34 ideas
| 22496 | Wisdom only implies the knowledge achievable in any normal lifetime [Foot] |
| Full Idea: Wisdom implies no more knowledge and understanding than anyone of normal capacity can and should acquire in the course of an ordinary life. | |
| From: Philippa Foot (Natural Goodness [2001], 5) | |
| A reaction: Have philosophers stopped talking about wisdom precisely because you now need three university degrees to be considered even remotely good at phillosophy? Hence wisdom is an inferior attainment, because Foot is right. |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| Full Idea: 'Analysis' is the theory of the real numbers. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |
| A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |
| A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |
| A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |
| A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
| 23694 | All criterions of practical rationality derive from goodness of will [Foot] |
| Full Idea: I want to say, baldly, that there is no criterion for practical rationality that is not derived from that of goodness of will. | |
| From: Philippa Foot (Natural Goodness [2001], 1) | |
| A reaction: Where does that put the successful and clever criminal? Presumably they are broadly irrational, but narrowly rational - but that is not very clear distinction. She says Kant's concept of the good will is too pure, and unrelated to human good. |
| 12165 | Romantics say music expresses ideas, or the Will, or intuitions, or feelings [Scruton] |
| Full Idea: According to the Romantic theory music was an expression of something, of an idea (Hegel), of the Will (Schopenhauer), of 'intuitions' (Croce), or of feelings (Collingwood). | |
| From: Roger Scruton (The Nature of Musical Expression [1981], p.54) | |
| A reaction: Deryck Cooke was the culmination of music as expression of feeling, and Stravinsky was the greatest rebel against the whole idea of expression in music. You can set out to create interesting music which does or does not grab the emotions. |
| 12164 | Expressing melancholy is a good thing, but arousing it is a bad thing [Scruton] |
| Full Idea: To describe a piece of music as expressive of melancholy is to give a reason for listening to it; to describe it as arousing or evoking melancholy is to give a reason for avoiding it. | |
| From: Roger Scruton (The Nature of Musical Expression [1981], p.49) | |
| A reaction: Expressing sexual desire, while avoiding arousing it, is the nice challenge for a particular type of art. Would Scruton say that expressing joy is a good thing, but arousing it is bad? It is a nice observation, though. |
| 23686 | Moral reason is not just neutral, because morality is part of the standard of rationality [Foot, by Hacker-Wright] |
| Full Idea: In her late period she again reverses her thoughts on moral rationalism; …rather than a neutral rationality which fulfils desires, she argues that morality ought to be thought of as part of the standard of rationality itself. | |
| From: report of Philippa Foot (Natural Goodness [2001]) by John Hacker-Wright - Philippa Foot's Moral Thought Intro | |
| A reaction: This comes much closer to the Greek and Aristotelian concept of logos. They saw morality as inseparable from our judgements about how the world is. All 'sensible' thinking will involve what is good for humanity. |
| 23693 | Practical rationality must weigh both what is morally and what is non-morally required [Foot] |
| Full Idea: Different considerations are on a par, in that judgement about what is required by practical rationality must take account of their interaction: of the weight of the ones we call non-moral as well as those we call moral. | |
| From: Philippa Foot (Natural Goodness [2001], 1) | |
| A reaction: Her final settled view of rationalism in morality, it seems. The point is that moral considerations are not paramount, because she sees possible justifications for ignoring moral rules (like 'don't lie') in certain practical situations. |
| 23687 | Moral virtues arise from human nature, as part of what makes us good human beings [Foot, by Hacker-Wright] |
| Full Idea: In her later work she offers a view of the relationship of morality to human nature, arguing that the moral virtues are part of what makes us good as human beings. | |
| From: report of Philippa Foot (Natural Goodness [2001]) by John Hacker-Wright - Philippa Foot's Moral Thought Intro | |
| A reaction: In this phase she talks explicitly of the Aristotelian idea that successful function is the grounding of what is good for any living being, including humans. |
| 22492 | Virtues are as necessary to humans as stings are to bees [Foot] |
| Full Idea: Virtues play a necessary part in the life of human beings as do stings in the life of a bee. | |
| From: Philippa Foot (Natural Goodness [2001], 2) | |
| A reaction: This presumably rests on the Aristotelian idea that humans are essentially social (as opposed to solitary humans who choose to be social, perhaps in a contractual way, as Plato implies). |
| 22493 | Sterility is a human defect, but the choice to be childless is not [Foot] |
| Full Idea: Lack of capacity to reproduce is a defect in a human being. But choice of childlessness and even celibacy is not thereby shown to be defective choice, because human good is not the same as plant or animal good. | |
| From: Philippa Foot (Natural Goodness [2001], 3) | |
| A reaction: Is failure to reproduce a defect in an animal? If goodness and virtue derive from function, it is hard to see how deliberate childlessness could be a human good, even if it is not a defect. Choosing to terminate a hereditary defect seems good. |
| 22491 | Moral evaluations are not separate from facts, but concern particular facts about functioning [Foot] |
| Full Idea: A moral evaluation does not stand over against the statement of a matter of fact, but rather has to do with facts about a particular subject matter, as do evaluations of such things as sight and hearing in animals. | |
| From: Philippa Foot (Natural Goodness [2001], 1) | |
| A reaction: She avoids the word 'function', and only deals with living creatures, but she uses a 'good knife' as an example, and this Aristotelian view clearly applies to any machine which has a function. |
| 22497 | Deep happiness usually comes from the basic things in life [Foot] |
| Full Idea: Possible objects of deep happiness seem to be things that are basic in human life, such as home, and family, and work, and friendship. | |
| From: Philippa Foot (Natural Goodness [2001], 6) | |
| A reaction: I've not encountered discussion of 'deep' happiness before. I heard of an old man in tears because he had just seen a Purple Emperor butterfly for the first time. She makes it sound very conservative. How about mountaineering achievements? |
| 22498 | Happiness is enjoying the pursuit and attainment of right ends [Foot] |
| Full Idea: In my terminology 'happiness' is understood as the enjoyment of good things, meaning the enjoyment in attaining, and in pursuing, right ends. | |
| From: Philippa Foot (Natural Goodness [2001], 6) | |
| A reaction: A modified version of Aristotle's view, which she contrasts with McDowell's identification of happiness with the life of virtue. They all seem to have an optimistic hope that the pleasure in being a bit wicked is false happiness. |
| 23695 | Good actions can never be justified by the good they brings to their agent [Foot] |
| Full Idea: There is no good case for assessing the goodness of human action by reference only to good that each person brings to himself. | |
| From: Philippa Foot (Natural Goodness [2001], 1) | |
| A reaction: She observes that even non-human animals often act for non-selfish reasons. The significance of this is its rejection of her much earlier view that virtues are justified by the good they bring their possessor. |
| 22499 | We all know that just pretending to be someone's friend is not the good life [Foot] |
| Full Idea: We know perfectly well that it is not true that the best life would consist in successfully pretending friendship: having friends to serve one but without being a real friend oneself. | |
| From: Philippa Foot (Natural Goodness [2001], 7) | |
| A reaction: For some skallywags the achieving of something for nothing seems to be very much the good life, but not many of them want to exploit people who are seen to be their friends. |
| 22495 | Someone is a good person because of their rational will, not their body or memory [Foot] |
| Full Idea: To speak of a good person is to speak of an individual not in respect of his body, or of faculties such as sight and memory, but as concerns his rational will (his 'will as controllable by reason'). | |
| From: Philippa Foot (Natural Goodness [2001], 5) | |
| A reaction: She more or less agrees with Kant that the only truly good moral thing is a good will, though she has plenty of other criticisms of his views. |
| 22502 | Refraining from murder is not made good by authenticity or self-fulfilment [Foot] |
| Full Idea: If a stranger should come on us when we are sleeping he will not think it all right to kill us. …In human life as it is, this kind of action is not made good by authenticity or self-fulfilment in the one who does it. | |
| From: Philippa Foot (Natural Goodness [2001], 7) | |
| A reaction: A rare swipe from Foot at existentialism, which she hardly ever mentions. I find it hard to see these existential virtues as in any way moral. It means nothing to other citizens whether one of their number is 'authentic'. |