116 ideas
| 19693 | There is practical wisdom (for action), and theoretical wisdom (for deep understanding) [Aristotle, by Whitcomb] |
| 1575 | For Aristotle logos is essentially the ability to talk rationally about questions of value [Roochnik on Aristotle] |
| 1589 | Aristotle is the supreme optimist about the ability of logos to explain nature [Roochnik on Aristotle] |
| 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
| 8200 | Aristotelian definitions aim to give the essential properties of the thing defined [Aristotle, by Quine] |
| 4385 | Aristotelian definition involves first stating the genus, then the differentia of the thing [Aristotle, by Urmson] |
| 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] |
| 13282 | Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki] |
| 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] |
| 4730 | For Aristotle, the subject-predicate structure of Greek reflected a substance-accident structure of reality [Aristotle, by O'Grady] |
| 10373 | Logical form can't dictate metaphysics, as it may propose an undesirable property [Schaffer,J] |
| 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] |
| 10367 | There is only one fact - the True [Schaffer,J] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 10275 | A blurry border is still a border [Shapiro] |
| 13276 | The unmoved mover and the soul show Aristotelian form as the ultimate mereological atom [Aristotle, by Koslicki] |
| 13277 | The 'form' is the recipe for building wholes of a particular kind [Aristotle, by Koslicki] |
| 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] |
| 5991 | For Aristotle, knowledge is of causes, and is theoretical, practical or productive [Aristotle, by Code] |
| 11239 | The notion of a priori truth is absent in Aristotle [Aristotle, by Politis] |
| 23312 | Aristotle is a rationalist, but reason is slowly acquired through perception and experience [Aristotle, by Frede,M] |
| 16111 | Aristotle wants to fit common intuitions, and therefore uses language as a guide [Aristotle, by Gill,ML] |
| 16971 | Plato says sciences are unified around Forms; Aristotle says they're unified around substance [Aristotle, by Moravcsik] |
| 11243 | Aristotelian explanations are facts, while modern explanations depend on human conceptions [Aristotle, by Politis] |
| 3320 | Aristotle's standard analysis of species and genus involves specifying things in terms of something more general [Aristotle, by Benardete,JA] |
| 12000 | Aristotle regularly says that essential properties explain other significant properties [Aristotle, by Kung] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 23300 | Aristotle and the Stoics denied rationality to animals, while Platonists affirmed it [Aristotle, by Sorabji] |
| 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] |
| 11240 | The notion of analytic truth is absent in Aristotle [Aristotle, by Politis] |
| 6559 | Aristotle never actually says that man is a rational animal [Aristotle, by Fogelin] |
| 11150 | It is the mark of an educated mind to be able to entertain an idea without accepting it [Aristotle] |
| 3037 | Aristotle said the educated were superior to the uneducated as the living are to the dead [Aristotle, by Diog. Laertius] |
| 8660 | There are potential infinities (never running out), but actual infinity is incoherent [Aristotle, by Friend] |
| 12058 | Aristotle's matter can become any other kind of matter [Aristotle, by Wiggins] |
| 10359 | In causation there are three problems of relata, and three metaphysical problems [Schaffer,J] |
| 10372 | Causation may not be transitive; the last event may follow from the first, but not be caused by it [Schaffer,J] |
| 10374 | There are at least ten theories about causal connections [Schaffer,J] |
| 10366 | Causation transcends nature, because absences can cause things [Schaffer,J] |
| 10377 | Causation may not be a process, if a crucial part of the process is 'disconnected' [Schaffer,J] |
| 10378 | A causal process needs to be connected to the effect in the right way [Schaffer,J] |
| 10382 | Causation can't be a process, because a process needs causation as a primitive [Schaffer,J] |
| 10375 | At least four rivals have challenged the view that causal direction is time direction [Schaffer,J] |
| 10389 | Causal order must be temporal, or else causes could be blocked, and time couldn't be explained [Schaffer,J] |
| 10390 | Causal order is not temporal, because of time travel, and simultanous, joint or backward causes [Schaffer,J] |
| 10380 | Causation is primitive; it is too intractable and central to be reduced; all explanations require it [Schaffer,J] |
| 10385 | If causation is just observables, or part of common sense, or vacuous, it can't be primitive [Schaffer,J] |
| 10388 | Causation is utterly essential for numerous philosophical explanations [Schaffer,J] |
| 10387 | The notion of causation allows understanding of science, without appearing in equations [Schaffer,J] |
| 10384 | If two different causes are possible in one set of circumstances, causation is primitive [Schaffer,J] |
| 10386 | If causation is primitive, it can be experienced in ourselves, or inferred as best explanation [Schaffer,J] |
| 10361 | Events are fairly course-grained (just saying 'hello'), unlike facts (like saying 'hello' loudly) [Schaffer,J] |
| 10360 | Causal relata are events - or facts, features, tropes, states, situations or aspects [Schaffer,J] |
| 10362 | One may defend three or four causal relata, as in 'c causes e rather than e*' [Schaffer,J] |
| 10368 | If causal relata must be in nature and fine-grained, neither facts nor events will do [Schaffer,J] |
| 10383 | The relata of causation (such as events) need properties as explanation, which need causation! [Schaffer,J] |
| 10393 | Our selection of 'the' cause is very predictable, so must have a basis [Schaffer,J] |
| 10394 | Selecting 'the' cause must have a basis; there is no causation without such a selection [Schaffer,J] |
| 10376 | The actual cause may make an event less likely than a possible more effective cause [Schaffer,J] |
| 10381 | All four probability versions of causation may need causation to be primitive [Schaffer,J] |
| 22729 | The concepts of gods arose from observing the soul, and the cosmos [Aristotle, by Sext.Empiricus] |