56 ideas
9847 | A contextual definition permits the elimination of the expression by a substitution [Dummett] |
9820 | In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical [Dummett] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
9896 | A prime number is one which is measured by a unit alone [Dummett] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
18255 | Addition of quantities is prior to ordering, as shown in cyclic domains like angles [Dummett] |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
9895 | A number is a multitude composed of units [Dummett] |
9852 | We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett] |
8764 | Categories are the best foundation for mathematics [Shapiro] |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
9829 | The identity of a number may be fixed by something outside structure - by counting [Dummett] |
9828 | Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett] |
9876 | Set theory isn't part of logic, and why reduce to something more complex? [Dummett] |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
9884 | The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy [Dummett] |
9869 | Realism is just the application of two-valued semantics to sentences [Dummett] |
4444 | One moderate nominalist view says that properties and relations exist, but they are particulars [Armstrong] |
4445 | If properties and relations are particulars, there is still the problem of how to classify and group them [Armstrong] |
4448 | Should we decide which universals exist a priori (through words), or a posteriori (through science)? [Armstrong] |
4446 | It is claimed that some universals are not exemplified by any particular, so must exist separately [Armstrong] |
9880 | Nominalism assumes unmediated mental contact with objects [Dummett] |
4440 | 'Resemblance Nominalism' finds that in practice the construction of resemblance classes is hard [Armstrong] |
4439 | 'Resemblance Nominalism' says properties are resemblances between classes of particulars [Armstrong] |
4431 | 'Predicate Nominalism' says that a 'universal' property is just a predicate applied to lots of things [Armstrong] |
4433 | Concept and predicate nominalism miss out some predicates, and may be viciously regressive [Armstrong] |
4432 | 'Concept Nominalism' says a 'universal' property is just a mental concept applied to lots of things [Armstrong] |
4436 | 'Class Nominalism' may explain properties if we stick to 'natural' sets, and ignore random ones [Armstrong] |
4434 | 'Class Nominalism' says that properties or kinds are merely membership of a set (e.g. of white things) [Armstrong] |
4435 | 'Class Nominalism' cannot explain co-extensive properties, or sets with random members [Armstrong] |
4437 | 'Mereological Nominalism' sees whiteness as a huge white object consisting of all the white things [Armstrong] |
4438 | 'Mereological Nominalism' may work for whiteness, but it doesn't seem to work for squareness [Armstrong] |
9885 | The existence of abstract objects is a pseudo-problem [Dummett] |
9858 | Abstract objects nowadays are those which are objective but not actual [Dummett] |
9859 | It is absurd to deny the Equator, on the grounds that it lacks causal powers [Dummett] |
9860 | 'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object [Dummett] |
9872 | Abstract objects need the context principle, since they can't be encountered directly [Dummett] |
9848 | Content is replaceable if identical, so replaceability can't define identity [Dummett, by Dummett] |
9842 | Frege introduced criteria for identity, but thought defining identity was circular [Dummett] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
9849 | Maybe a concept is 'prior' to another if it can be defined without the second concept [Dummett] |
9850 | An argument for conceptual priority is greater simplicity in explanation [Dummett] |
9873 | Abstract terms are acceptable as long as we know how they function linguistically [Dummett] |
9993 | There is no reason why abstraction by equivalence classes should be called 'logical' [Dummett, by Tait] |
9857 | We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus' [Dummett] |
9833 | To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too [Dummett] |
9836 | Fregean semantics assumes a domain articulated into individual objects [Dummett] |
18257 | Why should the limit of measurement be points, not intervals? [Dummett] |