41 ideas
| 16539 | A definition of a circle will show what it is, and show its generating principle [Lowe] |
| 16540 | Defining an ellipse by conic sections reveals necessities, but not the essence of an ellipse [Lowe] |
| 16548 | An essence is what an entity is, revealed by a real definition; this is not an entity in its own right [Lowe] |
| 16549 | Simple things like 'red' can be given real ostensive definitions [Lowe] |
| 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] |
| 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] |
| 16545 | The essence of lumps and statues shows that two objects coincide but are numerically distinct [Lowe] |
| 16546 | The essence of a bronze statue shows that it could be made of different bronze [Lowe] |
| 16551 | Grasping an essence is just grasping a real definition [Lowe] |
| 16542 | Explanation can't give an account of essence, because it is too multi-faceted [Lowe] |
| 16552 | If we must know some entity to know an essence, we lack a faculty to do that [Lowe] |
| 16533 | Logical necessities, based on laws of logic, are a proper sub-class of metaphysical necessities [Lowe] |
| 16531 | 'Metaphysical' necessity is absolute and objective - the strongest kind of necessity [Lowe] |
| 16532 | 'Epistemic' necessity is better called 'certainty' [Lowe] |
| 16543 | If an essence implies p, then p is an essential truth, and hence metaphysically necessary [Lowe] |
| 16544 | Metaphysical necessity is either an essential truth, or rests on essential truths [Lowe] |
| 16538 | We could give up possible worlds if we based necessity on essences [Lowe] |
| 16534 | 'Intuitions' are just unreliable 'hunches'; over centuries intuitions change enormously [Lowe] |
| 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] |
| 16535 | A concept is a way of thinking of things or kinds, whether or not they exist [Lowe] |
| 16550 | Direct reference doesn't seem to require that thinkers know what it is they are thinking about [Lowe] |
| 23565 | Our obedience to the king erases any crimes we commit for him [Shakespeare] |
| 16547 | H2O isn't necessary, because different laws of nature might affect how O and H combine [Lowe] |