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+--------- Thread: Three Infinities in Mathematics [Panza] (/showthread.php?tid=1815)
Three Infinities in Mathematics [Panza] - mklabgr - 09-03-2026
Three Infinities in Mathematics: Projective Geometry, Infinitesimal Analysis, Set Theory
Authors: Marco Panza & Daniele C. Struppa
Publication date: 25 May 2026 (eBook)
Publisher: Springer Cham
Series:Undergraduate Texts in Mathematics
Length: XV + 431 pages
DOI: 10.1007/978-3-032-00277-8 (Springer)
Summary
Three Infinities in Mathematics is a conceptual and historically oriented exploration of infinity as it appears in three major areas of mathematics: projective geometry, infinitesimal analysis, and set theory. Rather than presenting these subjects simply as collections of definitions and theorems, Marco Panza and Daniele C. Struppa explain why mathematicians were driven to introduce different forms of infinity and how those ideas gradually became mathematically rigorous. The authors combine mathematics with history and philosophy, making the book particularly useful for understanding the motivations behind theories that can otherwise appear highly abstract.
The first part examines projective geometry, beginning with Renaissance perspective and the geometry developed by painters. The idea that parallel lines can meet at a “point at infinity” leads naturally to projective space, where ordinary Euclidean geometry is enlarged by the addition of ideal points. The second part follows the development of the infinitesimal calculus, starting with ideas about infinity in ancient Greek mathematics and continuing through the emergence of derivatives, integrals, limits and infinitesimal reasoning. Here infinity appears primarily as a process—quantities becoming arbitrarily small or large—and the authors emphasize the historical difficulties mathematicians encountered in putting these concepts on rigorous foundations.
The final part turns to set theory, where infinity becomes an object that can itself be studied mathematically. It begins with naive set theory and the paradoxes that exposed its weaknesses, then moves to axiomatic set theory and the formal treatment of infinite collections. In this progression, the book shows that the apparently different infinities encountered in geometry, analysis and set theory are manifestations of a common mathematical problem: how to make the infinite precise enough to reason about consistently. Exercises and historical and philosophical remarks accompany the mathematical material throughout. The level is appropriate mainly for advanced undergraduate or beginning graduate students, but the conceptual presentation also makes much of the discussion accessible to mathematically sophisticated general readers.
Key takeaways
- Infinity does not play a single role in mathematics. In projective geometry it appears through ideal points and lines; in analysis through limiting and infinitesimal processes; and in set theory as genuinely infinite mathematical collections.
- The book emphasizes the historical motivation behind mathematical formalism, explaining why concepts such as projective points, limits and axiomatic set theory were introduced.
- A major theme is the transition from an intuitive or philosophical conception of infinity toward a rigorous mathematical treatment.
- It is especially valuable for readers interested in the intersection of mathematics, its history, and philosophy, rather than those looking only for a conventional problem-solving textbook. (Springer)
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