17 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. |
| 16648 | Accidents must have formal being, if they are principles of real action, and of mental action and thought [Duns Scotus] |
| Full Idea: Accidents are principles of acting and principles of cognizing substance, and are the per se objects of the senses. But it is ridiculous to say that something is a principle of acting (either real or intentional) and yet does not have any formal being. | |
| From: John Duns Scotus (Ordinatio [1302], IV.12.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 10.5 | |
| A reaction: Pasnau cites this as the key scholastic argument for accidental properties having some independent and real existence (as required for Transubstantiation). Rival views say accidents are just 'modes' of a thing's existence. Aquinas compromised. |
| 15386 | If only the singular exists, science is impossible, as that relies on true generalities [Duns Scotus, by Panaccio] |
| Full Idea: Scotus argued that if everything is singular, with no objective common feature, science would be impossible, as it proceeds from general concepts. General is the opposite of singular, so it would be inadequate to understand a singular reality. | |
| From: report of John Duns Scotus (Ordinatio [1302]) by Claude Panaccio - Medieval Problem of Universals 'John Duns' | |
| A reaction: [compressed] It is a fact that if you generalise about 'tigers', you are glossing over the individuality of each singular tiger. That is OK for 'electron', if they really are identical, but our general predicates may be imposing identity on electrons. |
| 15387 | If things were singular they would only differ numerically, but horse and tulip differ more than that [Duns Scotus, by Panaccio] |
| Full Idea: Scotus argued that there must be some non-singular aspects of things, since there are some 'less than numerical differences' among them. A horse and a tulip differ more from each other than do two horses. | |
| From: report of John Duns Scotus (Ordinatio [1302]) by Claude Panaccio - Medieval Problem of Universals 'John Duns' | |
| A reaction: This seems to treat being 'singular' as if it were being a singularity. Presumably he is contemplating a thing being nothing but its Scotist haecceity. A neat argument, but I don't buy it. |
| 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! |
| 16632 | We distinguish one thing from another by contradiction, because this is, and that is not [Duns Scotus] |
| Full Idea: What is it [that establishes distinctness of things]? It is, to be sure, that which is universally the reason for distinguishing one thing from another: namely, a contradiction…..If this is, and that is not, then they are not the same entity in being. | |
| From: John Duns Scotus (Ordinatio [1302], IV.11.3), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 08.2 | |
| A reaction: This is a remarkably intellectualist view of such things. John Wycliff, apparently, enquired about how animals were going to manage all this sort of thing. It should appeal to the modern logical approach to metaphysics. |
| 13094 | The haecceity is the featureless thing which gives ultimate individuality to a substance [Duns Scotus, by Cover/O'Leary-Hawthorne] |
| Full Idea: For Scotus, the haecceity of an individual was a positive non-quidditative entity which, together with a common nature from which it was formally distinct, played the role of the ultimate differentia, thus individuating the substance. | |
| From: report of John Duns Scotus (Ordinatio [1302]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 6.1.3 | |
| A reaction: Most thinkers seem to agree (with me) that this is a non-starter, an implausible postulate designed to fill a gap in a metaphysic that hasn't been properly worked out. Leibniz is the hero who faces the problem and works around it. |
| 16770 | It is absurd that there is no difference between a genuinely unified thing, and a mere aggregate [Duns Scotus] |
| Full Idea: It seems absurd …that there should be no difference between a whole that is one thing per se, and a whole that is one thing by aggregation, like a cloud or a heap. | |
| From: John Duns Scotus (Ordinatio [1302], III.2.2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 25.5 | |
| A reaction: Leibniz invented monads because he was driven crazy by the quest for 'true unity' in things. Objective unity may be bogus, but I suspect that imposing plausible unity on things is the only way we can grasp the world. |
| 10919 | What prevents a stone from being divided into parts which are still the stone? [Duns Scotus] |
| Full Idea: What is it in this stone, by which ...it is absolutely incompatible with the stone for it to be divided into several parts each of which is this stone, the kind of division that is proper to a universal whole as divided into its subjective parts? | |
| From: John Duns Scotus (Ordinatio [1302], II d3 p1 q2 n48) | |
| A reaction: This is the origin of the concept of haecceity, when Scotus wants to know what exactly individuates each separate entity. He may have been mistaken in thinking that such a question has an answer. |
| 16768 | Two things are different if something is true of one and not of the other [Duns Scotus] |
| Full Idea: If this is, and that is not, then they are not the same entity in being. | |
| From: John Duns Scotus (Ordinatio [1302], IV.11.3), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 25.3 | |
| A reaction: This is the contrapositive of the indiscernibility of identicals, expressed in terms of what is true about a thing, rather than what properties pertain to it. |
| 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? |