1960 | Logicism, Some Considerations (PhD) |
p.164 | p.164 | 13411 | If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation |
p.165 | p.165 | 13412 | Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order |
p.166 | p.166 | 13413 | We must explain how we know so many numbers, and recognise ones we haven't met before |
p.169 | p.169 | 13415 | An adequate account of a number must relate it to its series |
1965 | What Numbers Could Not Be |
p.18 | 9151 | Benacerraf says numbers are defined by their natural ordering |
p.83 | 8697 | Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them |
p.117 | 13891 | To understand finite cardinals, it is necessary and sufficient to understand progressions |
p.215 | 8304 | No particular pair of sets can tell us what 'two' is, just by one-to-one correlation |
I | p.274 | 9897 | The application of a system of numbers is counting and measurement |
I | p.274 | 9898 | We can count intransitively (reciting numbers) without understanding transitive counting of items |
I | p.275 | 17903 | Someone can recite numbers but not know how to count things; but not vice versa |
I | p.275 | 17904 | A set has k members if it one-one corresponds with the numbers less than or equal to k |
I n2 | p.275 | 17906 | To explain numbers you must also explain cardinality, the counting of things |
II | p.278 | 9899 | The successor of x is either x and all its members, or just the unit set of x |
II | p.278 | 9900 | For Zermelo 3 belongs to 17, but for Von Neumann it does not |
II | p.279 | 9901 | Numbers can't be sets if there is no agreement on which sets they are |
II | p.283 | 9903 | Number words are not predicates, as they function very differently from adjectives |
II | p.284 | 9904 | The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members |
III | p.286 | 9905 | Identity statements make sense only if there are possible individuating conditions |
IIIB | p.290 | 9906 | If ordinal numbers are 'reducible to' some set-theory, then which is which? |
IIIC | p.290 | 9908 | The job is done by the whole system of numbers, so numbers are not objects |
IIIC | p.290 | 9907 | If any recursive sequence will explain ordinals, then it seems to be the structure which matters |
IIIC | p.291 | 9909 | The number 3 defines the role of being third in a progression |
IIIC | p.292 | 9911 | Number words no more have referents than do the parts of a ruler |
IIIC | p.292 | 9910 | Number-as-objects works wholesale, but fails utterly object by object |
IIIC | p.294 | 9912 | There are no such things as numbers |
p.285 | p.581 | 8925 | Mathematical objects only have properties relating them to other 'elements' of the same structure |
p.301 | p.301 | 9938 | How can numbers be objects if order is their only property? |
1973 | Mathematical Truth |
p.8 | 17927 | Realists have semantics without epistemology, anti-realists epistemology but bad semantics |
Intro | p.403 | 9935 | Mathematical truth is always compromising between ordinary language and sensible epistemology |
III | p.412 | 9936 | The platonist view of mathematics doesn't fit our epistemology very well |