65 ideas
19275 | You cannot understand what exists without understanding possibility and necessity [Hale] |
19291 | A canonical defintion specifies the type of thing, and what distinguish this specimen [Hale] |
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] |
19297 | The two Barcan principles are easily proved in fairly basic modal logic [Hale] |
19301 | With a negative free logic, we can dispense with the Barcan formulae [Hale] |
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] |
19296 | If second-order variables range over sets, those are just objects; properties and relations aren't sets [Hale] |
19289 | Maybe conventionalism applies to meaning, but not to the truth of propositions expressed [Hale] |
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] |
19298 | Unlike axiom proofs, natural deduction proofs needn't focus on logical truths and theorems [Hale] |
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] |
19295 | Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers [Hale] |
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] |
19281 | Interesting supervenience must characterise the base quite differently from what supervenes on it [Hale] |
19278 | There is no gap between a fact that p, and it is true that p; so we only have the truth-condtions for p [Hale] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
19302 | If a chair could be made of slightly different material, that could lead to big changes [Hale] |
19290 | Absolute necessities are necessarily necessary [Hale] |
19286 | 'Absolute necessity' is when there is no restriction on the things which necessitate p [Hale] |
19288 | Logical and metaphysical necessities differ in their vocabulary, and their underlying entities [Hale] |
19285 | Logical necessity is something which is true, no matter what else is the case [Hale] |
19287 | Maybe each type of logic has its own necessity, gradually becoming broader [Hale] |
19282 | It seems that we cannot show that modal facts depend on non-modal facts [Hale] |
19276 | The big challenge for essentialist views of modality is things having necessary existence [Hale] |
19293 | Essentialism doesn't explain necessity reductively; it explains all necessities in terms of a few basic natures [Hale] |
19294 | If necessity derives from essences, how do we explain the necessary existence of essences? [Hale] |
19279 | What are these worlds, that being true in all of them makes something necessary? [Hale] |
19299 | Possible worlds make every proposition true or false, which endorses classical logic [Hale] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
19300 | The molecules may explain the water, but they are not what 'water' means [Hale] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |