55 ideas
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| 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] |
| 16643 | Accidents always remain suited to a subject [Bonaventura] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 16696 | Successive things reduce to permanent things [Bonaventura] |
| 6346 | The main epistemological theories are foundationalist, coherence, probabilistic and reliabilist [Pollock/Cruz] |
| 6351 | Most people now agree that our reasoning proceeds defeasibly, rather than deductively [Pollock/Cruz] |
| 6374 | To believe maximum truths, believe everything; to have infallible beliefs, believe nothing [Pollock/Cruz] |
| 6355 | Direct realism says justification is partly a function of pure perceptual states, not of beliefs [Pollock/Cruz] |
| 6359 | Phenomenalism offered conclusive perceptual knowledge, but conclusive reasons no longer seem essential [Pollock/Cruz] |
| 6366 | Perception causes beliefs in us, without inference or justification [Pollock/Cruz] |
| 6362 | Sense evidence is not beliefs, because they are about objective properties, not about appearances [Pollock/Cruz] |
| 6371 | Bayesian epistemology is Bayes' Theorem plus the 'simple rule' (believe P if it is probable) [Pollock/Cruz] |
| 6373 | Internalism says if anything external varies, the justifiability of the belief does not vary [Pollock/Cruz] |
| 6353 | People rarely have any basic beliefs, and never enough for good foundations [Pollock/Cruz] |
| 6361 | Foundationalism requires self-justification, not incorrigibility [Pollock/Cruz] |
| 6357 | Reason cannot be an ultimate foundation, because rational justification requires prior beliefs [Pollock/Cruz] |
| 6363 | Foundationalism is wrong, because either all beliefs are prima facie justified, or none are [Pollock/Cruz] |
| 6365 | Negative coherence theories do not require reasons, so have no regress problem [Pollock/Cruz] |
| 6354 | Coherence theories fail, because they can't accommodate perception as the basis of knowledge [Pollock/Cruz] |
| 6367 | Coherence theories isolate justification from the world [Pollock/Cruz] |
| 6370 | Externalism comes as 'probabilism' (probability of truth) and 'reliabilism' (probability of good cognitive process) [Pollock/Cruz] |
| 6358 | One belief may cause another, without being the basis for the second belief [Pollock/Cruz] |
| 6364 | We can't start our beliefs from scratch, because we wouldn't know where to start [Pollock/Cruz] |
| 6352 | Enumerative induction gives a universal judgement, while statistical induction gives a proportion [Pollock/Cruz] |
| 6372 | Since every tautology has a probability of 1, should we believe all tautologies? [Pollock/Cruz] |
| 6360 | Scientific confirmation is best viewed as inference to the best explanation [Pollock/Cruz] |