30 ideas
| 15557 | Verisimilitude has proved hard to analyse, and seems to have several components [Lewis] |
| Full Idea: The analysis of verisimilitude has been much debated. Some plausible analyses have failed disastrously, others conflict with one another. One conclusion is that verisimilitude seems to consist of several distinguishable virtues. | |
| From: David Lewis (Causal Explanation [1986], V n7) | |
| A reaction: Presumably if it is complex, you can approach truth in one respect while receding from it in another. It seems clear enough if you are calculating pi by some iterative process. |
| 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. |
| 16901 | The equivalent algebra model of geometry loses some essential spatial meaning [Burge] |
| Full Idea: Geometrical concepts appear to depend in some way on a spatial ability. Although one can translate geometrical propositions into algebraic ones and produce equivalent models, the meaning of the propositions seems to me to be thereby lost. | |
| From: Tyler Burge (Frege on Apriority (with ps) [2000], 4) | |
| A reaction: I think this is a widely held view nowadays. Giaquinto has a book on it. A successful model of something can't replace it. Set theory can't replace arithmetic. |
| 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. |
| 16902 | Peano arithmetic requires grasping 0 as a primitive number [Burge] |
| Full Idea: In the Peano axiomatisation, arithmetic seems primitively to involve the thought that 0 is a number. | |
| From: Tyler Burge (Frege on Apriority (with ps) [2000], 5) | |
| A reaction: Burge is pointing this out as a problem for Frege, for whom only the logic is primitive. |
| 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. |
| 15554 | A disposition needs a causal basis, a property in a certain causal role. Could the disposition be the property? [Lewis] |
| Full Idea: I take for granted that a disposition requires a causal basis: one has the disposition iff one has a property that occupies a certain causal role. Shall we then identify the disposition with its basis? That makes the disposition cause its manifestations. | |
| From: David Lewis (Causal Explanation [1986], III) | |
| A reaction: Introduce the concept of a 'power' and I see no problem with his proposal. Fundamental dispositions are powerful, and provide the causal basis for complex dispositions. Something had better be powerful. |
| 15560 | We can explain a chance event, but can never show why some other outcome did not occur [Lewis] |
| Full Idea: I think we are right to explain chance events, yet we are right also to deny that we can ever explain why a chance process yields one outcome rather than another. We cannot explain why one event happened rather than the other. | |
| From: David Lewis (Causal Explanation [1986], VI) | |
| A reaction: This misses out an investigation which slowly reveals that a 'chance' event wasn't so chancey after all. Failure to explain confirms chance, so the judgement of chance shouldn't block attempts to explain. |
| 16892 | Is apriority predicated mainly of truths and proofs, or of human cognition? [Burge] |
| Full Idea: Whereas Leibniz and Frege predicate apriority primarily of truths (or more fundamentally, proofs of truths), Kant predicates apriority primarily of cognition and the employment of representations. | |
| From: Tyler Burge (Frege on Apriority (with ps) [2000], 1) |
| 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. |
| 15559 | Does a good explanation produce understanding? That claim is just empty [Lewis] |
| Full Idea: It is said that a good explanation ought to produce understanding, ...but this just says that a good explanation produces possession of that which it provide, so this desideratum is empty. It adds nothing to our understanding of explanation. | |
| From: David Lewis (Causal Explanation [1986], V) | |
| A reaction: I am not convinced by this dismissal. If you are looking for a test of whether an explanation is good, the announcement that the participants feel they have achieved a good understanding sounds like success. |
| 15556 | Science may well pursue generalised explanation, rather than laws [Lewis] |
| Full Idea: The pursuit of general explanations may be very much more widespread in science than the pursuit of general laws. | |
| From: David Lewis (Causal Explanation [1986], IV) | |
| A reaction: Nice. I increasingly think that the main target of all enquiry is ever-widening generality, with no need to aspire to universality. |
| 15558 | A good explanation is supposed to show that the event had to happen [Lewis] |
| Full Idea: It is said that a good explanation ought to show that the explanandum event had to happen, given the laws and circumstances. | |
| From: David Lewis (Causal Explanation [1986], V) | |
| A reaction: I cautiously go along with this view. Given that there are necessities in nature (a long story), we should aim to reveal them. There is no higher aspiration open to us than successful explanation. Lewis says good explanations can reveal falsehoods. |
| 4809 | Lewis endorses the thesis that all explanation of singular events is causal explanation [Lewis, by Psillos] |
| Full Idea: Lewis endorses the thesis that all explanation of singular events is causal explanation. | |
| From: report of David Lewis (Causal Explanation [1986]) by Stathis Psillos - Causation and Explanation p.237 | |
| A reaction: It is hard to challenge this. The assumption is that only nomological and causal explanations are possible, and the former are unobtainable for singular events. |
| 14321 | To explain an event is to provide some information about its causal history [Lewis] |
| Full Idea: Here is my main thesis: to explain an event is to provide some information about its causal history. | |
| From: David Lewis (Causal Explanation [1986], II) | |
| A reaction: The obvious thought is that you might provide some tiny and barely relevant part of that causal history, such as a bird perched on the Titanic's iceberg. So how do we distinguish the 'important' causal information? |
| 15555 | Explaining match lighting in general is like explaining one lighting of a match [Lewis] |
| Full Idea: Explaining why struck matches light in general is not so very different from explaining why some particular struck match lit. ...We may generalize modestly, without laying claim to universality. | |
| From: David Lewis (Causal Explanation [1986], IV) | |
| A reaction: A suggestive remark, since particular causation and general causation seem far apart, but Lewis suggests that the needs of explanation bring them together. Lawlike and unlawlike explanations? |
| 15551 | Ways of carving causes may be natural, but never 'right' [Lewis] |
| Full Idea: There is no one right way - though there may be more or less natural ways - of carving up a causal history. | |
| From: David Lewis (Causal Explanation [1986], I) | |
| A reaction: This invites a distinction between the 'natural' causes and the 'real' causes. Presumably if any causes were 'real', they would have a better claim to be 'right'. Is an earthquake the 'real' (correct?) cause of a tsunami? |
| 15552 | We only pick 'the' cause for the purposes of some particular enquiry. [Lewis] |
| Full Idea: Disagreement about 'the' cause is only disagreement about which part of the causal history is most salient for the purposes of some particular inquiry. | |
| From: David Lewis (Causal Explanation [1986], I) | |
| A reaction: I don't believe this. In the majority of cases I see the cause of an event, without having any interest in any particular enquiry. It is just so obvious that there isn't even a disagreement. Maybe there is only one sensible enquiry. |
| 15553 | Causal dependence is counterfactual dependence between events [Lewis] |
| Full Idea: I take causal dependence to be counterfactual dependence, of a suitably back-tracking sort, between distinct events. | |
| From: David Lewis (Causal Explanation [1986], I) | |
| A reaction: He quotes Hume in support. 'Counterfactual dependence' strikes me as too vague, or merely descriptive, for the job of explanation. 'If...then' is a logical relationship; what is it in nature that justifies the dependency? |