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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  • Analysis of unpublished historical sources.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.