Purity in Arithmetic: some Formal and Informal Issues

Arana, Andrew

doi:10.1515/9781614518471.315

Cited by 6 publications

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References 12 publications

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“…At the most basic level, one might take this to mean 'prime natural numbers'. But it is not clear whether these are numbers in the structure (N, ×, 1), (N, ×, 1, <), (N , +, ×, 0, 1), or many other choices including (N , |, 1, S, <); see Arana (2011) for a number of variants. A natural response from a mathematician might be, 'In what ring?…”

Section: Axioms: Specify the Basic Properties Of The Situation In Quementioning

confidence: 99%

“…(c) Specify the notion of a formal deduction for these sentences. 3. Axioms: specify the basic properties of the situation in question by sentences of the logic.…”

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confidence: 99%

“…In general, context and topic are synonyms in this paper. See Subsection 2.3 3. For most logics there are only a set of formulas, but some infinitary languages have a proper class of formulas.…”

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confidence: 99%

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Formalization, Primitive Concepts, and Purity

Baldwin

2012

Rev. symb. logic

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We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are algebraic and even metamathematical. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem.Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity.

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Section: Axioms: Specify the Basic Properties Of The Situation In Quementioning

confidence: 99%

“…(c) Specify the notion of a formal deduction for these sentences. 3. Axioms: specify the basic properties of the situation in question by sentences of the logic.…”

mentioning

confidence: 99%