54 ideas
6575 | Philosophy may never find foundations, and may undermine our lives in the process [Fogelin] |
6585 | Rationality is threatened by fear of inconsistency, illusions of absolutes or relativism, and doubt [Fogelin] |
6557 | Humans may never be able to attain a world view which is both rich and consistent [Fogelin] |
6568 | A game can be played, despite having inconsistent rules [Fogelin] |
6560 | The law of noncontradiction is traditionally the most basic principle of rationality [Fogelin] |
6565 | The law of noncontradiction makes the distinction between asserting something and denying it [Fogelin] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
6574 | Legal reasoning is analogical, not deductive [Fogelin] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [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] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
6582 | Conventions can only work if they are based on something non-conventional [Fogelin] |
6576 | My view is 'circumspect rationalism' - that only our intellect can comprehend the world [Fogelin] |
6589 | Knowledge is legitimate only if all relevant defeaters have been eliminated [Fogelin] |
6596 | For coherentists, circularity is acceptable if the circle is large, rich and coherent [Fogelin] |
6597 | A rule of justification might be: don't raise the level of scrutiny without a good reason [Fogelin] |
6588 | Scepticism is cartesian (sceptical scenarios), or Humean (future), or Pyrrhonian (suspend belief) [Fogelin] |
6590 | Scepticism deals in remote possibilities that are ineliminable and set the standard very high [Fogelin] |
6583 | Radical perspectivism replaces Kant's necessary scheme with many different schemes [Fogelin] |
6555 | We are also irrational, with a unique ability to believe in bizarre self-created fictions [Fogelin] |
6605 | Critics must be causally entangled with their subject matter [Fogelin] |
6607 | The word 'beautiful', when deprived of context, is nearly contentless [Fogelin] |
6604 | Saying 'It's all a matter to taste' ignores the properties of the object discussed [Fogelin] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
6586 | Cynics are committed to morality, but disappointed or disgusted by human failings [Fogelin] |
6572 | Deterrence, prevention, rehabilitation and retribution can come into conflict in punishments [Fogelin] |
6573 | Retributivists say a crime can be 'paid for'; deterrentists still worry about potential victims [Fogelin] |