Ideas from 'Frege philosophy of mathematics' by Michael Dummett [1991], by Theme Structure

[found in 'Frege: philosophy of mathematics' by Dummett,Michael [Duckworth 1991,0-7156-2660-4]].

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1. Philosophy / F. Analytic Philosophy / 1. Analysis
Frege was the first to give linguistic answers to non-linguistic questions
2. Reason / D. Definition / 7. Contextual Definition
A contextual definition permits the elimination of the expression by a substitution
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
A prime number is one which is measured by a unit alone
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
Addition of quantities is prior to ordering, as shown in cyclic domains like angles
6. Mathematics / A. Nature of Mathematics / 3. Numbers / o. Units
A number is a multitude composed of units
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
We understand 'there are as many nuts as apples' as easily by pairing them as by counting them
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
The identity of a number may be fixed by something outside structure - by counting
Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Set theory isn't part of logic, and why reduce to something more complex?
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Frege was completing Bolzano's work, of expelling intuition from number theory and analysis
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy
7. Existence / D. Theories of Reality / 1. Realism
Realism is just the application of two-valued semantics to sentences
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
Nominalism assumes unmediated mental contact with objects
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
The existence of abstract objects is a pseudo-problem
9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
It is absurd to deny the Equator, on the grounds that it lacks causal powers
'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object
Abstract objects nowadays are those which are objective but not actual
9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
Abstract objects need the context principle, since they can't be encountered directly
9. Objects / F. Identity among Objects / 2. Defining Identity
Content is replaceable if identical, so replaceability can't define identity
Frege introduced criteria for identity, but thought defining identity was circular
18. Thought / D. Concepts / 4. Structure of Concepts / h. Conceptual priority
Maybe a concept is 'prior' to another if it can be defined without the second concept
An argument for conceptual priority is greater simplicity in explanation
18. Thought / E. Abstraction / 1. Abstract Thought
Abstract terms are acceptable as long as we know how they function linguistically
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus'
18. Thought / E. Abstraction / 8. Abstractionism Critique
To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too
19. Language / C. Assigning Meanings / 5. Fregean Semantics
Fregean semantics assumes a domain articulated into individual objects
27. Natural Reality / A. Space-Time / 1. Space / c. Points in space
Why should the limit of measurement be points, not intervals?