37 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. |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 19378 | Early modern possibility is what occurs sometime; for Leibniz, it is what is not contradictory [Arthur,R] |
| Full Idea: For Descartes, Hobbes and Spinoza, if a state of things is possible, it must occur at some time, whether past, present or future. For Leibniz possibility makes no reference to time; an individual is possible if its concept contains no contradiction. | |
| From: Richard T.W. Arthur (Leibniz [2014], 4 'Contingent') | |
| A reaction: It has always struck me as fallacious to say that anything that is possible must at some time occur. If '6' is possible on the die, what will constrain it to eventually come up when thrown? Mere non-contradiction doesn't imply possibility either. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 19380 | Occasionalism contradicts the Eucharist, which needs genuine changes of substance [Arthur,R] |
| Full Idea: The Jesuits rejected occasionalism ... because it is incompatible with the Catholic interpretation of the Eucharist, which there is genuine change of substance of the bread into the substance of Christ (transubstantiation). | |
| From: Richard T.W. Arthur (Leibniz [2014], 5 'Substance') | |
| A reaction: Not sure I understand this, but I take it that the Eucharist needs a real relation across the substance-spirit boundary, and not just a co-ordination. |
| 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. |
| 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… |
| 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. |
| 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. |