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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2. Reason / D. Definition / 7. Contextual Definition
9847
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A contextual definition permits the elimination of the expression by a substitution
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5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
9820
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In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
9896
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A prime number is one which is measured by a unit alone
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
18255
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Addition of quantities is prior to ordering, as shown in cyclic domains like angles
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
9895
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A number is a multitude composed of units
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
9852
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We understand 'there are as many nuts as apples' as easily by pairing them as by counting them
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9828
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Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
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9829
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The identity of a number may be fixed by something outside structure - by counting
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
9876
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Set theory isn't part of logic, and why reduce to something more complex?
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7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
9884
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The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy
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7. Existence / D. Theories of Reality / 2. Realism
9869
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Realism is just the application of two-valued semantics to sentences
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8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
9880
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Nominalism assumes unmediated mental contact with objects
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9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
9885
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The existence of abstract objects is a pseudo-problem
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9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
9858
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Abstract objects nowadays are those which are objective but not actual
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9859
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It is absurd to deny the Equator, on the grounds that it lacks causal powers
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9860
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'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object
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9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
9872
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Abstract objects need the context principle, since they can't be encountered directly
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9. Objects / F. Identity among Objects / 2. Defining Identity
9848
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Content is replaceable if identical, so replaceability can't define identity [Dummett]
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9842
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Frege introduced criteria for identity, but thought defining identity was circular
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18. Thought / D. Concepts / 4. Structure of Concepts / i. Conceptual priority
9850
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An argument for conceptual priority is greater simplicity in explanation
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9849
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Maybe a concept is 'prior' to another if it can be defined without the second concept
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18. Thought / E. Abstraction / 1. Abstract Thought
9873
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Abstract terms are acceptable as long as we know how they function linguistically
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18. Thought / E. Abstraction / 7. Abstracta by Equivalence
9993
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There is no reason why abstraction by equivalence classes should be called 'logical' [Tait]
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9857
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We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus'
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18. Thought / E. Abstraction / 8. Abstractionism Critique
9833
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To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too
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19. Language / C. Assigning Meanings / 5. Fregean Semantics
9836
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Fregean semantics assumes a domain articulated into individual objects
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27. Natural Reality / C. Space / 3. Points in Space
18257
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Why should the limit of measurement be points, not intervals?
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