50 ideas
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
5515 | Imaginary cases are good for revealing our beliefs, rather than the truth [Parfit] |
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] |
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] |
5516 | Reduction can be by identity, or constitution, or elimination [Parfit, by PG] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
19542 | It is nonsense that understanding does not involve knowledge; to understand, you must know [Dougherty/Rysiew] |
19543 | To grasp understanding, we should be more explicit about what needs to be known [Dougherty/Rysiew] |
19541 | Rather than knowledge, our epistemic aim may be mere true belief, or else understanding and wisdom [Dougherty/Rysiew] |
19540 | Don't confuse justified belief with justified believers [Dougherty/Rysiew] |
19539 | If knowledge is unanalysable, that makes justification more important [Dougherty/Rysiew] |
3539 | Personal identity is just causally related mental states [Parfit, by Maslin] |
5514 | Psychologists are interested in identity as a type of person, but philosophers study numerical identity [Parfit] |
1393 | One of my future selves will not necessarily be me [Parfit] |
5521 | If my brain-halves are transplanted into two bodies, I have continuity, and don't need identity [Parfit] |
5522 | Over a period of time what matters is not that 'I' persist, but that I have psychological continuity [Parfit] |
1392 | If we split like amoeba, we would be two people, neither of them being us [Parfit] |
5519 | It is fine to save two dying twins by merging parts of their bodies into one, and identity is irrelevant [Parfit] |
5520 | If two humans are merged surgically, the new identity is a purely verbal problem [Parfit] |
1391 | Concern for our own lives isn't the source of belief in identity, it is the result of it [Parfit] |
5518 | It doesn't matter whether I exist with half my components replaced (any more than an audio system) [Parfit] |
19538 | Entailment is modelled in formal semantics as set inclusion (where 'mammals' contains 'cats') [Dougherty/Rysiew] |
9762 | We should focus less on subjects of experience, and more on the experiences themselves [Parfit] |