how did renaissance intellectuals view mathematics

Renaissance intellectuals viewed mathematics as a critical and transformative discipline that bridged art, science, and the understanding of nature. They saw mathematics not only as a tool for practical problem-solving but also as a key to unlocking the beauty and order of the universe. Renaissance thinkers such as Leonardo da Vinci and Galileo Galilei advanced geometry, algebra, and perspective, integrating mathematical principles into art and science, which laid foundations for modern mathematics. The era was marked by a renewed interest in classical sources like Greek geometry and Arabic algebra, leading to innovations such as Cartesian geometry by René Descartes, which combined algebra and geometry to enhance spatial understanding. Moreover, Renaissance intellectuals debated the nature and certainty of mathematics, with some questioning whether mathematical demonstration qualified as scientific knowledge by Aristotelian standards, reflecting an evolving view of mathematics in the transition from medieval to modern science. Concepts like the golden ratio symbolized the aesthetic dimension of mathematics, linking it to architecture and natural patterns. Mathematics was also intertwined with humanism and was seen as a universal science capable of producing certain knowledge about nature. This period shaped the modern understanding of mathematics as both a theoretical and practical discipline fundamental to various scientific fields and the arts.