85 ideas
| 18495 | The best philosophers I know are the best people I know [Heil] |
| 18494 | Using a technical vocabulary actually prevents discussion of the presuppositions [Heil] |
| 18506 | Questions of explanation should not be confused with metaphyics [Heil] |
| 18535 | Without abstraction we couldn't think systematically [Heil] |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| 18534 | Truth relates truthbearers to truthmakers [Heil] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 18531 | Philosophers of the past took the truthmaking idea for granted [Heil] |
| 18509 | Not all truths need truthmakers - mathematics and logic seem to be just true [Heil] |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| 18518 | Infinite numbers are qualitatively different - they are not just very large numbers [Heil] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| 18500 | How could structures be mathematical truthmakers? Maths is just true, without truthmakers [Heil] |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| 18539 | Our categories lack the neat arrangement needed for reduction [Heil] |
| 18505 | Fundamental ontology aims at the preconditions for any true theory [Heil] |
| 18499 | Our quantifications only reveal the truths we accept; the ontology and truthmakers are another matter [Heil] |
| 18512 | Ontology aims to give the fundamental categories of being [Heil] |
| 18508 | Most philosophers now (absurdly) believe that relations fully exist [Heil] |
| 18532 | If causal relations are power manifestations, that makes them internal relations [Heil] |
| 18510 | We need properties to explain how the world works [Heil] |
| 18522 | Categorical properties were introduced by philosophers as actual properties, not if-then properties [Heil] |
| 18513 | Emergent properties will need emergent substances to bear them [Heil] |
| 18540 | Predicates only match properties at the level of fundamentals [Heil] |
| 18533 | In Fa, F may not be a property of a, but a determinable, satisfied by some determinate [Heil] |
| 18511 | Properties have causal roles which sets can't possibly have [Heil] |
| 18523 | Are all properties powers, or are there also qualities, or do qualities have the powers? [Heil] |
| 18524 | Properties are both qualitative and dispositional - they are powerful qualities [Heil] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 18498 | Abstract objects wouldn't be very popular without the implicit idea of truthmakers [Heil] |
| 18507 | Substances bear properties, so must be simple, and not consist of further substances [Heil] |
| 18515 | Spatial parts are just regions, but objects depend on and are made up of substantial parts [Heil] |
| 18516 | A 'gunky' universe would literally have no parts at all [Heil] |
| 18514 | Many wholes can survive replacement of their parts [Heil] |
| 18517 | Dunes depend on sand grains, but line segments depend on the whole line [Heil] |
| 18502 | If basic physics has natures, then why not reality itself? That would then found the deepest necessities [Heil] |
| 18496 | If possible worlds are just fictions, they can't be truthmakers for modal judgements [Heil] |
| 18525 | Mental abstraction does not make what is abstracted mind-dependent [Heil] |
| 18504 | Only particulars exist, and generality is our mode of presentation [Heil] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 18503 | You can think of tomatoes without grasping what they are [Heil] |
| 18537 | Linguistic thought is just as imagistic as non-linguistic thought [Heil] |
| 18538 | Non-conscious thought may be unlike conscious thought [Heil] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 18536 | The subject-predicate form reflects reality [Heil] |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| 18497 | Many reject 'moral realism' because they can't see any truthmakers for normative judgements [Heil] |
| 18519 | If there were infinite electrons, they could vanish without affecting total mass-energy [Heil] |
| 18526 | We should focus on actual causings, rather than on laws and causal sequences [Heil] |
| 18527 | Probabilistic causation is not a weak type of cause; it is just a probability of there being a cause [Heil] |
| 18520 | Electrons are treated as particles, but they lose their individuality in relations [Heil] |
| 18501 | Maybe the universe is fine-tuned because it had to be, despite plans by God or Nature? [Heil] |