79 ideas
13939 | No possible evidence could decide the reality of numbers, so it is a pseudo-question [Carnap] |
16252 | Metaphysics uses empty words, or just produces pseudo-statements [Carnap] |
18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
17824 | The master science is physical objects divided into sets [Maddy] |
8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
10594 | Henkin semantics is more plausible for plural logic than for second-order logic [Maddy] |
13342 | Carnap defined consequence by contradiction, but this is unintuitive and changes with substitution [Tarski on Carnap] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
13251 | Each person is free to build their own logic, just by specifying a syntax [Carnap] |
18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
18182 | The extension of concepts is not important to me [Maddy] |
18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
10718 | A natural number is a property of sets [Maddy, by Oliver] |
18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
13936 | Questions about numbers are answered by analysis, and are analytic, and hence logically true [Carnap] |
8748 | Logical positivists incorporated geometry into logicism, saying axioms are just definitions [Carnap, by Shapiro] |
18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
8960 | Internal questions about abstractions are trivial, and external ones deeply problematic [Carnap, by Szabó] |
13933 | Existence questions are 'internal' (within a framework) or 'external' (concerning the whole framework) [Carnap] |
13934 | To be 'real' is to be an element of a system, so we cannot ask reality questions about the system itself [Carnap] |
13938 | A linguistic framework involves commitment to entities, so only commitment to the framework is in question [Carnap] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
18435 | Resemblance Nominalists say that resemblance explains properties (not the other way round) [Rodriquez-Pereyra] |
18436 | Entities are truthmakers for their resemblances, so no extra entities or 'resemblances' are needed [Rodriquez-Pereyra] |
13935 | We only accept 'things' within a language with formation, testing and acceptance rules [Carnap] |
14305 | In the truth-functional account a burnt-up match was soluble because it never entered water [Carnap] |
13932 | Empiricists tend to reject abstract entities, and to feel sympathy with nominalism [Carnap] |
13937 | New linguistic claims about entities are not true or false, but just expedient, fruitful or successful [Carnap] |
18699 | Carnap tried to define all scientific predicates in terms of primitive relations, using type theory [Carnap, by Button] |
13940 | All linguistic forms in science are merely judged by their efficiency as instruments [Carnap] |
13048 | Good explications are exact, fruitful, simple and similar to the explicandum [Carnap, by Salmon] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
12131 | All concepts can be derived from a few basics, making possible one science of everything [Carnap, by Brody] |
11968 | The intension of a sentence is the set of all possible worlds in which it is true [Carnap, by Kaplan] |
18285 | All translation loses some content (but language does not create reality) [Carnap] |