70 ideas
| 15477 | Ontology is highly abstract physics, containing placeholders and exclusions [Martin,CB] |
| 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] |
| 15471 | Truth is a relation between a representation ('bearer') and part of the world ('truthmaker') [Martin,CB] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 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] |
| 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] |
| 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] |
| 15484 | A property is a combination of a disposition and a quality [Martin,CB] |
| 15478 | Properties are the respects in which objects resemble, which places them in classes [Martin,CB] |
| 15483 | Properties are ways particular things are, and so they are tied to the identity of their possessor [Martin,CB] |
| 15480 | Objects are not bundles of tropes (which are ways things are, not parts of things) [Martin,CB] |
| 15489 | A property that cannot interact is worse than inert - it isn't there at all [Martin,CB] |
| 15487 | If unmanifested partnerless dispositions are still real, and are not just qualities, they can explain properties [Martin,CB] |
| 15488 | Qualities and dispositions are aspects of properties - what it exhibits, and what it does [Martin,CB] |
| 15479 | Properties endow a ball with qualities, and with powers or dispositions [Martin,CB] |
| 15469 | Dispositions in action can be destroyed, be recovered, or remain unchanged [Martin,CB] |
| 15467 | Powers depend on circumstances, so can't be given a conditional analysis [Martin,CB] |
| 15466 | 'The wire is live' can't be analysed as a conditional, because a wire can change its powers [Martin,CB] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 15476 | Structural properties involve dispositionality, so cannot be used to explain it [Martin,CB] |
| 15465 | Structures don't explain dispositions, because they consist of dispositions [Martin,CB] |
| 15481 | I favour the idea of a substratum for properties; spacetime seems to be just a bearer of properties [Martin,CB] |
| 15474 | Properly understood, wholes do no more causal work than their parts [Martin,CB] |
| 15486 | Only abstract things can have specific and full identity specifications [Martin,CB] |
| 15475 | The concept of 'identity' must allow for some changes in properties or parts [Martin,CB] |
| 15472 | It is pointless to say possible worlds are truthmakers, and then deny that possible worlds exist [Martin,CB] |
| 15492 | Explanations are mind-dependent, theory-laden, and interest-relative [Martin,CB] |
| 15495 | Analogy works, as when we eat food which others seem to be relishing [Martin,CB] |
| 15493 | Memory requires abstraction, as reminders of what cannot be fully remembered [Martin,CB] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| 15485 | Instead of a cause followed by an effect, we have dispositions in reciprocal manifestation [Martin,CB] |
| 15491 | Causation should be explained in terms of dispositions and manifestations [Martin,CB] |
| 15468 | Causal counterfactuals are just clumsy linguistic attempts to indicate dispositions [Martin,CB] |
| 15470 | Causal laws are summaries of powers [Martin,CB] |
| 15482 | We can't think of space-time as empty and propertyless, and it seems to be a substratum [Martin,CB] |