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Single Idea 8304

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory ]

Full Idea

Hume's Principle can't tell us what a cardinal number is (this is one lesson of Benacerraf's well-known problem). An infinity of pairs of sets could actually be the number two (not just the simplest sets).

Clarification

See Idea 8302 for Hume's Principle

Gist of Idea

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

Source

report of Paul Benacerraf (What Numbers Could Not Be [1965]) by E.J. Lowe - The Possibility of Metaphysics 10.3

Book Ref

Lowe,E.J.: 'The Possibility of Metaphysics' [OUP 2001], p.215

A Reaction

The drift here is for numbers to end up as being basic, axiomatic, indefinable, universal entities. Since I favour patterns as the basis of numbers, I think the basis might be in a pre-verbal experience, which even a bird might have, viewing its eggs.

Related Idea

Idea 8302 Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe]

The 31 ideas from Paul Benacerraf

13411 If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [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]
13415 An adequate account of a number must relate it to its series [Benacerraf]
17927 Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan]
9935 Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf]
9936 The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf]
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]
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]
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]
17906 To explain numbers you must also explain cardinality, the counting of things [Benacerraf]
9901 Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
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]
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]
9906 If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
9910 Number-as-objects works wholesale, but fails utterly object by object [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]
9912 There are no such things as numbers [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]