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] |
12432 | Explanation of necessity must rest on something necessary or something contingent [Hale] |
12434 | Why is this necessary, and what is necessity in general; why is this necessary truth true, and why necessary? [Hale] |
12435 | The explanation of a necessity can be by a truth (which may only happen to be a necessary truth) [Hale] |
12433 | If necessity rests on linguistic conventions, those are contingent, so there is no necessity [Hale] |
12436 | Concept-identities explain how we know necessities, not why they are necessary [Hale] |
19553 | Commitment to 'I have a hand' only makes sense in a context where it has been doubted [Hawthorne] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
19551 | How can we know the heavyweight implications of normal knowledge? Must we distort 'knowledge'? [Hawthorne] |
19552 | We wouldn't know the logical implications of our knowledge if small risks added up to big risks [Hawthorne] |
19554 | Denying closure is denying we know P when we know P and Q, which is absurd in simple cases [Hawthorne] |