39 ideas
5896 | Speak the truth, for this alone deifies man [Pythagoras, by Porphyry] |
3051 | Pythagoras discovered the numerical relation of sounds on a string [Pythagoras, by Diog. Laertius] |
1765 | Diogenes said avoidance of philosophy is the lack of a desire to live properly [Diogenes of Sin., by Diog. Laertius] |
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] |
7485 | For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
1762 | When someone denied motion, Diogenes got up and walked away [Diogenes of Sin., by Diog. Laertius] |
7813 | Cynicism was open to anyone, and needed neither education nor sophistication [Diogenes of Sin., by Grayling] |
1763 | Diogenes said a plucked chicken fits Plato's definition of man [Diogenes of Sin., by Diog. Laertius] |
5071 | The Cynics rejected what is conventional as irrational, and aimed to live by nature [Taylor,R on Diogenes of Sin.] |
3053 | Pythagoras taught that virtue is harmony, and health, and universal good, and God [Pythagoras, by Diog. Laertius] |
7812 | For peace of mind, you need self-government, indifference and independence [Diogenes of Sin.] |
5244 | For Pythagoreans, justice is simply treating all people the same [Pythagoras, by Aristotle] |
1764 | Diogenes said he was a citizen of the world [Diogenes of Sin., by Diog. Laertius] |
5968 | Diogenes masturbated in public, wishing he could get rid of hunger so easily [Diogenes of Sin., by Plutarch] |
1766 | Diogenes said that the most excellent thing among men was freedom of speech [Diogenes of Sin., by Diog. Laertius] |
375 | When musical harmony and rhythm were discovered, similar features were seen in bodily movement [Pythagoras, by Plato] |
638 | Pythagoreans define timeliness, justice and marriage in terms of numbers [Pythagoras, by Aristotle] |
553 | Pythagoreans think mathematical principles are the principles of all of nature [Pythagoras, by Aristotle] |
554 | Pythagoreans say things imitate numbers, but Plato says things participate in numbers [Pythagoras, by Aristotle] |
644 | For Pythagoreans the entire universe is made of numbers [Pythagoras, by Aristotle] |
7467 | The modern idea of an immortal soul was largely created by Pythagoras [Pythagoras, by Watson] |