Article ID: CBB001213914

The Ways of Hilbert's Axiomatics: Structural and Formal (2014)

unapi

Sieg, Wilfried (Author)


Perspectives on Science
Volume: 22, no. 1
Issue: 1
Pages: 133-157


Publication Date: 2014
Edition Details: Part of a special issue, “Hilbert's Axiomatics---Geometry, Physics, Logic”
Language: English

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as one direction for fascinating work.

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Article Lupacchini, Rossella (2014) Hilbert's Axiomatics as “Symbolic Form”?. Perspectives on Science (pp. 1-34). unapi

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Authors & Contributors
Majer, Ulrich
Hallett, Michael
Zach, Richard
Willea, Matthias
Stillwell, John
Smith, James T.
Concepts
Mathematics
Logic
Geometry
Formalization (philosophy)
Philosophy of mathematics
Physics
Time Periods
20th century, early
19th century
20th century
20th century, late
Places
Italy
Germany
Great Britain
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