59 ideas
421 | Men who love wisdom must be inquirers into very many things indeed [Heraclitus] |
1491 | Everyone has the potential for self-knowledge and sound thinking [Heraclitus] |
5863 | Reason is eternal, but men are foolish [Heraclitus] |
414 | Logos is common to all, but most people live as if they have a private understanding [Heraclitus] |
416 | Beautiful harmony comes from things that are in opposition to one another [Heraclitus] |
425 | A thing can have opposing tensions but be in harmony, like a lyre [Heraclitus] |
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] |
1312 | If everything is and isn't then everything is true, and a midway between true and false makes everything false [Aristotle on Heraclitus] |
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] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
15658 | The hidden harmony is stronger than the visible [Heraclitus] |
13782 | Everything gives way, and nothing stands fast [Heraclitus] |
11853 | A mixed drink separates if it is not stirred [Heraclitus] |
427 | It is not possible to step twice into the same river [Heraclitus] |
11091 | You can bathe in the same river twice, but not in the same river stage [Quine on Heraclitus] |
2064 | If flux is continuous, then lack of change can't be a property, so everything changes in every possible way [Plato on Heraclitus] |
430 | Senses are no use if the soul is corrupt [Heraclitus] |
1500 | When we sleep, reason closes down as the senses do [Heraclitus, by Sext.Empiricus] |
417 | Donkeys prefer chaff to gold [Heraclitus] |
426 | Sea water is life-giving for fish, but not for people [Heraclitus] |
431 | Health, feeding and rest are only made good by disease, hunger and weariness [Heraclitus] |
1554 | Contradiction is impossible, since only one side of the argument refers to the true facts [Prodicus, by Didymus the Blind] |
429 | To God (though not to humans) all things are beautiful and good and just [Heraclitus] |
12294 | Good and evil are the same thing [Heraclitus, by Aristotle] |
419 | If one does not hope, one will not find the unhoped-for, since nothing leads to it [Heraclitus] |
415 | If happiness is bodily pleasure, then oxen are happy when they have vetch to eat [Heraclitus] |
5155 | It is hard to fight against emotion, but harder still to fight against pleasure [Heraclitus] |
433 | For man character is destiny [Heraclitus] |
422 | The people should fight for the law as if for their city-wall [Heraclitus] |
614 | Heraclitus said sometimes everything becomes fire [Heraclitus, by Aristotle] |
424 | Reason tells us that all things are one [Heraclitus] |
5096 | Heraclitus says that at some time everything becomes fire [Heraclitus, by Aristotle] |
17539 | The sayings of Heraclitus are still correct, if we replace 'fire' with 'energy' [Heraclitus, by Heisenberg] |
3054 | Heraclitus said fire could be transformed to create the other lower elements [Heraclitus, by Diog. Laertius] |
15660 | Logos is the source of everything, and my theories separate and explain each nature [Heraclitus] |
12269 | All things are in a state of motion [Heraclitus, by Aristotle] |
420 | The cosmos is eternal not created, and is an ever-living and changing fire [Heraclitus] |
1499 | Heraclitus says intelligence draws on divine reason [Heraclitus, by Sext.Empiricus] |
1555 | People used to think anything helpful to life was a god, as the Egyptians think the Nile a god [Prodicus] |
535 | The gods are just personified human benefits [Prodicus] |
1543 | He denied the existence of the gods, saying they are just exaltations of things useful for life [Prodicus] |
15659 | Purifying yourself with blood is as crazy as using mud to wash off mud [Heraclitus] |
1501 | In their ignorance people pray to statues, which is like talking to a house [Heraclitus] |