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
Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
The identity of a number may be fixed by something outside structure - by counting
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
Abstract objects nowadays are those which are objective but not actual
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
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
Frege introduced criteria for identity, but thought defining identity was circular
Content is replaceable if identical, so replaceability can't define identity
18. Thought / D. Concepts / 3. 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 / D. Concepts / 6. Abstract Concepts / a. Abstract concepts
Abstract terms are acceptable as long as we know how they function linguistically
18. Thought / D. Concepts / 6. Abstract Concepts / g. Abstracta by equivalence
We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus'
18. Thought / D. Concepts / 6. Abstract Concepts / h. Abstractionism critique
To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too
19. Language / C. Semantics / 2. Fregean Semantics
Fregean semantics assumes a domain articulated into individual objects
26. Natural Theory / B. Concepts of Nature / 3. Space / b. Points in space
Why should the limit of measurement be points, not intervals?