44 ideas
8623 | Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by Frege] |
13907 | If you pick an arbitrary triangle, things proved of it are true of all triangles [Euclid, by Lemmon] |
9935 | Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf] |
6297 | Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik] |
9912 | There are no such things as numbers [Benacerraf] |
13412 | Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf] |
13413 | We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf] |
9901 | Numbers can't be sets if there is no agreement on which sets they are [Benacerraf] |
9603 | An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR] |
13411 | If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf] |
9151 | Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K] |
13891 | To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C] |
17904 | A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf] |
17906 | To explain numbers you must also explain cardinality, the counting of things [Benacerraf] |
9894 | A unit is that according to which each existing thing is said to be one [Euclid] |
9898 | We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf] |
17903 | Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf] |
9897 | The application of a system of numbers is counting and measurement [Benacerraf] |
8738 | Postulate 2 says a line can be extended continuously [Euclid, by Shapiro] |
22278 | Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid] |
8673 | Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend] |
10250 | Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid] |
10302 | Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays] |
14157 | Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid] |
9899 | The successor of x is either x and all its members, or just the unit set of x [Benacerraf] |
9900 | For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf] |
1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |
8697 | Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend] |
8304 | No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe] |
9906 | If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf] |
13415 | An adequate account of a number must relate it to its series [Benacerraf] |
9908 | The job is done by the whole system of numbers, so numbers are not objects [Benacerraf] |
9907 | If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf] |
9909 | The number 3 defines the role of being third in a progression [Benacerraf] |
9911 | Number words no more have referents than do the parts of a ruler [Benacerraf] |
8925 | Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf] |
9938 | How can numbers be objects if order is their only property? [Benacerraf, by Putnam] |
9910 | Number-as-objects works wholesale, but fails utterly object by object [Benacerraf] |
17927 | Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan] |
9936 | The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf] |
9903 | Number words are not predicates, as they function very differently from adjectives [Benacerraf] |
9904 | The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf] |
9905 | Identity statements make sense only if there are possible individuating conditions [Benacerraf] |
5996 | Critolaus redefined Aristotle's moral aim as fulfilment instead of happiness [Critolaus, by White,SA] |