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Hilbert's problems, Kant, and decidability
1Universität Wien, Vienna, Austria.
Archive for History of Exact Sciences
|August 14, 2025
Summary
David Hilbert
Area of Science:
- History of Mathematics
- Philosophy of Mathematics
Background:
- David Hilbert's conviction regarding the solvability of all mathematical problems.
- Influence of Immanuel Kant's philosophy of mathematics on Hilbert's views.
- Kronecker's influence on the concept of decidability in mathematics.
Purpose of the Study:
- To investigate the origins of Hilbert's belief in universal mathematical problem solvability.
- To analyze Hilbert's evolving concepts of mathematical problem solvability and decidability.
- To distinguish the historical and conceptual roots of Hilbert's different notions of decidability.
Main Methods:
- Analysis of unpublished historical sources.
- Examination of Hilbert's philosophical and mathematical writings.
- Historical contextualization of Hilbert's ideas within 19th and 20th-century mathematics.
Main Results:
- Hilbert's conviction originated from his engagement with Kant's philosophy of mathematics.
- Identified distinct concepts of decidability: finite-step, finitistic, and the decision problem.
- These concepts possess unique historical and biographical origins.
Conclusions:
- Hilbert's early conviction on problem solvability is philosophically rooted in Kant.
- Later concepts of decidability evolved from Kronecker and Behmann, reflecting distinct concerns.
- Maintaining conceptual clarity between these different notions of decidability is crucial.
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