34 ideas
| 4036 | What matters is not how many entities we postulate, but how many kinds of entities [Armstrong, by Mellor/Oliver] |
| Full Idea: Armstrong argues that what matters is not how few entities we postulate (quantitative economy), but how few kinds of entities (qualitative economy). | |
| From: report of David M. Armstrong (Properties [1992]) by DH Mellor / A Oliver - Introduction to 'Properties' §9 | |
| A reaction: Is this what Ockham meant? Armstrong is claiming that the notion of a 'property' is needed to identify kinds. See also Idea 7038. |
| 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. |
| 15754 | Without properties we would be unable to express the laws of nature [Armstrong] |
| Full Idea: The ontological correlates of true law-statements must involve properties. How else can one pick our the uniformities which the law-statements entail? | |
| From: David M. Armstrong (Properties [1992], 1) | |
| A reaction: I'm unconvinced about the 'laws', but I have to admit that it is hard to know how to describe the relevant bits of nature without some family of concepts covered by the word 'property'. I'm in favour of taking some of the family into care, though. |
| 4034 | Whether we apply 'cold' or 'hot' to an object is quite separate from its change of temperature [Armstrong] |
| Full Idea: Evading properties by means of predicates is implausible when things change. If a cold thing becomes hot, first 'cold' applies, and then 'hot', but what have predicates to do with the temperature of an object? | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: A clear illustration of why properties are part of nature, not just part of language. But some applications of predicates are more arbitrary than this (ugly, cool) |
| 8535 | To the claim that every predicate has a property, start by eliminating failure of application of predicate [Armstrong] |
| Full Idea: Upholders of properties have been inclined to postulate a distinct property corresponding to each distinct predicate. We could start by eliminating all those properties where the predicate fails to apply, is not true, of anything. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: This would leave billions of conjunctional, disjunctional and gerrymandered properties where the predicate applies very well. We are all 'on the same planet as New York'. Am I allowed to say that I 'wish' that a was F? He aims for 'sparse' properties. |
| 8537 | Tropes fall into classes, because exact similarity is symmetrical and transitive [Armstrong] |
| Full Idea: Exact similarity is a symmetrical and transitive relation. (Less than exact similarity is not transitive, even for tropes). So the relation of exact similarity is an equivalence relation, partitioning the field of tropes into equivalence classes. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: Armstrong goes on the explore the difficulties for trope theory of less than exact similarity, which is a very good line of discussion. Unfortunately it is a huge problem for everyone, apart from the austere nominalist. |
| 8538 | Trope theory needs extra commitments, to symmetry and non-transitivity, unless resemblance is exact [Armstrong] |
| Full Idea: Trope theory needs extra ontological baggage, the Axioms of Resemblance. There is a principle of symmetry, and there is the failure of transitivity - except in the special case of exact resemblance. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: [see text for fuller detail] Is it appropriate to describe such axioms as 'ontological' baggage? Interesting, though I suspect that any account of properties and predicates will have a similar baggage of commitments. |
| 8539 | Universals are required to give a satisfactory account of the laws of nature [Armstrong] |
| Full Idea: A reason why I reject trope theory is that universals are required to give a satisfactory account of the laws of nature. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: This is the key thought in Armstrong's defence of universals. Issues about universals may well be decided on such large playing fields. I think he is probably wrong, and I will gradually explain why. Watch this space as the story unfolds... |
| 8529 | Deniers of properties and relations rely on either predicates or on classes [Armstrong] |
| Full Idea: The great deniers of properties and relations are of two sorts: those who put their faith in predicates and those who appeal to sets (classes). | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: This ignores the Quine view, which is strictly for ostriches. Put like this, properties and relations seem undeniable. Predicates are too numerous (gerrymandering) or too few (colour shades). Classes can have arbitrary members. |
| 8532 | Resemblances must be in certain 'respects', and they seem awfully like properties [Armstrong] |
| Full Idea: If a resembles b, in general, they resemble in certain respects, and fail to resemble in other respects. But respects are uncomfortably close to properties, which the Resemblance theory proposes to do without. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: This is a good objection. I think it is plausible to build a metaphysics around the idea of respects, and drop properties. Shall we just talk of 'respects' for categorising, and 'powers' for causation and explanation? Respects only exist in comparisons. |
| 8530 | Change of temperature in objects is quite independent of the predicates 'hot' and 'cold' [Armstrong] |
| Full Idea: To appreciate the implausibility of the predicate view, consider where a thing's properties change. 'Hot' becomes applicable when 'cold' ceases to, ..but the change in the object would have occurred if the predicates had never existed. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: They keep involving secondary qualities! Armstrong is taking a strongly realist view (fine by me), but anti-realists can ignore his argument. I take predicate nominalism to be a non-starter. |
| 8536 | We want to know what constituents of objects are grounds for the application of predicates [Armstrong] |
| Full Idea: The properties that are of ontological interest are those constituents of objects, of particulars, which serve as the ground in the objects for the application of predicates. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: Good. This is a reversal of the predicate nominalist approach, and is a much healthier attitude to the relationship between ontology and language. Value judgements will be an interesting case. Does this allow us to invent new predicates? |
| 8531 | In most sets there is no property common to all the members [Armstrong] |
| Full Idea: Most sets are uninteresting because they are utterly heterogeneous, that is, the members have nothing in common. For most sets there is no common property F, such that the set is the set of all the Fs. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: One might link the interesting sets together by resemblance, without invoking the actual existence of an item F which all the members carry (like freemasons' briefcases). Personally I am only really interested in 'natural' sets. |
| 15753 | Essences might support Resemblance Nominalism, but they are too coarse and ill-defined [Armstrong] |
| Full Idea: A sophisticated Resemblance theory can appeal to the natures of the resembling things, from which the resemblances flow. The natures are suitably internal, but are as coarse as the things themselves (and perhaps are the things themselves). | |
| From: David M. Armstrong (Properties [1992], 1) | |
| A reaction: Note that this is essentialism as an underpinning for Resemblance Nominalism. His objection is that he just can't believe in essences, because they are too 'coarse' - which I take to mean that we cannot distinguish the boundaries of an essence. |
| 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. |
| 18883 | Any equivalence relation among similar things allows the creation of an abstractum [Simons] |
| Full Idea: Whenever we have an equivalence relation among things - such as similarity in a certain respect - we can abstract under the equivalence and consider the abstractum. | |
| From: Peter Simons (Modes of Extension: comment on Fine [2008], p.19) | |
| A reaction: This strikes me as dressing up old-fashioned psychological abstractionism in the respectable clothing of Fregean equivalences (such as 'directions'). We can actually do what Simons wants without the precision of partitioned equivalence classes. |
| 18884 | Abstraction is usually seen as producing universals and numbers, but it can do more [Simons] |
| Full Idea: Abstraction as a cognitive tool has been associated predominantly with the metaphysics of universals and of mathematical objects such as numbers. But it is more widely applicable beyond this standard range. I commend its judicious use. | |
| From: Peter Simons (Modes of Extension: comment on Fine [2008], p.21) | |
| A reaction: Personally I think our view of the world is founded on three psychological principles: abstraction, idealisation and generalisation. You can try to give them rigour, as 'equivalence classes', or 'universal quantifications', if it makes you feel better. |
| 8533 | Predicates need ontological correlates to ensure that they apply [Armstrong] |
| Full Idea: Must there not be something quite specific about the thing which allows, indeed ensures, that predicates like 'underneath' and 'hot' apply? The predicates require ontological correlates. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: An interesting proposal, that in addition to making use of predicates, we should 'ensure that they apply'. Sounds verificationist. Obvious problem cases would be speculative, controversial or metaphorical predicates. "He's beneath contempt". |
| 4035 | There must be some explanation of why certain predicates are applicable to certain objects [Armstrong] |
| Full Idea: When we have said that predicates apply to objects, we have surely not said enough. The situation cries out for an explanation. Must there not be something specific about the things which allows, indeed ensures, that these predicates apply? | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: A nice challenge to any philosopher who places too much emphasis on language. A random and arbitrary (nominalist?) language simply wouldn't work. Nature has joints. |
| 8541 | Regularities theories are poor on causal connections, counterfactuals and probability [Armstrong] |
| Full Idea: Regularity theories make laws molecular, with no inner causal connections; also, only some cosmic regularities are manifestations of laws; molecular states can't sustain counterfactuals; and probabilistic laws are hard to accommodate. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: [very compressed] A helpful catalogue of difficulties. The first difficulty is the biggest one - that regularity theories have nothing to say about why there is a regularity. They offer descriptions instead of explanations. |
| 8540 | The introduction of sparse properties avoids the regularity theory's problem with 'grue' [Armstrong] |
| Full Idea: Regularity theories of laws face the grue problem. That, I think, can only be got over by introducing properties, sparse properties, into one's ontology. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: The problem is, roughly, that regularities have to be described in language, which is too arbitrary in character. Armstrong rightly tries to break the rigid link to language. See his Idea 8536, which puts reality before language. |