56 ideas
12330 | In ontology, logic dominated language, until logic was mathematized [Badiou] |
9808 | Philosophy aims to reveal the grandeur of mathematics [Badiou] |
12318 | The female body, when taken in its entirety, is the Phallus itself [Badiou] |
12325 | Philosophy has been relieved of physics, cosmology, politics, and now must give up ontology [Badiou] |
12324 | Consensus is the enemy of thought [Badiou] |
13520 | A 'tautology' must include connectives [Wolf,RS] |
13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
15375 | If terms change their designations in different states, they are functions from states to objects [Fitting] |
15376 | Intensional logic adds a second type of quantification, over intensional objects, or individual concepts [Fitting] |
15378 | Awareness logic adds the restriction of an awareness function to epistemic logic [Fitting] |
15379 | Justication logics make explicit the reasons for mathematical truth in proofs [Fitting] |
12337 | There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou] |
13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
12321 | The axiom of choice must accept an indeterminate, indefinable, unconstructible set [Badiou] |
13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
12342 | Topos theory explains the plurality of possible logics [Badiou] |
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] |
11026 | Classical logic is deliberately extensional, in order to model mathematics [Fitting] |
12341 | Logic is a mathematical account of a universe of relations [Badiou] |
11028 | λ-abstraction disambiguates the scope of modal operators [Fitting] |
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] |
9812 | In mathematics, if a problem can be formulated, it will eventually be solved [Badiou] |
12334 | There is no single unified definition of number [Badiou] |
12335 | Numbers are for measuring and for calculating (and the two must be consistent) [Badiou] |
12333 | Each type of number has its own characteristic procedure of introduction [Badiou] |
12322 | Must we accept numbers as existing when they no longer consist of units? [Badiou] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
9813 | Mathematics shows that thinking is not confined to the finite [Badiou] |
12327 | The undecidability of the Continuum Hypothesis may have ruined or fragmented set theory [Badiou] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
12329 | If mathematics is a logic of the possible, then questions of existence are not intrinsic to it [Badiou] |
12328 | Platonists like axioms and decisions, Aristotelians like definitions, possibilities and logic [Badiou] |
12331 | Logic is definitional, but real mathematics is axiomatic [Badiou] |
9809 | Mathematics inscribes being as such [Badiou] |
12340 | There is no Being as a whole, because there is no set of all sets [Badiou] |
12323 | Existence is Being itself, but only as our thought decides it [Badiou] |
12332 | The modern view of Being comes when we reject numbers as merely successions of One [Badiou] |
12326 | The primitive name of Being is the empty set; in a sense, only the empty set 'is' [Badiou] |
9811 | It is of the essence of being to appear [Badiou] |
12320 | Ontology is (and always has been) Cantorian mathematics [Badiou] |
15377 | Definite descriptions pick out different objects in different possible worlds [Fitting] |
12338 | We must either assert or deny any single predicate of any single subject [Badiou] |
9814 | All great poetry is engaged in rivalry with mathematics [Badiou] |
12316 | For Enlightenment philosophers, God was no longer involved in politics [Badiou] |
12317 | The God of religion results from an encounter, not from a proof [Badiou] |