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 1980,0-631-12911-1]].

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2. Reason / D. Definition / 2. Aims of Definition
Later Frege held that definitions must fix a function's value for every possible argument
2. Reason / D. Definition / 7. Contextual Definition
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
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
Cardinals say how many, and reals give measurements compared to a unit quantity
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Real numbers are ratios of quantities
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
A number is a class of classes of the same cardinality
Frege's biggest error is in not accounting for the senses of number terms
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Formalism misunderstands applications, metatheory, and infinity
Only applicability raises arithmetic from a game to a science
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
The first demand of logic is of a sharp boundary
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
The modern account of real numbers detaches a ratio from its geometrical origins
18. Thought / E. Abstraction / 8. Abstractionism Critique
If we abstract the difference between two houses, they don't become the same house