69 ideas
| 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] |
| 22317 | Truth does not admit of more and less [Frege] |
| 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] |
| 13455 | Frege did not think of himself as working with sets [Frege, by Hart,WD] |
| 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] |
| 16895 | The null set is indefensible, because it collects nothing [Frege, by Burge] |
| 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] |
| 3328 | Frege proposed a realist concept of a set, as the extension of a predicate or concept or function [Frege, by Benardete,JA] |
| 9179 | Frege frequently expressed a contempt for language [Frege, by Dummett] |
| 13473 | Frege thinks there is an independent logical order of the truths, which we must try to discover [Frege, by Hart,WD] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 6076 | For Frege, predicates are names of functions that map objects onto the True and False [Frege, by McGinn] |
| 3319 | Frege gives a functional account of predication so that we can dispense with predicates [Frege, by Benardete,JA] |
| 9871 | Frege always, and fatally, neglected the domain of quantification [Dummett on Frege] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 16884 | Basic truths of logic are not proved, but seen as true when they are understood [Frege, by Burge] |
| 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] |
| 3331 | If '5' is the set of all sets with five members, that may be circular, and you can know a priori if the set has content [Benardete,JA on Frege] |
| 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] |
| 16880 | Frege aimed to discover the logical foundations which justify arithmetical judgements [Frege, by Burge] |
| 8689 | Eventually Frege tried to found arithmetic in geometry instead of in logic [Frege, by 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] |
| 5657 | Frege's logic showed that there is no concept of being [Frege, by Scruton] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 3318 | Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x' [Frege, by Benardete,JA] |
| 16885 | To understand a thought, understand its inferential connections to other thoughts [Frege, by Burge] |
| 16887 | Frege's concept of 'self-evident' makes no reference to minds [Frege, by Burge] |
| 16894 | An apriori truth is grounded in generality, which is universal quantification [Frege, by Burge] |
| 21686 | Sense-data are dubious abstractions, with none of the plausibility of tables [Quine] |
| 21685 | Empiricism says evidence rests on the senses, but that insight is derived from science [Quine] |
| 16882 | The building blocks contain the whole contents of a discipline [Frege] |
| 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] |
| 5816 | Frege said concepts were abstract entities, not mental entities [Frege, by Putnam] |
| 7307 | A thought is not psychological, but a condition of the world that makes a sentence true [Frege, by Miller,A] |
| 7309 | Frege's 'sense' is the strict and literal meaning, stripped of tone [Frege, by Miller,A] |
| 7312 | 'Sense' solves the problems of bearerless names, substitution in beliefs, and informativeness [Frege, by Miller,A] |
| 7725 | 'P or not-p' seems to be analytic, but does not fit Kant's account, lacking clear subject or predicate [Frege, by Weiner] |
| 7316 | Analytic truths are those that can be demonstrated using only logic and definitions [Frege, by Miller,A] |
| 3307 | Frege put forward an ontological argument for the existence of numbers [Frege, by Benardete,JA] |