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| 10631 | If 'x is heterological' iff it does not apply to itself, then 'heterological' is heterological if it isn't heterological |
| Full Idea: If we stipulate that 'x is heterological' iff it does not apply to itself, we speedily arrive at the contradiction that 'heterological' is itself heterological just in case it is not. | |||
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2) |
| 10624 | The incompletability of formal arithmetic reveals that logic also cannot be completely characterized |
| Full Idea: The incompletability of formal arithmetic reveals, not arithmetical truths which are not truths of logic, but that logical truth likewise defies complete deductive characterization. ...Gödel's result has no specific bearing on the logicist project. | |||
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], §2 n5) | |||
| A reaction: This is the key defence against the claim that Gödel's First Theorem demolished logicism. |
| 10629 | If structures are relative, this undermines truth-value and objectivity |
| Full Idea: The relativization of ontology to theory in structuralism can't avoid carrying with it a relativization of truth-value, which would compromise the objectivity which structuralists wish to claim for mathematics. | |||
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2 n26) | |||
| A reaction: This is the attraction of structures which grow out of the physical world, where truth-value is presumably not in dispute. |
| 10628 | The structural view of numbers doesn't fit their usage outside arithmetical contexts |
| Full Idea: It is not clear how the view that natural numbers are purely intra-structural 'objects' can be squared with the widespread use of numerals outside purely arithmetical contexts. | |||
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2 n26) | |||
| A reaction: I don't understand this objection. If they refer to quantity, they are implicitly cardinal. If they name things in a sequence they are implicitly ordinal. All users of numbers have a grasp of the basic structure. |
| 10622 | The neo-Fregean is more optimistic than Frege about contextual definitions of numbers |
| Full Idea: The neo-Fregean takes a more optimistic view than Frege of the prospects for the kind of contextual explanation of the fundamental concepts of arithmetic and analysis (cardinals and reals), which he rejected in 'Grundlagen' 60-68. | |||
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], §1) |
| 10626 | Objects just are what singular terms refer to |
| Full Idea: Objects, as distinct from entities of other types (properties, relations or, more generally, functions of different types and levels), just are what (actual or possible) singular terms refer to. | |||
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.1) | |||
| A reaction: I find this view very bizarre and hard to cope with. It seems either to preposterously accept the implications of the way we speak into our ontology ('sakes'?), or preposterously bend the word 'object' away from its normal meaning. |
| 10630 | Abstracted objects are not mental creations, but depend on equivalence between given entities |
| Full Idea: The new kind of abstract objects are not creations of the human mind. ...The existence of such objects depends upon whether or not the relevant equivalence relation holds among the entities of the presupposed kind. | |||
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2) | |||
| A reaction: It seems odd that we no longer have any choice about what abstract objects we use, and that we can't evade them if the objects exist, and can't have them if the objects don't exist - and presumably destruction of the objects kills the concept? |
| 10627 | Many conceptual truths ('yellow is extended') are not analytic, as derived from logic and definitions |
| Full Idea: There are many statements which are plausibly viewed as conceptual truths (such as 'what is yellow is extended') which do not qualify as analytic under Frege's definition (as provable using only logical laws and definitions). | |||
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2) | |||
| A reaction: Presumably this is because the early assumptions of Frege were mathematical and logical, and he was trying to get away from Kant. That yellow is extended is a truth for non-linguistic beings. |