Principia Mathematica¶
Whitehead, A. N., & Russell, B. (1910). Principia Mathematica. Cambridge University Press.
Cited by¶
3 citations across 3 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Abstraction
- A precise abstraction can be rigorously operated on; a vague term cannot. Common misclassification. Treating an abstraction as if it were the thing it abstracts from (conflating map with territory), or treating it as if the dropped detail were absent from the underlying reality (conflating what the abstraction doesn't show with what isn't there).
This sourceDevelops the (ramified) theory of types as a hierarchy in which objects of a given type are built only from objects of lower types, blocking self-reference
- A precise abstraction can be rigorously operated on; a vague term cannot. Common misclassification. Treating an abstraction as if it were the thing it abstracts from (conflating map with territory), or treating it as if the dropped detail were absent from the underlying reality (conflating what the abstraction doesn't show with what isn't there).
- Deductive Reasoning
- Deductive reasoning is tightly bound to mathematical proof
This sourceLandmark logicist derivation of mathematics from logical axioms; introduces the theory of types (a hierarchy preventing self-reference) and renders mathematical proofs as logical proofs. WebSearch (SEP, Britannica) confirmed logicism and the theory of types.
Supported in partVerified against the work's full text
Principia Mathematica states as its first object "the complete enumeration of all the ideas and steps in reasoning employed in mathematics", carried out by presenting each proposition with the maximum of formality. A separate passage holds that deduction depends upon the relation of implication. Binding the steps of mathematical reasoning to formal proof in that way is the connection the sentence asserts. The work is a logicist reconstruction of mathematics rather than a treatise on deduction generally, and does not state the surrounding contention that validity is the fundamental criterion of correctness.
“the theory in question enables us to deduce ordinary mathematics”
- Deductive reasoning is tightly bound to mathematical proof
- Paradox
- Russell's paradox (set of all sets not members of themselves); Cantor's paradox of the largest cardinal; Burali-Forti paradox; the motivation for axiomatic set theory (ZF) and type theory
This sourceDevelops type theory and the theory of levels of abstraction in formal logic: types form a hierarchy to prevent self-reference, and each type is an abstraction level with its own properties. Establishes the mathematical formalization of level-of-abstraction.
- Russell's paradox (set of all sets not members of themselves); Cantor's paradox of the largest cardinal; Burali-Forti paradox; the motivation for axiomatic set theory (ZF) and type theory
Verification¶
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Links previously used in the corpus¶
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