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Ideas of Paul Benacerraf, by Text
[American, b.1931, Professor at Princeton University.]
1960

Logicism, Some Considerations (PhD)

p.164

p.164

13411

If numbers are basically the cardinals (FregeRussell 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 [Fine,K]


p.83

8697

Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Friend]


p.117

13891

To understand finite cardinals, it is necessary and sufficient to understand progressions [Wright,C]


p.215

8304

No particular pair of sets can tell us what 'two' is, just by onetoone correlation [Lowe]

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 oneone 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

9900

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

II

p.278

9899

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

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 settheory 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 settheory, 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

Numberasobjects 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? [Putnam]


p.8

17927

Realists have semantics without epistemology, antirealists epistemology but bad semantics [Colyvan]

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
