| 2001 | Intro to 'The Reason's Proper Study' |
| §1 | p.1 | 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) |
| §2 n5 | p.4 | 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. |
| 3.1 | p.8 | 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. |
| 3.2 | p.12 | 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. |
| 3.2 | p.17 | 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? |
| 3.2 | p.18 | 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) |
| 3.2 n26 | p.15 | 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. |
| 3.2 n26 | p.15 | 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. |
| 2007 | Logicism in the 21st Century |
| 1 | p.167 | 8786 | One first-order abstraction principle is Frege's definition of 'direction' in terms of parallel lines |
| Full Idea: An example of a first-order abstraction principle is Frege's definition of 'direction' in terms of parallel lines; a higher-order example (which refers to first-order predicates) defines 'equinumeral' in terms of one-to-one correlation (Hume's Principle). | |||
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 1) | |||
| A reaction: [compressed] This is the way modern logicians now treat abstraction, but abstraction principles include the elusive concept of 'equivalence' of entities, which may be no more than that the same adjective ('parallel') can be applied to them. |
| 1 | p.169 | 8784 | Neo-logicism founds arithmetic on Hume's Principle along with second-order logic |
| Full Idea: The result of joining Hume's Principle to second-order logic is a consistent system which is a foundation for arithmetic, in the sense that all the fundamental laws of arithmetic are derivable within it as theorems. This seems a vindication of logicism. | |||
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 1) | |||
| A reaction: The controversial part seems to be second-order logic, which Quine (for example) vigorously challenged. The contention against most attempts to improve Frege's logicism is that they thereby cease to be properly logical. |
| 1 n2 | p.167 | 8783 | Logicism might also be revived with a quantificational approach, or an abstraction-free approach |
| Full Idea: Two modern approaches to logicism are the quantificational approach of David Bostock, and the abstraction-free approach of Neil Tennant. | |||
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 1 n2) | |||
| A reaction: Hale and Wright mention these as alternatives to their own view. I merely catalogue them for further examination. My immediate reaction is that Bostock sounds hopeless and Tennant sounds interesting. |
| 3 | p.179 | 8787 | The Julius Caesar problem asks for a criterion for the concept of a 'number' |
| Full Idea: The Julius Caesar problem is the problem of supplying a criterion of application for 'number', and thereby setting it up as the concept of a genuine sort of object. (Why is Julius Caesar not a number?) | |||
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 3) | |||
| A reaction: One response would be to deny that numbers are objects. Another would be to derive numbers from their application in counting objects, rather than the other way round. I suspect that the problem only real bothers platonists. Serves them right. |
| 8 | p.196 | 8788 | Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology |
| Full Idea: It is only if logic is metaphysically and epistemologically privileged that a reduction of mathematical theories to logical ones can be philosophically any more noteworthy than a reduction of any mathematical theory to any other. | |||
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 8) | |||
| A reaction: It would be hard to demonstrate this privileged position, though intuitively there is nothing more basic in human rationality. That may be a fact about us, but it doesn't make logic basic to nature, which is where proper reduction should be heading. |
| 2009 | The Metaontology of Abstraction |
| §3 | p.182 | 12223 | It is a fallacy to explain the obscure with the even more obscure |
| Full Idea: The fallacy of 'ad obscurum per obscurius' is to explain the obscure by appeal to what is more obscure. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §3) | |||
| A reaction: Not strictly a fallacy, so much as an example of inadequate explanation, along with circularity and infinite regresses. |
| §4 | p.184 | 12224 | Are neo-Fregeans 'maximalists' - that everything which can exist does exist? |
| Full Idea: It is claimed that neo-Fregeans are committed to 'maximalism' - that whatever can exist does. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §4) | |||
| A reaction: [The cite Eklund] They observe that maximalism denies contingent non-existence (of the £20 note I haven't got). There seems to be the related problem of 'hyperinflation', that if abstract objects are generated logically, the process is unstoppable. |
| §4 n19 | p.186 | 12225 | Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment |
| Full Idea: A third way has been offered to 'make sense' of neo-Fregeanism: we should reject Quine's well-known criterion of ontological commitment in favour of one based on 'truth-maker theory'. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §4 n19) | |||
| A reaction: [The cite Ross Cameron for this] They reject this proposal, on the grounds that truth-maker theory is not sufficient to fix the grounding truth-conditions of statements. |
| §5 | p.187 | 12226 | The identity of Pegasus with Pegasus may be true, despite the non-existence |
| Full Idea: Identity is sometimes read so that 'Pegasus is Pegasus' expresses a truth, the non-existence of any winged horse notwithstanding. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §5) | |||
| A reaction: This would give you ontological commitment to truth, without commitment to existence. It undercuts the use of identity statements as the basis of existence claims, which was Frege's strategy. |
| §5 | p.188 | 12227 | Abstractionism needs existential commitment and uniform truth-conditions |
| Full Idea: Abstractionism needs a face-value, existentially committed reading of the terms occurring on the left-hand sides together with sameness of truth-conditions across the biconditional. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §5) | |||
| A reaction: They employ 'abstractionism' to mean their logical Fregean strategy for defining abstractions, not to mean the older psychological account. Thus the truth-conditions for being 'parallel' and for having the 'same direction' must be consistent. |
| §8 | p.195 | 12228 | Equivalence abstraction refers to objects otherwise beyond our grasp |
| Full Idea: Abstraction principles purport to introduce fundamental means of reference to a range of objects, to which there is accordingly no presumption that we have any prior or independent means of reference. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §8) | |||
| A reaction: There's the rub! They make it sound like a virtue, that we open up yet another heaven of abstract toys to play with. As fictions, they are indeed exciting new fun. As platonic discoveries they strike me as Cloud-Cuckoo Land. |
| §9 | p.197 | 18443 | A successful predicate guarantees the existence of a property - the way of being it expresses |
| Full Idea: The good standing of a predicate is already trivially sufficient to ensure the existence of an associated property, a (perhaps complex) way of being which the predicate serves to express. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §9) | |||
| A reaction: 'Way of being' is interesting. Is 'being near Trafalgar Sq' a way of being? I take properties to be 'features', which seems to give a clearer way of demarcating them. They say they are talking about 'abundant' (rather than 'sparse') properties. |
| §9 | p.198 | 12229 | Maybe we have abundant properties for semantics, and sparse properties for ontology |
| Full Idea: There is a compatibilist view which says that it is for the abundant properties to play the role of 'bedeutungen' in semantic theory, and the sparse ones to address certain metaphysical concerns. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §9) | |||
| A reaction: Only a philosopher could live with the word 'property' having utterly different extensions in different areas of discourse. They similarly bifurcate words like 'object' and 'exist'. Call properties 'quasi-properties' and I might join in. |
| §9 | p.207 | 12230 | Singular terms refer if they make certain atomic statements true |
| Full Idea: Anyone should agree that a justification for regarding a singular term as having objectual reference is provided just as soon as one has justification for regarding as true certain atomic statements in which it functions as a singular term. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §9) | |||
| A reaction: The meat of this idea is hidden in the word 'certain'. See Idea 10314 for Hale's explanation. Without that, the proposal strikes me as absurd. |
| §9 | p.208 | 12231 | Reference needs truth as well as sense |
| Full Idea: It takes, over and above the possession of sense, the truth of relevant contexts to ensure reference. | |||
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §9) | |||
| A reaction: Reference purely through sense was discredited by Kripke. The present idea challenges Kripke's baptismal realist approach. How do you 'baptise' an abstract object? But isn't reference needed prior to the establishment of truth? |