26 ideas
| 14092 | Philosophers are often too fussy about words, dismissing perfectly useful ordinary terms [Rosen] |
| Full Idea: Philosophers can sometimes be too fussy about the words they use, dismissing as 'unintelligible' or 'obscure' certain forms of language that are perfectly meaningful by ordinary standards, and which may be of some real use. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 01) | |
| A reaction: Analytic philosophers are inclined to drop terms they can't formalise, but there is more to every concept than its formalisation (Frege's 'direction' for example). I want to rescue 'abstraction' and 'essence'. Rosen says distinguish, don't formalise. |
| 14100 | Figuring in the definition of a thing doesn't make it a part of that thing [Rosen] |
| Full Idea: From the simple fact that '1' figures in the definition of '2', it does not follow that 1 is part of 2. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 10) | |
| A reaction: He observes that quite independent things can be mentioned on the two sides of a definition, with no parthood relation. You begin to wonder what a definition really is. A causal chain? |
| 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. |
| 14096 | Explanations fail to be monotonic [Rosen] |
| Full Idea: The failure of monotonicity is a general feature of explanatory relations. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 05) | |
| A reaction: In other words, explanations can always shift in the light of new evidence. In principle this is right, but some explanations just seem permanent, like plate-tectonics as explanation for earthquakes. |
| 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. |
| 14097 | Things could be true 'in virtue of' others as relations between truths, or between truths and items [Rosen] |
| Full Idea: Our relation of 'in virtue of' is among facts or truths, whereas Fine's relation (if it is a relation at all) is a relation between a given truth and items whose natures ground that truth. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 07 n10) | |
| A reaction: This disagreement between two key players in the current debate on grounding looks rather significant. I think I favour Fine's view, as it seems more naturalistic, and less likely to succumb to conventionalism. |
| 14095 | Facts are structures of worldly items, rather like sentences, individuated by their ingredients [Rosen] |
| Full Idea: Facts are structured entities built up from worldly items rather as sentences are built up from words. They might be identified with Russellian propositions. They are individuated by their constituents and composition, and are fine-grained. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 04) | |
| A reaction: I'm a little cautious about the emphasis on being sentence-like. We have Russell's continual warnings against imposing subject-predicate structure on things. I think we should happily talk about 'facts' in metaphysics. |
| 14093 | An 'intrinsic' property is one that depends on a thing and its parts, and not on its relations [Rosen] |
| Full Idea: One intuitive gloss on 'intrinsic' property is that a property is intrinsic iff whether or not a thing has it depends entirely on how things stand with it and its parts, and not on its relation to some distinct thing. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 02) | |
| A reaction: He offers this as a useful reward for reviving 'depends on' in metaphysical talk. The problem here would be to explain the 'thing' and its 'parts' without mentioning the target property. The thing certainly can't be a bundle of tropes. |
| 14094 | The excellent notion of metaphysical 'necessity' cannot be defined [Rosen] |
| Full Idea: Many of our best words in philosophy do not admit of definition, the notion of metaphysical 'necessity' being one pertinent example. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 03) | |
| A reaction: Rosen is busy defending words in metaphysics which cannot be pinned down with logical rigour. We are allowed to write □ for 'necessary', and it is accepted by logicians as being stable in a language. |
| 14101 | Are necessary truths rooted in essences, or also in basic grounding laws? [Rosen] |
| Full Idea: Fine says a truth is necessary when it is a logical consequence of the essential truths, but maybe it is a consequence of the essential truths together with the basic grounding laws (the 'Moorean connections'). | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 13) | |
| A reaction: I'm with Fine all the way here, as we really don't need to clog nature up with things called 'grounding laws', which are both obscure and inexplicable. Fine's story is the one for naturalistically inclined philosophers. |
| 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. |
| 14099 | 'Bachelor' consists in or reduces to 'unmarried' male, but not the other way around [Rosen] |
| Full Idea: It sounds right to say that Fred's being a bachelor consists in (reduces to) being an unmarried male, but slightly off to say that Fred's being an unmarried male consists in (or reduces to) being a bachelor. There is a corresponding explanatory asymmetry. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 10) | |
| A reaction: This emerging understanding of the asymmetry of the idea shows that we are not just dealing with a simple semantic identity. Our concepts are richer than our language. He adds that a ball could be blue in virtue of being cerulean. |
| 9284 | Reasons are 'internal' if they give a person a motive to act, but 'external' otherwise [Williams,B] |
| Full Idea: Someone has 'internal reasons' to act when the person has some motive which will be served or furthered by the action; if this turns out not to be so, the reason is false. Reasons are 'external' when there is no such condition. | |
| From: Bernard Williams (Internal and External Reasons [1980], p.101) | |
| A reaction: [compressed] An external example given is a family tradition of joining the army, if the person doesn't want to. Williams says (p.111) external reason statements are actually false, and a misapplication of the concept of a 'reason to act'. See Idea 8815. |
| 14098 | An acid is just a proton donor [Rosen] |
| Full Idea: To be an acid just is to be a proton donor. | |
| From: Gideon Rosen (Metaphysical Dependence [2010], 10) | |
| A reaction: My interest here is in whether we can say that we have found the 'essence' of an acid - so we want to know whether something 'deeper' explains the proton-donation. I suspect not. Being a proton donor happens to have a group of related consequences. |