62 ideas
8797 | The negation of all my beliefs about my current headache would be fully coherent [Sosa] |
8877 | We can't attain a coherent system by lopping off any beliefs that won't fit [Sosa] |
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] |
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] |
8884 | The phenomenal concept of an eleven-dot pattern does not include the concept of eleven [Sosa] |
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] |
8443 | Mereological essentialism says an entity must have exactly those parts [Sosa] |
8878 | It is acceptable to say a supermarket door 'knows' someone is approaching [Sosa] |
8880 | In reducing arithmetic to self-evident logic, logicism is in sympathy with rationalism [Sosa] |
8881 | Most of our knowledge has insufficient sensory support [Sosa] |
8794 | There are very few really obvious truths, and not much can be proved from them [Sosa] |
8882 | Perception may involve thin indexical concepts, or thicker perceptual concepts [Sosa] |
8883 | Do beliefs only become foundationally justified if we fully attend to features of our experience? [Sosa] |
8885 | Some features of a thought are known directly, but others must be inferred [Sosa] |
8876 | Much propositional knowledge cannot be formulated, as in recognising a face [Sosa] |
8796 | A single belief can trail two regresses, one terminating and one not [Sosa] |
8799 | If mental states are not propositional, they are logically dumb, and cannot be foundations [Sosa] |
8795 | Mental states cannot be foundational if they are not immune to error [Sosa] |
8879 | Fully comprehensive beliefs may not be knowledge [Sosa] |
8798 | Vision causes and justifies beliefs; but to some extent the cause is the justification [Sosa] |
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] |
6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
8442 | What law would explain causation in the case of causing a table to come into existence? [Sosa] |
8445 | The necessitated is not always a result or consequence of the necessitator [Sosa] |
8444 | Where is the necessary causation in the three people being tall making everybody tall? [Sosa] |