101 ideas
13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
13477 | The problems are the monuments of philosophy [Hart,WD] |
4194 | Metaphysics is concerned with the fundamental structure of reality as a whole [Lowe] |
4214 | Maybe such concepts as causation, identity and existence are primitive and irreducible [Lowe] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
4222 | If all that exists is what is being measured, what about the people and instruments doing the measuring? [Lowe] |
4217 | It is more extravagant, in general, to revise one's logic than to augment one's ontology [Lowe] |
13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
4229 | An infinite series of tasks can't be completed because it has no last member [Lowe] |
13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
4240 | It might be argued that mathematics does not, or should not, aim at truth [Lowe] |
13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
4241 | If there are infinite numbers and finite concrete objects, this implies that numbers are abstract objects [Lowe] |
13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
4239 | Nominalists deny abstract objects, because we can have no reason to believe in their existence [Lowe] |
4202 | Change can be of composition (the component parts), or quality (properties), or substance [Lowe] |
4201 | Four theories of qualitative change are 'a is F now', or 'a is F-at-t', or 'a-at-t is F', or 'a is-at-t F' [Lowe, by PG] |
4219 | Numerically distinct events of the same kind (like two battles) can coincide in space and time [Lowe] |
4221 | Maybe modern physics requires an event-ontology, rather than a thing-ontology [Lowe] |
4225 | Events are changes in the properties of or relations between things [Lowe] |
4220 | Maybe an event is the exemplification of a property at a time [Lowe] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
4196 | The main categories of existence are either universal and particular, or abstract and concrete [Lowe] |
4234 | Trope theory says blueness is a real feature of objects, but not the same as an identical blue found elsewhere [Lowe] |
4235 | Maybe a cushion is just a bundle of tropes, such as roundness, blueness and softness [Lowe] |
4236 | Tropes seem to be abstract entities, because they can't exist alone, but must come in bundles [Lowe] |
4197 | The category of universals can be sub-divided into properties and relations [Lowe] |
4232 | Nominalists believe that only particulars exist [Lowe] |
4205 | 'Is non-self-exemplifying' is a predicate which cannot denote a property (as it would be a contradiction) [Lowe] |
4233 | If 'blueness' is a set of particulars, there is danger of circularity, or using universals, in identifying the set [Lowe] |
4206 | Conventionalists see the world as an amorphous lump without identities, but are we part of the lump? [Lowe] |
4204 | Statues can't survive much change to their shape, unlike lumps of bronze, which must retain material [Lowe] |
4198 | If 5% replacement preserves a ship, we can replace 4% and 4% again, and still retain the ship [Lowe] |
4199 | A renovation or a reconstruction of an original ship would be accepted, as long as the other one didn't exist [Lowe] |
4200 | If old parts are stored and then appropriated, they are no longer part of the original (which is the renovated ship). [Lowe] |
4203 | Identity of Indiscernibles (same properties, same thing) ) is not Leibniz's Law (same thing, same properties) [Lowe] |
4195 | It is impossible to reach a valid false conclusion from true premises, so reason itself depends on possibility [Lowe] |
4207 | We might eliminate 'possible' and 'necessary' in favour of quantification over possible worlds [Lowe] |
13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
4223 | Unfalsifiability may be a failure in an empirical theory, but it is a virtue in metaphysics [Lowe] |
4193 | The behaviour of persons and social groups seems to need rational rather than causal explanation [Lowe] |
20429 | Most of us are too close to our own motives to understand them [Fry] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
4238 | The centre of mass of the solar system is a non-causal abstract object, despite having a location [Lowe] |
4237 | Concrete and abstract objects are distinct because the former have causal powers and relations [Lowe] |
20424 | Imaginative life requires no action, so new kinds of perception and values emerge in art [Fry] |
20427 | Everyone reveals an aesthetic attitude, looking at something which only exists to be seen [Fry] |
20433 | 'Beauty' can either mean sensuous charm, or the aesthetic approval of art (which may be ugly) [Fry] |
20430 | In life we neglect 'cosmic emotion', but it matters, and art brings it to the fore [Fry] |
20431 | Art needs a mixture of order and variety in its sensations [Fry] |
20423 | If graphic arts only aim at imitation, their works are only trivial ingenious toys [Fry] |
20428 | Popular opinion favours realism, yet most people never look closely at anything! [Fry] |
20432 | When viewing art, rather than flowers, we are aware of purpose, and sympathy with its creator [Fry] |
20425 | In the cinema the emotions are weaker, but much clearer than in ordinary life [Fry] |
20426 | For pure moralists art must promote right action, and not just be harmless [Fry] |
4210 | If the concept of a cause says it precedes its effect, that rules out backward causation by definition [Lowe] |
4209 | The theories of fact causation and event causation are both worth serious consideration [Lowe] |
4215 | It seems proper to say that only substances (rather than events) have causal powers [Lowe] |
4211 | Causal overdetermination is either actual overdetermination, or pre-emption, or the fail-safe case [Lowe] |
4213 | Causation may be instances of laws (seen either as constant conjunctions, or as necessities) [Lowe] |
4212 | Hume showed that causation could at most be natural necessity, never metaphysical necessity [Lowe] |
14581 | The normative view says laws show the natural behaviour of natural kind members [Lowe, by Mumford/Anjum] |
4208 | 'If he wasn't born he wouldn't have died' doesn't mean birth causes death, so causation isn't counterfactual [Lowe] |
4224 | If motion is change of distance between objects, it involves no intrinsic change in the objects [Lowe] |
4227 | Surfaces, lines and points are not, strictly speaking, parts of space, but 'limits', which are abstract [Lowe] |
4228 | If space is entirely relational, what makes a boundary, or a place unoccupied by physical objects? [Lowe] |