### Ideas from 'Ontology and Mathematical Truth' by Michael Jubien [1977], by Theme Structure

#### [found in 'Philosophy of Mathematics: anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,0-631-21870-x]].

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###### 4. Formal Logic / F. Set Theory ST / 1. Set Theory
 9967 'Impure' sets have a concrete member, while 'pure' (abstract) sets do not
 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.
###### 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
 9968 A model is 'fundamental' if it contains only concrete entities
 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.
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
 9965 There couldn't just be one number, such as 17
 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.
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
 9966 The subject-matter of (pure) mathematics is abstract structure
 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.
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
 9964 Since mathematical objects are essentially relational, they can't be picked out on their own
 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.
 9963 If we all intuited mathematical objects, platonism would be agreed
 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?
 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.
###### 9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
 9969 The empty set is the purest abstract object
 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!