16 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) |
| 8915 | How we refer to abstractions is much less clear than how we refer to other things [Rosen] |
| Full Idea: It is unclear how we manage to refer determinately to abstract entities in a sense in which it is not unclear how we manage to refer determinately to other things. | |
| From: Gideon Rosen (Abstract Objects [2001], 'Way of Ex') | |
| A reaction: This is where problems of abstraction overlap with problems about reference in language. Can we have a 'baptism' account of each abstraction (even very large numbers)? Will descriptions do it? Do abstractions collapse into particulars when we refer? |
| 19565 | How could the mind have a link to the necessary character of reality? [Devitt] |
| Full Idea: What non-experiential link to reality could support insights into its necessary character? | |
| From: Michael Devitt (There is No A Priori (and reply) [2005], 4) | |
| A reaction: The key to it, I think, is your theory of mind. If you are a substance dualist, then connecting to such deep things looks fine, but if you are a reductive physicalist then it looks absurdly hopeful. |
| 19564 | Some knowledge must be empirical; naturalism implies that all knowledge is like that [Devitt] |
| Full Idea: It is overwhelmingly plausible that some knowledge is empirical. The attractive thesis of naturalism is that all knowledge is; there is only one way of knowing. | |
| From: Michael Devitt (There is No A Priori (and reply) [2005], 1) | |
| A reaction: How many ways for us to know seems to depend on what faculties we have. We lump our senses together under a single heading. The arrival of data is not the same as the arrival of knowledge. I'm unconvinced that naturalists like me must accept this. |
| 8917 | The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting [Rosen] |
| Full Idea: The simplest version of the Way of Abstraction would be to say that an object is abstract if it is a referent of an idea that was formed by abstraction, but this is wedded to an outmoded philosophy of mind. | |
| From: Gideon Rosen (Abstract Objects [2001], 'Way of Abs') | |
| A reaction: This presumably refers to Locke, who wields the highly ambiguous term 'idea'. But if we sort out that ambiguity (by using modern talk of mental events, concepts and content?) we might reclaim the view. But do we have a 'genetic fallacy' here? |
| 8912 | Nowadays abstractions are defined as non-spatial, causally inert things [Rosen] |
| Full Idea: If any characterization of the abstract deserves to be regarded as the modern standard one, it is this: an abstract entity is a non-spatial (or non-spatiotemporal) causally inert thing. This view presents a number of perplexities... | |
| From: Gideon Rosen (Abstract Objects [2001], 'Non-spat') | |
| A reaction: As indicated in other ideas, the problem is that some abstractions do seem to be located somewhere in space-time, and to have come into existence, and to pass away. I like 'to exist is to have causal powers'. See Ideas 5992 and 8300. |
| 8913 | Chess may be abstract, but it has existed in specific space and time [Rosen] |
| Full Idea: The natural view of chess is not that it is a non-spatiotemporal mathematical object, but that it was invented at a certain time and place, that it has changed over the years, and so on. | |
| From: Gideon Rosen (Abstract Objects [2001], 'Non-spat') | |
| A reaction: This strikes me as being undeniable, and being an incredibly important point. Logicians seem to want to subsume things like games into the highly abstract world of logic and numbers. In fact the direction of explanation should be reversed. |
| 8914 | Sets are said to be abstract and non-spatial, but a set of books can be on a shelf [Rosen] |
| Full Idea: It is thought that sets are abstract, abstract objects do not exist in space, so sets must not exist in space. But it is not unnatural to say that a set of books is located on a certain shelf in the library. | |
| From: Gideon Rosen (Abstract Objects [2001], 'Non-spat') | |
| A reaction: The arguments against non-spatiality of abstractions seem to me to be conclusive. Not being able to assign a location to the cosine function is on a par with not knowing where my thoughts are located in my brain. |
| 8916 | Conflating abstractions with either sets or universals is a big claim, needing a big defence [Rosen] |
| Full Idea: The Way of Conflation account of abstractions (identifying them sets or with universals) is now relatively rare. The claim sets or universals are the only abstract objects would amount to a substantive metaphysical thesis, in need of defence. | |
| From: Gideon Rosen (Abstract Objects [2001], 'Way of Con') | |
| A reaction: If you produce a concept like 'mammal' by psychological abstraction, you do seem to end up with a set of things with shared properties, so this approach is not silly. I can't think of any examples of abstractions which are not sets or universals. |
| 8918 | Functional terms can pick out abstractions by asserting an equivalence relation [Rosen] |
| Full Idea: On Frege's suggestion, functional terms that pick out abstract expressions (such as 'direction' or 'equinumeral') have a typical form of f(a) = f(b) iff aRb, where R is an equivalence relation, a relation which is reflexive, symmetric and transitive. | |
| From: Gideon Rosen (Abstract Objects [2001], 'Way of Abs') | |
| A reaction: [Wright and Hale are credited with the details] This has become the modern orthodoxy among the logically-minded. Examples of R are 'parallel' or 'just as many as'. It picks out an 'aspect', which isn't far from the old view. |
| 8919 | Abstraction by equivalence relationships might prove that a train is an abstract entity [Rosen] |
| Full Idea: It seems possible to define a train in terms of its carriages and the connection relationship, which would meet the equivalence account of abstraction, but demonstrate that trains are actually abstract. | |
| From: Gideon Rosen (Abstract Objects [2001], 'Way of Abs') | |
| A reaction: [Compressed. See article for more detail] A tricky example, but a suggestive line of criticism. If you find two physical objects which relate to one another reflexively, symmetrically and transitively, they may turn out to be abstract. |