39 ideas
| 15169 | Metaphysics is clarifying how we speak and think (and possibly improving it) [Sidelle] |
| Full Idea: Metaphysics, for the conventionalist, is not a matter of trying to see deeply into the structure of mind-independent reality, but of trying to clarify the way we actually speak and think, and perhaps negotiating ways of doing this to our best advantage. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.1) | |
| A reaction: Note that he is still allowing space for 'revisionary' as well as for 'descriptive' metaphysics. I can't wholly accept this, as I really do think we can have some deep insights into reality, but Sidelle is articulating a large part of the truth. |
| 15164 | We seem to base necessities on thought experiments and imagination [Sidelle] |
| Full Idea: Judgments of necessity seem always to be based on thought experiments and appeals to what we can imagine. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.1) | |
| A reaction: That is, the denial of this thing seems inconceivable. I would say that they are also based on coherence. The idea that we can think without imagination is nonsense. |
| 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. |
| 8978 | Events are made of other things, and are not fundamental to ontology [Bennett] |
| Full Idea: Events are not basic items in the universe; they should not be included in any fundamental ontology...all the truths about them are entailed by and explained and made true by truths that do not involve the event concept. | |
| From: Jonathan Bennett (Events and Their Names [1988], p.12), quoted by Peter Simons - Events 3.1 | |
| A reaction: Given the variable time spans of events, their ability to coincide, their ability to contain no motion, their blatantly conventional component, and their recalcitrance to individuation, I say Bennett is right. |
| 15180 | There doesn't seem to be anything in the actual world that can determine modal facts [Sidelle] |
| Full Idea: Metaphysically, nothing in the actual world seems to be a candidate for determining what is necessarily the case. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.4) | |
| A reaction: I file this under 'Dispositions' to show what is at stake in the debate about dispositional and categorical properties. I take a commitment to dispositions to be a commitment to modal facts about the actual world. |
| 15184 | Causal reference presupposes essentialism if it refers to modally extended entities [Sidelle] |
| Full Idea: Even if the causal theory of reference proper does not presuppose essentialism, it does presuppose essentialism if it is to be an account of reference to modally extended entities. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.6) |
| 15172 | Clearly, essential predications express necessary properties [Sidelle] |
| Full Idea: It is clear, of course, that if there are true essential predications, then they express necessary properties. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.2) | |
| A reaction: I would certainly want to ask whether essences have to be analysed as properties, and also (more boldly) whether there might not be contingent essences. |
| 15181 | Being a deepest explanatory feature is an actual, not a modal property [Sidelle] |
| Full Idea: The property of being a deepest explanatory feature is a nonmodal property: it's an actual property. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.4) | |
| A reaction: I don't accept the existence of properties of the form 'being-F'. The possibility of securing a door may be the deepest explanatory feature of a lock. [To be fair to Sidelle, see context - just for once!] Dispositions are actual. |
| 15173 | That the essence of water is its microstructure is a convention, not a discovery [Sidelle] |
| Full Idea: The necessity to water of whatever is found out to be the water's microstructure is given by convention, and is not something which is discovered. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.2) | |
| A reaction: A powerful point. It shows the authority of science that we accept the microstructure as the essence. The essences of statues and people are definitely not their microstructures. One H2O molecule isn't water. Why not? Macro-properties count too! |
| 15185 | We aren't clear about 'same stuff as this', so a principle of individuation is needed to identify it [Sidelle] |
| Full Idea: Independent of conventions, no definite sense can be given to the notion of 'the same stuff as this'. So reference-fixing must include some principle of individuation to determine the aspects of sameness for the identity referred to. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.6) | |
| A reaction: Is he really saying that we don't understand 'same stuff as this'? Surely animals can manage that, and they are not famous for their conventions. Sidelle has fallen into the sortalist trap, I think. |
| 15175 | Evaluation of de dicto modalities does not depend on the identity of its objects [Sidelle] |
| Full Idea: In the evaluation of de dicto modal statements, whether some possible state of affairs is relevant to its truth does not depend on the identity of its objects, as in 'Necessarily, the President of the USA is male'. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.3) | |
| A reaction: This is a more clear-cut and easy to grasp criterion than most that are on offer. |
| 15032 | Necessary a posteriori is conventional for necessity and nonmodal for a posteriority [Sidelle, by Sider] |
| Full Idea: Sidelle defends conventionalism against a posteriori necessities by 'factoring' a necessary a posteriori truth into an analytic component and a nonmodal component. The modal force then comes from the analytic part, and the a posteriority from the other. | |
| From: report of Alan Sidelle (Necessity, Essence and Individuation [1989]) by Theodore Sider - Writing the Book of the World 12.8 | |
| A reaction: [I note that Sidelle refers, it seems, to the nonmodal component as a 'deep explanatory feature', which is exactly what I take an essence to be]. |
| 15179 | To know empirical necessities, we need empirical facts, plus conventions about which are necessary [Sidelle] |
| Full Idea: What we need to know, in order to know what is empirically necessary, is some empirical fact plus our conventions that tell us which truths are necessary given which empirical facts. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.4) | |
| A reaction: I take this attack on a posteriori necessities to be the most persuasive part of Sidelle's case, but you can't just put all of our truths down to convention. There are stabilities in the world, as well as in our conventions. |
| 15171 | The necessary a posteriori is statements either of identity or of essence [Sidelle] |
| Full Idea: The necessary a posteriori crudely divides into two groups - (synthetic) identity statements (between rigid designators), and statements of essential properties. The latter is either statements of property identity, or of the essences of natural kinds. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.2) | |
| A reaction: He cites Kripke's examples (Hesperus,Cicero,Truman,water,gold), and divides them into the two groups. Helpful. |
| 15167 | Empiricism explores necessities and concept-limits by imagining negations of truths [Sidelle] |
| Full Idea: In the traditional empiricist picture, we go about modal enquiry by trying to see whether we can imagine a situation in which it would be correct to assert the negation of a proposed necessary truth. Thus we can find out the limits of our concepts. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.1) |
| 15177 | Contradictoriness limits what is possible and what is imaginable [Sidelle] |
| Full Idea: Contradictoriness is the boundary both of what is possible and also of what is imaginable. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.4) | |
| A reaction: Of course we may see contradictions where there are none, and fail to grasp real hidden contradictions, so the two do not coincide in the practice. I think I would say it is 'a' boundary, not 'the' boundary. |
| 15176 | The individuals and kinds involved in modality are also a matter of convention [Sidelle] |
| Full Idea: It is not merely the modal facts that result from our conventions, but the individuals and kinds that are modally involved. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.3) | |
| A reaction: I am beginning to find Sidelle's views very sympathetic - going over to the Dark Side, I'm afraid. But conventions won't work at all if they don't correspond closely to reality. |
| 15174 | A thing doesn't need transworld identity prior to rigid reference - that could be a convention of the reference [Sidelle] |
| Full Idea: For a term to be rigid, it is said there must be real transworld identity prior to our use of the rigid term, ..but this may only be because we have conventional principles for individuating across worlds. 'Let's call him Fred' - perhaps explicitly rigid. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.3) | |
| A reaction: This seems right. An example might be a comic book character, who retains a perfect identity in all the comics, with no scars, weight change, or ageing. |
| 15183 | 'Dthat' operates to make a singular term into a rigid term [Sidelle] |
| Full Idea: 'Dthat' is Kaplan's indexical operator; it operates on a given singular term, φ, and makes it into a rigid designator of whatever φ designates in the original context. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.6 n11) | |
| A reaction: I like this idea a lot, because it strikes me that referring to something rigidly is a clear step beyond referring to it in actuality. I refer to 'whoever turns up each week', but that is hardly rigid. The germ of 2-D semantics is here. |
| 15165 | A priori knowledge is entirely of analytic truths [Sidelle] |
| Full Idea: The a priori method yields a priori knowledge, and the objects of this knowledge are not facts about the world, but analytic truths. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.1) | |
| A reaction: Are we not allowed any insights at all into how the world must be, independent of how we happen to conceptualise it? |
| 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. |
| 15168 | That water is essentially H2O in some way concerns how we use 'water' [Sidelle] |
| Full Idea: If water is essentially H2O, this is going to have something to do with our intentions in using 'water'. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.1) | |
| A reaction: This very simple point looks to be correct, and raises very important questions about the whole Twin Earth thing. When new discoveries are made, words shift their meanings. We're not quite sure what 'jade' means any more. |
| 15166 | Causal reference seems to get directly at the object, thus leaving its nature open [Sidelle] |
| Full Idea: The causal theory of reference appears to give us a way to get at an object while leaving it undetermined what its essence or necessary features might be. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.1) | |
| A reaction: This pinpoints why the direct/causal theory of reference seems to open the doors to scientific essentialism. Sidelle, of course, opposes the whole programme. |
| 15182 | Because some entities overlap, reference must have analytic individuation principles [Sidelle] |
| Full Idea: The phenomenon of overlapping entities requires that if our reference is to be determinate (as determinate as it is), then there must be analytic principles of individuation. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.5) | |
| A reaction: His point is that there is something inescapably conventional about the way in which our reference works. It isn't just some bald realist baptism. |
| 10364 | Facts are about the world, not in it, so they can't cause anything [Bennett] |
| Full Idea: Facts are not the sort of item that can cause anything. A fact is a true proposition (they say); it is not something in the world but is rather something about the world. | |
| From: Jonathan Bennett (Events and Their Names [1988], p.22), quoted by Jonathan Schaffer - The Metaphysics of Causation 1.1 | |
| A reaction: Compare 10361. Good argument, but maybe 'fact' is ambiguous. See Idea 10365. Events are said to be more concrete, and so can do the job, but their individuation also seems to depend on a description (as Davidson has pointed out). |
| 15178 | Can anything in science reveal the necessity of what it discovers? [Sidelle] |
| Full Idea: Is there anything in the procedures of scientists that could reveal to them that water is necessarily H2O or that gold necessarily has atomic number 79. | |
| From: Alan Sidelle (Necessity, Essence and Individuation [1989], Ch.4) | |
| A reaction: This is Leibniz's is view, that empirical evidence can never reveal necessities. Given that we know some necessities, you have an argument for rationalism. |