39 ideas
| 24032 | Clever scholars can obscure things which are obvious even to peasants [Descartes] |
| 24033 | Most scholastic disputes concern words, where agreeing on meanings would settle them [Descartes] |
| 24024 | The secret of the method is to recognise which thing in a series is the simplest [Descartes] |
| 24018 | One truth leads us to another [Descartes] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 24035 | Unity is something shared by many things, so in that respect they are equals [Descartes] |
| 24036 | I can only see the proportion of two to three if there is a common measure - their unity [Descartes] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 24029 | Among the simples are the graspable negations, such as rest and instants [Descartes] |
| 24030 | 3+4=7 is necessary because we cannot conceive of seven without including three and four [Descartes] |
| 24019 | If we accept mere probabilities as true we undermine our existing knowledge [Descartes] |
| 24020 | We all see intuitively that we exist, where intuition is attentive, clear and distinct rational understanding [Descartes] |
| 24031 | When Socrates doubts, he know he doubts, and that truth is possible [Descartes] |
| 24025 | Clear and distinct truths must be known all at once (unlike deductions) [Descartes] |
| 24022 | Our souls possess divine seeds of knowledge, which can bear spontaneous fruit [Descartes] |
| 24034 | If someone had only seen the basic colours, they could deduce the others from resemblance [Descartes] |
| 24021 | The method starts with clear intuitions, followed by a process of deduction [Descartes] |
| 24027 | Nerves and movement originate in the brain, where imagination moves them [Descartes] |
| 24026 | Our four knowledge faculties are intelligence, imagination, the senses, and memory [Descartes] |
| 24028 | The force by which we know things is spiritual, and quite distinct from the body [Descartes] |
| 7845 | When we need to do something, we depute an inner servant to remind us of it [Proust] |
| 24023 | All the sciences searching for order and measure are related to mathematics [Descartes] |