Innovation and Certainty

Mathematics has been displaced by non-Euclidean geometries in the nineteenth century, leading to questions about its status as a repository of necessary truth. This Element provides a sketch of this developmental history.

\n Format: Paperback / softback
\n Length: 75 pages
\n Publication date: 07 January 2021
\n Publisher: Cambridge University Press
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In the nineteenth century, a significant shift occurred in the traditional domains of mathematics, as the once-sacred verities of number and figure were displaced by altered settings that discarded them as no longer holding absolute truth. This transformative movement emerged not only in the realm of non-Euclidean geometries, but permeated various branches of mathematics. The question that arises is how mathematics can maintain its esteemed status as a repository of necessary truth in the face of these revisions.

To explore this issue, we will delve into the historical development of mathematics, examining how it has evolved and adapted to new perspectives and paradigms. We will examine the contributions of mathematicians such as Euclid, Pythagoras, and Newton, who laid the foundation for classical mathematics and its belief in the inherent truth of mathematical propositions.

However, as the nineteenth century progressed, mathematicians began to challenge the traditional assumptions and methodologies of classical mathematics. They introduced new ideas and theories, such as non-Euclidean geometries, set theory, and abstract algebra, that challenged the fundamental principles of classical mathematics. These innovations led to a reevaluation of what was considered true and necessary in mathematics.

One of the key challenges facing mathematics in this period was the development of modern mathematics. Modern mathematics is characterized by its emphasis on abstraction, logical reasoning, and mathematical structures. It seeks to develop a universal language that can be used to describe and analyze mathematical phenomena across different fields.

Despite the challenges posed by modern mathematics, many mathematicians argue that mathematics can retain its traditional status as a repository of necessary truth. They argue that while modern mathematics may introduce new concepts and theories, it does not necessarily undermine the fundamental principles of classical mathematics. Instead, it provides a deeper understanding of the underlying structures and principles that govern mathematics.

Another argument put forward by mathematicians is that mathematics is an ongoing and dynamic field of study. It is constantly evolving and adapting to new challenges and discoveries. As such, it is essential to be open to new ideas and perspectives and to be willing to revise and refine our understanding of mathematics as necessary.

In conclusion, the displacement of traditional domains of mathematics by altered settings in the nineteenth century has led to a reevaluation of what is considered true and necessary in mathematics. While modern mathematics has introduced new ideas and theories, it does not necessarily undermine the fundamental principles of classical mathematics. Instead, it provides a deeper understanding of the underlying structures and principles that govern mathematics. By remaining open to new ideas and perspectives and being willing to revise and refine our understanding of mathematics, mathematics can retain its traditional status as a repository of necessary truth.

\n Weight: 136g\n
Dimension: 152 x 230 x 9 (mm)\n
ISBN-13: 9781108742290\n \n