86 ideas
15477 | Ontology is highly abstract physics, containing placeholders and exclusions [Martin,CB] |
15471 | Truth is a relation between a representation ('bearer') and part of the world ('truthmaker') [Martin,CB] |
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] |
15484 | A property is a combination of a disposition and a quality [Martin,CB] |
15478 | Properties are the respects in which objects resemble, which places them in classes [Martin,CB] |
15483 | Properties are ways particular things are, and so they are tied to the identity of their possessor [Martin,CB] |
15480 | Objects are not bundles of tropes (which are ways things are, not parts of things) [Martin,CB] |
15489 | A property that cannot interact is worse than inert - it isn't there at all [Martin,CB] |
15487 | If unmanifested partnerless dispositions are still real, and are not just qualities, they can explain properties [Martin,CB] |
15479 | Properties endow a ball with qualities, and with powers or dispositions [Martin,CB] |
15488 | Qualities and dispositions are aspects of properties - what it exhibits, and what it does [Martin,CB] |
15469 | Dispositions in action can be destroyed, be recovered, or remain unchanged [Martin,CB] |
15467 | Powers depend on circumstances, so can't be given a conditional analysis [Martin,CB] |
15466 | 'The wire is live' can't be analysed as a conditional, because a wire can change its powers [Martin,CB] |
15465 | Structures don't explain dispositions, because they consist of dispositions [Martin,CB] |
15476 | Structural properties involve dispositionality, so cannot be used to explain it [Martin,CB] |
15481 | I favour the idea of a substratum for properties; spacetime seems to be just a bearer of properties [Martin,CB] |
15474 | Properly understood, wholes do no more causal work than their parts [Martin,CB] |
15486 | Only abstract things can have specific and full identity specifications [Martin,CB] |
15475 | The concept of 'identity' must allow for some changes in properties or parts [Martin,CB] |
15472 | It is pointless to say possible worlds are truthmakers, and then deny that possible worlds exist [Martin,CB] |
15492 | Explanations are mind-dependent, theory-laden, and interest-relative [Martin,CB] |
15495 | Analogy works, as when we eat food which others seem to be relishing [Martin,CB] |
15493 | Memory requires abstraction, as reminders of what cannot be fully remembered [Martin,CB] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
15485 | Instead of a cause followed by an effect, we have dispositions in reciprocal manifestation [Martin,CB] |
15491 | Causation should be explained in terms of dispositions and manifestations [Martin,CB] |
15468 | Causal counterfactuals are just clumsy linguistic attempts to indicate dispositions [Martin,CB] |
15470 | Causal laws are summaries of powers [Martin,CB] |
15482 | We can't think of space-time as empty and propertyless, and it seems to be a substratum [Martin,CB] |