Click on the Idea Number for the full details | back to texts | expand these ideas

8721 | An 'impredicative' definition seems circular, because it uses the term being defined |

8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects |

3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction |

8705 | Anti-realist see truth as our servant, and epistemically contrained |

8713 | In classical/realist logic the connectives are defined by truth-tables |

8708 | Double negation elimination is not valid in intuitionist logic |

8694 | Free logic was developed for fictional or non-existent objects |

8665 | A 'proper subset' of A contains only members of A, but not all of them |

8672 | A 'powerset' is all the subsets of a set |

8677 | Set theory makes a minimum ontological claim, that the empty set exists |

8666 | Infinite sets correspond one-to-one with a subset |

8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity |

8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false |

8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." |

8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' |

8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal |

8710 | The powerset of all the cardinal numbers is required to be greater than itself |

8670 | A number is 'irrational' if it cannot be represented as a fraction |

8667 | The 'integers' are the positive and negative natural numbers, plus zero |

8668 | The 'rational' numbers are those representable as fractions |

8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction |

8664 | Cardinal numbers answer 'how many?', with the order being irrelevant |

8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps |

8663 | Raising omega to successive powers of omega reveal an infinity of infinities |

8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega |

8669 | Between any two rational numbers there is an infinite number of rational numbers |

8676 | Is mathematics based on sets, types, categories, models or topology? |

8678 | Most mathematical theories can be translated into the language of set theory |

8701 | The number 8 in isolation from the other numbers is of no interest |

8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent |

8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? |

8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves |

8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential |

8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting |

8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? |

8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory |

8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research |

8706 | Constructivism rejects too much mathematics |

8707 | Intuitionists typically retain bivalence but reject the law of excluded middle |

8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' |

8685 | Studying biology presumes the laws of chemistry, and it could never contradict them |

8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) |

8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability |