Combining Texts

Ideas for 'Individuals without Sortals', 'The Rediscovery of the Mind' and 'Structuralism and the Notion of Dependence'

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9 ideas

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers]
     Full Idea: If we count the concept 'coin in this box', we could regard coin as the 'unit', while taking 'in this box' to limit the scope. Counting coins in two boxes would be not a difference in unit (kind of object), but in scope.
     From: M.R. Ayers (Individuals without Sortals [1974], 'Counting')
     A reaction: This is a very nice alternative to the Fregean view of counting, depending totally on the concept, and rests more on a natural concept of object. I prefer Ayers. Compare 'count coins till I tell you to stop'.
If counting needs a sortal, what of things which fall under two sortals? [Ayers]
     Full Idea: If we accepted that counting objects always presupposes some sortal, it is surely clear that the class of objects to be counted could be designated by two sortals rather than one.
     From: M.R. Ayers (Individuals without Sortals [1974], 'Realist' vii)
     A reaction: His nice example is an object which is both 'a single piece of wool' and a 'sweater', which had better not be counted twice. Wiggins struggles to argue that there is always one 'substance sortal' which predominates.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
'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.
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.
'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.
'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?
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
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.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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)
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'!