green numbers give full details | back to texts | unexpand these ideas
| 14085 | 'Deductivist' structuralism is just theories, with no commitment to objects, or modality |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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. |
| 14088 | An 'intrinsic' property is either found in every duplicate, or exists independent of all externals |
| 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? |