60 ideas
11051 | Frege's logical approach dominates the analytical tradition [Hanna] |
11054 | Scientism says most knowledge comes from the exact sciences [Hanna] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
11070 | 'Denying the antecedent' fallacy: φ→ψ, ¬φ, so ¬ψ [Hanna] |
11071 | 'Affirming the consequent' fallacy: φ→ψ, ψ, so φ [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] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
11058 | Logic is explanatorily and ontologically dependent on rational animals [Hanna] |
11072 | Logic is personal and variable, but it has a universal core [Hanna] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
11061 | Intensional consequence is based on the content of the concepts [Hanna] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
11063 | Logicism struggles because there is no decent theory of analyticity [Hanna] |
11055 | Supervenience can add covariation, upward dependence, and nomological connection [Hanna] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
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] |
11082 | Should we take the 'depictivist' or the 'descriptivist/propositionalist' view of mental imagery? [Hanna] |
11067 | Rational animals have a normative concept of necessity [Hanna] |
11068 | One tradition says talking is the essence of rationality; the other says the essence is logic [Hanna] |
11047 | Hegelian holistic rationality is the capacity to seek coherence [Hanna] |
11048 | Humean Instrumental rationality is the capacity to seek contingent truths [Hanna] |
11046 | Kantian principled rationality is recognition of a priori universal truths [Hanna] |
11045 | Most psychologists are now cognitivists [Hanna] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |