82 ideas
6627 | Radical pragmatists abandon the notion of truth [Stich, by Lowe] |
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] |
13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
13014 | Extensional sets are clearer, simpler, unique and expressive [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] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
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] |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [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] |
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] |
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] |
18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
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] |
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] |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
17827 | Sets exist where their elements are, but numbers are more like universals [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] |
18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
9240 | Love creates a necessity concerning what to care about [Frankfurt] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
9264 | Persons are distinguished by a capacity for second-order desires [Frankfurt] |
9266 | A person essentially has second-order volitions, and not just second-order desires [Frankfurt] |
9267 | Free will is the capacity to choose what sort of will you have [Frankfurt] |
4765 | Stich accepts eliminativism (labelled 'pragmatism') about rationality and normativity [Stich, by Engel] |
9265 | The will is the effective desire which actually leads to an action [Frankfurt] |
20015 | Freedom of action needs the agent to identify with their reason for acting [Frankfurt, by Wilson/Schpall] |
9228 | Ranking order of desires reveals nothing, because none of them may be considered important [Frankfurt] |
9270 | A 'wanton' is not a person, because they lack second-order volitions [Frankfurt] |
9269 | A person may be morally responsible without free will [Frankfurt] |
9238 | Morality isn't based on reason; moral indignation is quite unlike disapproval of irrationality [Frankfurt] |
9234 | If you don't care about at least one thing, you can't find reasons to care about anything [Frankfurt] |
9232 | It is by caring about things that we infuse the world with importance [Frankfurt] |
9233 | Our criteria for evaluating how to live offer an answer to the problem [Frankfurt] |
9229 | What is worthwhile for its own sake alone may be worth very little [Frankfurt] |
9235 | Rather than loving things because we value them, I think we value things because we love them [Frankfurt] |
9237 | The paradigm case of pure love is not romantic, but that between parents and infants [Frankfurt] |
9239 | I value my children for their sake, but I also value my love for them for its own sake [Frankfurt] |
9236 | Love can be cool, and it may not involve liking its object [Frankfurt] |
9227 | We might not choose a very moral life, if the character or constitution was deficient [Frankfurt] |
9230 | People want to fulfill their desires, but also for their desires to be sustained [Frankfurt] |
9241 | Loving oneself is not a failing, but is essential to a successful life [Frankfurt] |
9300 | Boredom is serious, not just uncomfortable; it threatens our psychic survival [Frankfurt] |
9231 | Freedom needs autonomy (rather than causal independence) - embracing our own desires and choices [Frankfurt] |