Doing Mathematics: Convention, Subject, Calculation, Analogy by Martin H. Krieger – A Rigorous Study of Mathematical Pra

Introduction

Doing Mathematics: Convention, Subject, Calculation, Analogy by Martin H. Krieger is a profound exploration of how mathematicians actually think, work, and create. This second edition, published by World Scientific Publishing Company, offers Indian students, researchers, and enthusiasts a rare glimpse into the living fabric of mathematical practice. Far from a dry textbook, it reveals the human and philosophical dimensions behind theorems, proofs, and discoveries.

Book Overview

This hardbound volume delves into the four pillars that shape mathematical work: the conventions we adopt, the subject areas we define, what we can prove and calculate about the physical world, and the analogies that drive insight. Krieger argues that mathematics is not a fixed edifice but a dynamic enterprise shaped by these forces. Through rich case studies—from the central limit theorem to the two-dimensional Ising model—the book shows how mathematical reality emerges from a delicate interplay of choice, constraint, and creativity.

Key Highlights

• Second Edition Updates: Each chapter has been deepened with new material on mathematical rigor, philosophy, and contemporary applications.
• Real-World Connections: Bridges abstract mathematics with physics, statistics, and number theory, making concepts tangible.
• Case Studies Include: The sound of a drum’s shape, stability of matter, the Langlands program, and Dedekind’s legacy.
• Accessible Yet Rigorous: Written for both advanced undergraduates and working mathematicians.

Inside the Book

The book opens with an examination of mathematical conventions—how notation, definitions, and axioms are chosen and why they matter. It then moves to subject matter, exploring how mathematicians delimit fields of study. The third section focuses on calculation and proof, using examples like rigorous proofs of matter stability. The final part investigates analogy, showing how connections between algebra and topology or number theory and function theory drive new discoveries. Each chapter weaves historical context with modern practice.

Key Topics

• Central limit theorem and statistical foundations
• Eigenvalues and the geometry of drums
• Algebraic topology and its analogies
• Stability of matter in quantum mechanics
• Two-dimensional Ising model and phase transitions
• Langlands program in number theory and representation theory
• Dedekind’s contributions to number theory and analysis

Reader Benefits

Indian readers will find this book invaluable for understanding the why behind mathematics. It sharpens critical thinking, reveals the philosophy underpinning mathematical practice, and provides a bridge between pure mathematics and its applications. Whether you are preparing for competitive exams, teaching, or pursuing research, this book enriches your perspective on how mathematics is actually done.

Learning Outcomes

• Grasp the role of convention and choice in mathematical reasoning
• Understand how subject boundaries influence discovery
• Appreciate the interplay between calculation and proof
• Recognize analogical thinking as a driver of mathematical progress
• Connect classical results to modern research programs

Who Should Read

This book is ideal for mathematics students at the postgraduate or advanced undergraduate level, researchers in pure and applied mathematics, physics students interested in mathematical foundations, and educators who wish to bring philosophical depth to their teaching. Anyone curious about the creative process behind theorems will find it rewarding.

About the Author

Martin H. Krieger is a distinguished scholar known for his interdisciplinary work in mathematics, physics, and philosophy. He has taught at leading institutions and brings decades of experience in making complex mathematical ideas accessible. His writing reflects a deep love for the subject and a commitment to showing mathematics as a living, evolving practice.

About the Publisher

World Scientific Publishing Company is a globally respected academic publisher, renowned for high-quality books in science, mathematics, and technology. Their rigorous editorial standards ensure that each title is authoritative, well-researched, and valuable to the scholarly community. This hardcover edition is built to last—perfect for library collections and personal study.

Conclusion

Doing Mathematics: Convention, Subject, Calculation, Analogy is more than a book—it is an invitation to think like a mathematician. For Indian readers seeking to deepen their understanding of mathematical practice, this second edition offers timeless insights. Order your hardcover copy from Bookshops.in today and embark on a journey into the heart of mathematical creation.