Ideas from 'Logic in Mathematics' by Gottlob Frege [1914], by Theme Structure

[found in 'Posthumous Writings' by Frege,Gottlob (ed/tr Hermes/Long/White etc) [Blackwell 1979,0-631-12835-2]].

Click on the Idea Number for the full details    |     back to texts     |     expand these ideas


2. Reason / D. Definition / 3. Types of Definition
A 'constructive' (as opposed to 'analytic') definition creates a new sign
2. Reason / D. Definition / 10. Stipulative Definition
Frege suggested that mathematics should only accept stipulative definitions
2. Reason / E. Argument / 6. Conclusive Proof
We must be clear about every premise and every law used in a proof
5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
Logic not only proves things, but also reveals logical relations between them
5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
Does some mathematical reasoning (such as mathematical induction) not belong to logic?
The closest subject to logic is mathematics, which does little apart from drawing inferences
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
'Theorems' are both proved, and used in proofs
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Tracing inference backwards closes in on a small set of axioms and postulates
The essence of mathematics is the kernel of primitive truths on which it rests
A truth can be an axiom in one system and not in another
Axioms are truths which cannot be doubted, and for which no proof is needed
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
To create order in mathematics we need a full system, guided by patterns of inference
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
If principles are provable, they are theorems; if not, they are axioms
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
Every concept must have a sharp boundary; we cannot allow an indeterminate third case
18. Thought / B. Mechanics of Thought / 5. Mental Files
We use signs to mark receptacles for complex senses
We need definitions to cram retrievable sense into a signed receptacle
19. Language / A. Nature of Meaning / 6. Meaning as Use
A sign won't gain sense just from being used in sentences with familiar components
19. Language / D. Propositions / 2. Abstract Propositions / a. Propositions as sense
Thoughts are not subjective or psychological, because some thoughts are the same for us all
A thought is the sense expressed by a sentence, and is what we prove
19. Language / D. Propositions / 5. Unity of Propositions
The parts of a thought map onto the parts of a sentence