33 ideas
| 18806 | Frege thought traditional categories had psychological and linguistic impurities [Frege, by Rumfitt] |
| Full Idea: Frege rejected the traditional categories as importing psychological and linguistic impurities into logic. | |
| From: report of Gottlob Frege (Function and Concept [1891]) by Ian Rumfitt - The Boundary Stones of Thought 1.2 | |
| A reaction: Resisting such impurities is the main motivation for making logic entirely symbolic, but it doesn't follow that the traditional categories have to be dropped. |
| 9193 | ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett] |
| Full Idea: ZF set theory is a first-order axiomatization. Variables range over sets, there are no second-order variables, and primitive predicates are just 'equals' and 'member of'. The axiom of extensionality says sets with the same members are identical. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 7) | |
| A reaction: If the eleven members of the cricket team are the same as the eleven members of the hockey team, is the cricket team the same as the hockey team? Our cricket team is better than our hockey team, so different predicates apply to them. |
| 9194 | The main alternative to ZF is one which includes looser classes as well as sets [Dummett] |
| Full Idea: The main alternative to ZF is two-sorted theories, with some variables ranging over classes. Classes have more generous existence assumptions: there is a universal class, containing all sets, and a class containing all ordinals. Classes are not members. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 7.1.1) | |
| A reaction: My intuition is to prefer strict systems when it comes to logical theories. The whole point is precision. Otherwise we could just think about things, and skip all this difficult symbolic stuff. |
| 21495 | Theoretical and practical politics are both concerned with the best lives for individuals [Russell] |
| Full Idea: Political ideals must be based upon ideals for the individual life. The aim of politics should be to make the lives of individuals as good as possible. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: Russell floats between socialism and anarchism, but this foundational remark is classic liberalism. |
| 9195 | Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett] |
| Full Idea: It must not be concluded from the rejection of excluded middle that intuitionistic logic operates with three values: true, false, and neither true nor false. It does not make use of true and false, but only with a construction being a proof. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 8.1) | |
| A reaction: This just sounds like verificationism to me, with all its problems. It seems to make speculative statements meaningless, which can't be right. Realism has lots of propositions which are assumed to be true or false, but also unknowable. |
| 8490 | First-level functions have objects as arguments; second-level functions take functions as arguments [Frege] |
| Full Idea: Just as functions are fundamentally different from objects, so also functions whose arguments are and must be functions are fundamentally different from functions whose arguments are objects. The latter are first-level, the former second-level, functions. | |
| From: Gottlob Frege (Function and Concept [1891], p.38) | |
| A reaction: In 1884 he called it 'second-order'. This is the standard distinction between first- and second-order logic. The first quantifies over objects, the second over intensional entities such as properties and propositions. |
| 8492 | Relations are functions with two arguments [Frege] |
| Full Idea: Functions of one argument are concepts; functions of two arguments are relations. | |
| From: Gottlob Frege (Function and Concept [1891], p.39) | |
| A reaction: Nowadays we would say 'two or more'. Another interesting move in the aim of analytic philosophy to reduce the puzzling features of the world to mathematical logic. There is, of course, rather more to some relations than being two-argument functions. |
| 9186 | First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett] |
| Full Idea: First-order logic is distinguished by generalizations (quantification) only over objects: second-order logic admits generalizations or quantification over properties or kinds of objects, and over relations between them, and functions defined over them. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1) | |
| A reaction: Second-order logic was introduced by Frege, but is (interestingly) rejected by Quine, because of the ontological commitments involved. I remain unconvinced that quantification entails ontological commitment, so I'm happy. |
| 9187 | Logical truths and inference are characterized either syntactically or semantically [Dummett] |
| Full Idea: There are two ways of characterizing logical truths and correct inference. Proof-theoretic or syntactic characterizations, if the formalization admits of proof or derivation; and model-theoretic or semantic versions, being true in all interpretations. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1) | |
| A reaction: Dummett calls this distinction 'fundamental'. The second one involves truth, and hence meaning, where the first one just responds to rules. ..But how can you have a notion of correctly following a rule, without a notion of truth? |
| 9191 | Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett] |
| Full Idea: It can be argued that the notion of ordinal numbers is more fundamental than that of cardinals. To count objects, we must count them in sequence. ..The theory of ordinals forms the substratum of Cantor's theory of cardinals. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 5) | |
| A reaction: Depends what you mean by 'fundamental'. I would take cardinality to be psychologically prior ('that is a lot of sheep'). You can't order people by height without first acquiring some people with differing heights. I vote for cardinals. |
| 9192 | The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett] |
| Full Idea: The number 4 cannot be characterized solely by its position in a system, because it has different positions in the system of natural numbers and that of the positive whole numbers, whereas these systems have the very same structure. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 6.1) | |
| A reaction: Dummett seems to think this is fairly decisive against structuralism. There is also the structure of the real numbers. We will solve this by saying that the wholes are abstracted from the naturals, which are abstracted from the reals. Job done. |
| 8487 | Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism [Frege] |
| Full Idea: I am of the opinion that arithmetic is a further development of logic, which leads to the requirement that the symbolic language of arithmetic must be expanded into a logical symbolism. | |
| From: Gottlob Frege (Function and Concept [1891], p.30) | |
| A reaction: This may the the one key idea at the heart of modern analytic philosophy (even though logicism may be a total mistake!). Logic and arithmetical foundations become the master of ontology, instead of the servant. The jury is out on the whole enterprise. |
| 18899 | Frege takes the existence of horses to be part of their concept [Frege, by Sommers] |
| Full Idea: Frege regarded the existence of horses as a property of the concept 'horse'. | |
| From: report of Gottlob Frege (Function and Concept [1891]) by Fred Sommers - Intellectual Autobiography 'Realism' |
| 4028 | Frege allows either too few properties (as extensions) or too many (as predicates) [Mellor/Oliver on Frege] |
| Full Idea: Frege's theory of properties (which he calls 'concepts') yields too few properties, by identifying coextensive properties, and also too many, by letting every predicate express a property. | |
| From: comment on Gottlob Frege (Function and Concept [1891]) by DH Mellor / A Oliver - Introduction to 'Properties' §2 | |
| A reaction: Seems right; one extension may have two properties (have heart/kidneys), two predicates might express the same property. 'Cutting nature at the joints' covers properties as well as objects. |
| 8489 | The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place [Frege] |
| Full Idea: I regard a regular definition of 'object' as impossible, since it is too simple to admit of logical analysis. Briefly: an object is anything that is not a function, so that an expression for it does not contain any empty place. | |
| From: Gottlob Frege (Function and Concept [1891], p.32) | |
| A reaction: Here is the core of the programme for deriving our ontology from our logic and language, followed through by Russell and Quine. Once we extend objects beyond the physical, it becomes incredibly hard to individuate them. |
| 9947 | Concepts are the ontological counterparts of predicative expressions [Frege, by George/Velleman] |
| Full Idea: Concepts, for Frege, are the ontological counterparts of predicative expressions. | |
| From: report of Gottlob Frege (Function and Concept [1891]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: That sounds awfully like what many philosophers call 'universals'. Frege, as a platonist (at least about numbers), I would take to be in sympathy with that. At least we can say that concepts seem to be properties. |
| 10319 | An assertion about the concept 'horse' must indirectly speak of an object [Frege, by Hale] |
| Full Idea: Frege had a notorious difficulty over the concept 'horse', when he suggests that if we wish to assert something about a concept, we are obliged to proceed indirectly by speaking of an object that represents it. | |
| From: report of Gottlob Frege (Function and Concept [1891], Ch.2.II) by Bob Hale - Abstract Objects | |
| A reaction: This sounds like the thin end of a wedge. The great champion of objects is forced to accept them here as a façon de parler, when elsewhere they have ontological status. |
| 8488 | A concept is a function whose value is always a truth-value [Frege] |
| Full Idea: A concept in logic is closely connected with what we call a function. Indeed, we may say at once: a concept is a function whose value is always a truth-value. ..I give the name 'function' to what is meant by the 'unsaturated' part. | |
| From: Gottlob Frege (Function and Concept [1891], p.30) | |
| A reaction: So a function becomes a concept when the variable takes a value. Problems arise when the value is vague, or the truth-value is indeterminable. |
| 9948 | Unlike objects, concepts are inherently incomplete [Frege, by George/Velleman] |
| Full Idea: For Frege, concepts differ from objects in being inherently incomplete in nature. | |
| From: report of Gottlob Frege (Function and Concept [1891]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This is because they are 'unsaturated', needing a quantified variable to complete the sentence. This could be a pointer towards Quine's view of properties, as simply an intrinsic feature of predication about objects, with no separate identity. |
| 4972 | I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false [Frege] |
| Full Idea: From sameness of meaning there does not follow sameness of thought expressed. A fact about the Morning Star may express something different from a fact about the Evening Star, as someone may regard one as true and the other false. | |
| From: Gottlob Frege (Function and Concept [1891], p.14) | |
| A reaction: This all gets clearer if we distinguish internalist and externalist theories of content. Why take sides on this? Why not just ask 'what is in the speaker's head?', 'what does the sentence mean in the community?', and 'what is the corresponding situation?' |
| 21517 | Individuals need creativity, reverence for others, and self-respect [Russell] |
| Full Idea: What we shall desire for individuals is now clear: strong creative impulses, overpowering and absorbing the instinct of possession; reverence for others; respect for the fundamental creative impulses in ourselves. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: Interesting that when Russell focuses on morality, he turns to virtues, rather than to rules. He uses 'reverence' where I would favour 'respect'. His concept of creativity is broad, and does not just concern art etc. |
| 21522 | Democracy is inadequate without a great deal of devolution [Russell] |
| Full Idea: Democracy is not at all an adequate device unless it is accompanied by a very great amount of devolution. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: This whole book of Russell's is an appeal for the devolution of power, and for workplace democracy. |
| 21523 | We would not want UK affairs to be settled by a world parliament [Russell] |
| Full Idea: We should none of us like the affairs of Great Britain to be settled by a parliament of the world. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: The UK is currently (Dec 2018) living with a plan to quit Europe, mainly on the grounds that a European parliament has some authority over Britain. In every country resentment of the government increases with distance from the capital city. |
| 21521 | Anarchy does not maximise liberty [Russell] |
| Full Idea: The greatest degree of liberty is not secured by anarchy. ...[22] The results of anarchy between states should suffice to persuade us that anarchism has no solution to offer for the evils of the world. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: I've heard Russell described as an anarchist, but this clearly wasn't true in 1917. Presumably liberty has to be protected. That we were watching anarchy between states in 1917 is a vivid observation. |
| 21528 | Groups should be autonomous, with a neutral authority as arbitrator [Russell] |
| Full Idea: For maximum freedom with minimum force: Autonomy within each politically important group, and a neutral authority for deciding questions involving relations between groups. | |
| From: Bertrand Russell (Political Ideals [1917], 3) | |
| A reaction: This is workplace democracy, and also considerable self-rule amongst minority groups such as religions. |
| 21526 | Unfortunately ordinary voters can't detect insincerity [Russell] |
| Full Idea: It is a painful fact that the ordinary voter, at any rate in England, is quite blind to insincerity. | |
| From: Bertrand Russell (Political Ideals [1917], 3) | |
| A reaction: Gor blimey yes! Well said, Bertie. Even in the age of television, when you can examine them in closeup, people seem to confuse superficial charm with genuine positive convictions. Why are people better at detecting it in private life? |
| 21527 | On every new question the majority is always wrong at first [Russell] |
| Full Idea: It is a mistake to suppose that the majority is necessarily right. On every new question the majority is always wrong at first. | |
| From: Bertrand Russell (Political Ideals [1917], 3) | |
| A reaction: Sounds like bitter experience. This is a good argument for taking time over decisions, and (topical) for a second referendum some time after the first one (if you must have a referendum). |
| 21525 | When the state is the only employer, there is no refuge from the prejudices of other people [Russell] |
| Full Idea: Under state socialism ...where the State is the only employer, there is no refuge from its prejudices such as may now accidentally arise through the differing opinions of men. | |
| From: Bertrand Russell (Political Ideals [1917], 2) | |
| A reaction: There is also a strong likelihood in full state socialism that the state will control housing as well as employment. This hadn't come to pass in 1917. |
| 21518 | Men unite in pursuit of material things, and idealise greed as part of group loyalty [Russell] |
| Full Idea: Men combine in groups to attain more strength in the scramble for material goods, and loyalty to the group spreads a halo of quasi-idealism round the central impulse of greed. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: See the 'greed is good' speech in the film 'Wall Street'. This sounds like a description of the USA, but Russell was very much in England at this stage. |
| 21519 | We need security and liberty, and then encouragement of creativity [Russell] |
| Full Idea: Security and liberty are only the negative conditions for good political institutions. When they have been won, we need also the positive condition: encouragement of creative energy. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: This sounds like some sort of liberal socialism. The nearest connection I can see is to the 'capabilities approach' of Martha Nussbaum. How do you intervene to encourage creativity? |
| 21524 | The right to own land gives a legal right to a permanent income [Russell] |
| Full Idea: There are many ways of becoming rich without contributing anything to the wealth of the community. Ownership of land or capital, whether acquired or inherited, gives a legal right to a permanent income. | |
| From: Bertrand Russell (Political Ideals [1917], 2) | |
| A reaction: I suspect that in the past land was the main source of this right, but now it is more likely to be capital. Land carries obligations of some sort, so income from capital is more fun. |
| 8491 | The Ontological Argument fallaciously treats existence as a first-level concept [Frege] |
| Full Idea: The ontological proof of God's existence suffers from the fallacy of treating existence as a first-level concept. | |
| From: Gottlob Frege (Function and Concept [1891], p.38 n) | |
| A reaction: [See Idea 8490 for first- and second-order functions] This is usually summarised as the idea that existence is a quantifier rather than a predicate. |
| 21520 | That our heaven is a dull place reflects the misery of excessive work in life [Russell] |
| Full Idea: It is a sad evidence of the weariness mankind has suffered from excessive toil that his heavens have usually been places where nothing ever happened or changed. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: Has any religion got an idea of heaven as a place full of lively activity and creative problem-solving? That is what suits us best. |