29 ideas
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
8717 | Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend] |
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] |
10113 | The grounding of mathematics is 'in the beginning was the sign' [Hilbert] |
10115 | Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman] |
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] |
10116 | Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman] |
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] |
7508 | Good reductionism connects fields of knowledge, but doesn't replace one with another [Pinker] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
7510 | Connectionists say the mind is a general purpose learning device [Pinker] |
7513 | Is memory stored in protein sequences, neurons, synapses, or synapse-strengths? [Pinker] |
7509 | Roundworms live successfully with 302 neurons, so human freedom comes from our trillions [Pinker] |
7511 | Neural networks can generalise their training, e.g. truths about tigers apply mostly to lions [Pinker] |
7512 | There are five types of reasoning that seem beyond connectionist systems [Pinker, by PG] |
7505 | Many think that accepting human nature is to accept innumerable evils [Pinker] |
7516 | In 1828, the stuff of life was shown to be ordinary chemistry, not a magic gel [Pinker] |
7515 | All the evidence says evolution is cruel and wasteful, not intelligent [Pinker] |
7514 | Intelligent Design says that every unexplained phenomenon must be design, by default [Pinker] |