63 ideas
| 9136 | The paradox of analysis says that any conceptual analysis must be either trivial or false [Sorensen] |
| 9131 | Two long understandable sentences can have an unintelligible conjunction [Sorensen] |
| 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] |
| 9139 | If nothing exists, no truthmakers could make 'Nothing exists' true [Sorensen] |
| 9140 | Which toothbrush is the truthmaker for 'buy one, get one free'? [Sorensen] |
| 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] |
| 9119 | No attempt to deny bivalence has ever been accepted [Sorensen] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 9135 | We now see that generalizations use variables rather than abstract entities [Sorensen] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 9125 | Denying problems, or being romantically defeated by them, won't make them go away [Sorensen] |
| 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] |
| 9137 | Banning self-reference would outlaw 'This very sentence is in English' [Sorensen] |
| 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] |
| 9116 | Vague words have hidden boundaries [Sorensen] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 9132 | An offer of 'free coffee or juice' could slowly shift from exclusive 'or' to inclusive 'or' [Sorensen] |
| 9128 | It is propositional attitudes which can be a priori, not the propositions themselves [Sorensen] |
| 9130 | Attributing apriority to a proposition is attributing a cognitive ability to someone [Sorensen] |
| 9118 | The colour bands of the spectrum arise from our biology; they do not exist in the physics [Sorensen] |
| 9124 | We are unable to perceive a nose (on the back of a mask) as concave [Sorensen] |
| 9126 | Bayesians build near-certainty from lots of reasonably probable beliefs [Sorensen] |
| 9121 | Illusions are not a reason for skepticism, but a source of interesting scientific information [Sorensen] |
| 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] |
| 9134 | The negation of a meaningful sentence must itself be meaningful [Sorensen] |
| 9133 | Propositions are what settle problems of ambiguity in sentences [Sorensen] |
| 2854 | Prescriptivism says 'ought' without commitment to act is insincere, or weakly used [Hooker,B] |
| 2856 | Universal moral judgements imply the Golden Rule ('do as you would be done by') [Hooker,B] |
| 9129 | I can buy any litre of water, but not every litre of water [Sorensen] |
| 9122 | God cannot experience unwanted pain, so God cannot understand human beings [Sorensen] |