| 1983 | Frege's Concept of Numbers as Objects |
| p.41 | 10140 | We derive Hume's Law from Law V, then discard the latter in deriving arithmetic | |
| Full Idea: Wright says the Fregean arithmetic can be broken down into two steps: first, Hume's Law may be derived from Law V; and then, arithmetic may be derived from Hume's Law without any help from Law V. | |||
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Kit Fine - The Limits of Abstraction I.4 | |||
| A reaction: This sounds odd if Law V is false, but presumably Hume's Law ends up as free-standing. It seems doubtful whether the resulting theory would count as logic. |
| p.55 | 10142 | The attempt to define numbers by contextual definition has been revived | |
| Full Idea: Frege gave up on the attempt to introduce natural numbers by contextual definition, but the project has been revived by neo-logicists. | |||
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Kit Fine - The Limits of Abstraction II |
| p.71 | 8692 | Frege has a good system if his 'number principle' replaces his basic law V | |
| Full Idea: Wright proposed removing Frege's basic law V (which led to paradox), replacing it with Frege's 'number principle' (identity of numbers is one-to-one correspondence). The new system is formally consistent, and the Peano axioms can be derived from it. | |||
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.7 | |||
| A reaction: The 'number principle' is also called 'Hume's principle'. This idea of Wright's resurrected the project of logicism. The jury is ought again... Frege himself questioned whether the number principle was a part of logic, which would be bad for 'logicism'. |
| p.189 | 17440 | Wright says Hume's Principle is analytic of cardinal numbers, like a definition | |
| Full Idea: Wright intends the claim that Hume's Principle (HP) embodies an explanation of the concept of number to imply that it is analytic of the concept of cardinal number - so it is an analytic or conceptual truth, much as a definition would be. | |||
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1 | |||
| A reaction: Boolos is quoted as disagreeing. Wright is claiming a fundamental truth. Boolos says something can fix the character of something (as yellow fixes bananas), but that doesn't make it 'fundamental'. I want to defend 'fundamental'. |
| p.189 | 9868 | An expression refers if it is a singular term in some true sentences | |
| Full Idea: For Wright, an expression refers to an object if it fulfils the 'syntactic role' of a singular term, and if we have fixed the truth-conditions of sentences containing it in such a way that some of them come out true. | |||
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Michael Dummett - Frege philosophy of mathematics Ch.15 | |||
| A reaction: Much waffle is written about reference, and it is nice to hear of someone actually trying to state the necessary and sufficient conditions for reference to be successful. So is it possible for 'the round square' to ever refer? '...is impossible to draw' |
| p.189 | 17441 | Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are | |
| Full Idea: Wright is claiming that HP is a special sort of truth in some way: it is supposed to be the fundamental truth about cardinality; ...in particular, HP is supposed to be more fundamental, in some sense than the Dedekind-Peano axioms. | |||
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1 | |||
| A reaction: Heck notes that although PA can be proved from HP, HP can be proven from PA plus definitions, so direction of proof won't show fundamentality. He adds that Wright thinks HP is 'more illuminating'. |
| p.226 | 9878 | Contextually defined abstract terms genuinely refer to objects | |
| Full Idea: Wright says we should accord to contextually defined abstract terms a genuine full-blown reference to objects. | |||
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |||
| A reaction: This is the punch line of Wright's neo-logicist programme. See Idea 9868 for his view of reference. Dummett regards this strong view of contextual definition as 'exorbitant'. Wright's view strikes me as blatantly false. |
| p.354 | 7804 | Wright has revived Frege's discredited logicism | |
| Full Idea: Crispin Wright has reactivated Frege's logistic program, which for decades just about everybody assumed was a lost cause. | |||
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by José A. Benardete - Logic and Ontology 3 | |||
| A reaction: [This opens Bernadete's section called "Back to Strong Logicism?"] |
| Intro | p.-10 | 13861 | Number theory aims at the essence of natural numbers, giving their nature, and the epistemology |
| Full Idea: In the Fregean view number theory is a science, aimed at those truths furnished by the essential properties of zero and its successors. The two broad question are then the nature of the objects, and the epistemology of those facts. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |||
| A reaction: [compressed] I pounce on the word 'essence' here (my thing). My first question is about the extent to which the natural numbers all have one generic essence, and the extent to which they are individuals (bless their little cotton socks). |
| Intro | p.-9 | 13862 | There are five Peano axioms, which can be expressed informally |
| Full Idea: Informally, Peano's axioms are: 0 is a number, numbers have a successor, different numbers have different successors, 0 isn't a successor, properties of 0 which carry over to successors are properties of all numbers. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |||
| A reaction: Each statement of the famous axioms is slightly different from the others, and I have reworded Wright to fit him in. Since the last one (the 'induction axiom') is about properties, it invites formalization in second-order logic. |
| Intro | p.-9 | 17853 | Number truths are said to be the consequence of PA - but it needs semantic consequence |
| Full Idea: The intuitive proposal is the essential number theoretic truths are precisely the logical consequences of the Peano axioms, ...but the notion of consequence is a semantic one...and it is not obvious that we possess a semantic notion of the requisite kind. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |||
| A reaction: (Not sure I understand this, but it is his starting point for rejecting PA as the essence of arithmetic). |
| Intro | p.-8 | 17854 | What facts underpin the truths of the Peano axioms? |
| Full Idea: We incline to think of the Peano axioms as truths of some sort; so there has to be a philosophical question how we ought to conceive of the nature of the facts which make those statements true. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |||
| A reaction: [He also asks about how we know the truths] |
| Intro | p.-1 | 13863 | Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms |
| Full Idea: Most would cite Russell's paradox, the non-logical character of the axioms which Russell and Whitehead's reconstruction of Frege's enterprise was constrained to employ, and the incompleteness theorems of Gödel, as decisive for logicism's failure. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) |
| Pref | p.-12 | 13860 | We can only learn from philosophers of the past if we accept the risk of major misrepresentation |
| Full Idea: We can learn from the work of philosophers of other periods only if we are prepared to run the risk of radical and almost inevitable misrepresentation of his thought. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Pref) | |||
| A reaction: This sounds about right, and a motto for my own approach to Aristotle and Leibniz, but I see the effort as more collaborative than this suggests. Professional specialists in older philosophers are a vital part of the team. Read them! |
| 1.i | p.2 | 13865 | 'Sortal' concepts show kinds, use indefinite articles, and require grasping identities |
| Full Idea: A concept is 'sortal' if it exemplifies a kind of object. ..In English predication of a sortal concept needs an indefinite article ('an' elm). ..What really constitutes the distinction is that it involves grasping identity for things which fall under it. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |||
| A reaction: This is a key notion, which underlies the claims of 'sortal essentialism' (see David Wiggins). |
| 1.i | p.3 | 13866 | A concept is only a sortal if it gives genuine identity |
| Full Idea: Before we can conclude that φ expresses a sortal concept, we need to ensure that 'is the same φ as' generates statements of genuine identity rather than of some other equivalence relation. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) |
| 1.i | p.3 | 13867 | Instances of a non-sortal concept can only be counted relative to a sortal concept |
| Full Idea: The invitation to number the instances of some non-sortal concept is intelligible only if it is relativised to a sortal. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |||
| A reaction: I take this to be an essentially Fregean idea, as when we count the boots when we have decided whether they fall under the concept 'boot' or the concept 'pair'. I also take this to be the traditional question 'what units are you using'? |
| 1.i | p.4 | 13869 | Number platonism says that natural number is a sortal concept |
| Full Idea: Number-theoretic platonism is just the thesis that natural number is a sortal concept. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |||
| A reaction: See Crispin Wright on sortals to expound this. An odd way to express platonism, but he is presenting the Fregean version of it. |
| 1.i | p.4 | 13870 | We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism |
| Full Idea: We may not be able to settle whether some general form of empiricism is correct independently of natural numbers. It might be precisely our grasp of the abstract sortal, natural number, which shows the hypothesis of empiricism to be wrong. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |||
| A reaction: A nice turning of the tables. In the end only coherence decides these things. You may accept numbers and reject empiricism, and then find you have opened the floodgates for abstracta. Excessive floodgates, or blockages of healthy streams? |
| 1.i | p.4 | 13868 | Sortal concepts cannot require that things don't survive their loss, because of phase sortals |
| Full Idea: The claim that no concept counts as sortal if an instance of it can survive its loss, runs foul of so-called phase sortals like 'embryo' and 'chrysalis'. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |||
| A reaction: The point being that those items only fall under that sortal for one phase of their career, and of their identity. I've always thought such claims absurd, and this gives a good reason for my view. |
| 1.iii | p.10 | 13873 | Treating numbers adjectivally is treating them as quantifiers |
| Full Idea: Treating numbers adjectivally is, in effect, treating the numbers as quantifiers. Frege observes that we can always parse out any apparently adjectival use of a number word in terms of substantival use. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.iii) | |||
| A reaction: The immediate response to this is that any substantival use can equally be expressed adjectivally. If you say 'the number of moons of Jupiter is four', I can reply 'oh, you mean Jupiter has four moons'. |
| 1.iii | p.14 | 13877 | Singular terms in true sentences must refer to objects; there is no further question about their existence |
| Full Idea: When a class of terms functions as singular terms, and the sentences are true, then those terms genuinely refer. Being singular terms, their reference is to objects. There is no further question whether they really refer, and there are such objects. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.iii) | |||
| A reaction: This seems to be a key sentence, because this whole view is standardly called 'platonic', but it certainly isn't platonism as we know it, Jim. Ontology has become an entirely linguistic matter, but do we then have 'sakes' and 'whereaboutses'? |
| 1.iv | p.17 | 17857 | We can accept Frege's idea of object without assuming that predicates have a reference |
| Full Idea: The heart of the problem is Frege's assumption that predicates have Bedeutungen at all; and no reason is at present evident why someone who espouses Frege's notion of object is contrained to make that assumption. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.iv) | |||
| A reaction: This seems like a penetrating objection to Frege's view of reference, and presumably supports the Kripke approach. |
| 1.vii | p.44 | 13882 | A milder claim is that understanding requires some evidence of that understanding |
| Full Idea: A mild version of the verification principle would say that it makes sense to think of someone as understanding an expression only if he is able, by his use of the expression, to give the best possible evidence that he understands it. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.vii) | |||
| A reaction: That doesn't seem to tell us what understanding actually consists of, and may just be the truism that to demonstrate anything whatsoever will necessarily involve some evidence. |
| 1.vii | p.49 | 13883 | The best way to understand a philosophical idea is to defend it |
| Full Idea: The most productive way in which to attempt an understanding of any philosophical idea is to work on its defence. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.vii) | |||
| A reaction: Very nice. The key point is that this brings greater understanding than working on attacking an idea, which presumably has the dangers of caricature, straw men etc. It is the Socratic insight that dialectic is the route to wisdom. |
| 2.x | p.83 | 13884 | The idea that 'exist' has multiple senses is not coherent |
| Full Idea: I have the gravest doubts whether any coherent account could be given of any multiplicity of senses of 'exist'. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 2.x) | |||
| A reaction: I thoroughly agree with this thought. Do water and wind exist in different senses of 'exist'? |
| 2.xi | p.88 | 13885 | If apparent reference can mislead, then so can apparent lack of reference |
| Full Idea: If the appearance of reference can be misleading, why cannot an apparent lack of reference be misleading? | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 2.xi) | |||
| A reaction: A nice simple thought. Analytic philosophy has concerned itself a lot with sentences that seem to refer, but the reference can be analysed away. For me, this takes the question of reference out of the linguistic sphere, which wasn't Wright's plan. |
| 3.xiv | p.112 | 13888 | If numbers are extensions, Frege must first solve the Caesar problem for extensions |
| Full Idea: Identifying numbers with extensions will not solve the Caesar problem for numbers unless we have already solved the Caesar problem for extensions. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xiv) |
| 3.xiv | p.114 | 13890 | Entities fall under a sortal concept if they can be used to explain identity statements concerning them |
| Full Idea: 'Tree' is not a sortal concept under which directions fall since we cannot adequately explain the truth-conditions of any identity statement involving a pair of tree-denoting singular terms by appealing to facts to do with parallelism between lines. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xiv) | |||
| A reaction: The idea seems to be that these two fall under 'hedgehog', because that is a respect in which they are identical. I like to notion of explanation as a part of this. |
| 3.xv | p.118 | 13892 | One could grasp numbers, and name sizes with them, without grasping ordering |
| Full Idea: Someone could be clear about number identities, and distinguish numbers from other things, without conceiving them as ordered in a progression at all. The point of them would be to make comparisons between sizes of groups. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv) | |||
| A reaction: Hm. Could you grasp size if you couldn't grasp which of two groups was the bigger? What's the point of noting that I have ten pounds and you only have five, if you don't realise that I have more than you? You could have called them Caesar and Brutus. |
| 3.xv | p.120 | 13894 | Sameness of number is fundamental, not counting, despite children learning that first |
| Full Idea: We teach our children to count, sometimes with no attempt to explain what the sounds mean. Doubtless it is this habit which makes it so natural to think of the number series as fundamental. Frege's insight is that sameness of number is fundamental. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv) | |||
| A reaction: 'When do children understand number?' rather than when they can recite numerals. I can't make sense of someone being supposed to understand number without a grasp of which numbers are bigger or smaller. To make 13='15' do I add or subtract? |
| 3.xv | p.120 | 13893 | It is 1-1 correlation of concepts, and not progression, which distinguishes natural number |
| Full Idea: What is fundamental to possession of any notion of natural number at all is not the knowledge that the numbers may be arrayed in a progression but the knowledge that they are identified and distinguished by reference to 1-1 correlation among concepts. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv) | |||
| A reaction: My question is 'what is the essence of number?', and my inclination to disagree with Wright on this point suggests that the essence of number is indeed caught in the Dedekind-Peano axioms. But what of infinite numbers? |
| 4.xix | p.168 | 13899 | The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals |
| Full Idea: The Peano Axioms are logical consequences of a statement constituting the core of an explanation of the notion of cardinal number. The infinity of cardinal numbers emerges as a consequence of the way cardinal number is explained. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xix) | |||
| A reaction: This, along with Idea 13896, nicely summarises the neo-logicist project. I tend to favour a strategy which starts from ordering, rather than identities (1-1), but an attraction is that this approach is closer to counting objects in its basics. |
| 4.xvi | p.131 | 13896 | The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes |
| Full Idea: We shall endeavour to see whether it is possible to follow through the strategy adumbrated in 'Grundlagen' for establishing the Peano Axioms without at any stage invoking classes. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xvi) | |||
| A reaction: The key idea of neo-logicism. If you can avoid classes entirely, then set theory paradoxes become irrelevant, and classes aren't logic. Philosophers now try to derive the Peano Axioms from all sorts of things. Wright admits infinity is a problem. |
| 4.xvi | p.131 | 13895 | The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems |
| Full Idea: The general view is that Russell's Paradox put paid to Frege's logicist attempt, and Russell's own attempt is vitiated by the non-logical character of his axioms (esp. Infinity), and by the incompleteness theorems of Gödel. But these are bad reasons. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xvi) | |||
| A reaction: Wright's work is the famous modern attempt to reestablish logicism, in the face of these objections. |
| 4.xviii | p.148 | 13898 | If we can establish directions from lines and parallelism, we were already committed to directions |
| Full Idea: The fact that it seems possible to establish a sortal notion of direction by reference to lines and parallelism, discloses tacit commitments to directions in statements about parallelism...There is incoherence in the idea that a line might lack direction. | |||
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xviii) | |||
| A reaction: This seems like a slippery slope into a very extravagant platonism about concepts. Are concepts like direction as much a part of the natural world as rivers are? What other undiscovered concepts await us? |
| 1986 | Inventing Logical Necessity |
| p.126 | 7320 | Holism cannot give a coherent account of scientific methodology | |
| Full Idea: Crispin Wright has argued that Quine's holism is implausible because it is actually incoherent: he claims that Quine's holism cannot provide us with a coherent account of scientific methodology. | |||
| From: report of Crispin Wright (Inventing Logical Necessity [1986]) by Alexander Miller - Philosophy of Language 4.5 | |||
| A reaction: This sounds promising, given my intuitive aversion to linguistic holism, and almost everything to do with Quine. Scientific methodology is not isolated, but spreads into our ordinary (experimental) interactions with the world (e.g. Idea 2461). |
| p.149 | 12189 | Logical necessity involves a decision about usage, and is non-realist and non-cognitive | |
| Full Idea: Wright espouses a non-realist, indeed non-cognitive account of logical necessity. Crucial to this is the idea that acceptance of a statement as necessary always involves an element of decision (to use it in a necessary way). | |||
| From: report of Crispin Wright (Inventing Logical Necessity [1986]) by Ian McFetridge - Logical Necessity: Some Issues §3 | |||
| A reaction: This has little appeal to me, as I take (unfashionably) the view that that logical necessity is rooted in the behaviour of the actual physical world, with which you can't argue. We test simple logic by making up examples. |