58 ideas
21474 | Metaphysics studies the inexplicable ends of explanation [Schopenhauer] |
18806 | Frege thought traditional categories had psychological and linguistic impurities [Frege, by Rumfitt] |
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] |
8490 | First-level functions have objects as arguments; second-level functions take functions as arguments [Frege] |
8492 | Relations are functions with two arguments [Frege] |
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] |
8487 | Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism [Frege] |
18899 | Frege takes the existence of horses to be part of their concept [Frege, by Sommers] |
21470 | For me the objective thing-in-itself is the will [Schopenhauer] |
4028 | Frege allows either too few properties (as extensions) or too many (as predicates) [Mellor/Oliver on Frege] |
8489 | The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place [Frege] |
21479 | Knowledge is not power! Ignorant people possess supreme authority [Schopenhauer] |
21476 | A priori propositions are those we could never be seriously motivated to challenge [Schopenhauer] |
17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
21473 | All knowledge and explanation rests on the inexplicable [Schopenhauer] |
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] |
21478 | Half our thinking is unconscious, and we reach conclusions while unaware of premises [Schopenhauer] |
17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
21477 | We don't control our own thinking [Schopenhauer] |
21475 | All of our concepts are borrowed from perceptual knowledge [Schopenhauer] |
9947 | Concepts are the ontological counterparts of predicative expressions [Frege, by George/Velleman] |
10319 | An assertion about the concept 'horse' must indirectly speak of an object [Frege, by Hale] |
8488 | A concept is a function whose value is always a truth-value [Frege] |
9948 | Unlike objects, concepts are inherently incomplete [Frege, by George/Velleman] |
4972 | I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false [Frege] |
21372 | Aesthetics concerns how we can take pleasure in an object, with no reference to the will [Schopenhauer] |
21488 | The beautiful is a perception of Plato's Forms, which eliminates the will [Schopenhauer] |
21483 | Man is essentially a dreadful wild animal [Schopenhauer] |
21466 | Pleasure is weaker, and pain stronger, than we expect [Schopenhauer] |
21484 | A man's character can be learned from a single characteristic action [Schopenhauer] |
21482 | The five Chinese virtues: pity, justice, politeness, wisdom, honesty [Schopenhauer] |
21481 | Buddhists wisely start with the cardinal vices [Schopenhauer] |
21480 | Boredom is only felt by those clever enough to need activity [Schopenhauer] |
21469 | Human life is a mistake, shown by boredom, which is direct awareness of the fact [Schopenhauer] |
21485 | The state only exists to defend citizens, from exterior threats, and from one another [Schopenhauer] |
21486 | Poverty and slavery are virtually two words for the same thing [Schopenhauer] |
21487 | The freedom of the press to sell poison outweighs its usefulness [Schopenhauer] |
21471 | If suicide was quick and easy, most people would have done it by now [Schopenhauer] |
21467 | Would humanity still exist if sex wasn't both desired and pleasurable? [Schopenhauer] |
8491 | The Ontological Argument fallaciously treats existence as a first-level concept [Frege] |
21472 | Only religion introduces serious issues to uneducated people [Schopenhauer] |
21468 | The Creator created the possibilities for worlds, so should have made a better one than this possible [Schopenhauer] |