85 ideas
| 3695 | Philosophy is a priori if it is anything [Bonjour] |
| 3651 | Perceiving necessary connections is the essence of reasoning [Bonjour] |
| 3700 | Coherence can't be validated by appeal to coherence [Bonjour] |
| 8893 | For any given area, there seem to be a huge number of possible coherent systems of beliefs [Bonjour] |
| 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] |
| 4261 | The Lottery Paradox says each ticket is likely to lose, so there probably won't be a winner [Bonjour, by PG] |
| 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] |
| 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] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 3697 | The concept of possibility is prior to that of necessity [Bonjour] |
| 8888 | The concept of knowledge is so confused that it is best avoided [Bonjour] |
| 8887 | It is hard to give the concept of 'self-evident' a clear and defensible characterization [Bonjour] |
| 8897 | The adverbial account will still be needed when a mind apprehends its sense-data [Bonjour] |
| 3704 | Moderate rationalists believe in fallible a priori justification [Bonjour] |
| 3707 | Our rules of thought can only be judged by pure rational insight [Bonjour] |
| 4255 | Externalist theories of knowledge are one species of foundationalism [Bonjour] |
| 4257 | The big problem for foundationalism is to explain how basic beliefs are possible [Bonjour] |
| 8896 | Conscious states have built-in awareness of content, so we know if a conceptual description of it is correct [Bonjour] |
| 3696 | A priori justification requires understanding but no experience [Bonjour] |
| 3703 | You can't explain away a priori justification as analyticity, and you can't totally give it up [Bonjour] |
| 3706 | A priori justification can vary in degree [Bonjour] |
| 4256 | The main argument for foundationalism is that all other theories involve a regress leading to scepticism [Bonjour] |
| 3699 | The induction problem blocks any attempted proof of physical statements [Bonjour] |
| 21506 | A coherence theory of justification can combine with a correspondence theory of truth [Bonjour] |
| 21509 | There will always be a vast number of equally coherent but rival systems [Bonjour] |
| 21503 | Empirical coherence must attribute reliability to spontaneous experience [Bonjour] |
| 21510 | The objection that a negated system is equally coherent assume that coherence is consistency [Bonjour] |
| 21511 | A well written novel cannot possibly match a real belief system for coherence [Bonjour] |
| 21505 | A coherent system can be justified with initial beliefs lacking all credibility [Bonjour] |
| 21504 | The best explanation of coherent observations is they are caused by and correspond to reality [Bonjour] |
| 8891 | My incoherent beliefs about art should not undermine my very coherent beliefs about physics [Bonjour] |
| 8892 | Coherence seems to justify empirical beliefs about externals when there is no external input [Bonjour] |
| 8894 | Coherentists must give a reason why coherent justification is likely to lead to the truth [Bonjour] |
| 4258 | Extreme externalism says no more justification is required than the truth of the belief [Bonjour] |
| 3701 | Externalist theories of justification don't require believers to have reasons for their beliefs [Bonjour] |
| 8889 | Reliabilists disagree over whether some further requirement is needed to produce knowledge [Bonjour] |
| 4259 | External reliability is not enough, if the internal state of the believer is known to be irrational [Bonjour] |
| 8890 | If the reliable facts producing a belief are unknown to me, my belief is not rational or responsible [Bonjour] |
| 4260 | Even if there is no obvious irrationality, it may be irrational to base knowledge entirely on external criteria [Bonjour] |
| 3702 | Externalism means we have no reason to believe, which is strong scepticism [Bonjour] |
| 21508 | Anomalies challenge the claim that the basic explanations are actually basic [Bonjour] |
| 3709 | Induction must go beyond the evidence, in order to explain why the evidence occurred [Bonjour] |
| 8895 | If neither the first-level nor the second-level is itself conscious, there seems to be no consciousness present [Bonjour] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 3708 | All thought represents either properties or indexicals [Bonjour] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 3698 | Indeterminacy of translation is actually indeterminacy of meaning and belief [Bonjour] |
| 20327 | Modern attention has moved from the intrinsic properties of art to its relational properties [Lamarque/Olson] |
| 20326 | Early 20th cent attempts at defining art focused on significant form, intuition, expression, unity [Lamarque/Olson] |
| 20330 | The dualistic view says works of art are either abstract objects (types), or physical objects [Lamarque/Olson] |