25 ideas
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
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] |
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] |
8915 | How we refer to abstractions is much less clear than how we refer to other things [Rosen] |
13074 | Only natural kinds and their members have real essences [Suárez, by Cover/O'Leary-Hawthorne] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
8917 | The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting [Rosen] |
8912 | Nowadays abstractions are defined as non-spatial, causally inert things [Rosen] |
8913 | Chess may be abstract, but it has existed in specific space and time [Rosen] |
8914 | Sets are said to be abstract and non-spatial, but a set of books can be on a shelf [Rosen] |
8916 | Conflating abstractions with either sets or universals is a big claim, needing a big defence [Rosen] |
8918 | Functional terms can pick out abstractions by asserting an equivalence relation [Rosen] |
8919 | Abstraction by equivalence relationships might prove that a train is an abstract entity [Rosen] |
7563 | The old 'influx' view of causation says it is a flow of accidental properties from A to B [Suárez, by Jolley] |