Ideas from 'Frege's Concept of Numbers as Objects' by Crispin Wright [1983], by Theme Structure
[found in 'Frege's Conception of Numbers' by Wright,Crispin [Scots Philosophical Monographs 1983,]].
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1. Philosophy / C. History of Philosophy / 1. History of Philosophy
13860

We can only learn from philosophers of the past if we accept the risk of major misrepresentation

2. Reason / C. Styles of Reason / 1. Dialectic
13883

The best way to understand a philosophical idea is to defend it

2. Reason / D. Definition / 7. Contextual Definition
10142

The attempt to define numbers by contextual definition has been revived [Fine,K]

5. Theory of Logic / F. Referring in Logic / 1. Naming / d. Singular terms
9868

An expression refers if it is a singular term in some true sentences [Dummett]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
13861

Number theory aims at the essence of natural numbers, giving their nature, and the epistemology

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
13892

One could grasp numbers, and name sizes with them, without grasping ordering

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
13867

Instances of a nonsortal concept can only be counted relative to a sortal concept

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17441

Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Heck]

13862

There are five Peano axioms, which can be expressed informally

17853

Number truths are said to be the consequence of PA  but it needs semantic consequence

17854

What facts underpin the truths of the Peano axioms?

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
13894

Sameness of number is fundamental, not counting, despite children learning that first

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
10140

We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Fine,K]

8692

Frege has a good system if his 'number principle' replaces his basic law V [Friend]

17440

Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Heck]

13893

It is 11 correlation of concepts, and not progression, which distinguishes natural number

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
13888

If numbers are extensions, Frege must first solve the Caesar problem for extensions

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
13869

Number platonism says that natural number is a sortal concept

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
13870

We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism

6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
13873

Treating numbers adjectivally is treating them as quantifiers

6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neologicism
13899

The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals

13896

The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes

7804

Wright has revived Frege's discredited logicism [Benardete,JA]

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13863

Logicism seemed to fail by Russell's paradox, Gödel's theorems, and nonlogical axioms

13895

The standard objections are Russell's Paradox, nonlogical axioms, and Gödel's theorems

7. Existence / A. Nature of Existence / 2. Types of Existence
13884

The idea that 'exist' has multiple senses is not coherent

7. Existence / D. Theories of Reality / 11. Ontological Commitment / b. Commitment of quantifiers
13877

Singular terms in true sentences must refer to objects; there is no further question about their existence

9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
9878

Contextually defined abstract terms genuinely refer to objects [Dummett]

9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind
13868

Sortal concepts cannot require that things don't survive their loss, because of phase sortals

18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
13866

A concept is only a sortal if it gives genuine identity

13865

'Sortal' concepts show kinds, use indefinite articles, and require grasping identities

18. Thought / D. Concepts / 4. Structure of Concepts / b. Analysis of concepts
13890

Entities fall under a sortal concept if they can be used to explain identity statements concerning them

18. Thought / E. Abstraction / 7. Abstracta by Equivalence
13898

If we can establish directions from lines and parallelism, we were already committed to directions

19. Language / A. Nature of Meaning / 5. Meaning as Verification
13882

A milder claim is that understanding requires some evidence of that understanding

19. Language / B. Reference / 1. Reference theories
13885

If apparent reference can mislead, then so can apparent lack of reference

19. Language / C. Assigning Meanings / 3. Predicates
17857

We can accept Frege's idea of object without assuming that predicates have a reference
