45 ideas
| 10468 | A metaphysics has an ontology (objects) and an ideology (expressed ideas about them) [Oliver] |
| Full Idea: A metaphysical theory hs two parts: ontology and ideology. The ontology consists of the entities which the theory says exist; the ideology consists of the ideas which are expressed within the theory using predicates. Ideology sorts into categories. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §02.1) | |
| A reaction: Say 'what there is', and 'what we can say about it'. The modern notion remains controversial (see Ladyman and Ross, for example), so it is as well to start crystalising what metaphysics is. I am enthusiastic, but nervous about what is being said. |
| 10471 | Ockham's Razor has more content if it says believe only in what is causal [Oliver] |
| Full Idea: One might give Ockham's Razor a bit more content by advising belief in only those entities which are causally efficacious. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §03) | |
| A reaction: He cites Armstrong as taking this line, but I immediately think of Shoemaker's account of properties. It seems to me to be the only account which will separate properties from predicates, and bring them under common sense control. |
| 15549 | If it were true that nothing at all existed, would that have a truthmaker? [Lewis] |
| Full Idea: If there was absolutely nothing at all, then it would have been true that there was nothing. Would there have been a truthmaker for this truth? | |
| From: David Lewis (A world of truthmakers? [1998], p.220) | |
| A reaction: This is a problem for Lewis's own claim that 'truth supervenes on being', as well as the more restricted truthmakers invoked by Armstrong. |
| 10749 | Necessary truths seem to all have the same truth-maker [Oliver] |
| Full Idea: The definition of truth-makers entails that a truth-maker for a given necessary truth is equally a truth-maker for every other necessary truth. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §24) | |
| A reaction: Maybe we could accept this. Necessary truths concern the way things have to be, so all realities will embody them. Are we to say that nothing makes a necessary truth true? |
| 10750 | Slingshot Argument: seems to prove that all sentences have the same truth-maker [Oliver] |
| Full Idea: Slingshot Argument: if truth-makers work for equivalent sentences and co-referring substitute sentences, then if 'the numbers + S1 = the numbers' has a truth-maker, then 'the numbers + S2 = the numbers' will have the same truth-maker. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §24) | |
| A reaction: [compressed] Hence every sentence has the same truth-maker! Truth-maker fans must challenge one of the premises. |
| 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. |
| 10747 | Accepting properties by ontological commitment tells you very little about them [Oliver] |
| Full Idea: The route to the existence of properties via ontological commitment provides little information about what properties are like. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §22) | |
| A reaction: NIce point, and rather important, I would say. I could hardly be committed to something for the sole reason that I had expressed a statement which contained an ontological commitment. Start from the reason for making the statement. |
| 10748 | Reference is not the only way for a predicate to have ontological commitment [Oliver] |
| Full Idea: For a predicate to have a referential function is one way, but not the only way, to harbour ontological commitment. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §22) | |
| A reaction: Presumably the main idea is that the predicate makes some important contribution to a sentence which is held to be true. Maybe reference is achieved by the whole sentence, rather than by one bit of it. |
| 10719 | There are four conditions defining the relations between particulars and properties [Oliver] |
| Full Idea: Four adequacy conditions for particulars and properties: asymmetry of instantiation; different particulars can have the same property; particulars can have many properties; two properties can be instantiated by the same particulars. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §09) | |
| A reaction: The distinction between particulars and universals has been challenged (e.g. by Ramsey and MacBride). There are difficulties in the notion of 'instantiation', and in the notion of two properties being 'the same'. |
| 10721 | If properties are sui generis, are they abstract or concrete? [Oliver] |
| Full Idea: If properties are sui generis entities, one must decide whether they are abstract or concrete. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §09) | |
| A reaction: A nice basic question! I take the real properties to be concrete, but we abstract from them, especially from their similarities, and then become deeply confused about the ontology, because our language doesn't mark the distinctions clearly. |
| 10716 | There are just as many properties as the laws require [Oliver] |
| Full Idea: One conception of properties says there are only as many properties as are needed to be constituents of laws. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §03) | |
| A reaction: I take this view to the be precise opposite of the real situation. The properties are what lead to the laws. Properties are internal to nature, and laws are imposed from outside, which is ridiculous unless you think there is an active deity. |
| 10720 | We have four options, depending whether particulars and properties are sui generis or constructions [Oliver] |
| Full Idea: Both properties and particulars can be taken as either sui generis or as constructions, so we have four options: both sui generis, or both constructions, or one of each. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §09) | |
| A reaction: I think I favour both being sui generis. God didn't make the objects, then add their properties, or make the properties then create some instantiations. There can't be objects without properties, or objectless properties (except in thought). |
| 10714 | The expressions with properties as their meanings are predicates and abstract singular terms [Oliver] |
| Full Idea: The types of expressions which have properties as their meanings may vary, the chief candidates being predicates, such as '...is wise', and abstract singular terms, such as 'wisdom'. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §02) | |
| A reaction: This seems to be important, because there is too much emphasis on predicates. If this idea is correct, we need some account of what 'abstract' means, which is notoriously tricky. |
| 10715 | There are five main semantic theories for properties [Oliver] |
| Full Idea: Properties in semantic theory: functions from worlds to extensions ('Californian'), reference, as opposed to sense, of predicates (Frege), reference to universals (Russell), reference to situations (Barwise/Perry), and composition from context (Lewis). | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §02 n12) | |
| A reaction: [compressed; 'Californian' refers to Carnap and Montague; the Lewis view is p,67 of Oliver]. Frege misses out singular terms, or tries to paraphrase them away. Barwise and Perry sound promising to me. Situations involve powers. |
| 10738 | Tropes are not properties, since they can't be instantiated twice [Oliver] |
| Full Idea: I rule that tropes are not properties, because it is not true that one and the same trope of redness is instantiated by two books. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §12) | |
| A reaction: This seems right, but has very far-reaching implications, because it means there are no properties, and no two things have the same properties, so there can be no generalisations about properties, let alone laws. ..But they have equivalence sets. |
| 10739 | The property of redness is the maximal set of the tropes of exactly similar redness [Oliver] |
| Full Idea: Using the predicate '...is exactly similar to...' we can sort tropes into equivalence sets, these sets serving as properties and relations. For example, the property of redness is the maximal set of the tropes of redness. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §12) | |
| A reaction: You have somehow to get from scarlet and vermilion, which have exact similarity within their sets, to redness, which doesn't. |
| 10740 | The orthodox view does not allow for uninstantiated tropes [Oliver] |
| Full Idea: It is usual to hold an aristotelian conception of tropes, according to which tropes are present in their particular instances, and which does not allow for uninstantiated tropes. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §12) | |
| A reaction: What are you discussing when you ask what colour the wall should be painted? Presumably we can imagine non-existent tropes. If I vividly imagine my wall looking yellow, have I brought anything into existence? |
| 10741 | Maybe concrete particulars are mereological wholes of abstract particulars [Oliver] |
| Full Idea: Some trope theorists give accounts of particulars. Sets of tropes will not do because they are always abstract, but we might say that particulars are (concrete) mereological wholes of the tropes which they instantiate. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §12) | |
| A reaction: Looks like a non-starter to me. How can abstract entities add up to a mereological whole which is concrete? |
| 10742 | Tropes can overlap, and shouldn't be splittable into parts [Oliver] |
| Full Idea: More than one trope can occupy the same place at the same time, and a trope occupies a place without having parts which occupy parts of the place. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §12) | |
| A reaction: This is the general question of the size of a spatial trope, or 'how many red tropes in a tin of red paint?' |
| 10472 | 'Structural universals' methane and butane are made of the same universals, carbon and hydrogen [Oliver] |
| Full Idea: The 'structural universals' methane and butane are each made up of the same universals, carbon and hydrogen. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §07) | |
| A reaction: He cites Lewis 1986, who is criticising Armstrong. If you insist on having universals, they might (in this case) best be described as 'patterns', which would be useful for structuralism in mathematics. They reduce to relations. |
| 10724 | Located universals are wholly present in many places, and two can be in the same place [Oliver] |
| Full Idea: So-called aristotelian universals have some queer features: one universal can be wholly present at different places at the same time, and two universals can occupy the same place at the same time. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §11) | |
| A reaction: If you want to make a metaphysical doctrine look ridiculous, stating it in very simple language will often do the job. Belief in fairies is more plausible than the first of these two claims. |
| 7963 | Aristotle's instantiated universals cannot account for properties of abstract objects [Oliver] |
| Full Idea: Properties and relations of abstract objects may need to be acknowledged, but they would have no spatio-temporal location, so they cannot instantiate Aristotelian universals, there being nowhere for such universals to be. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §11), quoted by Cynthia Macdonald - Varieties of Things | |
| A reaction: Maybe. Why can't the second-order properties be in the same location as the first-order ones? If the reply is that they would seem to be in many places at once, that is only restating the original problem of universals at a higher level. |
| 10730 | If universals ground similarities, what about uniquely instantiated universals? [Oliver] |
| Full Idea: If universals are to ground similarities, it is hard to see why one should admit universals which only happen to be instantiated once. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §11) | |
| A reaction: He is criticising Armstrong, who holds that universals must be instantiated. This is a good point about any metaphysics which makes resemblance basic. |
| 10727 | Uninstantiated universals seem to exist if they themselves have properties [Oliver] |
| Full Idea: We may have to accept uninstantiated universals because the properties and relations of abstract objects may need to be acknowledged. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §11) | |
| A reaction: This is the problem of 'abstract reference'. 'Courage matters more than kindness'; 'Pink is more like red than like yellow'. Not an impressive argument. All you need is second-level abstraction. |
| 7962 | Uninstantiated properties are useful in philosophy [Oliver] |
| Full Idea: Uninstantiated properties and relations may do some useful philosophical work. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §11), quoted by Cynthia Macdonald - Varieties of Things | |
| A reaction: Their value isn't just philosophical; hopes and speculations depend on them. This doesn't make universals mind-independent. I think the secret is a clear understanding of the word 'abstract' (which I don't have). |
| 10722 | Instantiation is set-membership [Oliver] |
| Full Idea: One view of instantiation is that it is the set-membership predicate. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §10) | |
| A reaction: This cuts the Gordian knot rather nicely, but I don't like it, if the view of sets is extensional. We need to account for natural properties, and we need to exclude mere 'categorial' properties. |
| 10744 | Nominalism can reject abstractions, or universals, or sets [Oliver] |
| Full Idea: We can say that 'Harvard-nominalism' is the thesis that there are no abstract objects, 'Oz-nominalism' that there are no universals, and Goodman's nominalism rejects entities, such as sets, which fail to obey a certain principle of composition. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §15 n46) | |
| A reaction: Personally I'm a Goodman-Harvard-Oz nominalist. What are you rebelling against? What have you got? We've been mesmerized by the workings of our own minds, which are trying to grapple with a purely physical world. |
| 10726 | Things can't be fusions of universals, because two things could then be one thing [Oliver] |
| Full Idea: If a particular thing is a bundle of located universals, we might say it is a mereological fusion of them, but if two universals can be instantiated by more than one particular, then two particulars can have the same universals, and be the same thing. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §11) | |
| A reaction: This and Idea 10725 pretty thoroughly demolish the idea that objects could be just bundles of universals. The problem pushes some philosophers back to the idea of 'substance', or some sort of 'substratum' which has the universals. |
| 10725 | Abstract sets of universals can't be bundled to make concrete things [Oliver] |
| Full Idea: If a particular thing is a bundle of located universals, we might say that it is the set of its universals, but this won't work because the thing can be concrete but sets are abstract. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §11) | |
| A reaction: This objection applies just as much to tropes (abstract particulars) as it does to universals. |
| 10745 | Science is modally committed, to disposition, causation and law [Oliver] |
| Full Idea: Natural science is up to its ears in modal notions because of its use of the concepts of disposition, causation and law. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §15) | |
| A reaction: This is aimed at Quine. It might be possible for an auster physicist to dispense with these concepts, by merely describing patterns of observed behaviour. |
| 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. |
| 10746 | Conceptual priority is barely intelligible [Oliver] |
| Full Idea: I find the notion of conceptual priority barely intelligible. | |
| From: Alex Oliver (The Metaphysics of Properties [1996], §19 n48) | |
| A reaction: I don't think I agree, though there is a lot of vagueness and intuition involved, and not a lot of hard argument. Can you derive A from B, but not B from A? Is A inconceivable without B, but B conceivable without A? |