24 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] |
22167 | Our images of bodies are not produced by the bodies, but by our own minds [Augustine, by Aquinas] |
22117 | Our minds grasp reality by direct illumination (rather than abstraction from experience) [Augustine, by Matthews] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
22118 | Augustine created the modern concept of the will [Augustine, by Matthews] |
4348 | Love, and do what you will [Augustine] |
7821 | Pagans produced three hundred definitions of the highest good [Augustine, by Grayling] |
22119 | Augustine said (unusually) that 'ought' does not imply 'can' [Augustine, by Matthews] |
6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
22116 | Augustine identified Donatism, Pelagianism and Manicheism as the main heresies [Augustine, by Matthews] |
19338 | Augustine said evil does not really exist, and evil is a limitation in goodness [Augustine, by Perkins] |