Ideas from 'Frege Philosophy of Language (2nd ed)' by Michael Dummett [1973], by Theme Structure
[found in 'Frege Philosophy of Language' by Dummett,Michael [Duckworth 1981,0715616498]].
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2. Reason / A. Nature of Reason / 5. Objectivity
17621

What matters in mathematics is its objectivity, not the existence of the objects

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
10537

The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}}

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10542

To associate a cardinal with each set, we need the Axiom of Choice to find a representative

5. Theory of Logic / G. Quantification / 2. Domain of Quantification
10536

Frege's domain for variables is all objects, but modern interpretations first fix the domain

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / g. Incompleteness of Arithmetic
10554

Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
10552

Intuitionism says that totality of numbers is only potential, but is still determinate

7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
10540

We can't say that light is concrete but radio waves abstract

10515

Ostension is possible for concreta; abstracta can only be referred to via other objects

10544

The concrete/abstract distinction seems crude: in which category is the Mistral?

10546

We don't need a sharp concrete/abstract distinction

7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
10548

The context principle for names rules out a special philosophical sense for 'existence'

10281

The objects we recognise the world as containing depends on the structure of our language

8. Modes of Existence / D. Universals / 1. Universals
10532

We can understand universals by studying predication

8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
10534

'Nominalism' used to mean denial of universals, but now means denial of abstract objects

9. Objects / A. Existence of Objects / 1. Physical Objects
10541

Concrete objects such as sounds and smells may not be possible objects of ostension

9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
10545

Abstract objects may not cause changes, but they can be the subject of change

9. Objects / A. Existence of Objects / 2. Abstract Objects / b. Need for abstracta
10555

If we can intuitively apprehend abstract objects, this makes them observable and causally active

9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
10543

Abstract objects must have names that fall within the range of some functional expression

9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
10320

If a genuine singular term needs a criterion of identity, we must exclude abstract nouns

10547

Abstract objects can never be confronted, and need verbal phrases for reference

9. Objects / A. Existence of Objects / 3. Objects in Thought
10531

There is a modern philosophical notion of 'object', first introduced by Frege

18. Thought / D. Concepts / 2. Ontology of Concepts / c. Fregean concepts
19168

Concepts only have a 'functional character', because they map to truth values, not objects

18. Thought / D. Concepts / 6. Abstract Concepts / g. Abstracta by equivalence
10549

Since abstract objects cannot be picked out, we must rely on identity statements

19. Language / D. Theories of Reference / 3. Direct Reference / b. Causal reference
10516

A realistic view of reference is possible for concrete objects, but not for abstract objects
