9935 | Mathematical truth is always compromising between ordinary language and sensible epistemology |

13412 | Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order |

13413 | We must explain how we know so many numbers, and recognise ones we haven't met before |

9912 | There are no such things as numbers |

9901 | Numbers can't be sets if there is no agreement on which sets they are |

13411 | If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation |

9151 | Benacerraf says numbers are defined by their natural ordering |

13891 | To understand finite cardinals, it is necessary and sufficient to understand progressions |

17904 | A set has k members if it one-one corresponds with the numbers less than or equal to k |

17906 | To explain numbers you must also explain cardinality, the counting of things |

9898 | We can count intransitively (reciting numbers) without understanding transitive counting of items |

17903 | Someone can recite numbers but not know how to count things; but not vice versa |

9897 | The application of a system of numbers is counting and measurement |

9900 | For Zermelo 3 belongs to 17, but for Von Neumann it does not |

9899 | The successor of x is either x and all its members, or just the unit set of x |

8697 | Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them |

8304 | No particular pair of sets can tell us what 'two' is, just by one-to-one correlation |

9906 | If ordinal numbers are 'reducible to' some set-theory, then which is which? |

13415 | An adequate account of a number must relate it to its series |

9908 | The job is done by the whole system of numbers, so numbers are not objects |

9907 | If any recursive sequence will explain ordinals, then it seems to be the structure which matters |

9909 | The number 3 defines the role of being third in a progression |

9911 | Number words no more have referents than do the parts of a ruler |

8925 | Mathematical objects only have properties relating them to other 'elements' of the same structure |

9938 | How can numbers be objects if order is their only property? |

9910 | Number-as-objects works wholesale, but fails utterly object by object |

17927 | Realists have semantics without epistemology, anti-realists epistemology but bad semantics |

9936 | The platonist view of mathematics doesn't fit our epistemology very well |

9903 | Number words are not predicates, as they function very differently from adjectives |

9904 | The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members |

9905 | Identity statements make sense only if there are possible individuating conditions |