48 ideas
| 4739 | In "if and only if" (iff), "if" expresses the sufficient condition, and "only if" the necessary condition [Engel] |
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| 4737 | Are truth-bearers propositions, or ideas/beliefs, or sentences/utterances? [Engel] |
| 4750 | The redundancy theory gets rid of facts, for 'it is a fact that p' just means 'p' [Engel] |
| 4744 | We can't explain the corresponding structure of the world except by referring to our thoughts [Engel] |
| 4738 | The coherence theory says truth is an internal relationship between groups of truth-bearers [Engel] |
| 4745 | Any coherent set of beliefs can be made more coherent by adding some false beliefs [Engel] |
| 4753 | Deflationism seems to block philosophers' main occupation, asking metatheoretical questions [Engel] |
| 4755 | Deflationism cannot explain why we hold beliefs for reasons [Engel] |
| 4751 | Maybe there is no more to be said about 'true' than there is about the function of 'and' in logic [Engel] |
| 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] |
| 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] |
| 4752 | Deflationism must reduce bivalence ('p is true or false') to excluded middle ('p or not-p') [Engel] |
| 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] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 4762 | The Humean theory of motivation is that beliefs may be motivators as well as desires [Engel] |
| 4754 | Our beliefs are meant to fit the world (i.e. be true), where we want the world to fit our desires [Engel] |
| 4763 | 'Evidentialists' say, and 'voluntarists' deny, that we only believe on the basis of evidence [Engel] |
| 4746 | Pragmatism is better understood as a theory of belief than as a theory of truth [Engel] |
| 4764 | We cannot directly control our beliefs, but we can control the causes of our involuntary beliefs [Engel] |
| 4759 | Mental states as functions are second-order properties, realised by first-order physical properties [Engel] |
| 16764 | The soul conserves the body, as we see by its dissolution when the soul leaves [Toletus] |