14 ideas
| 10185 | Set theory is the standard background for modern mathematics [Burgess] |
| Full Idea: In present-day mathematics, it is set theory that serves as the background theory in which other branches of mathematics are developed. | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
| A reaction: [He cites Bourbaki as an authority for this] See Benacerraf for a famous difficulty here, when you actually try to derive an ontology from the mathematicians' working practices. |
| 10184 | Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure [Burgess] |
| Full Idea: On the structuralist interpretation, theorems of analysis concerning the real numbers R are about all complete ordered fields. So R, which appears to be the name of a specific structure, is taken to be a variable ranging over structures. | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
| A reaction: Since I am beginning to think that nearly all linguistic expressions should be understood as variables, I find this very appealing, even if Burgess hates it. Terms slide and drift, and are vague, between variable and determinate reference. |
| 10189 | There is no one relation for the real number 2, as relations differ in different models [Burgess] |
| Full Idea: One might meet the 'Van Inwagen Problem' by saying that the intrinsic properties of the object playing the role of 2 will differ from one model to another, so that no statement about the intrinsic properties of 'the' real numbers will make sense. | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
| A reaction: There seems to be a potential confusion among opponents of structuralism between relations at the level of actual mathematical operations, and generalisations about relations, which are captured in the word 'patterns'. Call them 'meta-relations'? |
| 10186 | If set theory is used to define 'structure', we can't define set theory structurally [Burgess] |
| Full Idea: It is to set theory that one turns for the very definition of 'structure', ...and this creates a problem of circularity if we try to impose a structuralist interpretation on set theory. | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
| A reaction: This seems like a nice difficulty, especially if, like Shapiro, you wade in and try to give a formal account of structures and patterns. Resnik is more circumspect and vague. |
| 10187 | Abstract algebra concerns relations between models, not common features of all the models [Burgess] |
| Full Idea: Abstract algebra, such as group theory, is not concerned with the features common to all models of the axioms, but rather with the relationships among different models of those axioms (especially homomorphic relation functions). | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
| A reaction: It doesn't seem to follow that structuralism can't be about the relations (or patterns) found when abstracting away and overviewing all the models. One can study family relations, or one can study kinship in general. |
| 10188 | How can mathematical relations be either internal, or external, or intrinsic? [Burgess] |
| Full Idea: The 'Van Inwagen Problem' for structuralism is of explaining how a mathematical relation (such as set membership, or the ratios of an ellipse) can fit into one of the three scholastics types of relations: are they internal, external, or intrinsic? | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
| A reaction: The difficulty is that mathematical objects seem to need intrinsic properties to get any of these three versions off the ground (which was Russell's complaint against structures). |
| 7544 | Many people imagine that to experience is to understand [Goethe] |
| Full Idea: There are many people who imagine that what they experience they also understand. | |
| From: Wolfgang von Goethe (Maxims and Reflections [1825], 889) | |
| A reaction: This should be posted over the arrivals gate of every international airport, for returning holiday-makers. It seems to place Goethe on the rationalist side of the debate with empiricism. It is hard to explain 'understanding' in Humean terms. |
| 7541 | Man never understands how anthropomorphic he is [Goethe] |
| Full Idea: Man never understands how anthropomorphic he is. | |
| From: Wolfgang von Goethe (Maxims and Reflections [1825], 203) | |
| A reaction: Nice. It is true, even when it is pointed out to us. No matter how hard we try to realise how very different animals are from us, we can't help identifying with them. Religious people even do it with inanimate creation. |
| 7543 | We gain self-knowledge through action, not thought - especially when doing our duty [Goethe] |
| Full Idea: How can we learn self-knowledge? Never by taking thought, but rather by action. Try to do your duty and you'll soon discover what you're like. | |
| From: Wolfgang von Goethe (Maxims and Reflections [1825], 442) | |
| A reaction: Good! I even like the unfashionable bit about duty. If you just do what you want, you will discover your interests, but not so much about your capacities. However, when you have to do something less comfortable, it is very revealing. |
| 7540 | Beauty is a manifestation of secret natural laws [Goethe] |
| Full Idea: Beauty is a manifestation of secret natural laws which without this appearance would have remained eternally hidden from us. | |
| From: Wolfgang von Goethe (Maxims and Reflections [1825], 183) | |
| A reaction: An interesting defence of beauty as an objective feature of the world. I'm not sure. Much beauty is indeed the result of growth or erosion expressing underlying laws, but then I have always thought there was a sexual component to visual beauty. |
| 7538 | The happiest people link the beginning and end of life [Goethe] |
| Full Idea: The happiest man is one who can link the end of his life with its beginning. | |
| From: Wolfgang von Goethe (Maxims and Reflections [1825], 140) | |
| A reaction: [from 'Art and Antiquity']. A nice thought, which chimes in with the idea that a good life is like a complete story or a work of art (Idea 7501), or that it is 'eudaimon'. |
| 7542 | The best form of government teaches us to govern ourselves [Goethe] |
| Full Idea: You ask which form of government is the best? Whichever teaches us to govern ourselves. | |
| From: Wolfgang von Goethe (Maxims and Reflections [1825], 353) | |
| A reaction: Not a fashionable view, since the rise of freedom as the highest political ideal, but I identify with the idea that a good government should educate, and should try to facilitate virtue as well as pleasure. |
| 21097 | Modern monarchies are (like republics) rule by law, rather than by men [Hume] |
| Full Idea: In modern times monarchical government seems to have made the greatest advances towards perfection. It may now be affirmed of civilized monarchies, what was formerly said in praise of republics alone, that they are a government of laws, not of men. | |
| From: David Hume (Of Civil Liberty [1750], p.54) | |
| A reaction: Dreams of simple 'government by law' disappeared with the rise of modern media, which can be controlled by wealth. |
| 7539 | To get duties from people without rights, you must pay them well [Goethe] |
| Full Idea: If you demand duties from people and will not concede them rights, you have to pay them well. | |
| From: Wolfgang von Goethe (Maxims and Reflections [1825], 180) | |
| A reaction: [from 'Art and Antiquity']. ...or have great power over them. Goethe gives the optimistic liberal view, rather than the Marxist view. |