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Single Idea 16865
[filed under theme 5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
]
Full Idea
Usually a truth is only called a 'theorem' when it has not merely been obtained by inference, but is used in turn as a premise for a number of inferences in the science. ….Proofs use non-theorems, which only occur in that proof.
Gist of Idea
'Theorems' are both proved, and used in proofs
Source
Gottlob Frege (Logic in Mathematics [1914], p.204)
Book Ref
Frege,Gottlob: 'Posthumous Writings', ed/tr. Hermes/Long/White etc [Blackwell 1979], p.204
The
20 ideas
from 'Logic in Mathematics'
11219
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Frege suggested that mathematics should only accept stipulative definitions
[Frege, by Gupta]
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16863
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Does some mathematical reasoning (such as mathematical induction) not belong to logic?
[Frege]
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16862
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The closest subject to logic is mathematics, which does little apart from drawing inferences
[Frege]
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16864
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If principles are provable, they are theorems; if not, they are axioms
[Frege]
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16866
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Tracing inference backwards closes in on a small set of axioms and postulates
[Frege]
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16865
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'Theorems' are both proved, and used in proofs
[Frege]
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16867
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Logic not only proves things, but also reveals logical relations between them
[Frege]
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16868
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The essence of mathematics is the kernel of primitive truths on which it rests
[Frege]
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16870
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Axioms are truths which cannot be doubted, and for which no proof is needed
[Frege]
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16871
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A truth can be an axiom in one system and not in another
[Frege]
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16869
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To create order in mathematics we need a full system, guided by patterns of inference
[Frege]
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16873
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Thoughts are not subjective or psychological, because some thoughts are the same for us all
[Frege]
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16872
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A thought is the sense expressed by a sentence, and is what we prove
[Frege]
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16874
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The parts of a thought map onto the parts of a sentence
[Frege]
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16876
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We need definitions to cram retrievable sense into a signed receptacle
[Frege]
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16875
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We use signs to mark receptacles for complex senses
[Frege]
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16877
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A 'constructive' (as opposed to 'analytic') definition creates a new sign
[Frege]
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16878
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We must be clear about every premise and every law used in a proof
[Frege]
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16879
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A sign won't gain sense just from being used in sentences with familiar components
[Frege]
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9388
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Every concept must have a sharp boundary; we cannot allow an indeterminate third case
[Frege]
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