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

9641 | Definitions should be replaceable by primitives, and should not be creative |

9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) |

9613 | Naïve set theory assumed that there is a set for every condition |

9615 | Nowadays conditions are only defined on existing sets |

9617 | The 'iterative' view says sets start with the empty set and build up |

9642 | A flock of birds is not a set, because a set cannot go anywhere |

9605 | If a proposition is false, then its negation is true |

9649 | Axioms are either self-evident, or stipulations, or fallible attempts |

9638 | Berry's Paradox finds a contradiction in the naming of huge numbers |

9604 | Mathematics is the only place where we are sure we are right |

9622 | 'There are two apples' can be expressed logically, with no mention of numbers |

9648 | π is a 'transcendental' number, because it is not the solution of an equation |

9618 | Bolzano wanted to reduce all of geometry to arithmetic |

9646 | There is no limit to how many ways something can be proved in mathematics |

9647 | Computers played an essential role in proving the four-colour theorem of maps |

9621 | Mathematics represents the world through structurally similar models. |

9643 | Set theory may represent all of mathematics, without actually being mathematics |

9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs |

9625 | To see a structure in something, we must already have the idea of the structure |

9627 | Different versions of set theory result in different underlying structures for numbers |

9628 | Sets seem basic to mathematics, but they don't suit structuralism |

9606 | The irrationality of root-2 was achieved by intellect, not experience |

9610 | Numbers are not abstracted from particulars, because each number is a particular |

9612 | There is an infinity of mathematical objects, so they can't be physical |

9620 | Empiricists base numbers on objects, Platonists base them on properties |

9630 | The most brilliant formalist was Hilbert |

9639 | Does some mathematics depend entirely on notation? |

9629 | For nomalists there are no numbers, only numerals |

9608 | There are no constructions for many highly desirable results in mathematics |

9645 | Constructivists say p has no value, if the value depends on Goldbach's Conjecture |

9619 | David's 'Napoleon' is about something concrete and something abstract |

9607 | The greatest discovery in human thought is Plato's discovery of abstract objects |

9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars |

9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical |

9640 | A term can have not only a sense and a reference, but also a 'computational role' |

9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities |