more on this theme     |     more from this thinker

Single Idea 16869

[filed under theme 6. Mathematics / A. Nature of Mathematics / 1. Mathematics ]

Full Idea

We cannot long remain content with the present fragmentation [of mathematics]. Order can be created only by a system. But to construct a system it is necessary that in any step forward we take we should be aware of the logical inferences involved.

Gist of Idea

To create order in mathematics we need a full system, guided by patterns of inference

Source

Gottlob Frege (Logic in Mathematics [1914], p.205)

Book Ref

Frege,Gottlob: 'Posthumous Writings', ed/tr. Hermes/Long/White etc [Blackwell 1979], p.205

Related Ideas

Idea 16868 The essence of mathematics is the kernel of primitive truths on which it rests [Frege]

Idea 16871 A truth can be an axiom in one system and not in another [Frege]

The 20 ideas from 'Logic in Mathematics'

11219 Frege suggested that mathematics should only accept stipulative definitions [Frege, by Gupta]
16864 If principles are provable, they are theorems; if not, they are axioms [Frege]
16863 Does some mathematical reasoning (such as mathematical induction) not belong to logic? [Frege]
16862 The closest subject to logic is mathematics, which does little apart from drawing inferences [Frege]
16865 'Theorems' are both proved, and used in proofs [Frege]
16866 Tracing inference backwards closes in on a small set of axioms and postulates [Frege]
16867 Logic not only proves things, but also reveals logical relations between them [Frege]
16868 The essence of mathematics is the kernel of primitive truths on which it rests [Frege]
16871 A truth can be an axiom in one system and not in another [Frege]
16870 Axioms are truths which cannot be doubted, and for which no proof is needed [Frege]
16869 To create order in mathematics we need a full system, guided by patterns of inference [Frege]
16873 Thoughts are not subjective or psychological, because some thoughts are the same for us all [Frege]
16872 A thought is the sense expressed by a sentence, and is what we prove [Frege]
16874 The parts of a thought map onto the parts of a sentence [Frege]
16876 We need definitions to cram retrievable sense into a signed receptacle [Frege]
16875 We use signs to mark receptacles for complex senses [Frege]
16877 A 'constructive' (as opposed to 'analytic') definition creates a new sign [Frege]
16878 We must be clear about every premise and every law used in a proof [Frege]
16879 A sign won't gain sense just from being used in sentences with familiar components [Frege]
9388 Every concept must have a sharp boundary; we cannot allow an indeterminate third case [Frege]