40 ideas
| 23728 | Analysis aims to express the full set of platitudes surrounding a given concept [Smith,M] |
| Full Idea: The aim of analysis is to give us knowledge of all and only the platitudes surrounding our use of the concept that is up for analysis. | |
| From: Michael Smith (The Moral Problem [1994], 1.10) | |
| A reaction: His earlier specimen concept is 'redness'. For other concepts there might be considerable disagreement about which propositions are or are not the relevant platitudes. Smith emphasises that analysis need not be reductive. |
| 23744 | Defining a set of things by paradigms doesn't pin them down enough [Smith,M] |
| Full Idea: The discussion of colour concepts shows that permutation problems arise when a set of concepts, acquired inter alia via the presentation of paradigms, is largely interdefined. | |
| From: Michael Smith (The Moral Problem [1994], 5.9) | |
| A reaction: Smith says that our normative moral concepts are largely interdefined in this way. The 'permutation' problem is that they can change places in the definition set, and so their intrinsic individual character is not pinned down. Sounds 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. |
| 23743 | Capturing all the common sense facts about rationality is almost impossible [Smith,M] |
| Full Idea: It would be a superhuman task just to write down an explicit, non-summary style, statement of the platitudes that capture our idea of what it is to be fully rational. | |
| From: Michael Smith (The Moral Problem [1994], 5.9) | |
| A reaction: Well said. Philosophers are inclined to make simplistic binary judgements about whether persons or animals are rational. A visit to YouTube will show fish acting extremely rationally. |
| 23739 | Goals need desires, and so only desires can motivate us [Smith,M] |
| Full Idea: Only an agent's desires may constitute her having certain goals, and it follows from this that only her desires may constitute her motivating reasons. | |
| From: Michael Smith (The Moral Problem [1994], 4.8) | |
| A reaction: We might distinguish between reasons which direct us towards certain ends, and reasons which motivate us to pursue those ends. Most mornings I have a reason to get out of bed, which precedes my motivation to actually do it. |
| 23724 | A pure desire could be criticised if it were based on a false belief [Smith,M] |
| Full Idea: There is a minor proviso to Hume's view, which is that desires are subject to rational criticism, but only insofar as they are based on beliefs that are subject to rational criticism. | |
| From: Michael Smith (The Moral Problem [1994], 1.3) | |
| A reaction: He says this is not a refutation of the basic Humean claim. He has in mind a desire such as to consume cyanide because you believe it will be good for you. |
| 23736 | A person can have a desire without feeling it [Smith,M] |
| Full Idea: We should concede that a desire may be had in the absence of its being felt. | |
| From: Michael Smith (The Moral Problem [1994], 4.5) | |
| A reaction: A nice observation. An example he gives is a father's desire that his child does well. Smith is discussing Hume's account of motivation in terms of desires and beliefs. |
| 23723 | In the Humean account, desires are not true/false, or subject to any rational criticism [Smith,M] |
| Full Idea: According to the standard picture of human psychology that we get from Hume, not only are desires not assessable in terms of truth and falsehood, they are not subject to any sort of rational criticism at all. | |
| From: Michael Smith (The Moral Problem [1994], 1.3) | |
| A reaction: This is where action theory meets metaethics. The separation of facts from values underlies this, because a desire is a fact, but the wickedness of a desire is not. Surely a desire could be a failure of practical reason? |
| 23735 | Subjects may be fallible about the desires which explain their actions [Smith,M] |
| Full Idea: It is an adequacy constraint on any conception of desire that the epistemology of desire it recommends allows that subjects may be fallible about the desires they have. | |
| From: Michael Smith (The Moral Problem [1994], 4.5) | |
| A reaction: [I do wish authors would write my short versions instead of their rambling sentences!] Even after the event we may be unsure why we did something. If someone observes self-interest when I thought my action was altruistic, I don't know how to respond. |
| 23738 | Humeans (unlike their opponents) say that desires and judgements can separate [Smith,M] |
| Full Idea: Humeans claim that agents who believe they should act may nevertheless lack the desire to do so, where anti-Humeans must say the two go together, and someone with the belief thereby has the desire. | |
| From: Michael Smith (The Moral Problem [1994], 4.7) | |
| A reaction: [very compressed] A very helpful distinction about the classic debates over the motivations of action. Smith defends the Humean view, and makes it very plausible. No mere sense of rightness or duty can compel us to act. |
| 23742 | If first- and second-order desires conflict, harmony does not require the second-order to win [Smith,M] |
| Full Idea: Even if we assume that reason prefers harmony between first- and second-order desires, there is no reason to assume that reason is on the side of achieving that harmony by changing first-order desires to suit second-order, rather than vice versa. | |
| From: Michael Smith (The Moral Problem [1994], 5.7) | |
| A reaction: [Smith is discussing David Lewis 1989 on second-order desires] Smith says that on the Humean view the rational winner should simply be the stronger of the two. Since this sounds like an endorsement for weakness of will, Smith relies on beliefs. |
| 23746 | Objective reasons to act might be the systematic desires of a fully rational person [Smith,M] |
| Full Idea: One way to decide what we have normative reasons to do …is by trying to find a set of desires that is systematically justifiable, which is our best assessment of the desires we would have under conditions of full rationality. | |
| From: Michael Smith (The Moral Problem [1994], 5.9) | |
| A reaction: This is Smith accepting the Humean view that desires are essential for motivation, but trying to find a marriage of desires with reason to produce the more objective aspects of morality. An interesting aspiration… |
| 23733 | Motivating reasons are psychological, while normative reasons are external [Smith,M] |
| Full Idea: There are motivating reasons for action, which are psychological states, and normative reasons, which are propositions of the general form 'a person's doing this is desirable or required'. | |
| From: Michael Smith (The Moral Problem [1994], 4.2) | |
| A reaction: Motivating reasons are locatable entities in minds, whereas normative reasons are either abstract, or perhaps motivating reasons expressed by other people. Smith says the two types are unconnected. |
| 23740 | Humeans take maximising desire satisfaction as the normative reasons for actions [Smith,M] |
| Full Idea: The distinctive Humean view of normative reasons for action is that the rational thing for an agent to do is simply to act so as to maximally satisfy her desires, whatever the content of those desires. | |
| From: Michael Smith (The Moral Problem [1994], 5.1) | |
| A reaction: Smith disagrees with this view (though he agrees with Hume about motivating reasons). An obvious problem for the Humean view would be a strong desire to do something excessively dangerous. |
| 23745 | We cannot expect even fully rational people to converge on having the same desires for action [Smith,M] |
| Full Idea: We cannot expect that, even under conditions of full rationality, agents would all converge on the same desires about what is to be done in the various circumstances they might face. | |
| From: Michael Smith (The Moral Problem [1994], 5.9) | |
| A reaction: A very good argument in favour of the Humean view that desires are an essential part of moral motivation. Possible convergence of view is a standard hallmark of communal rationality. |
| 23731 | 'Externalists' say moral judgements are not reasons, and maybe not even motives [Smith,M] |
| Full Idea: The 'externalist' view of morality says either that judgements of rightness are motives but not reasons, or (more strongly) that they are neither, meaning that moral judgements do not have practical implications. | |
| From: Michael Smith (The Moral Problem [1994], 3.1) | |
| A reaction: [Philippa Foot's untypical 1972 article is cited for the strong view. Hare and Blackburn are typical of the first view]. I would say that such judgements are both reasons and motives - but not necessarily for me! 'Someone should do something about this!'. |
| 23732 | A person could make a moral judgement without being in any way motivated by it [Smith,M] |
| Full Idea: Amoralists make moral judgements without being motivated accordingly, and without suffering any sort of practical irrationality either; the practicality requirement of moral judgement is thus false. | |
| From: Michael Smith (The Moral Problem [1994], 3.3) | |
| A reaction: It is hard to imagine an immoralist with this nihilistic attitude bothering to make any moral judgements at all. Why would someone indifferent to art make aesthetic judgements? What could a 'judgement of rightness' mean to an amoralist? |
| 23729 | Moral internalism says a judgement of rightness is thereby motivating [Smith,M] |
| Full Idea: Moral 'internalism' says if an agent judges an action as right in some circumstance, then they are either thereby motivated to do it, or they are irrational (e.g. their will is weak). | |
| From: Michael Smith (The Moral Problem [1994], 3.1) | |
| A reaction: [Somewhat reworded] So the motivation comes from an internal judgement, not from external factors. Is it not tautological that 'this is the right thing to do' means it should be done (ceteris paribus)? |
| 23730 | 'Rationalism' says the rightness of an action is a reason to perform it [Smith,M] |
| Full Idea: Moral 'rationalism' says if an action is right for agents in some circumstances, then there is a reason for the agents to do it. | |
| From: Michael Smith (The Moral Problem [1994], 3.1) | |
| A reaction: That is, there is not merely a motivation to act (the 'internalist' view), but there is a reason to act. Smith calls both views the 'practicality requirement' of normal moral judgements. Smith defends the rationalist view. |
| 23727 | Expressivists count attitudes as 'moral' if they concern features of things, rather than their mere existence [Smith,M] |
| Full Idea: The pro- and con- attitudes of the expressivists count as 'moral' only if they are had towards particular people, actions or states of affairs in virtue of their natural features, ….rather than in virtue of being the particulars that they are. | |
| From: Michael Smith (The Moral Problem [1994], 2.4) | |
| A reaction: So whereas emotivists don't have to have any reasons for their moral feelings, other expressivists seem to require reasons (i.e. indicating features of things) to endorse their attitudes. What of reasonless emotionless attitudes? |
| 23741 | Is valuing something a matter of believing or a matter of desiring? [Smith,M] |
| Full Idea: What is it to value something? That is, equivalently, what is it to accept that we have a normative reason to do something? In Hume's terms, is it a matter of believing? Or is it a matter of desiring? We seem to face a dilemma. | |
| From: Michael Smith (The Moral Problem [1994], 5.4) | |
| A reaction: Smith is discussing moral motivation, and there is obviously more to valuing something than acting on it. Nice question, though. Personally I value St Paul's Cathedral, but I don't desire it. I value heart surgeons, but don't want to emulate them. |
| 9111 | God is not wise, but more-than-wise; God is not good, but more-than-good [William of Ockham] |
| Full Idea: God is not wise, but more-than-wise; God is not good, but more-than-good. | |
| From: William of Ockham (Reportatio [1330], III Q viii) | |
| A reaction: [He is quoting 'Damascene'] I quote this for interest, but I very much doubt whether Damascene or William knew what it meant, and I certainly don't. There seems to have been a politically correct desire to invent super-powers for God. |
| 9112 | We could never form a concept of God's wisdom if we couldn't abstract it from creatures [William of Ockham] |
| Full Idea: What we abstract is said to belong to perfection in so far as it can be predicated of God and can stand for Him. For if such a concept could not be abstracted from a creature, then in this life we could not arrive at a cognition of God's wisdom. | |
| From: William of Ockham (Reportatio [1330], III Q viii) | |
| A reaction: This seems to be the germ of an important argument. Without the ability to abstract from what is experienced, we would not be able to apply general concepts to things which are beyond experience. It is a key idea for empiricism. |