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? |
| 8122 | True works of art transmit completely new feelings [Tolstoy] |
| Full Idea: Only that is a true work of art which transmits fresh feelings not previously experienced by man. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.9) | |
| A reaction: I think a great composer will probably not have any new feelings at all, but will discover new expressions which contain feelings by which even they are surprised (e.g. the Tristan chord). |
| 8121 | Art is when one man uses external signs to hand on his feelings to another man [Tolstoy] |
| Full Idea: Art is a human activity in which one man consciously by means of external signs, hands on to others feelings he has lived through, and other are infected by those feelings, and also experience them. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.5) | |
| A reaction: Such definitions always work better for some art forms than for others. This may fit 'Anna Karenin' quite well, but probably not Bach's 'Art of Fugue'. Writing obscenities on someone's front door would fit this definition. |
| 8124 | The highest feelings of mankind can only be transmitted by art [Tolstoy] |
| Full Idea: The highest feelings to which mankind has attained can only be transmitted from man to man by art. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.17) | |
| A reaction: We are much more nervous these days of talking about 'highest' feelings. Tolstoy obviously considers religion to be an ingredient of the highest feelings, but that prevents us from judging them purely as feelings. Music is the place to rank feelings. |
| 8123 | The purpose of art is to help mankind to evolve better, more socially beneficial feelings [Tolstoy] |
| Full Idea: The evolution of feeling proceeds by means of art - feelings less kind and less necessary for the well-being of mankind being replaced by others kinder and more needful for that end. That is the purpose of art. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.16) | |
| A reaction: Underneath his superficially expressivist view of art, Tolstoy is really an old-fashioned moralist about it, like Dr Johnson. This is the moralism of the great age of the nineteenth century novel (which was, er, the greatest age of the novel!). |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 22710 | People estimate art according to their moral values [Tolstoy] |
| Full Idea: The estimation of the value of art …depends on men's perception of the meaning of life; depends on what they hold to be the good and evil of life. | |
| From: Leo Tolstoy (What is Art? [1898]), quoted by Iris Murdoch - The Sublime and the Good p.206 | |
| A reaction: [No ref given] This is put to the test by the insightful depiction of wickedness. We condemn the wickedness and admire the insight. Every reading of a novel is a moral journey, though I'm not sure how the true psychopath reads a novel. |
| 8125 | The upper classes put beauty first, and thus freed themselves from morality [Tolstoy] |
| Full Idea: The people of the upper class, more and more frequently encountering the contradictions between beauty and goodness, put the ideal of beauty first, thus freeing themselves from the demands of morality. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.17) | |
| A reaction: The rich are a great deal freer to pursue the demands of beauty than are the poor. They also have a tradition of 'immorality' (such as duels and adultery) which was in place long before they discovered art. |
| 8064 | We separate the concept of beauty from goodness, unlike the ancients [Tolstoy] |
| Full Idea: The ancients had not that conception of beauty separated from goodness which forms the basis and aim of aesthetics in our time. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.3) | |
| A reaction: This is written at around the time of the Aesthetic Movement, but Tolstoy's own novels are intensely moral. This separation makes abstract painting possible. |