118 ideas
| 19275 | You cannot understand what exists without understanding possibility and necessity [Hale] |
| 10308 | Questions about objects are questions about certain non-vacuous singular terms [Hale] |
| 19291 | A canonical defintion specifies the type of thing, and what distinguish this specimen [Hale] |
| 10314 | An expression is a genuine singular term if it resists elimination by paraphrase [Hale] |
| 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] |
| 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] |
| 19296 | If second-order variables range over sets, those are just objects; properties and relations aren't sets [Hale] |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| 19289 | Maybe conventionalism applies to meaning, but not to the truth of propositions expressed [Hale] |
| 18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
| 10316 | We should decide whether singular terms are genuine by their usage [Hale] |
| 10312 | Often the same singular term does not ensure reliable inference [Hale] |
| 10313 | Plenty of clear examples have singular terms with no ontological commitment [Hale] |
| 10322 | If singular terms can't be language-neutral, then we face a relativity about their objects [Hale] |
| 19298 | Unlike axiom proofs, natural deduction proofs needn't focus on logical truths and theorems [Hale] |
| 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] |
| 10632 | The real numbers may be introduced by abstraction as ratios of quantities [Hale, by Hale/Wright] |
| 18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [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] |
| 19295 | Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers [Hale] |
| 19281 | Interesting supervenience must characterise the base quite differently from what supervenes on it [Hale] |
| 10512 | The abstract/concrete distinction is based on what is perceivable, causal and located [Hale] |
| 10517 | Colours and points seem to be both concrete and abstract [Hale] |
| 10519 | The abstract/concrete distinction is in the relations in the identity-criteria of object-names [Hale] |
| 10520 | Token-letters and token-words are concrete objects, type-letters and type-words abstract [Hale] |
| 10524 | There is a hierarchy of abstraction, based on steps taken by equivalence relations [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] |
| 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
| 10521 | If F can't have location, there is no problem of things having F in different locations [Hale] |
| 10511 | It is doubtful if one entity, a universal, can be picked out by both predicates and abstract nouns [Hale] |
| 10318 | Realists take universals to be the referrents of both adjectives and of nouns [Hale] |
| 10310 | Objections to Frege: abstracta are unknowable, non-independent, unstatable, unindividuated [Hale] |
| 10518 | Shapes and directions are of something, but games and musical compositions are not [Hale] |
| 10513 | Many abstract objects, such as chess, seem non-spatial, but are not atemporal [Hale] |
| 10514 | If the mental is non-spatial but temporal, then it must be classified as abstract [Hale] |
| 10523 | Being abstract is based on a relation between things which are spatially separated [Hale] |
| 10307 | The modern Fregean use of the term 'object' is much broader than the ordinary usage [Hale] |
| 10315 | We can't believe in a 'whereabouts' because we ask 'what kind of object is it?' [Hale] |
| 19302 | If a chair could be made of slightly different material, that could lead to big changes [Hale] |
| 10522 | The relations featured in criteria of identity are always equivalence relations [Hale] |
| 10321 | We sometimes apply identity without having a real criterion [Hale] |
| 15086 | Absolute necessity might be achievable either logically or metaphysically [Hale] |
| 19290 | Absolute necessities are necessarily necessary [Hale] |
| 8261 | Maybe not-p is logically possible, but p is metaphysically necessary, so the latter is not absolute [Hale] |
| 15081 | A strong necessity entails a weaker one, but not conversely; possibilities go the other way [Hale] |
| 15080 | 'Relative' necessity is just a logical consequence of some statements ('strong' if they are all true) [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] |
| 15082 | Metaphysical necessity says there is no possibility of falsehood [Hale] |
| 15085 | 'Broadly' logical necessities are derived (in a structure) entirely from the concepts [Hale] |
| 15088 | Logical necessities are true in virtue of the nature of all logical concepts [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] |
| 12432 | Explanation of necessity must rest on something necessary or something contingent [Hale] |
| 12434 | Why is this necessary, and what is necessity in general; why is this necessary truth true, and why necessary? [Hale] |
| 12435 | The explanation of a necessity can be by a truth (which may only happen to be a necessary truth) [Hale] |
| 19282 | It seems that we cannot show that modal facts depend on non-modal facts [Hale] |
| 12433 | If necessity rests on linguistic conventions, those are contingent, so there is no necessity [Hale] |
| 15087 | Conceptual necessities are made true by all concepts [Hale] |
| 12436 | Concept-identities explain how we know necessities, not why they are necessary [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] |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| 19300 | The molecules may explain the water, but they are not what 'water' means [Hale] |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |