38 ideas
| 14480 | Maybe analytic truths do not require truth-makers, as they place no demands on the world [Thomasson] |
| Full Idea: It is a venerable view that analytic claims do not require truth-makers, as they place no demands on the world, but this claim has often been challenged. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03.4) | |
| A reaction: She offers two challenges (bottom p.68), but I would have thought that the best response is that the meanings of the words themselves constitute truthmakers - perhaps via the essence of each word, as Fine suggests. |
| 14471 | Analytical entailments arise from combinations of meanings and inference rules [Thomasson] |
| Full Idea: 'Analytically entail' means entail in virtue of the meanings of the expressions involved and rules of inference. So 'Jones bought a house' analytically entails 'Jones bought a building'. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 01.2) | |
| A reaction: Quine wouldn't like this, but it sounds OK to me. Thomasson uses this as a key tool in her claim that common sense objects must exist. |
| 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. |
| 14493 | Existence might require playing a role in explanation, or in a causal story, or being composed in some way [Thomasson] |
| Full Idea: A higher standard for saying that entities exist might require that they play an essential role in explanation, or must figure in any complete causal story, or exist according to some uniform and nonarbitrary principle of composition. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 11.2) | |
| A reaction: I am struck by the first of these three. If I am defending the notion that essence depends on Aristotle's account of explanation, then if we add that existence also depends on explanation, we get a criterion for the existence of essences. Yay. |
| 14491 | Rival ontological claims can both be true, if there are analytic relationships between them [Thomasson] |
| Full Idea: Where there are analytic interrelations among our claims, distinct ontological claims may be true without rivalry, redundancy, or reduction. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 10) | |
| A reaction: Thus we might, I suppose, that it is analytically necessary that a lump of clay has a shape, and that a statue be made of something. Interesting. |
| 14489 | Theories do not avoid commitment to entities by avoiding certain terms or concepts [Thomasson] |
| Full Idea: A theory does not avoid commitment to any entities by avoiding use of certain terms or concepts. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09.4) | |
| A reaction: This is a salutary warning to those who apply the notion of ontological commitment rather naively. |
| 14485 | Ordinary objects may be not indispensable, but they are nearly unavoidable [Thomasson] |
| Full Idea: I do not argue that ordinary objects are indispensable, but rather that they are (nearly) unavoidable. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09) | |
| A reaction: Disappointing, given the blurb and title of the book, but put in those terms it will be hard to disagree. Clearly ordinary objects figure in the most useful way for us to talk. I wonder whether we have a clear ontology of 'simples' in which they vanish. |
| 14487 | The simple existence conditions for objects are established by our practices, and are met [Thomasson] |
| Full Idea: The existence conditions for ordinary objects are established by our practices, and they are quite minimal, so it is rather obvious that they are fulfilled, and so there are such things. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09.3) | |
| A reaction: This is one of her main arguments. The same argument would have worked for witches or ghosts in certain cultures. |
| 21651 | It is analytic that if simples are arranged chair-wise, then there is a chair [Thomasson, by Hofweber] |
| Full Idea: Thomasson argues that the existence of ordinary objects follows analytically from the distribution of simples, assuming that there are any simples. It is an analytic truth that if there are simples arranged chair-wise, then there is a chair. | |
| From: report of Amie L. Thomasson (Ordinary Objects [2007]) by Thomas Hofweber - Ontology and the Ambitions of Metaphysics 07.3 | |
| A reaction: But how do you distinguish when simples are arranged nearly chair-wise from the point where they click into place as actually chair-wise? What is the criterion? |
| 14467 | Ordinary objects are rejected, to avoid contradictions, or for greater economy in thought [Thomasson] |
| Full Idea: Objections to ordinary objects are the Causal Redundancy claim (objects lack causal powers), the Anti-Colocation view (statues and lumps overlap), Sorites arguments, a more economical ontology, or a more scientific ontology. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], Intro) | |
| A reaction: [my summary of two paragraphs] The chief exponents of these views are Van Inwagen and Merricks. Before you glibly accept ordinary objects, you must focus on producing a really strict ontology. These arguments all have real force. |
| 14479 | To individuate people we need conventions, but conventions are made up by people [Thomasson] |
| Full Idea: The conventionalist faces paradox if they hold that conventions are logically prior to people (since this plurality requires conventions of individuation), and people are logically prior to conventions (if they make up the conventions). | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03.3) | |
| A reaction: [Sidelle is the spokesman for conventionalism] The best defence would be to deny the second part, and say that conventions emerge from whatever is there, but only conventions can individuate the bits of what is there. |
| 14486 | Eliminativists haven't found existence conditions for chairs, beyond those of the word 'chair' [Thomasson] |
| Full Idea: The eliminativist cannot claim to have 'discovered' some real existence conditions for chairs beyond those entailed by the semantic rules associated with ordinary use of the word 'chair'. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09.3) | |
| A reaction: It is difficult to understand atoms arranged 'chairwise' or 'baseballwise' if you don't already know what a chair or a baseball are. |
| 14481 | Wherever an object exists, there are intrinsic properties instantiating every modal profile [Thomasson] |
| Full Idea: In a 'modally plenitudinous' ontology, wherever there is an object at all, there are objects with intrinsic modal properties instantiating every consistent modal profile. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03.5) | |
| A reaction: [She cites K.Bennett, Hawley, Rea, Sidelle] I love this. At last a label for the view I have been espousing. I am a Modal Plenitudinist. I must get a badge made. |
| 14482 | If the statue and the lump are two objects, they require separate properties, so we could add their masses [Thomasson] |
| Full Idea: An objection to the idea that statues are not identical to material lumps of stuff is the proliferation of instances of properties shared by those objects. If the mass of the statue is 500kg, and the mass of the lump is 500kg, do we have 1000kg? | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 04.3) | |
| A reaction: [compressed; she cites Rea 1997 and Zimmerman 1995] To wriggle out of this we would have to understand 'object' rather differently, so that an independent mass is not intrinsic to it. I leave this as an exercise for the reader. |
| 14483 | Given the similarity of statue and lump, what could possibly ground their modal properties? [Thomasson] |
| Full Idea: The 'grounding problem' is that given all that the statue and the lump have in common, what could possibly ground their different modal properties? | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 04.4) | |
| A reaction: Their modal properties are, of course, different, because only one of them could survive squashing. Thomasson suggests their difference of sort, but I'm not sure what that means, separately from what they actually are. |
| 14476 | Identity claims between objects are only well-formed if the categories are specified [Thomasson] |
| Full Idea: Identity claims are only well-formed and truth-evaluable if the terms flanking the statement are associated with a certain category of entity each is to refer to, which disambiguates the reference and identity-criteria. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03) | |
| A reaction: The first of her two criteria for identity. She is buying the full Wiggins package. |
| 14477 | Identical entities must be of the same category, and meet the criteria for the category [Thomasson] |
| Full Idea: Identity claims are only true if the entities referred to are of the same category, and meet the criteria of identity appropriate for things of that category. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03) | |
| A reaction: This may be a little too optimistic about having a set of clear-cut and reasonably objective categories to work with, but attempts at establishing metaphysical categories have not gone especially well. |
| 14478 | Modal Conventionalism says modality is analytic, not intrinsic to the world, and linguistic [Thomasson] |
| Full Idea: Modal Conventionalism has at least three theses: 1) modal truths are either analytic truths, or combine analytic and empirical truths, 2) modal properties are not intrinsic features of the world, 3) modal propositions depend on linguistic conventions. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03.2) | |
| A reaction: [She cites Alan Sidelle 1989 for this view] I disagree mainly with number 2), since I take dispositions to be key intrinsic features of nature, and I interpret dispositions as modal properties. |
| 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. |
| 14466 | A chief task of philosophy is making reflective sense of our common sense worldview [Thomasson] |
| Full Idea: Showing how, reflectively, we can make sense of our unreflective common sense worldview is arguably one of the chief tasks of philosophy. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], Intro) | |
| A reaction: Maybe. The obvious problem is that when you look at weird and remote cultures like the Aztecs, what counts as 'common sense' might be a bit different. She is talking of ordinary objects, though, where her point is reasonable. |
| 14475 | How can causal theories of reference handle nonexistence claims? [Thomasson] |
| Full Idea: Pure causal theories of reference have problems in handling nonexistence claims | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 02.3) | |
| A reaction: This is a very sound reason for shifting from a direct causal baptism view to one in which the baptism takes place by a social consensus. So there is a consensus about 'unicorns', but obviously no baptism. See Evans's 'Madagascar' example. |
| 14474 | Pure causal theories of reference have the 'qua problem', of what sort of things is being referred to [Thomasson] |
| Full Idea: Pure causal theories of reference face the 'qua problem' - that it may be radically indeterminate what the term refers to unless there is some very basic concept of what sort of thing is being referred to. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 02.3) | |
| A reaction: She cites Dummett and Wiggins on this. There is an obvious problem that when I say 'look at that!' there are all sorts of conventions at work if my reference is to succeed. |
| 14488 | Analyticity is revealed through redundancy, as in 'He bought a house and a building' [Thomasson] |
| Full Idea: The analytic interrelations among elements of language become evident through redundancy. It is redundant to utter 'He bought a house and a building', since buying a house analytically entails that he bought a building. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09.4) | |
| A reaction: This appears to concern necessary class membership. It is only linguistically redundant if the class membership is obvious. Houses are familiar, uranium samples are not. |
| 22817 | Citizenship involves a group of mutually supporting rights, which create community and equality [Miller,D] |
| Full Idea: The idea of citizenship is that rights support each other. Protective and welfare rights provide a basis for a political role. This underpins a sense of membership, and an obligation to provide welfare. Rights confer equal status and self-respect. | |
| From: David Miller (Community and Citizenship [1989], 3) | |
| A reaction: A helpful eludation of what a richer concept of citizenship than mere membership might look like. Communitarians have a different concept of rights from that of liberals. |
| 22816 | Socialists reject nationality as a false source of identity [Miller,D] |
| Full Idea: The socialist tradition has been overwhelmingly hostile to nationality as a source of identity, usually regarding it merely as an artificially created impediment to the brotherhood of man. | |
| From: David Miller (Community and Citizenship [1989], 2) | |
| A reaction: I have some sympathy with this, especially when nationalism is expressed in terms of enemies, but the question of what community a person can plausibly identify with is difficult. We start in hunter gather tribes of several hundred. |