Ideas of Paul Benacerraf, by Theme

[American, b.1931, Professor at Princeton University.]

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6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematical truth is always compromising between ordinary language and sensible epistemology
6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers
There are no such things as numbers
Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order
We must explain how we know so many numbers, and recognise ones we haven't met before
Numbers can't be sets if there is no agreement on which sets they are
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation
Benacerraf says numbers are defined by their natural ordering
6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
To understand finite cardinals, it is necessary and sufficient to understand progressions
A set has k members if it one-one corresponds with the numbers less than or equal to k
To explain numbers you must also explain cardinality, the counting of things
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
Someone can recite numbers but not know how to count things; but not vice versa
We can count intransitively (reciting numbers) without understanding transitive counting of items
6. Mathematics / A. Nature of Mathematics / 7. Application of Mathematics
The application of a system of numbers is counting and measurement
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
The successor of x is either x and all its members, or just the unit set of x
For Zermelo 3 belongs to 17, but for Von Neumann it does not
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them
No particular pair of sets can tell us what 'two' is, just by one-to-one correlation
If ordinal numbers are 'reducible to' some set-theory, then which is which?
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
An adequate account of a number must relate it to its series
The number 3 defines the role of being third in a progression
Number words no more have referents than do the parts of a ruler
Mathematical objects only have properties relating them to other 'elements' of the same structure
How can numbers be objects if order is their only property?
The job is done by the whole system of numbers, so numbers are not objects
If any recursive sequence will explain ordinals, then it seems to be the structure which matters
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Number-as-objects works wholesale, but fails utterly object by object
Realists have semantics without epistemology, anti-realists epistemology but bad semantics
The platonist view of mathematics doesn't fit our epistemology very well
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Number words are not predicates, as they function very differently from adjectives
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
9. Objects / F. Identity among Objects / 6. Identity between Objects
Identity statements make sense only if there are possible individuating conditions