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 1981,0-7156-1649-8]].

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2. Reason / A. Nature of Reason / 5. Objectivity
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
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
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
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
Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
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
Ostension is possible for concreta; abstracta can only be referred to via other objects
We can't say that light is concrete but radio waves abstract
The concrete/abstract distinction seems crude: in which category is the Mistral?
We don't need a sharp concrete/abstract distinction
7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
The context principle for names rules out a special philosophical sense for 'existence'
The objects we recognise the world as containing depends on the structure of our language
8. Modes of Existence / D. Universals / 1. Universals
We can understand universals by studying predication
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
'Nominalism' used to mean denial of universals, but now means denial of abstract objects
9. Objects / A. Existence of Objects / 1. Physical Objects
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
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
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
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
If a genuine singular term needs a criterion of identity, we must exclude abstract nouns
Abstract objects can never be confronted, and need verbal phrases for reference
9. Objects / A. Existence of Objects / 3. Objects in Thought
There is a modern philosophical notion of 'object', first introduced by Frege
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
Concepts only have a 'functional character', because they map to truth values, not objects
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Since abstract objects cannot be picked out, we must rely on identity statements
19. Language / C. Reference / 3. Direct Reference / b. Causal reference
A realistic view of reference is possible for concrete objects, but not for abstract objects