86 ideas
11051 | Frege's logical approach dominates the analytical tradition [Hanna] |
11054 | Scientism says most knowledge comes from the exact sciences [Hanna] |
11071 | 'Affirming the consequent' fallacy: φ→ψ, ψ, so φ [Hanna] |
11070 | 'Denying the antecedent' fallacy: φ→ψ, ¬φ, so ¬ψ [Hanna] |
11088 | We can list at least fourteen informal fallacies [Hanna] |
11059 | Circular arguments are formally valid, though informally inadmissible [Hanna] |
11089 | Formally, composition and division fallacies occur in mereology [Hanna] |
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] |
11058 | Logic is explanatorily and ontologically dependent on rational animals [Hanna] |
11072 | Logic is personal and variable, but it has a universal core [Hanna] |
10594 | Henkin semantics is more plausible for plural logic than for second-order logic [Maddy] |
11061 | Intensional consequence is based on the content of the concepts [Hanna] |
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] |
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] |
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] |
11063 | Logicism struggles because there is no decent theory of analyticity [Hanna] |
11055 | Supervenience can add covariation, upward dependence, and nomological connection [Hanna] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
11083 | A sentence is necessary if it is true in a set of worlds, and nonfalse in the other worlds [Hanna] |
11086 | Metaphysical necessity can be 'weak' (same as logical) and 'strong' (based on essences) [Hanna] |
11084 | Logical necessity is truth in all logically possible worlds, because of laws and concepts [Hanna] |
11085 | Nomological necessity is truth in all logically possible worlds with our laws [Hanna] |
11077 | Intuition includes apriority, clarity, modality, authority, fallibility and no inferences [Hanna] |
11080 | Intuition is more like memory, imagination or understanding, than like perception [Hanna] |
11078 | Intuition is only outside the 'space of reasons' if all reasons are inferential [Hanna] |
11053 | Explanatory reduction is stronger than ontological reduction [Hanna] |
11081 | Imagination grasps abstracta, generates images, and has its own correctness conditions [Hanna] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
11082 | Should we take the 'depictivist' or the 'descriptivist/propositionalist' view of mental imagery? [Hanna] |
11046 | Kantian principled rationality is recognition of a priori universal truths [Hanna] |
11048 | Humean Instrumental rationality is the capacity to seek contingent truths [Hanna] |
11067 | Rational animals have a normative concept of necessity [Hanna] |
11047 | Hegelian holistic rationality is the capacity to seek coherence [Hanna] |
11068 | One tradition says talking is the essence of rationality; the other says the essence is logic [Hanna] |
11045 | Most psychologists are now cognitivists [Hanna] |
1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |