107 ideas
| 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] |
| 4098 | The theory of descriptions supports internalism, since they are thinkable when the object is non-existent [Crane] |
| 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] |
| 10245 | One geometry cannot be more true than another [Poincaré] |
| 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] |
| 4077 | Aesthetic properties of thing supervene on their physical properties [Crane] |
| 4078 | Constitution (as in a statue constituted by its marble) is supervenience without identity [Crane] |
| 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] |
| 4082 | The distinction between 'resultant' properties (weight) and 'emergent' properties is a bit vague [Crane] |
| 4083 | If mental properties are emergent they add a new type of causation, and physics is not complete [Crane] |
| 4079 | Properties are causes [Crane] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 4068 | Traditional substance is separate from properties and capable of independent existence [Crane] |
| 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] |
| 4096 | Maybe beliefs don't need to be conscious, if you are not conscious of the beliefs guiding your actions [Crane] |
| 4097 | Maybe there are two kinds of belief - 'de re' beliefs and 'de dicto' beliefs [Crane] |
| 4093 | Many cases of knowing how can be expressed in propositional terms (like how to get somewhere) [Crane] |
| 4108 | Phenol-thio-urea tastes bitter to three-quarters of people, but to the rest it is tasteless, so which is it? [Crane] |
| 4104 | One can taste that the wine is sour, and one can also taste the sourness of the wine [Crane] |
| 4105 | The traditional supports for the sense datum theory were seeing double and specks before one's eyes [Crane] |
| 4101 | If we smell something we are aware of the smell separately, but we don't perceive a 'look' when we see [Crane] |
| 4102 | The problems of perception disappear if it is a relation to an intentional state, not to an object or sense datum [Crane] |
| 4109 | If perception is much richer than our powers of description, this suggests that it is non-conceptual [Crane] |
| 4103 | The adverbial theory of perceptions says it is the experiences which have properties, not the objects [Crane] |
| 4065 | Is knowledge just a state of mind, or does it also involve the existence of external things? [Crane] |
| 4092 | The core of the consciousness problem is the case of Mary, zombies, and the Hard Question [Crane] |
| 4087 | Intentionalism does not require that all mental states be propositional attitudes [Crane] |
| 4095 | Object-directed attitudes like love are just as significant as propositional attitudes [Crane] |
| 4106 | If someone removes their glasses the content of experience remains, but the quality changes [Crane] |
| 4089 | Pains have a region of the body as their intentional content, not some pain object [Crane] |
| 4090 | Weak intentionalism says qualia are extra properties; strong intentionalism says they are intentional [Crane] |
| 4107 | With inverted qualia a person's experiences would change, but their beliefs remain the same [Crane] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 4069 | Descartes did not think of minds as made of a substance, because they are not divisible [Crane] |
| 4074 | Functionalism defines mental states by their causal properties, which rules out epiphenomenalism [Crane] |
| 4091 | The problems of misrepresentation and error have dogged physicalist reductions of intentionality [Crane] |
| 4070 | Properties dualism says mental properties are distinct from physical, despite a single underlying substance [Crane] |
| 4084 | Non-reductive physicalism seeks an explanation of supervenience, but emergentists accept it as basic [Crane] |
| 4080 | If mental supervenes on the physical, then every physical cause will be accompanied by a mental one [Crane] |
| 4075 | Identity theory is either of particular events, or of properties, depending on your theory of causation [Crane] |
| 4085 | Physicalism may be the source of the mind-body problem, rather than its solution [Crane] |
| 4073 | Overdetermination occurs if two events cause an effect, when each would have caused it alone [Crane] |
| 4072 | The completeness of physics must be an essential component of any physicalist view of mind [Crane] |
| 4094 | Experience teaches us propositions, because we can reason about our phenomenal experience [Crane] |
| 4100 | The Twin Earth argument depends on reference being determined by content, which may be false. [Crane] |
| 4067 | Broad content entails the existence of the object of the thought [Crane] |
| 4063 | In intensional contexts, truth depends on how extensions are conceived. [Crane] |
| 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] |
| 4071 | Causation can be seen in counterfactual terms, or as increased probability, or as energy flow [Crane] |
| 4076 | Causes are properties, not events, because properties are what make a difference in a situation [Crane] |
| 4066 | It seems that 'exists' could sometimes be a predicate [Crane] |