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10237 | Coherence is a primitive, intuitive notion, not reduced to something formal |

10204 | An 'implicit definition' gives a direct description of the relations of an entity |

10206 | Modal operators are usually treated as quantifiers |

10208 | Axiom of Choice: some function has a value for every set in a given set |

10252 | The Axiom of Choice seems to license an infinite amount of choosing |

10207 | Anti-realists reject set theory |

10259 | The two standard explanations of consequence are semantic (in models) and deductive |

10257 | Intuitionism only sanctions modus ponens if all three components are proved |

10253 | Either logic determines objects, or objects determine logic, or they are separate |

10251 | The law of excluded middle might be seen as a principle of omniscience |

10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' |

10209 | A function is just an arbitrary correspondence between collections |

10268 | Maybe plural quantifiers should be understood in terms of classes or sets |

10235 | A sentence is 'satisfiable' if it has a model |

10240 | Model theory deals with relations, reference and extensions |

10239 | The central notion of model theory is the relation of 'satisfaction' |

10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' |

10238 | The set-theoretical hierarchy contains as many isomorphism types as possible |

10234 | Any theory with an infinite model has a model of every infinite cardinality |

10201 | Virtually all of mathematics can be modeled in set theory |

10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts |

18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis |

18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational |

10256 | For intuitionists, proof is inherently informal |

10236 | There is no grounding for mathematics that is more secure than mathematics |

10202 | Natural numbers just need an initial object, successors, and an induction principle |

10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) |

10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem |

10222 | Mathematical foundations may not be sets; categories are a popular rival |

10218 | Baseball positions and chess pieces depend entirely on context |

10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 |

10228 | Could infinite structures be apprehended by pattern recognition? |

10230 | The 4-pattern is the structure common to all collections of four objects |

10249 | The main mathematical structures are algebraic, ordered, and topological |

10273 | Some structures are exemplified by both abstract and concrete |

10276 | Mathematical structures are defined by axioms, or in set theory |

10270 | The main versions of structuralism are all definitionally equivalent |

10221 | Is there is no more to structures than the systems that exemplify them? |

10248 | Number statements are generalizations about number sequences, and are bound variables |

10220 | Because one structure exemplifies several systems, a structure is a one-over-many |

10223 | There is no 'structure of all structures', just as there is no set of all sets |

8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics |

10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? |

10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) |

10210 | If mathematical objects are accepted, then a number of standard principles will follow |

10215 | Platonists claim we can state the essence of a number without reference to the others |

10233 | Platonism must accept that the Peano Axioms could all be false |

10244 | Intuition is an outright hindrance to five-dimensional geometry |

10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location |

10254 | Can the ideal constructor also destroy objects? |

10255 | Presumably nothing can block a possible dynamic operation? |

10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? |

10227 | The abstract/concrete boundary now seems blurred, and would need a defence |

10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract |

10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) |

10277 | Structuralism blurs the distinction between mathematical and ordinary objects |

10272 | The notion of 'object' is at least partially structural and mathematical |

10275 | A blurry border is still a border |

10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models |

10266 | Why does the 'myth' of possible worlds produce correct modal logic? |

10203 | We apprehend small, finite mathematical structures by abstraction from patterns |

10229 | Simple types can be apprehended through their tokens, via abstraction |

10217 | We can apprehend structures by focusing on or ignoring features of patterns |

9554 | We can focus on relations between objects (like baseballers), ignoring their other features |

10231 | Abstract objects might come by abstraction over an equivalence class of base entities |