Final answer:
René Descartes is the mathematician and philosopher regarded as the founder of modern philosophy, distinguishing his approach from classical ones set by Plato and Aristotle, whose influences were carried forward by scientists like Newton and Galileo.
Step-by-step explanation:
The mathematician and philosopher considered to be the founder of modern philosophy is René Descartes. René Descartes (1596-1650), often known as the father of modern Western philosophy, made significant contributions to the fields of mathematics and philosophy. He coined the famous phrase 'cogito, ergo sum' ('I think, therefore I am') and substantially influenced rationalism and the scientific method. His philosophical musings laid the groundwork for modern philosophical inquiries into the nature of existence and knowledge.
Descartes' approach was a significant break from classical philosophers such as Aristotle and Plato, who established reasoning patterns and philosophical frameworks in ancient times. Scientists and mathematicians like Newton and Galileo Galilei also built upon and advanced the ideas of these early thinkers, leading to a flourish in natural philosophy later evolving into specialized sciences.