17 ideas
| 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. |
| 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? |
| 5662 | Maybe induction could never prove the existence of something unobservable [Ayer] |
| Full Idea: Some people hold that no inductive argument can give us any reason to believe in the existence of something which could not even in principle be observed. | |
| From: A.J. Ayer (The Concept of a Person [1963], §I) | |
| A reaction: I see nothing illogical in inferring the existence of a poltergeist from the recurrent flight of objects around my lounge. Only an excessive empiricism (which used to afflict Ayer) could lead to this claim. |
| 5664 | Consciousness must involve a subject, and only bodies identify subjects [Ayer] |
| Full Idea: It may not make sense to talk of states of consciousness except as the experiences of some conscious subject; and it may well be that this conscious subject can not be identified except by reference to his body. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: It strikes me that Ayer deserves more credit as a pioneer of this view. It tracks back to what may turn out to be the key difficulty for Descartes - how do you individuate a mental substance? I may identify me, but how do I identify you? |
| 5668 | People own conscious states because they are causally related to the identifying body [Ayer] |
| Full Idea: I think personal identity depends on the identity of the body, and that a person's ownership of states of consciousness consists in their standing in a special causal relation to the body by which he is identified. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: I think with this is right, with the slight reservation that Ayer talks as if there were two things which have a causal relationship, implying that the link is contingent. Better to think of the whole thing as a single causal network. |
| 5661 | We identify experiences by their owners, so we can't define owners by their experiences [Ayer] |
| Full Idea: Normally we identify experiences in terms of the persons whose experiences they are; but this will lead to a vicious circle if persons themselves are to be analysed in terms of their experiences. | |
| From: A.J. Ayer (The Concept of a Person [1963], §I) | |
| A reaction: This (from a leading empiricist) is a nice basic challenge to all empiricist accounts of personal identity. One might respond my saying that the circle is not vicious. There are two interlinked concepts (experience and persons), like day and night. |
| 5665 | Memory is the best proposal as what unites bundles of experiences [Ayer] |
| Full Idea: The most promising suggestion is that the bundles are tied together by means of memory. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: This is interesting for showing how Locke was essentially trying to meet (in advance) Hume's 'bundle' scepticism. Hume proposed associations as the unifying factor, instead of memories. Ayer proposes concepts as a candidate. |
| 5666 | Not all exerience can be remembered, as this would produce an infinite regress [Ayer] |
| Full Idea: Not every experience can be remembered; otherwise each piece of remembering, which is itself an experience, would have to be remembered, and each remembering of a remembering and so ad infinitum. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: See Idea 5667. Ayer takes for granted two sorts of consciousness - current awareness, and memory. Ayer brings out a nice difficulty for Locke's proposal, but also draws attention to what may be a very basic misunderstanding about the mind. |
| 5669 | Personal identity can't just be relations of experiences, because the body is needed to identify them [Ayer] |
| Full Idea: A Humean theory, in which a person's identity is made to depend upon relations between experiences ..is not tenable unless the experiences themselves can be identified, and that is only possible through their association with the body. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: This seems to me a very fruitful response to difficulties with the 'bundle' view of a person - a better response than the a priori claims of Butler and Reid, or the transcendental argument of Kant. Only a philosopher could ignore the body. |
| 1590 | The just man does not harm his enemies, but benefits everyone [Plato] |
| Full Idea: First, Socrates, you told me justice is harming your enemies and helping your friends. But later it seemed that the just man, since everything he does is for someone's benefit, never harms anyone. | |
| From: Plato (Clitophon [c.372 BCE], 410b) | |
| A reaction: Socrates certainly didn't subscribe to the first view, which is the traditional consensus in Greek culture. In general Socrates agreed with the views later promoted by Jesus. |