96 ideas
| 24047 | An account is either a definition or a demonstration [Aristotle] |
| 24052 | From one thing alone we can infer its contrary [Aristotle] |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| 1729 | We perceive number by the denial of continuity [Aristotle] |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| 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] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 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] |
| 1725 | Why can't we sense the senses? And why do senses need stimuli? [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] |
| 16723 | Perception of sensible objects is virtually never wrong [Aristotle] |
| 1724 | Perception necessitates pleasure and pain, which necessitates appetite [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] |
| 24048 | Demonstrations move from starting-points to deduced conclusions [Aristotle] |
| 16647 | Demonstration starts from a definition of essence, so we can derive (or conjecture about) the properties [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] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 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] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 24060 | Self-controlled follow understanding, when it is opposed to desires [Aristotle] |
| 22489 | 'Good' is an attributive adjective like 'large', not predicative like 'red' [Geach, by Foot] |
| 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] |
| 24044 | Movement can be intrinsic (like a ship) or relative (like its sailors) [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] |
| 24064 | If something is pushed, it pushes back [Aristotle] |
| 24063 | What is born has growth, a prime, and a withering away [Aristotle] |