From mathematical axioms to mathematical rules of proof: recent developments in proof analysis

Area of Science:

• Mathematical Logic
• Foundations of Mathematics

Background:

• David Hilbert's 1900 list of mathematical problems.
• A newly discovered text by Hilbert reveals consideration of a 24th problem.
• The 24th problem focused on criteria for proof simplicity and a general theory of proof methods.

Purpose of the Study:

• To evaluate the progress of proof theory in addressing Hilbert's 24th problem.
• To assess the development of a general theory of proof methods in mathematics.

Main Methods:

• Analysis of historical mathematical texts.
• Review of developments in proof theory.

Main Results:

• Proof theory has made progress towards a general theory of proof methods.
• The extent of this achievement is discussed in relation to Hilbert's original aim.

Conclusions:

• The paper explores the connection between proof theory and Hilbert's 24th problem.
• It highlights the ongoing relevance of Hilbert's ideas on proof simplicity and methods.