15 ideas
| 9967 | 'Impure' sets have a concrete member, while 'pure' (abstract) sets do not [Jubien] |
| Full Idea: Any set with a concrete member is 'impure'. 'Pure' sets are those that are not impure, and are paradigm cases of abstract entities, such as the sort of sets apparently dealt with in Zermelo-Fraenkel (ZF) set theory. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.116) | |
| A reaction: [I am unclear whether Jubien is introducing this distinction] This seems crucial in accounts of mathematics. On the one had arithmetic can be built from Millian pebbles, giving impure sets, while logicists build it from pure sets. |
| 9968 | A model is 'fundamental' if it contains only concrete entities [Jubien] |
| Full Idea: A first-order model can be viewed as a kind of ordered set, and if the domain of the model contains only concrete entities then it is a 'fundamental' model. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.117) | |
| A reaction: An important idea. Fundamental models are where the world of logic connects with the physical world. Any account of relationship between fundamental models and more abstract ones tells us how thought links to world. |
| 9965 | There couldn't just be one number, such as 17 [Jubien] |
| Full Idea: It makes no sense to suppose there might be just one natural number, say seventeen. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.113) | |
| A reaction: Hm. Not convinced. If numbers are essentially patterns, we might only have the number 'twelve', because we had built our religion around anything which exhibited that form (in any of its various arrangements). Nice point, though. |
| 9966 | The subject-matter of (pure) mathematics is abstract structure [Jubien] |
| Full Idea: The subject-matter of (pure) mathematics is abstract structure per se. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.115) | |
| A reaction: This is the Structuralist idea beginning to take shape after Benacerraf's launching of it. Note that Jubien gets there by his rejection of platonism, whereas some structuralist have given a platonist interpretation of structure. |
| 9963 | If we all intuited mathematical objects, platonism would be agreed [Jubien] |
| Full Idea: If the intuition of mathematical objects were general, there would be no real debate over platonism. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.111) | |
| A reaction: It is particularly perplexing when Gödel says that his perception of them is just like sight or smell, since I have no such perception. How do you individuate very large numbers, or irrational numbers, apart from writing down numerals? |
| 9962 | How can pure abstract entities give models to serve as interpretations? [Jubien] |
| Full Idea: I am unable to see how the mere existence of pure abstract entities enables us to concoct appropriate models to serve as interpretations. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.111) | |
| A reaction: Nice question. It is always assumed that once we have platonic realm, that everything else follows. Even if we are able to grasp the objects, despite their causal inertness, we still have to discern innumerable relations between them. |
| 9964 | Since mathematical objects are essentially relational, they can't be picked out on their own [Jubien] |
| Full Idea: The essential properties of mathematical entities seem to be relational, ...so we make no progress unless we can pick out some mathematical entities wihout presupposing other entities already picked out. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.112) | |
| A reaction: [compressed] Jubien is a good critic of platonism. He has identified the problem with Frege's metaphor of a 'borehole', where we discover delightful new properties of numbers simply by reaching them. |
| 9969 | The empty set is the purest abstract object [Jubien] |
| Full Idea: The empty set is the pure abstract object par excellence. | |
| From: Michael Jubien (Ontology and Mathematical Truth [1977], p.118 n8) | |
| A reaction: So a really good PhD on the empty set could crack the whole nature of reality. Get to work, whoever you are! |
| 22971 | In moving from capitalism to communism a revolutionary dictatorship of the proletariat is needed [Marx] |
| Full Idea: Between the capitalist and communist society lies the revolutionary transformation of the one into the other. Corresponding to this is a political transition period in which the state can be nothing but the revolutionary dictatorship of the proletariat. | |
| From: Karl Marx (Critique of the Gotha Program [1875], IV) | |
| A reaction: This hugely influential idea was catastrophic for the twentieth century, because the leaders of the proletarian dictatorship adored and abused the power, and wouldn't give it up for some feeble next stage. |
| 22000 | From each according to his ability, to each according to his need [Marx] |
| Full Idea: From each according to his ability, to each according to his need. | |
| From: Karl Marx (Critique of the Gotha Program [1875]), quoted by Peter Singer - Marx 9 | |
| A reaction: Singer says this was not original to Marx, and he placed little emphasis on it. The obvious capitalist response is to ask how you will motivate someone who has huge abilities but few needs. It implies huge inequalities of altruism. |
| 22969 | People who only have their labour power are the slaves of those permitting them to work [Marx] |
| Full Idea: The man who possesses no other property than his labour power must, in all conditions of society and culture, be the slave of other men who have made themselves the owners of the material conditions of labour. He can only work with their permission. | |
| From: Karl Marx (Critique of the Gotha Program [1875], I) | |
| A reaction: In a world of vast multinationals, the person giving the permission to work is nearly always dependent on some higher level permission. In any sort of society people can only work with the consensus of other people. |
| 22970 | Freedom is making the state subordinate to its society [Marx] |
| Full Idea: Freedom consists in converting the state from an organ superimposed on society into one completely subordinate to it. | |
| From: Karl Marx (Critique of the Gotha Program [1875], IV) | |
| A reaction: The intermediate stage is dictatorship of the proletariat (presumably exercised by the communist leadership). No twentieth century marxist state ever got near the freedom which Marx was seeking. A liberal society might achieve it! |
| 22972 | Bourgeois 'freedom of conscience' just tolerates all sorts of religious intolerance [Marx] |
| Full Idea: Bourgeois 'freedom of conscience' is just the toleration of all possible kinds of religious unfreedom of conscience, and the workers' party should endeavour to liberate the conscience from the witchery of religion. | |
| From: Karl Marx (Critique of the Gotha Program [1875], IV) | |
| A reaction: We see this in modern 'faith' schools in the UK, which do not seem to be required to live up to the standards of freedom of belief expected in the rest of a liberal society. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |