71 ideas
1627 | Any statement can be held true if we make enough adjustment to the rest of the system [Quine] |
1623 | Definition rests on synonymy, rather than explaining it [Quine] |
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] |
9204 | Quine's arguments fail because he naively conflates names with descriptions [Fine,K on Quine] |
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] |
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] |
17738 | Quine blurs the difference between knowledge of arithmetic and of physics [Jenkins on Quine] |
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] |
19492 | Quine is hopeless circular, deriving ontology from what is literal, and 'literal' from good ontology [Yablo on Quine] |
1628 | If physical objects are a myth, they are useful for making sense of experience [Quine] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
10929 | Aristotelian essence of the object has become the modern essence of meaning [Quine] |
12188 | Contrary to some claims, Quine does not deny logical necessity [Quine, by McFetridge] |
15090 | Quine's attack on the analytic-synthetic distinction undermined necessary truths [Quine, by Shoemaker] |
19440 | How do you know you have conceived a thing deeply enough to assess its possibility? [Vaidya] |
9383 | Metaphysical analyticity (and linguistic necessity) are hopeless, but epistemic analyticity is a priori [Boghossian on Quine] |
12424 | Quine challenges the claim that analytic truths are knowable a priori [Quine, by Kitcher] |
9338 | Quine's objections to a priori knowledge only work in the domain of science [Horwich on Quine] |
9337 | Science is empirical, simple and conservative; any belief can hence be abandoned; so no a priori [Quine, by Horwich] |
9340 | Logic, arithmetic and geometry are revisable and a posteriori; quantum logic could be right [Horwich on Quine] |
1620 | Empiricism makes a basic distinction between truths based or not based on facts [Quine] |
1629 | Our outer beliefs must match experience, and our inner ones must be simple [Quine] |
19488 | The second dogma is linking every statement to some determinate observations [Quine, by Yablo] |
1625 | Statements about the external world face the tribunal of sense experience as a corporate body [Quine] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
1626 | It is troublesome nonsense to split statements into a linguistic and a factual component [Quine] |
7317 | 'Renate' and 'cordate' have identical extensions, but are not synonymous [Quine, by Miller,A] |
1621 | Once meaning and reference are separated, meaning ceases to seem important [Quine] |
9371 | Analytic statements are either logical truths (all reinterpretations) or they depend on synonymy [Quine] |
1622 | Did someone ever actually define 'bachelor' as 'unmarried man'? [Quine] |
9366 | Quine's attack on analyticity undermined linguistic views of necessity, and analytic views of the a priori [Quine, by Boghossian] |
14473 | Quine attacks the Fregean idea that we can define analyticity through synonyous substitution [Quine, by Thomasson] |
7321 | The last two parts of 'Two Dogmas' are much the best [Miller,A on Quine] |
8803 | Erasing the analytic/synthetic distinction got rid of meanings, and saved philosophy of language [Davidson on Quine] |
17737 | The analytic needs excessively small units of meaning and empirical confirmation [Quine, by Jenkins] |
1624 | If we try to define analyticity by synonymy, that leads back to analyticity [Quine] |