |
| 8411 | Instead of localised events, I take enduring and extended processes as basic to causation [Salmon] |
| Full Idea: I propose to approach causality by taking processes rather than events as basic entities. Events are relatively localised in space and time, while processes have much greater temporal duration, and, in many cases, much greater spatial extent. | |
| From: Wesley Salmon (Causality: Production and Propagation [1980], §2) | |
| A reaction: This strikes me as an incredibly promising proposal, not just in our understanding of causation, but for our general metaphysics and understanding of nature. See Idea 4931, for example. Vague events and processes blend into one another. |