84 ideas
| 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] |
| 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] |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 4229 | An infinite series of tasks can't be completed because it has no last member [Lowe] |
| 4240 | It might be argued that mathematics does not, or should not, aim at truth [Lowe] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 4241 | If there are infinite numbers and finite concrete objects, this implies that numbers are abstract objects [Lowe] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 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] |
| 4220 | Maybe an event is the exemplification of a property at a time [Lowe] |
| 4225 | Events are changes in the properties of or relations between things [Lowe] |
| 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] |
| 4200 | If old parts are stored and then appropriated, they are no longer part of the original (which is the renovated ship). [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] |
| 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] |
| 2986 | Belief is the most important propositional attitude [Lyons] |
| 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] |
| 2978 | Consciousness no longer seems essential to intentionality [Lyons] |
| 2984 | Perceptions could give us information without symbolic representation [Lyons] |
| 2979 | Propositional attitudes require representation [Lyons] |
| 2987 | Folk psychology works badly for alien cultures [Lyons] |
| 2977 | All thinking has content [Lyons] |
| 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] |
| 4210 | If the concept of a cause says it precedes its effect, that rules out backward causation by definition [Lowe] |
| 4215 | It seems proper to say that only substances (rather than events) have causal powers [Lowe] |
| 4209 | The theories of fact causation and event causation are both worth serious consideration [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] |