74 ideas
24047 | An account is either a definition or a demonstration [Aristotle] |
24052 | From one thing alone we can infer its contrary [Aristotle] |
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] |
1729 | We perceive number by the denial of continuity [Aristotle] |
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] |
24057 | What is prior is always potentially present in what is next in order [Aristotle] |
16752 | Sight is the essence of the eye, fitting its definition; the eye itself is just the matter [Aristotle] |
24058 | The substance is the cause of a thing's being [Aristotle] |
24055 | Matter is potential, form is actual [Aristotle] |
24040 | Scientists explain anger by the matter, dialecticians by the form and the account [Aristotle] |
5051 | The intellect has potential to think, like a tablet on which nothing has yet been written [Aristotle] |
16723 | Perception of sensible objects is virtually never wrong [Aristotle] |
1724 | Perception necessitates pleasure and pain, which necessitates appetite [Aristotle] |
1730 | Why do we have many senses, and not just one? [Aristotle] |
17711 | Our minds take on the form of what is being perceived [Aristotle, by Mares] |
1725 | Why can't we sense the senses? And why do senses need stimuli? [Aristotle] |
1732 | Sense organs aren't the end of sensation, or they would know what does the sensing [Aristotle] |
1728 | Many objects of sensation are common to all the senses [Aristotle] |
1727 | Some objects of sensation are unique to one sense, where deception is impossible [Aristotle] |
1734 | In moral thought images are essential, to be pursued or avoided [Aristotle] |
1726 | We may think when we wish, but not perceive, because universals are within the mind [Aristotle] |
16647 | Demonstration starts from a definition of essence, so we can derive (or conjecture about) the properties [Aristotle] |
24048 | Demonstrations move from starting-points to deduced conclusions [Aristotle] |
16646 | To understand a triangle summing to two right angles, we need to know the essence of a line [Aristotle] |
1714 | Mind involves movement, perception, incorporeality [Aristotle] |
5507 | Aristotle led to the view that there are several souls, all somewhat physical [Aristotle, by Martin/Barresi] |
24051 | Soul is seen as what moves, or what is least physical, or a combination of elements [Aristotle] |
12086 | Psuché is the form and actuality of a body which potentially has life [Aristotle] |
16754 | The soul is the cause or source of movement, the essence of body, and its end [Aristotle] |
24046 | Understanding is impossible, if it involves the understanding having parts [Aristotle] |
1717 | If the soul is composed of many physical parts, it can't be a true unity [Aristotle] |
24053 | If a soul have parts, what unites them? [Aristotle] |
1721 | What unifies the soul would have to be a super-soul, which seems absurd [Aristotle] |
1735 | In a way the soul is everything which exists, through its perceptions and thoughts [Aristotle] |
24061 | If we divide the mind up according to its capacities, there are a lot of them [Aristotle] |
24062 | Self-moving animals must have desires, and that entails having imagination [Aristotle] |
1710 | Emotion involves the body, thinking uses the mind, imagination hovers between them [Aristotle] |
24039 | All the emotions seem to involve the body, simultaneously with the feeling [Aristotle] |
24056 | The soul (or parts of it) is not separable from the body [Aristotle] |
24050 | If soul is separate from body, why does it die when the body dies? [Aristotle] |
24049 | Thinkers place the soul within the body, but never explain how they are attached [Aristotle] |
1514 | Early thinkers concentrate on the soul but ignore the body, as if it didn't matter what body received the soul [Aristotle] |
2683 | Aristotle has a problem fitting his separate reason into the soul, which is said to be the form of the body [Ackrill on Aristotle] |
1718 | Does the mind think or pity, or does the whole man do these things? [Aristotle] |
13275 | The soul and the body are inseparable, like the imprint in some wax [Aristotle] |
1733 | Thinking is not perceiving, but takes the form of imagination and speculation [Aristotle] |
23307 | Aristotle makes belief a part of reason, but sees desires as separate [Aristotle, by Sorabji] |
17722 | The concept 'red' is tied to what actually individuates red things [Peacocke] |
24060 | Self-controlled follow understanding, when it is opposed to desires [Aristotle] |
4376 | Pleasure and pain are perceptions of things as good or bad [Aristotle] |
1740 | Nature does nothing in vain [Aristotle] |
1739 | If all movement is either pushing or pulling, there must be a still point in between where it all starts [Aristotle] |
24045 | Movement is spatial, alteration, withering or growth [Aristotle] |
1738 | Practical reason is based on desire, so desire must be the ultimate producer of movement [Aristotle] |
24044 | Movement can be intrinsic (like a ship) or relative (like its sailors) [Aristotle] |
24064 | If something is pushed, it pushes back [Aristotle] |
24063 | What is born has growth, a prime, and a withering away [Aristotle] |