24 ideas
| 5182 | Claims about 'the Absolute' are not even verifiable in principle [Ayer on Bradley] |
| Full Idea: Such a metaphysical pseudo-proposition as 'the Absolute enters into, but is itself incapable of, evolution and progress' (F.H.Bradley) is not even in principle verifiable. | |
| From: comment on F.H. Bradley (Appearance and Reality [1893]) by A.J. Ayer - Language,Truth and Logic Ch.1 | |
| A reaction: One may jeer at the Verification Principle for either failing to be precise, or for failing to pass its own test, but Ayer still has a point here. When we drift off into sustained abstractions, we must keeping asking if we are still saying anything real. |
| 6864 | Metaphysics is finding bad reasons for instinctive beliefs [Bradley] |
| Full Idea: Metaphysics is the finding of bad reasons for what we believe upon instinct; but to find these reasons is no less an instinct. | |
| From: F.H. Bradley (Appearance and Reality [1893]), quoted by Robin Le Poidevin - Interview with Baggini and Stangroom p.165 | |
| A reaction: A famous and very nice remark. The idea of believing things on instinct sounds more like David Hume than an idealist. Personally I am not so pessimistic about the enterprise. I think metaphysics is capable of changing what we believe. |
| 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) |
| 10999 | Names need a means of reidentifying their referents [Bradley, by Read] |
| Full Idea: Unless a name has associated with it a means of reidentifying its referent, we cannot use it. | |
| From: report of F.H. Bradley (Appearance and Reality [1893]) by Stephen Read - Thinking About Logic Ch.4 | |
| A reaction: Brilliant! This point is totally undeniable. It is not enough that someone be 'baptised'. We need to hang onto both the name and what it refers to, and how are we going to do that? |
| 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) |
| 14085 | 'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo] |
| Full Idea: The 'deductivist' version of eliminativist structuralism avoids ontological commitments to mathematical objects, and to modal vocabulary. Mathematics is formulations of various (mostly categorical) theories to describe kinds of concrete structures. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], 1) | |
| A reaction: 'Concrete' is ambiguous here, as mathematicians use it for the actual working maths, as opposed to the metamathematics. Presumably the structures are postulated rather than described. He cites Russell 1903 and Putnam. It is nominalist. |
| 14084 | Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo] |
| Full Idea: The 'non-eliminative' version of mathematical structuralism takes it to be a fundamental insight that mathematical objects are really just positions in abstract mathematical structures. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], I) | |
| A reaction: The point here is that it is non-eliminativist because it is committed to the existence of mathematical structures. I oppose this view, since once you are committed to the structures, you may as well admit a vast implausible menagerie of abstracta. |
| 14086 | 'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo] |
| Full Idea: The 'modal' version of eliminativist structuralism lifts the deductivist ban on modal notions. It studies what necessarily holds in all concrete models which are possible for various theories. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], I) | |
| A reaction: [He cites Putnam 1967, and Hellman 1989] If mathematical truths are held to be necessary (which seems to be right), then it seems reasonable to include modal notions, about what is possible, in its study. |
| 14087 | 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo] |
| Full Idea: 'Set-theoretic' structuralism rejects deductive nominalism in favour of a background theory of sets, and mathematics as the various structures realized among the sets. This is often what mathematicians have in mind when they talk about structuralism. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], I) | |
| A reaction: This is the big shift from 'mathematics can largely be described in set theory' to 'mathematics just is set theory'. If it just is set theory, then which version of set theory? Which axioms? The safe iterative conception, or something bolder? |
| 14089 | Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo] |
| Full Idea: Structuralism can be distinguished from traditional Platonism in that it denies that mathematical objects from the same structure are ontologically independent of one another | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], III) | |
| A reaction: My instincts strongly cry out against all versions of this. If you are going to be a platonist (rather as if you are going to be religious) you might as well go for it big time and have independent objects, which will then dictate a structure. |
| 14083 | Structuralism is right about algebra, but wrong about sets [Linnebo] |
| Full Idea: Against extreme views that all mathematical objects depend on the structures to which they belong, or that none do, I defend a compromise view, that structuralists are right about algebraic objects (roughly), but anti-structuralists are right about sets. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], Intro) |
| 14090 | In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo] |
| Full Idea: If objects depend on the other objects, this would mean an 'upward' dependence, in that they depend on the structure to which they belong, where the physical realm has a 'downward' dependence, with structures depending on their constituents. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], III) | |
| A reaction: This nicely captures an intuition I have that there is something wrong with a commitment primarily to 'structures'. Our only conception of such things is as built up out of components. Not that I am committing to mathematical 'components'! |
| 14091 | There may be a one-way direction of dependence among sets, and among natural numbers [Linnebo] |
| Full Idea: We can give an exhaustive account of the identity of the empty set and its singleton without mentioning infinite sets, and it might be possible to defend the view that one natural number depends on its predecessor but not vice versa. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], V) | |
| A reaction: Linnebo uses this as one argument against mathematical structuralism, where the small seems to depend on the large. The view of sets rests on the iterative conception, where each level is derived from a lower level. He dismisses structuralism of sets. |
| 6422 | Internal relations are said to be intrinsic properties of two terms, and of the whole they compose [Bradley, by Russell] |
| Full Idea: The doctrine of internal relations held that every relation between two terms expresses, primarily, intrinsic properties of the two terms and, in ultimate analysis, a property of the whole which the two compose. | |
| From: report of F.H. Bradley (Appearance and Reality [1893]) by Bertrand Russell - My Philosophical Development Ch.5 | |
| A reaction: Russell's first big campaign was to reject this view, and his ontology from then on included relations among the catalogue of universals. The coherence theory of truth also gets thrown out at the same time. Russell seems right. |
| 7966 | Relations must be linked to their qualities, but that implies an infinite regress of relations [Bradley] |
| Full Idea: If a relation between qualities is to be something, then clearly we will now require a new connecting relation. The links are united by a link, and this link has two ends, which require a fresh link to connect them to the old. | |
| From: F.H. Bradley (Appearance and Reality [1893], p.28), quoted by Cynthia Macdonald - Varieties of Things Ch.6 | |
| A reaction: That is: external relations generates an infinite regress, so relations must be internal. Russell launched his own philosophy with an attack on Bradley's idea. Personally I take how two things 'relate' to one another as one of the deepest of mysteries. |
| 14088 | An 'intrinsic' property is either found in every duplicate, or exists independent of all externals [Linnebo] |
| Full Idea: There are two main ways of spelling out an 'intrinsic' property: if and only if it is shared by every duplicate of an object, ...and if and only if the object would have this property even if the rest of the universe were removed or disregarded. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], II) | |
| A reaction: [He cites B.Weatherson's Stanford Encyclopaedia article] How about an intrinsic property being one which explains its identity, or behaviour, or persistence conditions? |
| 6404 | British Idealists said reality is a single Mind which experiences itself [Bradley, by Grayling] |
| Full Idea: The idealism of Green and Bradley, both of whom were much influenced by the German Idealists, espoused the thesis that the universe ultimately consists of a single Mind which, so to speak, experiences itself. | |
| From: report of F.H. Bradley (Appearance and Reality [1893]) by A.C. Grayling - Russell Ch.2 | |
| A reaction: This looks now like the last (extreme) throw by the religious view of the world, which collapsed in the face of the empirical realism of Russell and Moore. It is all Kant's fault, for cutting us off from his 'noumenon'. |
| 22299 | Bradley's objective idealism accepts reality (the Absolute), but says we can't fully describe it [Bradley, by Potter] |
| Full Idea: Objective idealists such as Bradley (rather than Berkeley's subjective view) accepted the substantial existence of reality (which they called the 'Absolute') but held that thought cannot fully describe it. | |
| From: report of F.H. Bradley (Appearance and Reality [1893]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 23 'Abs' | |
| A reaction: That thought can't 'fully' describe it seems obvious, so I suspect Bradley's view are stronger than that. This sounds like modern strong and weak anti-realists; strong ones deny reality, but weak ones just deny we know where the joints are. |
| 21343 | Qualities and relations are mere appearance; the Absolute is a single undifferentiated substance [Bradley, by Heil] |
| Full Idea: In Bradley's view, qualities and relations belong to the realm of appearance. We are left with a single, undifferentiated substance: the Absolute. | |
| From: report of F.H. Bradley (Appearance and Reality [1893]) by John Heil - Relations 'Internal' | |
| A reaction: I've not read Bradley, but I can't distinguish this proposal from Parmenides's belief in The One. Or maybe Spinoza's monist view of God and Nature (but that is 'differentiated'). It doesn't sound like Hegel. |
| 6406 | Reality is one, because plurality implies relations, and they assert a superior unity [Bradley] |
| Full Idea: Reality is one. It must be simple because plurality, taken as real, contradicts itself. Plurality implies relations, and, through its relations it unwillingly asserts always a superior unity. | |
| From: F.H. Bradley (Appearance and Reality [1893], p.519), quoted by A.C. Grayling - Russell Ch.2 | |
| A reaction: This argument depends on a belief in 'internal' relations, which Russell famously attacked. If an internal feature of every separate item was its relation to other things, then I suppose Bradley would be right. But it isn't, and he isn't. |