89 ideas
3240 | There is more insight in fundamental perplexity about problems than in their supposed solutions [Nagel] |
3242 | Philosophy is the childhood of the intellect, and a culture can't skip it [Nagel] |
3241 | It seems mad, but the aim of philosophy is to climb outside of our own minds [Nagel] |
3248 | Realism invites scepticism because it claims to be objective [Nagel] |
20989 | Views are objective if they don't rely on a person's character, social position or species [Nagel] |
22354 | Things cause perceptions, properties have other effects, hence we reach a 'view from nowhere' [Nagel, by Reiss/Sprenger] |
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
10206 | Modal operators are usually treated as quantifiers [Shapiro] |
10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
10207 | Anti-realists reject set theory [Shapiro] |
10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
10253 | Either logic determines objects, or objects determine logic, or they are separate [Shapiro] |
10251 | The law of excluded middle might be seen as a principle of omniscience [Shapiro] |
10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro] |
10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
10240 | Model theory deals with relations, reference and extensions [Shapiro] |
10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro] |
18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro] |
18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
10256 | For intuitionists, proof is inherently informal [Shapiro] |
10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
10218 | Baseball positions and chess pieces depend entirely on context [Shapiro] |
10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro] |
10228 | Could infinite structures be apprehended by pattern recognition? [Shapiro] |
10230 | The 4-pattern is the structure common to all collections of four objects [Shapiro] |
10249 | The main mathematical structures are algebraic, ordered, and topological [Shapiro] |
10273 | Some structures are exemplified by both abstract and concrete [Shapiro] |
10276 | Mathematical structures are defined by axioms, or in set theory [Shapiro] |
10270 | The main versions of structuralism are all definitionally equivalent [Shapiro] |
10221 | Is there is no more to structures than the systems that exemplify them? [Shapiro] |
10248 | Number statements are generalizations about number sequences, and are bound variables [Shapiro] |
10220 | Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro] |
10223 | There is no 'structure of all structures', just as there is no set of all sets [Shapiro] |
8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend] |
10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro] |
10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro] |
10254 | Can the ideal constructor also destroy objects? [Shapiro] |
10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
10227 | The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro] |
10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro] |
10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro] |
10277 | Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro] |
10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
10275 | A blurry border is still a border [Shapiro] |
10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
3249 | Modern science depends on the distinction between primary and secondary qualities [Nagel] |
22429 | We achieve objectivity by dropping secondary qualities, to focus on structural primary qualities [Nagel] |
3247 | Epistemology is centrally about what we should believe, not the definition of knowledge [Nagel] |
3252 | Scepticism is based on ideas which scepticism makes impossible [Nagel] |
3251 | Observed regularities are only predictable if we assume hidden necessity [Nagel] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
3244 | Personal identity cannot be fully known a priori [Nagel] |
3245 | The question of whether a future experience will be mine presupposes personal identity [Nagel] |
3246 | I can't even conceive of my brain being split in two [Nagel] |
10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
3257 | Total objectivity can't see value, but it sees many people with values [Nagel] |
3265 | We don't worry about the time before we were born the way we worry about death [Nagel] |
3263 | If our own life lacks meaning, devotion to others won't give it meaning [Nagel] |
3256 | Pain doesn't have a further property of badness; it gives a reason for its avoidance [Nagel] |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
3261 | Something may be 'rational' either because it is required or because it is acceptable [Nagel] |
3258 | If cockroaches can't think about their actions, they have no duties [Nagel] |
3254 | If we can decide how to live after stepping outside of ourselves, we have the basis of a moral theory [Nagel] |
3264 | We should see others' viewpoints, but not lose touch with our own values [Nagel] |
3255 | We find new motives by discovering reasons for action different from our preexisting motives [Nagel] |
3262 | Utilitarianism is too demanding [Nagel] |