40 ideas
| 10041 | Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] |
| Full Idea: Impredicative Definitions are definitions of an object by reference to the totality to which the object itself (and perhaps also things definable only in terms of that object) belong. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], n 13) |
| 23755 | Genius and love of truth are always accompanied by great humility [Weil] |
| Full Idea: Love of truth is always accompanied by humility, and real genius is nothing else but the supernatural virtue of humility in the domain of thought. | |
| From: Simone Weil (Human Personality [1943], p.87) | |
| A reaction: A striking and attractive thought, true of all the lovers of truth I have ever encountered. Socrates is the role model. She likens truth to an inarticulate plaintiff stammering before a judge who fluently manipulates opinions. |
| 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. |
| 21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
| Full Idea: In the superior realist and simple theory of types, the place of the axiom of reducibility is not taken by the axiom of classes, Zermelo's Aussonderungsaxiom. | |
| From: report of Kurt Gödel (Russell's Mathematical Logic [1944], p.140-1) by Bernard Linsky - Russell's Metaphysical Logic 6.1 n3 | |
| A reaction: This is Zermelo's Axiom of Separation, but that too is not an axiom of standard ZFC. |
| 10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] |
| Full Idea: 'Mathematical Logic' is a precise and complete formulation of formal logic, and is both a section of mathematics covering classes, relations, symbols etc, and also a science prior to all others, with ideas and principles underlying all sciences. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.447) | |
| A reaction: He cites Leibniz as the ancestor. In this database it is referred to as 'theory of logic', as 'mathematical' seems to be simply misleading. The principles of the subject are standardly applied to mathematical themes. |
| 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. |
| 10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
| Full Idea: One may, on good grounds, deny that reference to a totality necessarily implies reference to all single elements of it or, in other words, that 'all' means the same as an infinite logical conjunction. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.455) |
| 10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
| Full Idea: In order to be sure that new expression can be translated into expressions not containing them, it is necessary to have a survey of all possible expressions, and this can be furnished only by syntactical considerations. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.448) | |
| A reaction: [compressed] |
| 10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel] |
| Full Idea: The generalized Continuum Hypothesis says that there exists no cardinal number between the power of any arbitrary set and the power of the set of its subsets. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.464) |
| 10039 | Some arithmetical problems require assumptions which transcend arithmetic [Gödel] |
| Full Idea: It has turned out that the solution of certain arithmetical problems requires the use of assumptions essentially transcending arithmetic. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.449) | |
| A reaction: A nice statement of the famous result, from the great man himself, in the plainest possible English. |
| 10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
| Full Idea: Classes and concepts may be conceived of as real objects, ..and are as necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions, with neither case being about 'data'. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.456) | |
| A reaction: Note that while he thinks real objects are essential for mathematics, be may not be claiming the same thing for our knowledge of logic. If logic contains no objects, then how could mathematics be reduced to it, as in logicism? |
| 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. |
| 10045 | Impredicative definitions are admitted into ordinary mathematics [Gödel] |
| Full Idea: Impredicative definitions are admitted into ordinary mathematics. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.464) | |
| A reaction: The issue is at what point in building an account of the foundations of mathematics (if there be such, see Putnam) these impure definitions should be ruled out. |
| 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. |
| 23747 | What is sacred is not a person, but the whole physical human being [Weil] |
| Full Idea: There is something sacred in every man, but it is not his person. Nor yet is it the human personality. It is this man; no more and no less. …It is he. The whole of him. The arms, they eyes, the thoughts, everything. | |
| From: Simone Weil (Human Personality [1943], p,70) | |
| A reaction: I take her to be referring to exactly the concept of a 'person' which Locke introduced. It is important to remember that his concept is mainly forensic - as a concept of ownership and contracts. A person is an abstraction. Even a corpse is a human. |
| 23756 | The mind is imprisoned and limited by language, restricting our awareness of wider thoughts [Weil] |
| Full Idea: At the very best, a mind is enclosed in language is in a prison. It is limited to the number of relations which words can make simultaneously present to it; and remains in ignorance of thoughts which involve the combination of a greater number. | |
| From: Simone Weil (Human Personality [1943], p.89) | |
| A reaction: This seems to be a germ of the type of view of language which blossoms in Derrida. But she is on to something. None of us grasp fully, I think, the non-linguistic nature of good thinking. |
| 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?' |
| 23758 | Beauty is an attractive mystery, leaving nothing to be desired [Weil] |
| Full Idea: Beauty is the supreme mystery of the world. It is a gleam which attracts the attention and yet does nothing to sustain it. …While exciting desire, it makes clear that there is nothing in it to be desired, because what we want is that it should not change. | |
| From: Simone Weil (Human Personality [1943], p.92) | |
| A reaction: She attributes beauty to a supernatural source. I catalogue this idea under 'the sublime', rather than 'beauty'. It may be better to say that beauty inspires love, rather than desire. |
| 23760 | All we need are the unity of justice, truth and beauty [Weil] |
| Full Idea: Justice, truth, and beauty are sisters and comrades. With three such beautiful words we have no need to look for any others. | |
| From: Simone Weil (Human Personality [1943], p.93) | |
| A reaction: The embodiment of platonist values. Without the platonist ontology, I like the identification of a few core values, and have always thought that Beauty, Goodness and Truth were a well chosen trio. Swapping 'justice' for 'goodness' is interesting. |
| 23748 | The sacred in every human is their expectation of good rather than evil [Weil] |
| Full Idea: At the bottom of every human heart …there is something that goes on indomitably expecting, in the teeth of all crimes committed, suffered and witnessed, that good and not evil will be done to him. It is this above all that is sacred in every human being. | |
| From: Simone Weil (Human Personality [1943], p.71) | |
| A reaction: I'm thinking that this expectation may come from having at least one loving parent, and failing that there are people who have no such expectation as adults. Simone obviously thinks the hope runs deeper than that. |
| 23759 | Everything which originates in love is beautiful [Weil] |
| Full Idea: Everything which originates from pure love is lit with the radiance of beauty. | |
| From: Simone Weil (Human Personality [1943], p.93) | |
| A reaction: I suppose if I found a counterexample, she would say that is not 'pure' love. This sentence leaves open the possibility of beauty in the absence of love (such as a beautiful face noticed in the street). In her case, can beauty and love be separated? |
| 23762 | Evil is transmitted by comforts and pleasures, but mostly by doing harm to people [Weil] |
| Full Idea: One may transmit evil to a human being by flattering him or giving him comforts and pleasures; but most often men transmit evil to other men by doing them harm. | |
| From: Simone Weil (Human Personality [1943], p.94) | |
| A reaction: Some people receive harm very passively, especially if it is normal. What of tough love, which is erroneously seen as harm? |
| 23750 | It is not more money which the wretched members of society need [Weil] |
| Full Idea: Suppose the devil were bargaining for the soul of some wretch, and some pitying person said to the devil 'Shame on you, that commodity is worth twice as much'. Such is the sinister farce played by the working class unions, parties and intellectuals. | |
| From: Simone Weil (Human Personality [1943], p.80) | |
| A reaction: A striking thought. It is paradoxical when the working classes despise the middle classes, and yet aspire to be like them. It's hard to know what a mystic like Weil has in mind. An obvious thought is that the aspiration should be freedom, not money. |
| 23749 | The problem of the collective is not suppression of persons, but persons erasing themselves [Weil] |
| Full Idea: The chief danger does not lie in the collectivity's tendency to circumscribe the person, but in the person's tendency to immolate himself in the collective. | |
| From: Simone Weil (Human Personality [1943], p.78) | |
| A reaction: I'm guessing that in 1943 she had in mind both Nazis and Communists. She seems to articulate a strong form of liberalism in an interesting way. It sounds like a form of Bad Faith. |
| 23753 | People absurdly claim an equal share of things which are essentially privileged [Weil] |
| Full Idea: To the dimmed understanding of our age there seems nothing odd in claiming an equal share of privilege for everybody - an equal share in things whose essence is privilege. | |
| From: Simone Weil (Human Personality [1943], p.84) | |
| A reaction: Not sure what she has in mind. Probably not the finest food and drink. I suppose she is attacking the modern egalitarian view of democratic society. What things have privilege as their 'essence'? Being a 'winner'? Interesting, though. |
| 23751 | Rights are asserted contentiously, and need the backing of force [Weil] |
| Full Idea: Rights are always asserted in a tone of contention; and when this tone is adopted, it must rely upon force in the background, or else it will be laughed at. | |
| From: Simone Weil (Human Personality [1943], p.81) | |
| A reaction: This is the sort of observation which leads on to Foucault's account of all-pervasive power. Her observation may not be so sinister. It is obvious that introductions of new rights go against the grain of a conservative society - and so need a push. |
| 23752 | Giving centrality to rights stifles all impulses of charity [Weil] |
| Full Idea: To place the notion of rights at the centre of social conflicts is to inhibit any possible impulse of charity on both sides. | |
| From: Simone Weil (Human Personality [1943], p.83) | |
| A reaction: I think she exaggerates. To place personal charity at the centre of social conflicts strikes me as extremely conservative, and unlikely to improve the situation very much. I'm unsure how to reconcile this with Idea 23750. What sort of charity? |
| 23757 | The spirit of justice needs the full attention of truth, and that attention is love [Weil] |
| Full Idea: Because affliction and truth need the same kind of attention …the spirit of justice and the spirit of truth are one. The spirit of justice and truth is nothing else be a certain kind of attention, which is pure love. | |
| From: Simone Weil (Human Personality [1943], p.92) | |
| A reaction: I'm not sure about this as an observation, but as an inspiration it is very appealing, and (as so often with Weil) strikingly and attractively independent. I prefer love to arise naturally, rather than be a product of exhortation. |
| 23761 | Justice (concerning harm) is distinct from rights (concerning inequality) [Weil] |
| Full Idea: Justice is seeing that no harm is done to men. When a man cries inwardly 'Why am I being hurt?' he is being harmed. The other cry of 'Why have others got more than me?' refers to rights. We must distinguish them, and hush the second with law. | |
| From: Simone Weil (Human Personality [1943], p.93) | |
| A reaction: Her great passion is for justice, and so she downplays rights. The simple 'why am I being hurt?' has a horrible resonance in 1943. What of the hurts of disease? Are they unjust? |
| 23764 | The only thing in society worse than crime is repressive justice [Weil] |
| Full Idea: There is one, and only one, thing in society more hideous than crime - namely, repressive justice. | |
| From: Simone Weil (Human Personality [1943], p.95) | |
| A reaction: Presumably fans of 'repressive' justice would describe it as 'reformative' justice. In general, one of the most hideous parts of historical human societies has been the punishments they dished out (simply because they had the power to do it). |
| 23763 | Punishment aims at the good for men who don't desire it [Weil] |
| Full Idea: Punishment is solely a method of procuring pure good for men who do not desire it. The art of punishing is the art of awakening in a criminal, by pain or even death, the desire for pure good. | |
| From: Simone Weil (Human Personality [1943], p.95) | |
| A reaction: I know Weil is seen as some sort of saint, but this remark could have come from the Inquisition. I'm always alarmed by talk of 'pure' good and 'pure' evil, which seem to need a superior insight the rest of us lack. But see Idea 23764. |
| 23754 | The only choice is between supernatural good, or evil [Weil] |
| Full Idea: In all the crucial problems of human existence the only choice is between supernatural good on the one hand and evil on the other. | |
| From: Simone Weil (Human Personality [1943], p.86) | |
| A reaction: This idea strikes me as absurd, but I include it for a fuller picture of Simone Weil. Aristotle (my hero) is referred to, and labelled as more stupid than a village idiot. |
| 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. |