Ideas from 'Grundgesetze der Arithmetik 2 (Basic Laws)' by Gottlob Frege [1903], by Theme Structure
[found in 'Translations from the Writings of Gottlob Frege' by Frege,Gottlob (ed/tr Geach,P. /Black,Max) [Blackwell 1980,0631129111]].
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2. Reason / D. Definition / 2. Aims of Definition
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Later Frege held that definitions must fix a function's value for every possible argument

2. Reason / D. Definition / 7. Contextual Definition
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We can't define a word by defining an expression containing it, as the remaining parts are a problem

2. Reason / D. Definition / 11. Ostensive Definition
10019

Only what is logically complex can be defined; what is simple must be pointed to

6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
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Cardinals say how many, and reals give measurements compared to a unit quantity

6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
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Real numbers are ratios of quantities

6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
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A number is a class of classes of the same cardinality

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Frege's biggest error is in not accounting for the senses of number terms

6. Mathematics / C. Sources of Mathematics / 7. Formalism
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Formalism misunderstands applications, metatheory, and infinity

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Only applicability raises arithmetic from a game to a science

9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
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The first demand of logic is of a sharp boundary

18. Thought / E. Abstraction / 3. Abstracta by Ignoring
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The modern account of real numbers detaches a ratio from its geometrical origins

18. Thought / E. Abstraction / 8. Abstractionism Critique
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If we abstract the difference between two houses, they don't become the same house
