From Magnitudes to Geometry and Back: De Zolt's Postulate

Area of Science:

• History of Mathematics
• Foundations of Geometry
• Mathematical Logic

Background:

• Explores the 19th-century mathematical trend of seeking pure foundations.
• Focuses on De Zolt's postulate as a geometric expression of the 'whole is greater than the part' principle.
• Connects geometric purity with early axiomatizations of magnitude.

Purpose of the Study:

• Examine the trend of pure foundations in mathematics, specifically in plane area theory.
• Analyze David Hilbert's proof of De Zolt's postulate.
• Investigate the link between geometric problems and the axiomatization of magnitude.

Main Methods:

• Analysis of David Hilbert's classical proof of De Zolt's postulate.
• Connection of De Zolt's postulate to the first axiomatizations of magnitude.
• Logical analysis of the concept of magnitude.

Main Results:

• Illustrates the striving for purity in geometric foundations through Hilbert's proof.
• Highlights the relationship between geometric principles and the abstract concept of magnitude.
• Presents a recent result in logical analysis that illuminates Hilbert's proof.

Conclusions:

• De Zolt's postulate serves as a key example of foundational purity in geometry.
• The study provides new insights into Hilbert's proof via modern logical analysis.
• An alternative abstract theory of magnitude is proposed, including a proof of De Zolt's postulate.