41 ideas
17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
13134 | We negate predicates but do not negate names [Westerhoff] |
17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
17928 | Ordinal numbers represent order relations [Colyvan] |
17923 | Intuitionists only accept a few safe infinities [Colyvan] |
17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
13117 | How far down before we are too specialised to have a category? [Westerhoff] |
13116 | Maybe objects in the same category have the same criteria of identity [Westerhoff] |
13118 | Categories are base-sets which are used to construct states of affairs [Westerhoff] |
13125 | Categories are held to explain why some substitutions give falsehood, and others meaninglessness [Westerhoff] |
13126 | Categories systematize our intuitions about generality, substitutability, and identity [Westerhoff] |
13130 | Categories as generalities don't give a criterion for a low-level cut-off point [Westerhoff] |
13124 | Categories can be ordered by both containment and generality [Westerhoff] |
13131 | The aim is that everything should belong in some ontological category or other [Westerhoff] |
13123 | All systems have properties and relations, and most have individuals, abstracta, sets and events [Westerhoff] |
13115 | Ontological categories are like formal axioms, not unique and with necessary membership [Westerhoff] |
13119 | Categories merely systematise, and are not intrinsic to objects [Westerhoff] |
13135 | A thing's ontological category depends on what else exists, so it is contingent [Westerhoff] |
13129 | Essential kinds may be too specific to provide ontological categories [Westerhoff] |
22868 | The value and truth of knowledge are measured by success in activity [Dewey] |
17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
22865 | Habits constitute the self [Dewey] |
22871 | The good people are those who improve; the bad are those who deteriorate [Dewey] |
22876 | Democracy is the development of human nature when it shares in the running of communal activities [Dewey] |
22875 | Democracy is not just a form of government; it is a mode of shared living [Dewey] |
22874 | Individuality is only developed within groups [Dewey] |