Philosophy of mathematics

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This book serves as an introduction to the philosophy of mathematics, exploring fundamental concepts, constructions, and methods from the perspectives of mathematical research and education. It seeks answers through both mathematics and its philosophical underpinnings, tracing developments from their origins to the present. A pivotal focus is the introduction of real numbers in the 19th century, which marked a significant shift in the foundations of mathematics. Key topics include the nature of numbers, infinity, the continuum, infinitesimals, the applicability of mathematics, sets, logic, provability, truth, and the axiomatic approach. Chapter 6 specifically addresses the role of infinitesimals in mathematics and analysis. The authors, who are mathematicians, aim to engage mathematicians, mathematics educators, and students in philosophical discourse. The text is also relevant for professional philosophers and philosophy students, as it approaches philosophical questions through a mathematical lens. An elementary understanding of mathematics suffices for comprehension. The book reflects on historical perspectives, contemporary mathematical practices, and recent advancements, based on the third, revised German edition. It is intended for mathematicians, students, teachers, researchers, and anyone interested in the intersection of mathematics and philosophy.