43 ideas
| 16440 | I don't think Lewis's cost-benefit reflective equilibrium approach offers enough guidance [Stalnaker] |
| Full Idea: Lewis articulated and made fashionable the cost-benefit reflective equilibrium methodology, but I have my reservations as it does not offer much guidance. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1.1) | |
| A reaction: Stalnaker suggests that this approach has 'run amok' in Lewis's case, giving reality to possible worlds. He spends much effort on showing the 'benefits' of a profoundly implausible view. The same can be said of 4D Perdurantism. |
| 16468 | Non-S5 can talk of contingent or necessary necessities [Stalnaker] |
| Full Idea: One can make sense of necessary versus contingent necessities in a non-S5 modal semantics. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 4.3 n17) | |
| A reaction: In S5 □φ → □□φ, so all necessities are necessary. Does it make any sense to say 'I suppose this might have been necessarily true'? |
| 16449 | In modal set theory, sets only exist in a possible world if that world contains all of its members [Stalnaker] |
| Full Idea: One principle of modal set theory should be uncontroversial: a set exists in a given possible world if and only if all of its members exist at that world. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 2.4) | |
| A reaction: Does this mean there can be no set containing all of my ancestors and future descendants? In no world can we coexist. |
| 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. |
| 16464 | We regiment to get semantic structure, for evaluating arguments, and understanding complexities [Stalnaker] |
| Full Idea: The point of regimentation is to give a perspicuous representation of the semantic structure of an expression, making it easier to evaluate the validity of arguments and to interpret complex statements. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 4.2) | |
| A reaction: This is an authoritative summary from an expert of why all philosophers must take an interest in logical form. |
| 16465 | In 'S was F or some other than S was F', the disjuncts need S, but the whole disjunction doesn't [Stalnaker] |
| Full Idea: In 'either Socrates was a philosopher or someone other than Socrates was a philosopher', both propositions expressed by the disjuncts depend for their existence on the existence of Socrates, but the whole disjunction does not. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 4.2) | |
| A reaction: Nice example, just the sort of thing we pay philosophers to come up with. He is claiming that propositions can exist in possible worlds in which the individuals mentioned do not exist. |
| 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. |
| 16434 | Some say what exists must do so, and nothing else could possible exist [Stalnaker] |
| Full Idea: Some philosophers deny there could have been anything other than what in fact exists, or that anything that exists could have failed to exist. This is developed in very different ways by Wittgenstein (in 'Tractatus'), Lewis and Williamson. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1) | |
| A reaction: This could come in various strengths. A weak version would say that, empirically, that all talk of what doesn't exist is vacuous. A strong necessity (Williamson?) that totally rules out other possible existence is a very odd view. |
| 16439 | A nominalist view says existence is having spatio-temporal location [Stalnaker] |
| Full Idea: A nominalist definition of existence is 'having spatio-temporal location'. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1.1) | |
| A reaction: This would evidently be physicalist as well as nominalist. Presumably it fits the 'mosaic' of reality Lewis refers to. I find this view sympathetic. A process of abstraction is required to get the rest of the stuff we talk about. |
| 16443 | Properties are modal, involving possible situations where they are exemplified [Stalnaker] |
| Full Idea: I take properties and relations to be modal notions. Properties are to be understood in terms of what it would be for them to be exemplified, which means understanding them in terms of a range of possible situations. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1.2) | |
| A reaction: I can't make head or tail of a property as anything other than a feature of some entity. Treating properties as a 'range of situations' is just as baffling to me as treating them as sets of objects. |
| 16471 | I accept a hierarchy of properties of properties of properties [Stalnaker] |
| Full Idea: I myself am prepared to accept higher-order properties and relations. There is the property of being Socrates, …and the property of being the property of being Socrates, ..and so on. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 4.4) | |
| A reaction: Elsewhere I have quoted such a hierarchy of vacuous properties as an absurdity that arises if all predicates are treated as properties. Logicians can live with such stuff, given their set hierarchy and so on, but in science and life this is a nonsense. |
| 16452 | Dispositions have modal properties, of which properties things would have counterfactually [Stalnaker] |
| Full Idea: Dispositional properties deserve special mention since they seem to be properties that have modal consequences - consequences for what properties the individuals that instantiate them would have in counterfactual circumstances. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 3.4) | |
| A reaction: I take this to be the key idea in trying to understand modality, but Stalnaker makes this point and then moves swiftly on, because it is so far away from his possible worlds models, in which he has invested a lifetime. |
| 16467 | 'Socrates is essentially human' seems to say nothing could be Socrates if it was not human [Stalnaker] |
| Full Idea: It seems natural to paraphrase the claim that Socrates is essentially human as the claim that nothing could be Socrates if it was not human. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 4.3) | |
| A reaction: In ordinary speech it would be emphasising how very human Socrates was (in comparison with Frege, for example). By this token Socrates essentially breathes oxygen, but that is hardly part of his essence. |
| 16453 | The bundle theory makes the identity of indiscernibles a necessity, since the thing is the properties [Stalnaker] |
| Full Idea: On the bundle theory, the identity of indiscernibles (for 'individuals') is a necessary truth, since an individual is just the co-instantiation of all the properties represented by a point in the space of properties. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 3.6) | |
| A reaction: So much the worse for the bundle theory, I presume. Leibniz did not, I think, hold a bundle theory, but his belief in the identity of indiscernibles seems to have had a theologicial underpinning. |
| 16466 | Strong necessity is always true; weak necessity is cannot be false [Stalnaker] |
| Full Idea: Prior had a strong and a weak reading of necessity, where strong necessity is truth in all possible worlds, while weak necessity is falsity in no possible world. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 4.3) | |
| A reaction: [K.Fine 2005:Ch.9 is also cited] The point of the weak one is that in some worlds there might not exist the proposition which is the candidate for truth or falsehood. |
| 16438 | Necessity and possibility are fundamental, and there can be no reductive analysis of them [Stalnaker] |
| Full Idea: My view is that if there were a nonmodal analysis of the modal concepts, that would be a sure sign that we were on the wrong track. Necessity and possibility are fundamental concepts, like truth and existence. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1.1) | |
| A reaction: The mystery of modality is tied up with the mystery of time (which is a very big mystery indeed). You get a nice clear grip on the here and now, but time and motion whisk you away to something else. Modality concerns the something else. |
| 16436 | Modal concepts are central to the actual world, and shouldn't need extravagant metaphysics [Stalnaker] |
| Full Idea: Modal concepts are central to our understanding of the world - the actual world - and understanding them should not require extravagant metaphysical commitments. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1) | |
| A reaction: I agree. Personally I think powers and dispositions do the job nicely. You just have to embrace Leibniz's emphasis on the active nature of reality, and the implausible metaphysics starts to recede. |
| 16433 | Given actualism, how can there be possible individuals, other than the actual ones? [Stalnaker] |
| Full Idea: My main focus is on how, on an actualist interpretation of possible worlds as ways a world might be, one is to account for the possibility that there be individuals other than those that actually exist. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], Pref) | |
| A reaction: The obvious thought would be that they are constructions from components of actual individuals, such as the chimaera, or fictional characters. We need some psychology here, which is not Stalnaker's style. |
| 16437 | Possible worlds are properties [Stalnaker] |
| Full Idea: Possible worlds are (to a first approximation) properties. [p.12] They are properties of the total universe. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1) |
| 16444 | Possible worlds don't reduce modality, they regiment it to reveal its structure [Stalnaker] |
| Full Idea: It is not reduction (of modality) but regimentation that the possible-worlds framework provides - a procedure for representing modal discourse, using primitive modal notions, in a way that helps reveal its structure. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1.2) | |
| A reaction: I think this is exactly my view. All discussion of the ontology of possible worlds is irrelevant. They no more exist than variables in logic exist. They're good when they clarify, but dubious when they over-simplify. |
| 16445 | I think of worlds as cells (rather than points) in logical space [Stalnaker] |
| Full Idea: I prefer to think of the possible worlds not as points in logical space but as cells of a relatively fine-grained partition of logical space - a partition that makes all the distinctions we need. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1.2) | |
| A reaction: Since he regards possible worlds as simply a means of regimenting our understanding of modality, he can think of possible worlds in any way that suits him. I find it hard work tuning in to his vision. |
| 16454 | Modal properties depend on the choice of a counterpart, which is unconstrained by metaphysics [Stalnaker] |
| Full Idea: Things have modal properties only relative to the choice of a counterpart relation, and the choice between alternative counterpart relations is not constrained by the metaphysics. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 3.6) | |
| A reaction: Stalnaker is sympathetic to counterparts, but this strikes me as a powerful objection to the theory. I take the modal properties of something to be fixed by its actuality. |
| 16450 | Anti-haecceitism says there is no more to an individual than meeting some qualitative conditions [Stalnaker] |
| Full Idea: The anti-haecceitist strategy holds that a purely qualitative characterisation of a possible world would be a complete characterisation; there is, on this view, nothing to being a particular individual other than meeting certain qualitative conditions. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 3) | |
| A reaction: Not quite the same as the bundle theory of objects, which says the objects are the qualities. This is about individuation, not about ontology (I think). I don't like anti-haecceitism, but I also don't like haecceitism. Hmm. |
| 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. |
| 23549 | We treat testimony with a natural trade off of belief and caution [Reid, by Fricker,M] |
| Full Idea: Reid says we naturally operate counterpart principles of veracity and credulity in our testimonial exchanges. | |
| From: report of Thomas Reid (An Enquiry [1764], 6.24) by Miranda Fricker - Epistemic Injustice 1.3 n11 | |
| A reaction: What you would expect from someone who believed in common sense. Fricker contrasts this with Tyler Burge's greater confidence, and then criticises both (with Reid too cautious and Burge over-confident). She defends a 'low-level' critical awareness. |
| 16474 | How can we know what we are thinking, if content depends on something we don't know? [Stalnaker] |
| Full Idea: How can we know what we ourselves are thinking if the very existence of the content of our thought may depend on facts of which we are ignorant? | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 5) | |
| A reaction: This has always been my main doubt about externalism. I may defer to experts about what I intend by an 'elm' (Putnam's example), but what I mean by elm is thereby a fuzzy tall tree with indeterminate leaves. I don't know the meaning of 'elm'! |
| 16461 | We still lack an agreed semantics for quantifiers in natural language [Stalnaker] |
| Full Idea: We still do not know how to give a direct semantics for the quantifiers of a natural language; that is something that we still do not know how to do (or at least how it is done remains controversial). | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 4) | |
| A reaction: I am struck by how rapidly the domain of quantification changes, even in mid-sentence, in the course of an ordinary conversation. This is decided almost entirely by context, not by pure ('direct'?) semantics. |
| 16448 | Possible world semantics may not reduce modality, but it can explain it [Stalnaker] |
| Full Idea: Most theorists agree that possible worlds semantics cannot provide an analysis of modal concepts which is an eliminative reduction, but it can still provide an explanation of the meanings of modal expressions. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 2.2) | |
| A reaction: Stalnaker cites Kit Fine for the view that there is no reduction of modality, which Fine takes to be primitive. Stalnaker defends the semantics, while denying the reduction which Lewis thought possible. |
| 16442 | I take propositions to be truth conditions [Stalnaker] |
| Full Idea: I will defend the view that propositions are truth conditions. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 1.2) | |
| A reaction: This sounds close to the Russellian view, which I take to equate propositions (roughly) with facts or states of affairs. But are 'truth conditions' in the world or in the head? |
| 16447 | A theory of propositions at least needs primitive properties of consistency and of truth [Stalnaker] |
| Full Idea: A minimal theory of propositions can make do with just two primitive properties: a property of consistency applied to sets of propositions, and a property of truth applied to propositions. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 2) | |
| A reaction: I would have thought a minimal theory would need some account of what a proposition is supposed to be (since there seems to be very little agreement about that). Stalnaker goes on to sketch a theory. |
| 16446 | Propositions presumably don't exist if the things they refer to don't exist [Stalnaker] |
| Full Idea: It seems plausible that singular propositions are object-dependent in the sense that the proposition would not exist if the individual did not. It is also plausible that some objects exist contingently, and there are singular propositions about them. | |
| From: Robert C. Stalnaker (Mere Possibilities [2012], 2) | |
| A reaction: This replies to the view that possible worlds are maximal sets of propositions, and so must exist for the worlds to exist; e.g. Lowe 1999:248. That is yet another commonplace of contemporary philosophy which I find utterly bewildering. |